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Mathematics Work Book First form secondary

Second term

Mathematics has Practical applications in various fields including road construction, bridges and urban planning and preparing their maps which depend on parallel lines and their transversals according to the proportion between the real length and the drawing length.

Elsalam bridge connecting between the two shores of the Suez canel


Authors Mr. Omar Prof.Dr. Afaf Dr. Essam

Fouad Gaballa

Abo-ElFoutoh Saleh

Wasfy Rouphaiel Mr. Kamal

Prof.Dr. Nabil

Tawfik Eldhabai

Mr. Serafiem

Elias Skander

yones kabsha

All rights reserved no part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission of the Publishers.

SAKKARA Publishing Company 5 Messaha Square - Al nada Building - Dokki - Giza. Tel.: 02/ 37618469

Issued: 2013 D.N: 2013 / 14854 ISPN: 975 - 977 - 706 - 008 - 0


Name:

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School:

Class:

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INTRODUCTION º«MôdG øªMôdG ˆG º°ùH We are pleased to offer this book to clarify the philosophy that has been constructed of educational material and can be summarized as follows:

1. To emphasise that the main purpose of these books is to help the learner to solve problems and make decisions in their daily lives, which help them to participate in society.

2. Emphasis on the principle of continuity of learning through work so that students rely on systematic scientific thinking, and practice learning mixed with fun and suspense, relying on the development of problem- solving skills and develop the skills of the conclusion and reasoning, and the use of methods of self-learning, active learning and collaborative learning team spirit, and discussion and dialogue, and accept the opinions of others, and objectivity in sentencing, in addition to some definition of national activities and accomplishments.

3. Provide comprehensive coherent visions of the relationship between science, technology and society (STS) reflect the role scientific progress in the development of the local community, in addition to focusing on the practice of conscious students to act effectively about to use technological instruments.

4. The development of positive attitudes towards the study of mathematics and aspect of its scientists.

5. To provide students with a comprehensive culture to use the available environmental resource.

6. Rely on the fundamentals of knowledge and develop methods of thinking, the development of scientific skills and stay away of the details and educational memorization, that’s concern directed to bring concepts and general principles and research methods, problem solving and methods of thinking about the fundamental distinction mathematics from the others. In light of the previous, we have taken into consideration the folowing:

 Provide exercises start from easy to difficult and include a variety of levels of thinking.  There are general exercises at the end of each unit, unit test, and accumulative test includes many of the questions that varied between objectives and Essay questions, short answers questions and deal with the previous study units, the book included first term examinations.



It is also taken into consideration the use of appropriate language in the mathematical and life problems, relying on what studded before in the previous years, and also in the linguistic achievement to the students of the first form secondary. Eventually, we hope getting the right track for the benefits of our students as well as for our dearest Egypt hopping bright future to our dearest students. And the god of the intent behind, which leads to either way.


Contents Unit One

1-1 1-2 1-3 1-4 1-5

Matrices Organizing Data in Matrices Adding and subtracting Matrices Matrix Multiplication. Determinants Multiplicative Inverse of a Matrix. General Exercises Unit test Accumulative test

2 4 5 6 8 10 12 13

.....................................................................................................................................................................

.................................................................................................................................................

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...........................................................................................................................................................................................................................................

.....................................................................................................................................................................................................

Unit Two

Linear Programing

2 - 1 Linear Inequalities

16

..................................................................................................................................................................................................

2 - 2 Solving systems of Linear Inequalities Graphically

18

2 - 3 Linear programing and optimization

19 21 22 23

.....................................................................

General Exercises Unit test Accumulative test

..............................................................................................................................

.....................................................................................................................................................................................................

...........................................................................................................................................................................................................................................

Unit Three

.....................................................................................................................................................................................................

Vectors

3 - 1 Scalars, Vectors and Directed Line Segment 3 - 2 Vectors

26

................................................................................................

28

..................................................................................................................................................................................................................................................

3 - 3 Operations on Vectors

3 - 4 Applications on Vectors General Exercises Unit test Accumulative test

30

........................................................................................................................................................................................

32 35 36 37

.................................................................................................................................................................................

.....................................................................................................................................................................................................

...........................................................................................................................................................................................................................................

.....................................................................................................................................................................................................


Unit Four

Straight Line

4 - 1 Dividing a line segment

40

.................................................................................................................................................................................

4 - 2 Equation of the straight line

41

..............................................................................................................................................................

4 - 3 Measure the angle between two straight lines

43

.......................................................................................

4 - 4 Length of the perpendicular drawn from a point to a straight line 45 .........

4 - 5 General equation of the straight line passing through the point of intersection of two lines General Exercises

47 48 49 50

...........................................................................................................................

.....................................................................................................................................................................................................

Unit test ................................................................................................................................................................................................................................................ Accumulative test ................................................................................................................................................................................................................

Unit Five

Trigonometry

5 - 1 Trigonometric Identities

52

.............................................................................................................................................................................

5 - 2 Solving Trigonometric Equations

53

...........................................................................................................................................

5 - 3 Solving the Right Angled Triangle

5 - 4 Angles of Elevation and Angles of Depression 5 - 5 Circular Sector

54

.......................................................................................................................................

56

.........................................................................................

57

...................................................................................................................................................................................................................

5 -6 5 -7

Circular Segment

58

.........................................................................................................................................................................................................

Areas General Exercises Unit test Accumulative test General Tests Model Answer

59 61 62 64 65 71

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.....................................................................................................................................................................................................

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Algebra

Unit

1

Matrices

Lessons of the unit

Lesson (1 - 1):

Organizing Data in Matrices.

Lesson (1 - 2):

Adding and subtracting Matrices.

Lesson (1 - 3):

Linear programing and optimization.

Lesson (1 - 4):

Determinants.

Lesson (1 - 5):

Multiplicative Inverse of a Matrix.


1-1

Organizing Data in Matrices

1 A, B and C are three cities , if the distance in kilometres between any two cities is shown in the table opposite. A Write a matrix represent these data. A B C B Let X be the required matrix in "A" find the following: A 0 75 80 1- What does it mean by X23? ................................................................................ B 75 0 56 2- What does it mean by X32? .................................................................................. 3- what is the relation between x23, x32? ......................................................

C

80

56

0

C

Write all elements of the second row in the matrix X. .................................................................................. D Write all elements of the second column in the matrix X. What do you deduce from C and D ? ......................................................................................................................................................................................................................................

E

Find X k k when K = 1, 2, 3 What do you notice? ................................................................................................ F Complete each of the following: 1- X is a matrix of order ................................................................................................................................................................. 2- Xi j = Xj i for all values ...............................................................................................................................................................

2 What is the number of elements in each of the following matrices: A

Matrix of order 2 * 3 ...........................................................................................................................................................................

B

Matrix of order 2 * 2 ...........................................................................................................................................................................

C

Matrix of order 3 * 2 ...........................................................................................................................................................................

3 Find the values of a, b, c and d if: 3 -5 l = b a - 2 A b a - 3 3d - 2 c B

b

15 0

2b l 2a + c

= b

3a 2b-d

2b + 1 l ........................................................................................................................ 16 10 10

l ........................................................................................................................

4 Industry: The table opposite shows the number of general national factories of food and leather industry in three different cities in Egypt. A Organize the data in a matrix. Food Leather City

..........................................................................................................................

2

Mathematics - first form secondary

industry industry

6 October

44

68

Sadat

28

52

Alasher min ramadan

37

14


Organizing Data in Matrices

B

Add the elements in each columan, what's your interpretation of the results obtained? ......................................................................................................................................................................................................................................

C

Add the elements in each row. Are the results you obtained possible to help us with an extra data. Explain your answer. .............................................................................................................................................................................................................................................

4

-1 l = b 4 2a - 1 3b + 1 3

5 Find the value of each of a and b if b

-1 l 7

.................................................................................................................................................................................................................................................

6 If A = b 2 -1

-3 l , B = 2d b 4 3e

-1 l where A = Bt 4

then find the value of each of d and e. .................................................................................................................................................................................................................................................

7 If A = f

1 2 4

0 1 5

2 -3 -1

Find A + B , A - B

3

p , B = f -2 -4 , A + 2B ,

0 3 -5

-2 3 5

p

B - 3A

8 Critical thinking: If A = (Axy) ∀ x, y ∈ { 1, 2, 3 } Write the matrix A, given that Ax y = y - x, then find At .................................................................................................................................................................................................................................................

Activity

Form a matrix from life data , such that the sum of the elements in its columns are meaning ful and the sum of the elements in its rows are meaningless. Enter the matrix's data in the spreadsheet (Excel) and verify that the sum of elements you obtained are correct, then interpretet what does it mean by the sum of elements in the columns. ................................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

Work book - Second term

3


1-2

Adding and subtracting Matrices

1 If A = b -2 0 -1 l and K1 = 2 , K2 = -1 then find each of the following matrices: 4

5

0

K1A and K2A .................................................................................................................................................................................................................................................

-8 4 2 If A = b -7 0 -5 l , B = f 0 7 p then find the result of the following operations if 4 7 5 -6 -5 "possible" , give reasons in case of impossible solution. A

B

A+B

A + Bt

C

At + B

-4 -2 -2 -4 5 1 3 If X = f 3 6 p , Y = f 0 -2 p , Z = f -3 2 p then find the matrix 3X - Y + Z 0 4 6 0 4 -3 .................................................................................................................................................................................................................................................

4 8 -6 2 -6 2 4 If A = f 2 -4 8 p , B = f 4 -10 0 p , then find the matrix X where X = 2A - 3 B 6 12 0 -1 8 -4 .................................................................................................................................................................................................................................................

5 Critical thinking : Find the values of a, b, c and d which satisfy the equation: 2b a

6

3 l b

= 3b a

d l c -2

- 4b c

3 l a

0

................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

6 Open problem : Choose two matrices A and B have the same order, then prove that: A

A - B = A + (-B)

B

(A + B)t = At + Bt

C

(A - B)t = At - Bt

................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

Activity

1 Write a life problem, such that it is possible to solve it using adding or subtracting matrices. 2 Search in your school library or internet about the applications of matrices in other sciences.

4

Mathematics - first form secondary


1-3

Matrix Multiplication 1 If A = b 3 6

1 l 2

, B = b -1 3

3 l , then find each of the following: 9

A

AB

.....................................................................................................................................................................................

B

BA

.....................................................................................................................................................................................

C

(A + B)A

.....................................................................................................................................................................................

2 If b 2 4

3 l b x 5 3

7 l y

=b 7

11

8 l 18

find the value of each of x, y:

.................................................................................................................................................................................................................................................

3 Critical thinking: If A and B are two matrices and, is the zero matrix, A B = Does it aways mean that A = or B = . Take A = b -2 3 l , B = b 3 6 l , then show your 2 -3 2 4 opinion. ................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

4 Tourism: A hotel in the tourist town of hurghada consumes the following quantities of meat, vegetables, and fruits in kilogram for lunch dinner meals as in the following table: Meat

Vegetables

Fruits

Lunch meal

200

100

150

Dinner meal

120

80

100

If the average of the price of one kilogram of meat was 65 pounds, the average of the price of one kilogram of vegetables was 4 pounds, and the average of the price of one kilogram of fruits was 5 pounds. Find using multiplication of matrices the total cost of the two meals. ................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

Work book - Second term

5


Determinants

1-4

1 Find the value of each of the following determinants: A

7 3

5 2

B

......................................

D

a +x b+y

a b

0 42 3 2 18 7 0 28 3

2

C

-1

......................................

E

......................................

G

1 3

x2 + 1 y2 + 1

x+1 y+1

......................................

3 2 5

-4 -3 0 -31 0

-3 -7

......................................

F

1

2

-1 4 0

......................................

H

6 19

7

3 4 8

......................................

13 3 23 30 7 5 0 0 1

I

2

......................................

......................................

2 Solve the following linear equations by Cramar's rule: A

2x-3y=5

B

3 x + 4 y = -1 D

3x+2y=5

2x +y - 2z =10

C

2 x + 5 y = 16 E

2x+y=3 G

x+y=5

3x=1-4y

2x+5y=8 F

5 x + 12 = 7 y H

x + 2y - 3z = 6

x+3y=5

2x=3+7y y=5-x

I

y +2x +3z = 6

3x + 2y + 2z =1

2x-y-4z=2

2x - y +z = -3

5x + 4y + 3z = 4

4x + 3y - 2 z = 14

x - 2y + 2z= -11

3 Geometry: Find the area of the triangle A B C in which A(2, 4), B (-2, 4), and C(0, -2). .................................................................................................................................................................................................................................................

4 Find the area of triangle X Y Z in which X (3, 3), Y (-4, 2), and Z ( 1, -4). .................................................................................................................................................................................................................................................

5 Use the determinants to prove that the points (3, 5), (4, -1), (5, 7). are collinear ................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

6

Mathematics - first form secondary


Determinants

Activity

6 To find the value of the determinant

5 7 4 6 25 11 7 52 21

Notice the following method of solution ✔ Notice that, we wrote the three columns of the determinant and repreated the first two columns. ✔ Draw diagonal lines across every three elements as shown in dotted arrows, then the resulting terms from each line are terms in the expansion, and the directed arrows down, its corresponding terms are positive while the directed arrows up, its corresponding terms are negative.

5

7

4

5

7

6

25

11

6

25

7

52

21

7

52

The determinant = 5 * 25 * 21 + 7 * 11 * 7 + 4 * 6 * 52 - 7 * 25 * 4 - 52 * 11 * 5 - 21 * 6 * 7 = 2625 + 539 + 1248 - 700 - 2860 - 882 = -30 Try to solve Use the previous method to expand the determinant . Find the value of each of the following: A

9=

3 5 7 11 9 13 15 17 19

B

9=

13 3 23 30 7 53 39 9 70

C

9=

1 1 6

D

9=

3 2 5

-2 1 2 4

3 3

-4 -3 7 -31 -9 2

Check your answer by finding the value of each determinant using the usual method then compare the two results. ................................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

Work book - Second term

7


1-5

Multiplicative Inverse of a Matrix

1 Show the matrices which have inverses, and the matrices which have not inverses in the following. if there is an inverse find it. A

b 2 1 l -1 1

B

.............................

E

b 4 3

2 l 1

b 2 6 l -1 3

C

.............................

F

.............................

b 2 0

b -1 0 l 3 4

D

.............................

0 l 1

G

.............................

b 3 2

9 l 6

b 2 2

2 l 2

.............................

H

.............................

b 2 5

3 l 6

.............................

2 What are the values of a which make to each of the following matrices a multiplicative inverse. A

b a 6

1 l 3

.............................

2 3 If X = b 0

0 l 2

B

b a 4

9 l a

C

.............................

b a 2

4 l a-2

.............................

1

then prove that X -1 = b 2 1

0 1 2

D

b a-1 -2 l 1 a-1 .............................

l

.................................................................................................................................................................................................................................................

-2 0 1 4 Find the matrix A if:A b 3 1 l = b 0

0 l 1

.................................................................................................................................................................................................................................................

2

5 If X = b 0

1 -1 -3 l , Y = b -1 3 l 1

,

Prove that = (XY) -1 = Y-1X-1

.................................................................................................................................................................................................................................................

6 Solve the following linear equations using matrices, then check your answer: A

C

8

4 x + 3 y = 26 , 5 x - y = 4

B

2x-7y=3,x-3y=2

..............................................................................

..............................................................................

..............................................................................

..............................................................................

2 x = 3 + 7 y, y = 5 - x

D

2y=5-x,2x=3-y

..............................................................................

..............................................................................

..............................................................................

..............................................................................

Mathematics - first form secondary


Multiplicative Inverse of a Matrix

7 Geometry: The straight line whose equation y + a x = c passes through the two points (1, 5) and (3, 1), use the matrix to find the constants a and c . ................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

8 Life: A driver of a motorcycle buys 24 litres of gasoline and 5 litres of oil for 56 pounds to fill his motorcycle. While a driver of another motorcycle buys 18 litres of Gasoline and 10 litres of oil for 67 pounds to fill his motorcycle, use matrices to find the price of each litre of Gasoline and the price of each litre of oil, given that they use the same type of Gasoline and oil. ................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

9 Geometry: the curve whose equation y = a x2 + bx passes through the two points (2, 0) , (4, 8), use the matrices to find the constants a and b. ................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

10

Criticel thinking: Half the difference of two numbers is 2, the sum of the greater number

and double the smaller number is 13, use matrices to find the two numbers. ................................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

Activity

Write a problem such that its solution needs to form a system of linear equations then solve it using matrices. ................................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

Work book - Second term

9


General Exercises 1 If A =

b 5 4

2 -1 l 0 1

A

What is the order of the matrix A?

B

Write the elements of the first row in the matrix A............................................................................................

C

Write the elements of the third column in the matrix A.

D

Write the elements: A11, A22, A13, A12 .................................................................................................................................

........................................................................................................................

.............................................................................

2 Express the coordinates of the following points by a matrix of order 1 * 2 A (2, 3) , B (-1, 0), C(-2, -3) .................................................................................................................................................................................................................................................

3 What is the number of elements in each of the following matrices? A

matrix of order 3 * 2

........................................................................................................................

B

Square matrix of order 2 * 2.

........................................................................................................................

4 Solve each of the following equations: A B

b x +5 l = b 9 l x-y 4 b 3b a l = b 0 b b+a 2 4

........................................................................................................................

1 l 2

........................................................................................................................

5 Industry: A factory produces three types of television screens: 32 inches, 42 inches, and 48 inches, the factory has 2 branches A and B, and the number of screens produced in each branch from each type during January and February 2013, the following matrices show these data, express the production of the two months together using matrices. 32inch

A b B

850 750

42 inch

48 inch

700 600

600 550

32inch

l

A b B

January 2013

800 840

42 inch

48 inch

650 550 700 600 February 2013

l

................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

2 -1 l 6 Solve the equation: A + b 1 -2 l = b . 0 5 3 -4 ................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

10

Mathematics - first form secondary


General Exercises 7 Find a, b, c and d where A B

b -3 -2 l 1 5

f

1 0 -1

a +b c

b l d

a

-2 1 = b 4 -3 l

........................................................................................................................

0

p - f b p = f -1 p c 1

2 3 4 -2 0 -1 -6

8 If: X = f 6

........................................................................................................................

1 1

2

3

0

2

1

p , Y = f -2 -3 -1 p , then find:

A

2X+3Y-2I

........................................................................................................................

B

X - (Y - 5 I )

........................................................................................................................

3 9 Find the values of x and y where: b 1 10

b 1 3

2 l 4

B

.............................

b 0 3

2 l 0

C

.............................

b a a

-b l b

D

.............................

b 3 2

9 l 6

.............................

Solve each system of the following linear equations using matrices A

12

0 = b 1 l

Show which of the following matrices has a multiplicative inverse , then find it A

11

2 l b x l 4 y

x+y=3

B

C

2x-3y-3=0

y = 11 - 5x

x-y=5

5 x + 4 y - 19 = 0

x=3-5y

............................................

............................................

............................................

1 -2 4 If (AB)-1 = 1 b 1 3 l and A = b 1 5

3 l 2

then find B - 1.

.................................................................................................................................................................................................................................................

13

Solve the following equation using Cramar's rule: 16 x + 20 y = 69 , 12 x + 4 y =27. ................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

Work book - Second term

11


Unit test 1 If A = b 3 1 l , B = b -1 2 l , C = b 2 -1 l Find: -2 5 6 -3 -3 4 A

A+2B-3C C Verify that: A (B + C) = A B + A C

B

each of the two matrices: A B , B A D verify that: (AB)t = Bt At

2 A stationery sells 3 types of CD's, the table opposite selling shows the cost of each set and its selling price, if the Type of set of CD's Cost price (in pounds) stationery sold 40 set of CD's of scientific topics, Scientific 80 100 64 set of CD's of cultural issues, and 45 set of CD's Cultural 65 85 of Global stories. Scientific stories 90 110 A Write these data in two matrices, then use multiplication of matrices to find the total cost of CD's. B Use multiplication of materices to find the total sum of money that the stationery will get from the selling of sets of CD's. C Use the operations on matrices to find the profit of the stationery from the selling of sets of CD's. 3 Use Cramar's rule to solve each of the following systems: A x+y=4 B 3x-y+5=0 2x - y = 1 x + 2y +1 = 0

C

2x+y-z=3 -3x + 2 y + z = 4 4x+2y-z=8

4 Show which of the following matrices has a multiplicative inverse ? Find the multiplicative inverse if possible: A

b 2 3

-1 l 2

B

b 3 4

5 l 6

C

b 1 2

2 l 1

D

b 1 1 l -1 -1

5 Solve each of the following systems using matrices: A x + 2y = 8 B 3x-y=0 C 4 x = - 6y 2 x - y = -9 5 x + 2 y = 22 8x-7=2y 6

12

A

Solve the system of the following equations using Cramar's rule. 3y = 1 + z - x , 2 x = -2y + z , 3 x + y + 2 z = -1

B

Khalid has 25 pieces of a 25pt note, a 50pt note, If he had 8.5 pounds, use the matrices to find the number of a 25pt note and number of a 50pt note.

Mathematics - first form secondary


Accumulative test First: Complete each of the following: 2 l t 1 If A = b -2 , B = (2 5) then (BA) = ........................................................................................................................................... 2 If b 4 1 l b 1 -1 l = I, then x = .............................................................................................................................................. 3 1 -3 x 1 2 l then A2 = ......................................................................................................................................................................... 3 If A = b 3

2

Second: Multiple choice questions: 3 -1 l 4 If A and B are two matrices where AB = b then Bt At = 4

3 A b 4

5 If: A

-1 l 5

2-x 2 -3 x + 2

3 4 l B b -1 5

= 1 then x equals B

-3

5

5 C b 4

..................................................................

5 4 l D b -1 3

-1 l 3

.................................................................................................................................................

C !3

3

D !4

Third: Answer the following questions 6 Find the value of each of the following determinants: 3 1 -4 -2 3 D 5 0 -1 1 -1 4 2 2 0 2 -1 7 Solve each of the following systems of equations using Cramar's rule A 2k - 4 = -y B 3x + y = 5 C x-2y-z=0 D 3 x = 2 y + 3 +z y = 3x - 6 2x + 3 y = 8 x + y - 2z = -1 2x-y+4=z -x + 4y + 7z = 6 y + z = -x + 3 A

-3 4 2 -5

B

2 5

-2

C

2 4 1

8 Use the matrices A, B and C to determine which of the following is correct. A = b 2 3 l , B = b 0 5 l, C = b 4 1 l 6 -1 -1 -3 3 2 A A(B C) = (A B) C. B A(B + C) = AB + AC. 9 Solve the following system of simultaneous equations using matrices 5Y = 1 - 2X , 3X = 2 - 7Y 10

If A = b 2 0

1 l

-1

, B = b 1 3 l , then find each of the following: -1 2

A

AB

B

BA

C

A2

D

B2

E

AtB

F

ABt

Work book - Second term

13


Accumulative test 11

12 13

Carryout the following operations , if possible , give reasons in case of impossible solution: -4 -6 l -x -y 4 6 x y l - b A b B b l + b -z -l l 3 9 3 2 z l 1 -2 3 1 -2 3 1 0 3 2 -3 l l b C b D b 0 1 4 l b 4 5 6 l 2 1 0 4 5 -3 2 1 -1 2 1 1 If x = b 0 3 l , y = b -1 1 l , prove that: x y ! y x a

0

3 0 If x = b 0 4 l , y = b 0 b l where A B ! 0 Prove that for each of the matrices X , Y , X Y has a multiplicative inverse and find it.

14

Karim bought 10 CD's from a stationery, 6 pens, and 4 pencils. His colleague samy bought 8CD's, 5 pens, and 3 pencils from the same kinds of what karim bought. If the price of a CD was 2 pounds, 1.5 pounds for each pen, and 0.75 pound for each pencil. A Write a matrix representing karim and samy purchases, then write a matrix showing the prices of each item. B Write a matrix showing the payment of karim and samy. C How much money did karim and samy pay for purchases?

15

If the production of three different sections Month Month Month of a factory to produce kids toys in three (1) (2) (3) consecutive months is as in the table opposite. Section (A) 360 400 380 The table shows the selling price of each toy, Section (B) 480 500 450 what is the matrix which represent the income Section (C) 570 600 550 of the factory from the production of the three sections in the three months supposing that the factory sold all its productions? What is the amount of total income to the factory?

Price of each toy 28 Pounds 37 Pounds 32 Pounds

14

6

7

8

9 10 11 12 13 14 15

No of lesson

1-3

1-3

1-2

1-2

1-4

1-4

1-3

1-4

Mathematics - first form secondary

1-3

5

1-3

4

1-5

3

1-5

2

1-3

1

1-3

No of question

1-3

Now, if you can not solve any one of the previous questions , you can use the following table:


Algebra

2 Unit

1

-

2

0

O

Linear Programing 3

O

O

4

Lessons of the unit

Lesson (2 - 1):

Linear Inequalities.

Lesson (2 - 2):

Solving systems of Linear Inequalities Graphcally.

Lesson (2 - 3):

Linear Programing and Optimization.

Work book - Second term

15


2-1

Linear Inequalities

1 Join each inequality with the graph which represents its solution set (test the point (0, 0) in each inequality). A

B y

x‫ﹶ‬

x

yGx+1

D

y

x‫ﹶ‬

y‫ﹶ‬

1-

C y

x

x‫ﹶ‬

x x‫ﹶ‬

y‫ﹶ‬

2-

y<x+1

y

y‫ﹶ‬

3-

x

y‫ﹶ‬

4-

y>x+1

yHx+1

2 Test which of the points is the solution of the inequality: A yH2x+3 [ (0, 1) , (3, 9) , (–1, 0) ] ................................................................................................. B y<2x+3 [ (0, 1) , (3, 9) , (–1, 0) ] ................................................................................................. 3 Find the solution set of each of the following inequalities: A yGx+2 B y>2x–3

C

x+3yG6

................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

4 Consumer: Suppose you want to buy decoration paper to decorate your class for a party to the excellent students. If the price of a roll of the golden decoration paper was 5 pounds and the price of a roll of the blue decoration paper was 3 pounds. If you pay at most 48 pounds to buy the decoration paper. How many rolls of each type can you buy? Explain your answer. .................................................................................................................................................................................... .................................................................................................................................................................................................................................................

Activity (1) Describe to your classmate who was absent during the explanation of this lesson because of illness. How can you represent the inequality x - y H 2 graphically and check your answer.

16

Mathematics − first form secondary


Linear Inequalities

Activity (2) Technology You can use the property

Learning tools: Graphic calculator

in the graphic calculator to draw the inequalities where the order of input data depends on what you will shade above or under the boundary line. If the shade was down the boundary line y1 use the shade (ymin, y1), when you will shade above y1 you will use the shade (y1, ymax) you do not need to use the closed brackets before pressing the key ENTER Draw

Example

5 Represent each inequality graphically: A y < 2x + 3

B

the shade is down the boundary line because there is the sign less than y1 Y= 2nd

Draw

1

VARS

2nd

X,T,i

2

+

7 4

Y-VARS

1

3

y > 0.5x – 1 the shade is above the boundary line y1 because there is the sign greater than

# Enter the equation of the boundary Y= line $ 2nd # Enter the shade property $

, 1

2nd ENTRER

#inter

(ymin, y1)

inter (y1, ymax)$

0

.

Draw

1

1

,

1

7 1

Y-VARS

VARS

X,T,i

5

5

ENTRER

½ You can control the degree of shading by entering the whole number from (Dark) 1 to (Light) 8, add comma and the whole number before pressing the key ENTRER ½ The graphic calculator does not distinguish between the boundary dotted or solid line, thus you have to determine the type of the line (solid or dotted) when drawing the inequality in your note book.

Try to solve 1 Use the graphic calculator, to draw each of the following inequalities: A

y<x

B

y > 2x + 1

C

y H -x + 3

D

yG5

E

x-yH4

F

2x + 3y G 12

Work book - Second term

17


Solving systems of Linear Inequalities Graphically

2-2

1 Which of the following systems has the shown shaded region as a solution in the figure y opposite: 4 A

x+yG3

B

yGx-3

y>x-3 C

x+yH3 y<x-3

D

3

x+y>3

x+y<3 yHx-3

2 1

x/ -2

-1

-1

x 1

2

3

4 5

6

7

-2 -3 -4

y/

2 Solve each system of the following linear inequalities graphically: A xH4 B y-x>0 C x+4y>4 y>x+2 2 x + 2 y H 12 4x+yH2 x + 2 y H -2 y<6+2x x-y<1 ................................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

3 Mr. Karim gave 60 minutes to his students to solve a test in mathematics. The students have to solve at least 4 questions from section "A" and at least 3 questions from section"B" such that the number of the answered questions is at least 10 questions from both sections. If Hanaa answered her questions in 4 minutes to every question in section (A), 5 minutes to every questions in section "B". How many questions in each section did Hanaa try to solve?................................................................................................................................................................................................................................ 4 Critical thinking: A Write a system of linear inequalities in which their solution is a straight line. ................ B Without graphic representation , explain why the point of intersection of the two boundary lines in the system: 2 X + Y > 2, X - Y H 3 is not a solution of this system? ...... Activity 5 Write a problem such that its solution needs to write a system of two linear inequalities in two variables, then solve this system. ................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

18

Mathematics â&#x2C6;&#x2019; first form secondary


Linear programing and optimization

2-3

1 Choose the correct answer from the given answers: A The point which belongs to the solution of the enqualities: x > 2 , y > 1 , x + y H 3 is: ................................................................................................................................................................................................... [(3, 1) , (1, 2) , (3, 2) , (1, 3)] B The point at which the function P = 40x + 20y has a maximum value is : .............................. [(0, 0) , (0, -4) , (15, 10) , (25, 0)] C The point at which the function P = 35x + 10y has a minimum value is: ................................. [(0, 0) , (0, 10) , (0, 40) , (20, 10)] y B 6

2 Use the graph opposite to find the values of x, y which make the value of the objective function p = 3x + 2y has a minimum value , then find this value .

C

4

A

..................................................................................................................................................................

2

D

..................................................................................................................................................................

x

/

x

3 Represent each of the following systems graphically, then find the maximum or minimum value to the objective function according to the given. A

-1

2

4

6

y/

x+yG5 B 2x + y G 6 yH1 xH1 xH2 yH2 has minimum value to the objective has maximum value to the objective function P = 2x + 3y function P = 2x + 3y

4 Industry: Suppose you manufacture and sell skin moisturizer, if manufacturing a unit of the normal moisturizer requires 2cm3 of oil, 1cm3 of cocoa butter, and manufacturing a unit of the excellent moisturizer requires 1cm3 of oil, 2cm3 of coca butter. You will gain 10 pounds for every unit of the normal kind, 8 pounds for every unit of the excellent kind. If you had 24 cm3 of oil, 18cm3 of cocoa butter. What is the number of units you can manufacture from each kind to get a maximum possible profit and what is this profit? ................................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

Work book - Second term

19


5 Occupations: A tailor has 10 metres of linen cloth and 6 metres of cotton cloth, he wants to make two types of clothing from his own available materials. The first kind of clothing, the tailor needs 1 metre of linen, 1 metre of collon to gain 3 pounds profit, while the second kind of clothing he needs 2 metres of linen and 1 metre of cotton to gain 4 pounds profit. Find the number of clothing that should be made from each type such that the tailor gets the maximum profit ...................................................................................................................................... .................................................................................................................................................................................................................................................

6 Music: One of the factories of musical instruments produces two types of blowing instruments, the first kind needs 25 units of copper, 4 units of nickel and the second type needs 15 units of copper, 8 units of nickel. If the available quantities in the factory on a day were 95 units of copper, 32 units of nickel and the profit of the factory from the first types was 60 pounds and 48 pounds from the second type. Find the number of instruments which the factory should produce from each type to get the maximum profit. ................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

7 Tourism: One of the tourism companies set up an air bridge to transport 1600 tourists and 90 tons of luggage with the least cost. If there are two types of aeroplanes A and B and the number of available aeroplanes from the first type was 12 aeroplanes, and 9 aeroplanes from the second type and the full load to the aeroplane of the first type A was 200 persons, 6 tons of luggage and the full load to the aeroplane of the second type B was 100 persons, 15 tons of luggage. The rent of the aeroplane of the type A was 320000 pounds, and the rent of the type B was 150000 pounds. How many aeroplanes from each type could the company rent? ................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

Activity 8 Refer to your school library or the international network for information. Write an example to show the uses of linear programming in each of the following subjects: A Economy C Time management B Industry D Operations research .................................................................................................................................................................................................................................................

9 Write a problem, whose solution requires forming 4 linear inequalities, then represent the region of solution graphically. Write the objective function to your problem and determine when it has a maximum value or a minimum value, then find these values . ................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

20

Mathematics â&#x2C6;&#x2019; first form secondary


General Exercises 1 Put (✔) for the correct answer and (✘) for the incorrect one, starting resons, give an example for the correct answer. A

If a, b ∈ R , a > b and a, b are both positive, then a2 > b2.

(

)

................................

B

If a, b ∈ R , a > b and a, b are both negative, then a2 > b2.

(

)

................................

C

If a, b ∈ R , a > b and a, b have the same sign , then

)

.............................

D

If a, b ∈ R , a > b and

)

..............................

1 1 < . ( a b 1 1 a, b have the different sign , then < . ( a b

2 Solve each of the following inequalities, then represent the solution on the number line: A 2x + 4 < 5x - 5 ..................................................................................................................................................................................... ......................................................................................................................................................................................................................................

B

7 < 5x + 2 G 12

.....................................................................................................................................................................................

......................................................................................................................................................................................................................................

3 Represent graphically the solution set of each of the following inequalities: B y H -x - 4 C y < -2x + 3 A y G 2x + 1 4 Solve each system in the following inequalities graphically: A y<x+3 B x+y>2 y>6+x 2x - y < 1

5 In the figure opposite: find the values of x, y which make the function P = 1 x + y has a minimum value. 2

............................................................................................................................................................. ............................................................................................................................................................. .............................................................................................................................................................

C

2x + 3y G 12 xH2 yH1

y 8 7 6 5 4 3 2 1 0

x 1 2 3 4 5 6 7 8

6 Industry: A small factory of making metal containers produces two types of containers A and B. It takes 10 minutes to produce container A from the first machine and 12 minutes from the second machine. While it takes 15 minutes to produce container B from the first machine and 10 minutes from the second machine. If the profit of container A was 4 pounds and the profit of container B was 5 pounds. How many containers could be produced from each type such that the profit is to be maximum given that each machine works at most 8 hours. ................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

Work book - Second term

21


Unit Test 1 Solve each system of the following inequalities graphically: A

x - 2y < 3 , 2x + y > 8

B

x H 0 , y H 0 , x + y G 5 , 4x + y G 8

................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

2 Find the maximum and minimum values of the objective function P = 4x + y where: x + y G 6 , 2x + y H 10 , x H 0 , y H 0 ................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

3 Health: Sami burns 4 calories/min when walking on foot, 10 calories/min when running, Sami walks 10 - 20 minutes daily, and runs 30 - 45 minutes daily. He does not take more than an hour a day to walk and run. What time does it take in each exercise to burn the maximum amount of calories? ................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

4 In the figure opposite: Find the values of x, y which make the objective function: P = 2x + 3y is greatest as possible , then find this value.

y 10

B

8 6

............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................

C

4 x/ 2

D

-2 O

2 4 6

x 8 10

y/

5 Industry: One of the small factories produces two types of office chairs, one of them is a beech wood and the other is a white wood. It is required to produce each of them two types of machine A and B. If the production of the beach wood chair requires machine A to work for 3 hours, and machine B for 2 hours, and the production of the white wood chair reuires machine B to work for 3 hours. If the factory gains 20 pounds for the beech wood chair, and 12 pounds for the white wood chair, find the number of chairs of each type that must be produced daily to gain a maximum profi, given that the factory does not work more than 15 hours daily. ................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

22

Mathematics â&#x2C6;&#x2019; first form secondary


Accumulative test y

1 Multiple choice:

4 3

Choose the inequality which represents the shaded region

C

yH1x 2

yG

1 2

x

1

x‫ﹶ‬

as its solution: A

2

B

-4 -3 -2 -1 -1

y>1x 2

D y<1 2

x 1

2

3 4

-2 -3

x

-4

y‫ﹶ‬

2 True or false: Is the following statement true or false? explain your answer:" the system formed from two linear inequalities has no solution , or has an infinite number of solutions". 3 Represent graphically the solution set of each of the following inequalities: A x H -3 B x-3yH6 4 Solve each system of the following linear inequalities graphically: A 3y > 4 x , 2x -3y > -6 B x + y >6 , y < - 5x+ 12 , y < 5 x + 12 5 Find the maximum value of the function P = 2 x + y where: x H 0 , y H 0 , 2x + 3y G 18 , - 4x + y H -8. 6 Represent the solution set of each of the following inequalities graphically: A x+yH2 B x + 3y G 6 C 1x+3yH3 2 2 4 D 4y > 6 x +2 E 2x + 3y < 6 F 4x - 2y H 8 7 Solve each system of the following linear inequalities graphically: A y H 2 x -1 B 3 x + 2 y G 12 C y > 4 x -1 xH2 x-yG3 yG-x+4 D

xH0 yH0 x + 2y > 4 4x + y > 9

E

xH0 yH0 y-xG1 4x + 3y G 12

Work book - Second term

F

xH0 yH0 x+y<8 x + 2y < 11

23


Accumulative test 8 Sports: A merchant of sporting goods wants to buy two types of balls. The price of a ball of type A is 30 pounds and the price of a ball of type B is 60 pounds. If the merchant does not pay more than 1800 pounds to buy the balls. The gallery in his shop does not allow to show more than 50 balls. How many balls of each type could the merchant buy and show? 9 A factory produces two types of embroidered doilies, the first type needs 1 metre of cloth and 4 hours of work, and is sold for 12 pounds profit, the second type needs 1.5 metres of cloth and 2 hours of work and is sold for 8 pounds profit. If the available quantity of cloth in the factory is 150 metres and the available hours are 45 hours of work. The factory can sell all its production of doilies, find the number of doilies from each type which the factory produces, to gain a maximum profit. One of the tourism companies set up an air bridge to transport 1600 tourists and 90 tons luggage with a least cost, there were two types of airplanes A and B and the number of available airplanes of type A was 12 and the number of available airplanes of type B was 9 and the full load to the airplane of type A was 200 persons, 6 tons luggage, the full load to the airplane of type B was 100 persons, 15 tons luggage. If the rent of the airplane of type A was 32000 pounds and of type B was 150000 pounds. How many airplanes of each type could the company rent?

10 Tourism:

11

A factory produces two different kinds of alloys consisting of a mixture of iron and cast iron. The first kind has 2 units of iron and 2 units of cast iron. While the second has one unit of iron and 3 units of cast iron, given that the factory has 10 kg. of iron and 18kg of cast iron, and the selling price of an alloy of the first kind is LE 15 and the second is LE 10, then find the number of alloys of each kind the factory should produce to get the maximum income.

Do you need any Help? If you do not solve a poblem from the previous exercises, use the following table:

No of question No of lesson

24

1

2

3

4

5

6

7

8

9

10

11

2-1

2-2

2-1

2-2

2-2

2-2

2-2

2-3

2-3

2-3

2-3

Mathematics â&#x2C6;&#x2019; first form secondary


Analytic Geometry

Unit

3

Vectors

Lessons of the unit

Lesson (3 - 1): Scalars, Vectors and Directed Line Segment Lesson (3 - 2): Vectors. Lesson (3 - 3): Operations on Vectors. Lesson (3 - 4): Applications on Vectors.


Scalars, Vectors and Directed Line Segment

3-1

1 Complete the following statements to be true: A To defined Scalor quantity completely you should know ................................. B To defined Vector quantity completely you should know .................................... C The directed line segment is a line segment which has ......................, ......................,....................... D Two directed line segments are equivalent if they have................................... 2 In the figure opposite: AB is parallel to CE and each of them is not parallel to XY . join each of the following

A

C

X

statements with a suitable one .

D F

A AD

, AB

1- have the same

B

E

FX , XY

direction C DA

, ER

2- have different

D CE

, AB

Y

B

R

direction E BD

, YF

3- have opposite

F

F

E

CR , XY

direction 3 In the figure opposite ABCDEF, is a regular hexagon, its centre is M, Complete the following:

D

C

A AB

A

M

B

is equivalent to ................. and equivalent to ................. and equivalent to .................

B MD is equivalent to ................. and equivalent to ................. and equivalent to ................. C CD

is equivalent to ................. and equivalent to ................. and equivalent to .................

4 ABCD is a square , its diagonals are intersecting at M, write all directed equivalent line segments determined in the figure. .................................................................................................................................................................................................................................................

5 In a coordinate orthogonal plane : If A( 4, -3) , B (4, 4), C (-3, -1), BA , CD , OM, NO are equivalent directed line segments , where O is the origin. Find the coordinates of each of D, M , and N. .................................................................................................................................................................................................................................................

26

Mathematics - first form secondary


Scalars, Vectors and Directed Line Segment

6 On the lattice : A(3, -2), B (6, 2) , C (1, 3) , D (4, 7) : A

Find || AB || , || CD || ...................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................

B

Prove that: AB is equivalent to CD ...................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................

C

If each of the directed line segments BC , AM , ND , OR are equivalent, Find the coordinates of each of M, N and R where O is the origin. ...................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................

7 On the lattice: A(2, 3) , B (-3, 1), C (5, -1) A

Draw CD , such that it is equivalent to AB and determine the coordinates of D.

................................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

B

Determine the coordinates of the point M which is the mid-point of BC , then determine the directed line segments which are equivalent to each of : First: BM

Second: AM

Third: AC

Fourth: DB

................................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

C

Is the figure ACDB a parallelogram? Explain your answer.

................................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

Work book - Second term

27


3-2

Vectors

1 On the lattice : A( 3, -4) , B ( -12, 5) , C ( -3, -6), Find the position vector for each of the points A , B , C with respect to the origin O (0, 0), then find the norm of each of them. .................................................................................................................................................................................................................................................

2 Express each of the following vectors in terms of the fundamental unit vectors. Then find the norm of each of them: A M

= (-4, -3)

...........................................

B N

= (8, -6)

C

F

= (-5, -12)

...........................................

D A

=(0, 2

E

B

=(-3

...........................................

F C

=(

3

, 0)

2

............................................................

2

, -3

)

...........................................................

2

)

............................................................

3 Find the polar form of each of the following vectors: A M

=8

3

B N

=3

2

+8 j

i i

+3

........................................................................................................................

2

j

........................................................................................................................

4 If A = (3, -2) , B = (-2, 5) , C = (0, 11): A

Write each of the following vectors in terms of the fundamental unit vectors 1 2 B ,3 C , A + B - C , ( B + C ) 2

................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

B

Express C in terms of A

and

B

......................................................................................................................................................................................................................................

5 Find in terms of the fundamental unit vectors, the vector which expresses: A

A uniform speed of magnitude 60 km/h in west direction.

B

A force of magnitude 20 kgm wt . acts on a body in the direction 30c south of east .....

...........................................................

........................................................................................................................................................................................................................................

C

28

A displacement of a body a distance 40cm in the direction north west . .....................................

Mathematics - first form secondary


Vectors

6 If M = (5, 1), N = (4, -20), L = (-10, -2) Prove that: A

M

B M

= N

7 If M = 3 i + 4 j L A

=a i

,

-8 i

N

,

F

C N

// L

= L

= -6 i - 8 j =4 I +b j

Prove that M // N ......................................................................................................................................................................................................................................

B

Find a â&#x2C6;&#x2C6; R, if M // L ......................................................................................................................................................................................................................................

C

Find b â&#x2C6;&#x2C6; R, if F = N ......................................................................................................................................................................................................................................

D

Is F = M ? explain your answer ......................................................................................................................................................................................................................................

8 The lattice opposite of congruent parallelograms. Express each of the following directed line segments in terms of M and N . A

AB

............................

B

BY

............................

M

L C

EC

E

XE

............................

D

DE

F

XY

............................

N

G YM I

BM

K

FL

............................

............................

H

LM

............................

............................

J

EF

............................

L

FD

............................

X

Y

............................

A

D

E

F

M

B

C

............................

Activity

Draw M = (2, r )in the orthogonal coordinate plane, then represent geometrically each of 4 the following position vectors by directed line segments in the same plane: , B = -M , C = -2M A = 3M .................................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................................

Work book - Second term

29


Operations on Vectors

3-3

1 In the figure opposite: ABCD is a parallelogram , M is the intersection point of its diagonals. Complete: A AB

=

..............................

B BC =

C AB

+ BC =

..............................

D AC + CD

=

.............................

M

E BD + DC

=

..............................

F

CA

+ AD =

.............................

G AB + AD

=

..............................

H BC

+ DC =

.............................

..............................

J AM + MC =

.............................

I

DA

+ DC =

K AB + AD =

2.........................

L AD

+ CD =

2.........................

M

............................

N AB

+ 2 BM =

.............................

MA + MB =

A

D

.............................

C

B

2 In any triangle XYZ, Prove that: XY + YZ + ZX = 0 ................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

3 In any quadrilateral A B C D, prove that : AB + BC + CD = AD . ................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

A D

4 In the figure opposite: ABCD is a quadrilateral, E ∈ AB , F ∈ CD . prove that: EB + BC + CF = EA + AD + DF F

.......................................................................................................................................................................................

E

.......................................................................................................................................................................................

C

5 ABCD is a quadrilateral, if AC + DB = 2 AB prove that : ABCD is a parallelogram. ................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

30

Mathematics - first form secondary

B


Operations on Vectors

6 ABC is a triangle in which D is the mid-point of AB , E is the mid-point of AC . Prove that : AE + CD = EB + DA. ................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

7 In the triangle ABC : D , E and F are the mid-points of AB , reprectively .Prove that : AE + BF = DC

BC

,

CA

................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

8 ABCD is a trapezium in which AD // BC , AD = 2 . Prove that: AC + BD = 5 AD 3 2 BC 9 If A = 3 i - 2 j , B =- i Find:

10

-4j

A

A

+B

.....................................................................................................................................................................................

B

A

-B

.....................................................................................................................................................................................

C

|| A + B ||

.....................................................................................................................................................................................

D

2 A + 3B

.....................................................................................................................................................................................

E

A

F

- 3A

- 3B

.....................................................................................................................................................................................

.....................................................................................................................................................................................

ABCD is a parallelogram, in which A(3, 0) , B (0, 4) , D( -2, -1). Find the coordinates of the point C. .................................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................................

11

ABCD is a trapezium in which A( -2, -3) , B (4, -1) , C ( 2, 5), D( -1 , k). A

If AB // DC , find the value of k. ......................................................................................................................................................................................................................................

B

Prove that: CB = AB . ......................................................................................................................................................................................................................................

C

Find the area of the trapezium ABCD. ......................................................................................................................................................................................................................................

Work book - Second term

31


3-4

Applications on Vectors A

1 In the figure opposite: ABCD, ABXY are parallelograms. prove using vectors that the figure CXYD is a parallelogram.

D

Y C

B

......................................................................................................................................

......................................................................................................................................

......................................................................................................................................

2 In the figure opposite: A B C is a triangle in which D ∈ AB , E ∈ AC . DA = M , EA = N , BD = 2 M , CE = 2 N . Find BC in terms of M and N then prove that: BC // DE

X

A N E

2N

M D

2M

C

B

.................................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................................

3 In the triangle AB C , D ∈ BC where B D : D C = 3 : 2 prove that: 2 AB + 3 AC = 5 AD .................................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................................

4 ABCD is a quadrilateral , if AC + BD = 2 DC , prove that: ABCD is a parallelogram .................................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................................

32

Mathematics - first form secondary


Applications on Vectors

A

5 ABCD is a trapezium in which AD // BC , BC = 3AD , AD = M , AB = N

N

M

D X

First: Express in terms of M and N each of: BC

, AC , BD , DC

C

B

................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

Second: If Xâ&#x2C6;&#x2C6; DB where DX = 1 XB 3 Prove that: the points A , X , C are collinear . ................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

B

6 In the figure opposite: FAB is a triangle in which FA = 7 cm , FB = 5 2 cm, m (d A F B) = 135c. 5 2 cm Find using vectors the length of AB

135c

F

7cm

.................................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

7 If A(5, 1) , B (2, 5) , C (-2, 3) , D (-5 , -4) Prove using vectors that the figure ABCD is a trapezium. ................................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

8 If A(6, 5), B(8, -3), C (-2, -5) are vertices of the triangle ABC, find using vectors the coordinates of the point of intersection of its medians. . ................................................................................................................................................................................................................................................ ................................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

Work book - Second term

33

A


Activity (1)

First: Write in terms of the unit vector U the resultant of the forces shown in each figure A

B

u

u 12 newtons

C u

20 newtons

20 kg wt

80 gm.wt

25 kg wt 30 gm.wt

......................................................................................................................................................................................................................................

Second: In each of the following: the two forces F1 , F2 act at a particle, show the magnitude and the direction of the resultant of each two forces. A

F1 = 15 Newtons acts in the east direction, F2 = 40 newtons acts in the west direction. ......................................................................................................................................................................................................................................

B

F1 = 34 g m wt acts in the north east direction, F2 = 34 gm. wt. acts in the south west direction. ......................................................................................................................................................................................................................................

C

F1 = 50 Dyne acts in 60c west of the north direction, F2 = 50 Dyne acts in 30c south of the east direction. ......................................................................................................................................................................................................................................

D

F1 = 30 newtons acts in 20c east of the north direction , F2 = 30 newtons acts in 70c north of the east direction north. ......................................................................................................................................................................................................................................

Third: Forces: F1 = 7 i - 5 j

, F2 = a i + 3 j , F3 = -4 i + (b - 3) j act at a particle - Find the values of a , b If: A the resulatant of a set of forces equals 4 i - 7 j B Set of forces are in equilibrium. ...................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................

Activity (2)

9 A controlling speed car (Radar) moves on the desert road at 40 km/h. It watched a car coming from the other opposite road which seemed to be moving at 135 km/h. If the maximum available speed on the road is 100 km/h. Is the coming car in violation of the prescribed speed ? Explain your answer. .................................................................................................................................................................................................................................................

34

Mathematics - first form secondary


General Exercises 1 On the lattice O (0, 0)is the origin point. Determine the points A(-4, 0) , B (0, -3), C (3, 1) , D (2, 8), then find: A

The position vector of the points A, B, C in terms of the fundamental unit vectors . B The position vector of the point D with respect to the origin point O in the polar form. C The norm of the directed line segment AB . D The value of K which makes AD = K BC . ................................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

2 Find in terms of the fundamental unit vectors the vector which expresses: A

A force of magnitude 20 newtons acts at a particle vertically in the north direction. ............. B A displacement of a body moves a distance 50 cm in 30c north of the west direction . .......... C The uniform velocity of a car covers a distance 70 km/h in the west direction................... 3 If A = (4, -6) , B = (-6, 9) , C = (-3, -2) A

Prove that: A // B , B = C , C = A B Find: 2 A + B , B - 2 C , 1 A + B - 3 C 2 ......................................................................................................................................................................................................................................

4 If ABCD is a parallelogram, in which A (2, -2) , B (4, -2) , C (2, 3) Find the coordinates of the point D. .................................................................................................................................................................................................................................................

5 Using vectors prove that : the diagonals of the parallelogram bisect each other. .................................................................................................................................................................................................................................................

6 On the lattice , AB = (-2, 3) , CB = (-6, -4), 2 B + AC = (6, 11). Find: First: the coordinates of each of the points A , B and C Second: Area of the triangle ABC (using vectors). ................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

Work book - Second term

35


Unit Test 1 On the lattice, O (0, 0) is the origin point , if A (1, -4) , B(4, 0) , C (-2, 1), D(1, 5). A

Find || AB ||, || CD ||. ................................................................................... B Prove that: AB is equivalent to CD . ............................................................................................................. C If BC is equivalent to ED , find the coordinates of E. .................................................................................. 2 If A = (4, 1) , B = (-1, 6) , C = (1, 12): Find in terms of the fundamental unit vectors each of B - C , 1 ( B + C ). 2 B Express C in terms of A and B . A

......................................................................................................................................................................................................................................

3 If M = i + 2 j , N = - 5 i - 10 j , L = a i + 10 j , and F =2 i + b j A

Prove that: M // N B Find a â&#x2C6;&#x2C6;R, If M // L C Find b â&#x2C6;&#x2C6;R, If F = N . D Is F = M ? Explain your answer

........................................................................................................................ ........................................................................................................................ ........................................................................................................................

4 In the figure opposite: ABCD a parallelogram. Complete : A

AB

+ BC

= .............................

B

DA

+ DC

= .............................

C

AM + CM

= .............................

D

AB

+ 2 BM

= .............................

E

AB

+ ....................

= 2 AM

F

AB

- AM

= .............................

D

A M

C

B

5 If AB = (1, -4), A (2, 3) , C ( -1, 15), find the values of l , m if: l A - m B = C ................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

6 ABCD is a rectangle in which A (-1, -2), B (5, 1) , C (3, K). find: First: the value of K

Second: the coordinates of the point D

................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

36

Mathematics - first form secondary


Accumulative Test Multiple choice questions 1 The slope of the line passing through the two points A (3, 4) , B ( -1, 2) equals A

B

-2

2

C 1 2

D

2 In the triangle A B C: m(d A) = 90c, cos C = 0.6 then tan B equals: A

B 3 4

0.4

C 4 3

D

4

...................................................

-4 3

9cm

B

8 2 C 4 2

-1

A

3 In the figure opposite: AD = BC , BC = 12cm then AB equals ........................................................................................................................................ A

........................

6 3 D 10

C

B

D

4 In the figure opposite: All the following statements express AC Except the statement: A

D A

B

2 AM C AB + BD

+ DC D BC + DC AD

M

C

B

5 The vector M = (12 2 , r ) in terms of the fundamental unit vectors expresses the 4 vector: ............................................................................................................................................................................................................................ A

B

6 i +6 j C -6 i - 12 j

12 i -12 j D 12 i +12 j

Short answer questions: A

6 In the figure opposite: DE // BC Find the numerical values of K , L, M , N if : A BD = K DA ........................................................................................................ B CE = L CA ........................................................................................................ C BC = M ED ........................................................................................................ C D AD + DE = N AC ........................................................................................................

3 cm E

2 cm D 3 cm B

7 If A = (3, -2) , B = (2, 7) find : A

+B B B - A C 2A - 3 C A

..................................................................................................................................................................................... ..................................................................................................................................................................................... .....................................................................................................................................................................................

Work book - Second term

37


Accumulative Test 8 ABC is a triangle. Prove that AB + BC + CA = 0 ................................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

9 ABCD is a parallelogram, its diagonals are intersecting at M. if Nâ&#x2C6;&#x2C6; AD prove that : NA + NB + NC + ND = 4 NM. ................................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

10

On the lattice: the origin point is O (0, 0), If A (-1, -4) , B (1, 1) , C (6, -1) A

Find each of AB , BC in terms of the fundamental unit vectors ......................................................................................................................................................................................................................................

B

Prove that: AB = BC ...................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................

Long answers questions 11

On the lattice: If L = 2 i - 2 3 j , M = - i +4 3 j , N = 3 i +2 3 j . Find C in the polar form where C = L + M + N .................................................................................................................................................................................................................................................

12

Prove that: the points A (5, 1), B (7, 2), C (4, -4) and D(-1, -2) are vertices of a square , and find its area

No of question

1

2

3

4

5

6

7

8

9

10

11

12

No of lesson

skills

skills

skills

3-3

3-2

3-4

3-2

3-3

3-3

3-2

3-2

3-4

If you can not solve any problem from the previous exercises , use the following table:

38

Mathematics - first form secondary


Analytic Geometry

Unit -

4

Straight Line

Lessons of the unit

Lesson (4 - 1):

Dividing a line segment.

Lesson (4 - 2):

Equation of the straight line.

Lesson (4 - 3):

Measure the angle between two straight lines.

Lesson (4 - 4):

Length of the perpendicular drawn from a point to a straight line.

Lesson (4 - 5):

General equation of the straight line passing through the point of intersection of two lines.

Work book - Second term

39


4-1

Dividing a line segment

First : complete each of the following 1 In the figure opposite: AD is a median in 9 ABC , M is the intersection point of the medians, where A(0, 8), B(3, 2), C(-3, 5)

(0, 8)

A

M A

The coordinates of the point D is (................., ................) B The coordinates of the point M is (................, ................)

B C (-3, 5)

E

(3, 2)

y 2 In the figure opposite: If A (2, 3), B (-1, 7), C and D are B (-1, 7) points on the coordinate axes E (0, y) A C divides AB ...................... by the ratio ...................... : ...................... B D divides AB ...................... by the ratio ...................... : ...................... A (2, 3) C The coordinates of the point C is (..................., ..................) C (x, 0) D The coordinates of the point D is (..................., ..................) O x

Second

: Answer the following questions

3 If A(8, -4), B(-1, 2), find the coordinates of the points which divide AB into 3 equal parts .......................................... , .......................................... 4 If A(3, 1) , B(- 2, 5), find the coordinates of the point C which divides AB internally by the ratio 2 : 3. ...................................................................................................................................................................................................... 5 If A(1, 3), B(-4, -2), find the coordinates of the point C where C ∈ AB where 3AC = 2CB .................................................................................................................................................................................................................................................

6 If A(2, 5) , B(7, -1), find the coordinates of the point C which divides AB externally by the ratio 3 : 2 .................................................................................................................................................................................................................................................

7 If C ∈ BA , C∉ AB and A(3, 1), B(4, 2) , AC = 2AB. find the coordinates of the point C. .......... 8 If A, B and C are three collinear points where A(2, 5), B(5, 2), C(4, y). Find the ratio by which the point C divides the directed line segment AB showing the type of division, then find the value of y. ........................................................................................................... , .................................................................................................................................

40

Mathematics - first form secondary


Equation of the straight line

4-2

First: Complete each of the following 1 If the straight line passes through the points (3, 0) , (0, 2) is parallel to the straight line whose equation y = ax – 3, then a equals ................................................................................................................................ 2 The vector equation of the straight line passes through the point (3, 5) and parallel to the x-axis is ............................................................................................................................................................................................................. 3 The cartesian equation of the straight line passes through the point (-2, 7) and parallel to the y-axis is ...................................................................................................................................................................................................... 4 The vector equation of the straight line passes through the origin point and the point (1, 2) is .......................................................................................................................................................................................................................... 5 The equation of the straight line which makes 45c with the positive direction of the x-axis and cuts 5 units from the positive part of the y-axis is ............................................................................ 6 The cartesian equation of the striaight line which cuts the positive parts of the x-axis and the y-axis with magnitudes 2, 3 respectively is .................................................................................................... 7 Area of the triangle enclosed by the x-axis and the y - axis and the striaight line 2x + 3y = 6 equals ..................................................................................................................................................................................... Second

: Answer the following questions

8 If A (3, -2), B (5, 6), C (1, -2) Find the slope of each of the following straight lines: AB , AC , BC ...................................................., .............................................................., ........................................................................... 9 If the equations of the straight lines L1 , L2 are 2x – 3y + a = 0 and 3x + by – 6 = 0 respectively , Find: A The slope of the straight line L ............................................................................................................................................. 1 B The value of b which makes the two lines L , L parallel equals ....................................................... 1 2 C The value of b which makes the two lines L , L perpendicular equals ....................................... 1 2 D If the straight line passes through the point (1, 3), then find the value of a. .......................... 10

If the straight line whose equation ax – 4y + 5 = 0 makes an angle whose tangent is 0.75 with the positive direction of the x-axis, then find the value of a . ............................................................. Work book - Second term

41


11

Find the vector equation of the straight line whose slope 1 and passes through the point 3 (2, -1). ............................................................................................................................................................................................................................

12

Find the parametric equations of the straifht line which makes 45cwith the positive direction of the x-axis and passes through the point (3, -5). .............................................................................

13

Find the vector equation of the straight line which passes through the points (2, -3), (5, 1) .................................................................................................................................................................................................................................................

14

Find the general equation of the straight line which passes through the points (5, 0), (0, -7) .................................................................................................................................................................................................................................................

15

If A(0, 2), B(2, 1), C(-2, 3) are three points in the plane, find the vector equation of the straight line AB , then prove that the points A, B and C are collinear . .................................................................................................................................................................................................................................................

16

If A(5, -6), B(3, 7), C(1, -3), find the equation of the straight line which passes through the point A and bisects BC . .................................................................................................................................................................................................................................................

17

Find the cartesian equation of the straight line which passing through the point (3, -5) and parallel to the straight line whose equation is x + 2y – 7 = 0 .................................................................................................................................................................................................................................................

18

Find the vector equation of the straight line which passing through the point (5, 7) and is perpendicular to the straight line whose equation is r = (3 , 0) + K(4 , 3) .................................................................................................................................................................................................................................................

19

If A(1, 4) , B(- 4, 6), find the equation of the straight line which passing through the point of division of AB internally by the ratio 2 : 3 and is perpendicular to the straight line whose equation is 5x – 4y – 12 = 0 .................................................................................................................................................................................................................................................

is a diameter in the circle M, if B(- 7 , 11) , M(- 2 , 3), find the equation of the tangent to the circle at the point A.

20 Geometry: AB

.................................................................................................................................................................................................................................................

21 Geometry: If the straight line whose equation 3x + 4y – 12 = 0 intersects the x-axis and

the y-axis at A and B respectively. Find: A Area of 9 OAB where O is the origin point. ........................................................................................................... B The equation of the straight line which is perpendicular to AB and passes through its midpoint. .................................................................................................................................................................................................

42

Mathematics - first form secondary


Measure the angle between two straight lines First

4-3

: Complete each of the following

1 Measure of the angle between the straight lines whose slopes 2 , - 1 equals .................................... 2

2 Measure of the angle between the straight lines whose equation x = 3 , y = 4 equals

...........

3 Measure of the acute angle between the straight line whose equation r = (2, 2) + t(1,1) and the straight line whose equation x = 0 equals ......................................................................................................... 4 If the two straight lines whose equations ax + 3y – 7 = 0 and 2x – 3y + 5 = 0 are parallel, then a equals ............................................................................................................................................................................................................ 5 If the two straight lines whose equations ax + 7y – 9 = 0 and 7x – 2y + 12 = 0 are perpendicular then a equals .................................................................................................................................................................... Second : Activity The figure opposite shows a triangular piece of land, its vertices are A (6, 0) , B (-6, 0) , C (0, 6). Complete each of the following: 6 Measure of the acute angle between AC and the x-axis equals

7 6 5 4 3 2 1 B ................... -6 -5 -4 -3 -2-1

Y C

D

A 1 2 3 4 5 6

7 Measure of the angle between the straight lines AC and BC equals ............................... 8 The vector equation of the straight line AC is ............................................................................................ 9 The vector equation of the straight line BC is

..................................................................................................................

10

The cartesian equation of the straight line which passes through the point C and parallel to AB is ......................................................................................................................................................................................................................

11

Area of the triangle ABC equals

Third

.......................................................................................................................................................

: Choose the correct answer from the given answers

12

Measure of the acute angle between the straight line which passes through the points (0, 1) , (-1, 0) and the positive direction of the x-axis equals: ........................................................................ A zeroc B 45c C 60c D 90c

13

Measure of the acute angle between the straight line whose equation r = (0, 3) + t(1, 1) and the straight line whose equation x = 0 equals: .......................................................................................................... A 30c B 45c C 60c D 90c Work book - Second term

43

X


14

Measure of the acute angle between the straight lines whose equation 3 x – y = 4 and y = 3 equals .............................................................................................................................................................................................................. A 30c B 45c C 60c D 90c

15

The straight line perperpendicular to the straight line whose equation r = (0, 5) + t( 3 , 1) and makes an angle of measure .................... with the positive direction of the x-axis A 30c B 60c C 120c D 150c

Fourth

: Answer the following questions

16

Find the measure of the acute angle between each of the following pairs of the straight lines whose equations are: A r = (5, 0) , x – y + 4 = 0 ............................................................................................................. B r = (0, 1) + t(1, 1) , 2x – y – 3 = 0 ............................................................................................................. C y- 3 x–5=0,x- 3 y–6=0 .............................................................................................................

17

Prove that the triangle ABC is right at B where A(5 , 2) , B(2 , -2) , C(- 2, 1), then calculate the area of its surface. .........................................................................................................................................................

18

If i is the measure of the acute angle between the straight lines whose equations x – 6y + 6 = 0 , ax – 2y + 4 = 0 and tan i = 3 , then find the value of a. ................................................. 4

x

y

19

If L1: ax – 3y + 7 = 0 , L2 : 4x + 6y – 5 = 0 , L3 : 3 - 2 = 3, then find the value of a which makes: A L // L ................................................................................................................................................................................................................. 1 3 B L = L .............................................................................................................................................................................................................. 1 2

20

If the measure of the acute angle between the straight lines whose equations x + ky – 8 = 0 , 2x – y – 5 = 0 equals r , then find the value of K. ............................................................. 4

21

If the triangle ABC is right at B where A(2, 3) , B(5, 7) , C(1, y) , find the value of y , then find the measure of the other two angles. .............................................................................................................

22

9 ABC is a triangle in which A(5, 7) , B(1, 5) , C (4, 2) A Find the coordinates of the point D which divides BC internally by the ratio 1 : 2 B Prove that AD = BC ........................................................................................................................ C Prove that AD = BC ........................................................................................................................ D Find the measure of the acute angle B ........................................................................................................................ E Find area of the triangle ABC. ........................................................................................................................

44

Mathematics - first form secondary


Length of the perpendicular drawn from a point to a straight line

4-4

Activity

First

y

: Complete each of the following:

C

6 5 4

1 The figure opposite shows karim's house A (2, 0) and the 3 school B (7, 0) and the mosque C (4, 6): Complete each of 2 the following: 1 A The equation of AB is .......................................................................................... B A 1 2 3 4 5 6 7 O x B The length of AB equals ................................................................................... C Shortest distaence between the Mosque C and the road from the house to the school equals .................................................................................................................................................................................................................. D Measure of the acute angle between the straight lines AC and Y = 0 equals ....................... E Area of (9 ABC) equals ................................................................................................................................................................. Second

: Multiple choice

2 Length of perpendicular from the point (- 3 , 5) on the y-axis equals A

B

2

3

C

D

5

....................................................

8

3 The distance between the straight lines whose equations y â&#x20AC;&#x201C; 3 = 0 , y + 2 = 0 equals A 1 B 2 C 3 D 5

...........

4 Length of perpendicular from the point (1 ,1) to the straight line whose equation x + y = 0 equals ................................................................................................................................................................................................................................... A 1 B C 2 D 2 2 2 5 If the length of perpendicular drawn fron (3 , 1) to the straight line whose equation 3x - 4y + c = 0 equals 2 unit of length, then C equals ............................................................................................... A

B

Zero

3

C

D

5

7

6 Find the length of the perpendicular drawn from A to the straight line L in exercises A - D A

A(0,0)

,

L : r = (0 , 5) + t (3 , 4)

.................................................................................................

B

A ( 2 , - 4)

,

L : 12x + 5y â&#x20AC;&#x201C; 43 = 0

................................................................................................

Work book - Second term

45


C

A(5,2)

,

L : 8x + 15y – 19 = 0

D

A (- 2 , - 1)

,

L : r = (0 , - 7) + t (1 , 2) ................................................................................................

................................................................................................

7 Find the length of the radius of the circle whose centre (- 2 , 5), and touches the straight line whose equation 3x + 4y + 1 = 0 .................................................................................................................................................................................................................................................

8 Find the distance between (1, 5) and the straight line passing through the points (5, -3) , (1, 0) .................................................................................................................................................................................................................................................

9 Prove that the straight lines whose equations 3x – 4y – 12 = 0 and 6x – 8y + 21 = 0 are parallel , then find the distance between them. ................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

10

If A(4 , 3) , B(-2 , 5) , C(- 1 , -2) are vertices of the triangle ABC , Draw BD = AC . A

Prove that 9 ABC is an isosceles triangle

B

Find the equation of BD

C

Find the length of BD

................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

11

ABCD is a parallelogram, if A(- 3 , 2), B(2 , 3) , C(5, 7). Find the coordinates of the vertex D, then find the area of the parallelogram . ................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

12 Geometry: Acircle of centre the origin point in which two chords whose equations

4x – 3y + 10 = 0, 5x – 12y + 26 = 0. Prove that the two chords are equal in length. .................................................................................................................................................................................................................................................

13 Geometry: ABCD is a trapezium in which AD

// BC , if A(2, 1), B(5, 3), C(6, 1), D(4, y). Find the value of y, then find the area of the trapezium ABCD.

46

Mathematics - first form secondary


4-5

General equation of the straight line passing through the point of intersection of two lines

1 Find the vector equation of the straight line which passes through the origin point and the two straight lines whose equation x = 3 , y = 4 .................................................................................................................................................................................................................................................

2 Find the vector equation of the straight line which passes through the point (3 , 1), and the point of intersection of the two lines whose equations 3x + 2y – 7 = 0, x + 3y = 7 ............................. 3 Find the equation of the straight line passes through the point of intersection of the two straight lines whose equations r = k(-3 , 2), 3x - 2y = 13 and parallel to the y-axis. ........ 4 Find the equation of the straight line passes through the point of intersection of the two lines whose equations 2x + y = 5 , x + 5y = 16 and perpendicular to the line whose equation x – y = 8 .............................................................................................................................................................................................. 5 Find the equation of the straight line passes through the point of intersection of the two lines whose equations 2x – 7y + 9 = 0 , 3x + 2y – 4 = 0 and perpendicular to the first line.

.

6 Find the equation of the line passes through the point of intersection of the two lines whose equations 2x + 3y – 2 = 0, 3x – y – 14 = 0 and makes an angle of measure 135c with the positive direction of the y-axis ............................. Activity

7 Life: Two straight roads, the equation of the first is 3x - 4y – 14 = 0 , equation of the second is 4x + 3y - 2 = 0 Prove that the two roads are perpendicular, then find :

y 4 3 2 1 -3 -2 -1 -1

A

1

2 3 4

5 6 x

Their point of intersection ................................................................................... -2 B The equation of the line which passes through the point -3 of intersection and the point (3, -2) .......................................................... -4 y´ C The shortest distance from the point of intersection of the two roads and other road whose equation 4x + 3 = 0 ...................................................................................... D The area of the triangular region enclosed by the two roads and the y-axis .........................

Work book - Second term

47


General Exercises First : Complete each of the following: 1 The slope of the line whose equation 2x - 3y + 5 = 0 equals ............................................................................. 2 The direction vector perpendicular to the line whose equation r = (2, -1) + t(3, -5) is

..

3 The equation of the line which makes an angle of measure 135c with the positive direction of the x-axis and passes through the point (4, 0) is ................................. 4 The line whose equation 3x + 4y - 12 = 0 intersects the two axes at the points

...........................

5 Measure of the acute angle between the line passes through (0, 2), (-2, 0) and the line whose equation y = 0 equals ................................................................................................................................................................. Second; Multiple choice: 6 Measure of the acute angle between the two lines whose equations: x - 3y + 5 = 0, x + 2y - 7 = 0 equals ........................................................................................................................................................................................ A

15c

B

30c

C

45c

D

60c

7 The equation of the line which passes through the point (2, -3) and parallel to the x-axis is: ...................................................................................................................................................................................................................... A x+3=0 B y+3=0 C x-2=0 D y-3=0 8 Length of perpendicular drawn from the origin to the line whose equation 3x - 4y - 15 = 0 equals ................................................................................................................................................................................................................................... A 3 B 4 C 5 D 15 9 All the following equations represent the equation of the line which passes through the points (3, 0), (0, 2) except the equation: ................................................................................................................................... A r = (3, 0) + t (3, -2) B r = (0, 2) + t (3, -2) C r = (3, 0) + t (2, 3) D r = (0, 2) + t (-6, 4) 10

Length of perpendicular drawn from the point (0, -5) to the straight line whose equation x + 7 = 0 equals .................................................................................................................................................................................................... A 2 B 5 C 7 D 12

Third: 11

If A(5, 2), B(-3, 1), find the ratio by which AB is divided by the x-axis, showing the type of division. .........................................................................................................................................................................

12

Find the length of perpendicular drawn from A(2, 5) to the line whose eqaution ......................................................................................................................................................................... r = (1, -2) + t (3, 4).

13

Prove that the triangle ABC is right at B where A(5, 2), B(2, -2), C(-2, 1) then calculate its area .....................................................................................................................................................................................

48

Mathematics - first form secondary


Unit test First : complete questions 1 The direction vector perpendicular to the line whose equation 8x - 7y + 12 = 0 is .................. 2 If the two lines whose equations: 2x + by + 3 = 0 , r = (0, 2) + t (1, 3) are parallel , then b equals ........................................................................................................................................................................................................... 3 The vector equation of the line passes through the origin and the point (2, 1) is ........................ 4 The parametric equations of the line which makes an angle of measure 45c with the positive direction of the x-axis and passes through (0, 5) is .............................................................................. 5 Equation of the line which passes through the origin and the point of intersection of the two lines whose equations x = 3 , y = 3 is ................................................................................................................... Second : multiple choice questions: 1 Mid point of AB where A(3, 7), B(1, 5) is: ............................................................................................................................ A (3, 5) B (5, -2) C (2, 5) D (2, 6) 2 Equation of the line passes through the point (2, -3) and parallel to the x-axis is: ................... A x = -3 B y = -3 C x=2 D y=3 3 Measure of the acute angle between the two lines whose equaiton x - 3y + 5 = 0, x + 2y - 7 = 0 equals: ..................................................................................................................................................................................... A 45c B 60c C 120c D 135c 4 Length of perpendicular from the origin to the line whose equation r = (5, 0) + t (4, 3) equals .............................................................................................................................................................................................................................. A 3 B 4 C 5 D 15 5 Equation of the line passing through the points (3, 0), (0, 2) is: A 2x + 3y = 6 B 3x + 2y = 6 C 2x + 3y = 0

....................................................................

D

3y + 2x = 0

Third : Answer the following questions: 1 A ABCD is a parallelogram in which A(2, 2), B(3, 8), C(8, 10), find the coordinates of the point D. B Find the equation of the line passes through the point (7, 4) and parallel to the line whose equation 3x = 2y. 2

A

Prove that the points A (1, 4), B (3, -2), C (-3, 16) are collinear, then find the ratio by which the point B divides AC , showing the type of division . B Prove that the two lines L : 2x + y - 4 = 0, L : 2x + y + 2 = 0 are parallel, and find 1 2 the distance between them .

3 A Find the length of perpendicular from (8, -2) to the line whose equation: r = (0, 1 ) + t(-3, 4). 3 B

4

If A (-1, 4), B (5, -2), find the coordinates of the point C which divides AB internally by the ratio 1 : 2.

A

If A (2, 4), B (-7, 5), C (-3, 0), D (-2, 9) are the vertices of a square. Find the area of its surface. B Find the equation of the line passing through the point of intersection of the two lines 2x + y = 5, x - y = 1 and the point (5, 3). Work book - Second term

49


Accumulative Test 1 Find the measure of the acute angle between the two lines x + 2y - 5 = 0 , x - 3y + 2 = 0. 2 If A(3, -2), B(-2, 4), find the ratio by which C(8, y) divides AB , then find the value of y. 3 Prove that the triangle whose vertices A(-4, 2), B(1, -2), C(1, 6) is an isosceles, then find its area. 4 Find the equation of the line passes through the point (2, 5) and perpendicular to the line whose equation 2x + 3y + 7 = 0. 5 Prove that the two lines whose equations r = (0, 7 ) + t (3, 4) and 4x - 3y + 17 = 0 are 3 parallel , then find the distance between them. 6 Find the equation of the line passing through the point of intersection of the two lines 2x + 3y - 5 = 0 , x + 3y = 7 and the point (2, 4). 7 Prove that the points A (-2, 5), B(3, 3), C(-4, 2) are non collinear. If D (-9, 4), prove that the figure ABCD is a parallelogram. 8 If L1: 3x + 4y - 5 = 0, L2 : r = (0, 3) + t (1, -1), find each of : A Measure of the acute angle between L , L . 1 2 B Find the length of perpendicular drawn from (0, -2) to the line L . 1 9 Prove that the points A (2, 3), B (-1, -1), C (3, -4), D (6, 0) are the vertices of a square, then the area of its surface. 10

ABC is a triangle in which A (1, 2), B (5, 2), C (7, 5), find A The cartesian equation to the line BC . B The length of perpendicular from A to BC . C The area of the triangle ABC.

11

Prove that the line: 3x - 4y + 3 = 0 touches each of the two circles whose centres are N1 (5, 2) , N2 (-2, 3) and whose radii are of the lengths 2, 3 unit of length respectively. Are the two circles lie on one side or on two sides of the straight line?

Do you need any help? If you can not solve a problem from the previous excercies, you can use the following table:

No of question No of lesson

50

1

2

3

4

5

6

7

4-3

4-1

4-4

4-2

4-4

4-2

4-2

8

9

4-3

4-4

4-4

Mathematics - first form secondary

4-2

A

10 B

C

4-2

4-3

5-7

11 4-3 4-4


Unit

5

Trigonometry

Lessons of the unit Lesson (5 - 1): Trigonometric Identities. Lesson (5 - 2): Solving Trigonometric Equations. Lesson (5 - 3): Solving the Right Angled Triangle. Lesson (5 - 4): Angles of Elevation and Angles of Depression. Lesson (5 - 5): Circular Sector. Lesson (5 - 6): Circular Segment. Lesson (5 - 7): Areas.


5-1

Trigonometric Identities

First: Multiple choice 1 tani coti equals: ......................................................................................................................................... in the simplest form: seci

A

sin i

B

cos i

C

sec i

D

csc i

2 sin i cos i tan i equals....................................................................................................................... in the simplest form A

sin2 i

B

cos2 i

C

tan2i

D

1 â&#x20AC;&#x201C; sin2 i

3 sin(90c - i) csc(90c - i) equals : ............................................................................................... in the simplest form A

B

1

sin2 i

C

cos2 i

D

sin i cos i

2 4 1 -2cos b equals : ........................................................................................................................................ in the simplest form sin b - 1

A

- tan2 b

B

- cot2 b

C

tan2 b

D

cot2b

Second: Answer the following questions 5 Prove the validity of the following identities: A

tann + cot n = sec n csc n

B

csca - sina = cos a cot a

C

cot2n - cos2 n = cot2 n cos2 n

D

tan2a - sin2a = tan2a sin2a

E

sin2a + tan2a sin2a = tan2a

F

sin (90c â&#x20AC;&#x201C; n) cosn = 1â&#x20AC;&#x201C; sin2n

6 Prove the validity of the following identities: A

csc i cos i

C

1 1 + tan2a

E

G

52

B

1 sin2(90c- i)

- 1 + tan2b = cos2a - cos2b

D

1 + tan2 i sec4i

1 - sinz 1 + sinz

F

1 1 + cot i

(1 â&#x20AC;&#x201C; sin2i) = coti 1

(sec z - tan z)2 =

cos2 i - sin2i sini cos2i + cosi sin2i

= csc i - sec i

Mathematics - first form secondary

- tan2 i = 1

= 1 - sin2i =

tani 1 + tani


Solving Trigonometric Equations

5-2

First : Complete each of the following 1 The general solution of the equation cos i = 1 for all values of i is ........................................................ 2 The general solution of the equation sin i = 1 where i ∈ [r, 2r[ is..................................................... 3 The general solution of the equation sin i = cos i for all values of i is .............................................. 4 The solution set of the equation cot i =

3

where i ∈ [r, 2r[ is ..........................................................

Second : Multiple choice 5 If 0c G i < 360c , sin i + 1 = 0, then i equals ................................................................................................................ A 0c B 90c C 180c D 270c 6 If 0c G i < 360c , cosi + 1 = 0, then i equals .................................................................................................................. A 90c B 180c C 270c D 360c 7 If 0c G i < 180c , 3 tan i - 1 = 0, then i equals ....................................................................................................... A 30c B 60c C 120c D 150c 8 If 180c G i < 360c , 2 cosi + 1 = 0, then i equals ...................................................................................................... A 210c B 240c C 300c D 330c Third: Answer the following questions 9 Find the general solution for each of the following equations.

10

A

sini = 1

B

2cosi -

C

3

........................................................................................................................

2 3

=0

........................................................................................................................

tan i – 1 = 0

........................................................................................................................

3r Solve each of the following equations in the interval [0, 2 [: A

tan2 i - tani = 0

........................................................................................................................

B

2sin i cos i - cos i = 0

........................................................................................................................

C

2sin2 i - 3sini - 2 = 0

........................................................................................................................

Work book - Second term

53


5-3

Solving the Right Angled Triangle

1 Find the value of x, y in each of the following figures A

B

27째

12 cm

7 cm

C

y

x

y

y

56째

x

8 cm

47째

x ....................................................

....................................................

...............................................

....................................................

....................................................

...............................................

2 Find the value of each of the angles a, b in degree measure in each of the following figures: B 5 cm

b

a

13 cm

a

C

a

b

5 cm

A

5 cm 6 cm

b 3 cm

....................................................

....................................................

...............................................

....................................................

....................................................

...............................................

3 Solve the triangle ABC right at B approximating the angles to the nearest degree and the length to the nearest cm where: A

AB = 4 cm, BC = 6 cm

B

AB = 12.5 cm, BC = 17.6 cm

C

AB = 5.3 cm, AC = 12.2 cm

D

BC = 31 cm, AC = 42 cm

................................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

54

Mathematics - first form secondary


Solving the Right Angled Triangle

4 Solve the triangle ABC right at B approximating the angles to the nearest thousandth in radian measure, and the length to the nearest thousandth cm where: A

m (dA) = 0.925rad . BC = 8 cm

B

m (dA) = 1.169rad . AB = 18 cm

C

m (dC) = 0.646rad . AC = 15.7 cm

D

m (dC) = 1.082rad . AC = 35.8 cm

................................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

5 ABC is a triangle, draw AB = BC , if AD = 6 cm, m(dB) = 52c , m (dC) = 28c, find the length of BC to the nearest centimetre. ................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

6 Geometry: AB is the radius of a circle of length 20 cm draw the chord AC of length 12 cm . Find measures of angles of the triangle ABC. ................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

7 Geometry: Apiece of land is in the shape of a rhombus ABCD of side length 12 metres, m (dABC) = 100c. find the length of each of its diagonals AC and BD to the nearest metre. ................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

8 Geometry: ABCD is an isosceles trapezium in which AD // BC , AB = CD = 5cm, AD = 4cm, BC = 10cm. Find the measure of each of its four angles. ................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

Work book - Second term

55


Angles of Elevation and Angles of Depression

5-4

1 The length of the thread of a kite is 42 metres, if the angle which the thread makes with the horizontal ground equals 63c, Find to the nearest metre the height of the kite from the surface of the ground. ......................................................................................................................................................................... 2 A man found that the angle of elevation of the top of a minaret 42m distance from its base was 52c. What is the height of the minaret to the nearest metre? ..................................................... 3 A mountain of height 1820 metres, It is observed from its top that the measure of depression angle of a point on the ground was 68c. What is the distance between the point and the observer to the nearest metre? ........................................................................................................................ 4 The upper end of a ladder rests on a vertical wall, it is 3.8m from the surface of the ground, the lower end rests on a horizontal ground. If the angle of inclination of the ladder to the ground is 64c. Find to the nearest hundredth each of: A The distance between the lower end and the wall B The length of the ladder ................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................

5 From the top surface of a house 8 metres high, a person found that the elevation angle of the top of an opposite building was of measure 63c, and observed the depression angle of its base, it was 28c. Find the height of the building to the nearest metre. .................................................................................................................................................................................................................................................

6 If the measure of the elevation angle of the top of a minaret from the point 140 metres distance from its base was 26c 46' . What is the height of the minaret to the nearest metre? If it is measured from 110 metres distance from its base, find to the nearest minute, the measure of its elevation angle at that distance. ................................................................................ 7 An observer measured the angle of elevation of a fixed ballon to be r , when he walked 6 in a horizontal plane towards the ballon a distance 800 metres, he measures its angle of elevation to be r . Find the height of the ballon to the nearest metre. .................................................... 4

8 A ship approaches a light house 50 metres high at a moment, it was found that the elevation angle of the top of the light house to be 0.11rad, after 15 minutes, it was found again that its elevation angle to be 0.22rad. Calculate the uniform velocity of the ship. .................................................................................................................................................................................................................................................

56

Mathematics - first form secondary


Circular Sector

5-5

First : Complete each of the following 1 Area of the circular sector, in which l = 6 cm, r = 4 cm equals ....................................................................... 2 Area of the circular sector, in which r = 4 cm, and its perimeter 20 cm equals ................ cm2. 3 Perimeter of the circular sector in which , its area 24 cm2, length of its arc 8 cm equals .................... Second : Multiple choice 1 Area of the circular sector in which , measure of its angle 1.2rad and length of the radius of its circle 4 cm equals ................................. A 4.8 cm2 B 9.6 cm2 C 12.8 cm2 D 19.6 cm2 2 Perimeter of the circular sector in which length of its arc 4 cm and length of the diameter of its circle 10 cm equals ................................. A 14 cm B 20 cm C 30 cm D 40 cm 3 Area of the circular sector in which, measure of its angle 120c, length of the radius of its circle 3 cm equals ................................. A 3 r cm2 B 6 r cm2 C 9 r cm2 D 12 r cm2 4 Area of the circular sector in which, its perimeter 12 cm, length of its arc 6 cm equals .................. A 6 cm2 B 9 cm2 C 12 cm2 D 18 cm2 5 If the area of the circular sector equals 110 cm2, measure of its angle equals 2.2rad, then length of the radius of its circle equals: ............................... A 2 cm B 5 cm C 10 cm D 20 cm Third : Answer the following questions 1 Find the area of the circular sector in which, length of the diameter of its circle is 20 cm and measure of its angle is 120c. ........................................................................................................................................................ 2 Find the area of the circular sector in which length of its arc is 16cm, and length of the radius of its circle is 9cm. .......................................................................................................................................................................... 3 Find the area of the circular sector in which , length of its arc is 7cm and its perimeter equals 25cm. ........................................................................................................................................................................................................... 4 Agriculture: Abasin flowers is in the shape of a circular sector, its area equals 48 m2 , length of its arc equals 6m. Find its perimeter and the length of the radius of its circle. .................................................................................................................................................................................................................................................

5 The perimeter of a circular sector equals 24 cm and length of its arc equals 10 cm. Find the area of its surface. .................................................................................................................................................................................................................................................

Work book - Second term

57


5-6

Circular Segment

F 1 In the figure drawn: M is a circle of radius 6 cm, MC = A B , M C = 3 cm complete: 6 cm A The height of the circular segment ADB = ................................. cm B The height of the major circular segment AFB = ........................ cm B C Measure of the angle of the minor circular segment ADB = ..................째 D Measure of the angle of the major circular segment AFB = ..................째

M

6 cm

3 cm

C

A

D

E

Area of the triangle MAB = ................................. cm2. F Area of the circular sector M A D B in terms of r equals ................................. cm2. G Area of the minor circular segment in terms of r equals ................................. cm2. 2 Find the area of the circular segment in which, A Length of the radius of its circle is 12 cm and measure of its angle equals 1.4rad. .......... B The length of the radius of its circle equals 8 cm, and measure of its angle equals 135c. ......................................................................................................................................................................................................................................

C

The length of the radius of its circle equals 14 cm and the length of its arc equals 22 cm.

.................................................................................................................................................................................................................................................

A

3 In the figure drawn: ABC is an equilateral triangle drawn in the circle M in which, the length of its radius equals 8 cm. find the area of each shaded circular segments. ........................................................................................................................... 4 Find the area of the major circular segment in which the length of its chord equals the length of the radius of its circle equals 12 cm.

M C

B

.................................................................................................................................................................................................................................................

5 Find the area of the circular segment in which: A The length of its chord equals 6 cm , and the length of the radius of its circle equals 5 cm. ................................................................................................................................................................................................ B Its height equals 5 cm , and the length of the radius of its circle equals 10 cm. .............. 6 A chord of length 8cm in a circle is at a distance 3cm from its centre. Find the area of the minor circular segment resulting from the intersection of this chord with the surface of the circle . ............................................................................................................................................................................................................................................................................

58

Mathematics - first form secondary


Areas

5-7

1 Find the area of each of the following figures, given that

expresses a unit of the area.

A

B

C

D

E

F

G

H

D

2 Find the area of the triangle ABC in each of the following cases: A

A B = 6cm, B C = 8cm, m(dB) = 90c

B

A C = 12cm, length of perpendicular drawn from B to AC equals 7 cm.

.............................................................................................................

.................................................................................................................................................................................................................................................

C

A B = 16cm, B C = 20 cm, m(dB) = 46c

.............................................................................................................

D

A B = 8cm, B C = 7cm, A C = 11cm.

.............................................................................................................

3 Find the area of the figure ABCD in each of the following cases: A Parallelogram in which A B = 8cm, B C = 11cm, m(dB) = 60c ...................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................

B

Trapezium in which, length of its parallel bases AD and BC are equals to 7cm , 11cm respectively, the length of the perpendicular drawn from D to BC equals 6cm. ...................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................

Work book - Second term

59


C

A rhombus in which AB = 8cm, and measure of the included angle between two adjacent sides in it equals 58c . ....................................................................................................................................................................................................................................... .......................................................................................................................................................................................................................................

4 Find the area of each of the following regular polygons approximating the result to the nearest tenth A A regular pentagon of side length equals 16cm. ................................................. B A regular hexagon of side length equals 12cm. ................................................. 3 metres

5 Constructions: the figure opposite represents a set of bicycles lead to the entrance of the residential compound is in the shape of an isosceles trapezium, its larger base is down and its length equals 7 metres, its smaller base is up and its length equals 3 metres, each leg inclines by an angle of measure 75c to the larger base. Find: A Length of its base at the middle.

75째

75째 7 metres

..................................................................................................................................................................................................................................................

B

Length of each of its legs to the nearest tenth .

.................................................................................................................................................................................................................................................. ..................................................................................................................................................................................................................................................

C

Area of the trapezium to the nearest metre.

.................................................................................................................................................................................................................................................. ..................................................................................................................................................................................................................................................

6 Basins decorations: Basin is designed to fish decoration , its base is in the shape of a regular pentagon, the length of its diagonal equals 72 cm. Find to the nearest square centimetre the area of its base. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. ..................................................................................................................................................................................................................................................

7 Folowers: Karim designs a Garden to his house, and hope to determine a special part for flowers, is in the form of a regular hexagon of area 54 3 m2. Find the length of its side. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. ..................................................................................................................................................................................................................................................

60

Mathematics - first form secondary


General Exercises

1 Simplify each of the following:

2

B (sini + cosi)2 sin(-i) csc (i) ..................................................... .................................... seci csci 2 Prove the validity of the following identities: A cos(180c+ i) csc (90째 + i) = -1 B sec i - sini tani = cos i C tan2i + cot2i - (sec2i + csc2i) = -2 D (sini + cosi)2 + (sini - cosi)2 = 2 A

3 Find the solution set of each of the following equation in [0, 2r[ A

2 sini - 1 = 0

B

......................................

3

coti = 1

C

......................................

2cos2i + cosi = 0 ..................................................

4 Find the general solution of each of the following equations : A

tani -

3

=0

..................................................

B

cos5 i = sin4i ..................................................

C

sec4i = csc2i ..................................................

5 Solve the triangle A B C right at B in which: A m(dA) = 43c , B C = 12 cm B A B = 12.8 cm , B C = 19.2 cm .................................................................................................................................................................................................................................................

6 From the top of a rock 40 metres high , two ships were observed in one ray on the sea with the rock and were measured to be 35c 12' and 53c 6'. Find the distance between the two ships. .................................................................................................................................................................................................................................................

7 A circle M of radius length 7.5 cm, MA , MB are radii where A B = 12cm . Find the area of the minor circular sector M A B to the nearest square centimetres . .................................................................................................................................................................................................................................................

8 Find the area of the circular segment in which the length of the radius of its circle equals 10cm and the length of its arc equals 26.19 cm. .................................................................................................................................................................................................................................................

9 ABC is an equilateral triangle of side length 24 cm , A circle is drawn passing through its vertices. Find the length of the radius of the circle , then Find the area of the minor circular segment in which its chord is BC . .................................................................................................................................................................................................................................................

10

Find the area of the quadrilateral A B C D in which AC = 14 cm , BD = 18 cm and measure of the angle between its diagonals is 78c approximating the result to the nearest hundredth. .................................................................................................................................................................................................................................................

11

Find the area of a regular polygon of 8-sides of side length 10cm approximating the result to the nearest tenth. .................................................................................................................................................................................................................................................

Work book - Second term

61


Unit Test First : Complete each of the following: 1 (sin i + cos i)2 - 2sin i cos i equals ............................ in the simplest form 2 If 2 sin X - 3 = 0 and X â&#x2C6;&#x2C6; ]0, r[, then m(dX) = ............................. 3 If cos (90c - i) = 1, then the general solution to the equation is ............................. 4 Area of the circular sector whose perimeter 12cm, and the length of its arc 4cm equals .............. 5 If x is the length of the side of an equilateral triangle whose area 9 3 cm2 then x equals ........... cm. Second : Multiple choice 6 sin (90c - i) csc (180c - i) equals: ....................................................................................... in the simplest form A -1 B 1 C sin i D tani 7 sin2 i + cos2 i - csc2 i in the simplest form equals: A

B

Zero

1

C

-cot2 i

D

tan2i

8 If i â&#x2C6;&#x2C6; [0, r[, cosi + 1 = 0, then i equals: .............................................................................................................................. A Zero B r C r D 2r 2 9 Area of the circular sector is 45cm2 , length of its arc equals 3cm , then the length of the radius of its circle equals: A 15cm B 30cm C 22.5 cm D 90cm 10 The general solution of the equation: cot( r - i) = 3 is .................................................................................. A r +nr 3 C 4r + 2 n r 3

B D

2 r 3 + 2n r r 4r 2 + 2n r or 3 + 2 n r

Third : Answer the following questions: 11 A Prove that (1 - sini)(seci + tani) = cos i B Find the solution set of the equation 2sin2 i - 5sin i + 2 = 0 in the interval [0, r[ A

12

Solve the triangle ABC right at B , AB = 6cm, AC = 8cm, m(dC) = 42c. B Find the area of the circular segment in which the length of its radius 12cm and the length of its arc equals 20cm. ...................................................................................................................................................

13

Find the general solution of the equation: cos5 i = sin i ........................................................................ B From the top of a rock 100 metres high, a person was observed a car on the road with an angle of depression 25c find to the nearest metre the distance between the observer and the car. ...........................................................................................................................................................................

14

A

62

A

The central angle of a circular sector is 48c and the length of the radius of its circle is 6cm. Find the area of the sector to the nearest cm2. B Find the area of the quadrilateral in which its diagonals lengths are 12cm, 16cm, measure of the angle between them is 62c approximating the result to the nearest square centimetre. .................................................................................................................................................................................. Mathematics - first form secondary


Accumulative test

First : Multiple choice: 1 - sin2 i 1 equals ...................................................................................................................................... in the simplest form: 1 - cos2 i A

B

-1

C

1

tan2 i

D

cot2 i

2 The solution set of the equation 3 tani = 1 where 0c < i < 270c is: ................................................. A 30c B 150c C 210c D 240c 3 It is possible to solve the right angled triangle in all of the following cases except : A Length of two sides in it B length of two sides and measure of included angle C Measure of two given angles D lengths of a side and the hypotenuse 4 If the area of a circular sector equals 48cm2 and the length of its arc equals 12cm, then the length of the radius of its circle equals: .......................................................................................................................... A 4cm B 8cm C 12cm D 16cm 5 The solution set of the inequality x2 > 4 is: .......................................................................................................................... A [-2, 2] B ]-2, 2[ C R - [-2, 2] D R - ]-2, 2[ 6 The two roots of the equation x2 - bx + 4 = 0 are complex numbers if b equals .......................... A [-4, 4] B ]-4, 4[ C R - [-4, 4] D R - ]-4, 4[ Second : Answer the following questions: 7 Put each of the following statements in the simplest form: A

sini csci - cos2 i

B

........................................................

cos2 i sec0 csci

C

........................................................

sin( r + i) sec(-i) 2

....................................................

8 Prove the validity of the following statements in the simplest form. A sin(90c - i) cot(90c - i) csc (180c - i) = 1 B

cosi 1 + sin2 i

+ tani = seci

9 Find in the form of a rational number the value of each of:

10

A

sini if cosi = 4 ,

B

tani if csci = 13 , 90c < i < 180c

5

0c < i < 90c

5

.............................................................................................................

.............................................................................................................

Find the general solution of each of the following equation: A

2sini +

3

=0

....................................................

B

2sin i = 1 ....................................................

Work book - Second term

C

sec2 i = csc4 i ....................................................

63


Accumulative test 11 TV: TV screens are measured according to the length of its digonal. Use the figure

opposite to answer each of the following: A Find the value of x in inch. B Find i in degree and radian measure. C Calculate in square inch the area of the screen.

40

24 inch

inc

h

x inch

i

12

From a point 8 metres apart of the base of a tree, it was found that measure of elevation angle of the top of the tree 22c, find the height of the tree to the nearest hundredth.

13

A circular sector of area 270cm2, the length of the radius of its circle equals 15cm, find the length of the arc of the sector and measure of its central angle in radian measure.

14

Find the area of the major circular segment in which the length of the radius of its circle equals 5cm, and the length of its chord equals 8cm.

15 Analytic geometry:

First: Find the area of the figure opposite.

6

Second: Find to the nearest tenth the area of each

5

of the following figures: A an isosceles triangle, the length of one of its legs equals 12cm and the measure of the included angle between them equals 64c. B A regular figure of 12 sides and the length of its side equals 19cm.

4 3 2 1 0

1

2

3

4

5

6

7

8

64

4

5

6

7

8

9

10

11

12

13

14

15

1-6

1-2

5-1

5-1

4-3

5-2

5-3

5-4

5-5

5-6

5-7

First term

First term

Mathematics - first form secondary

First term

3

5-5

No of lesson

2

5-3

1 5-1

No of question

5-2

The following table shows the number of question in the test and the number of question in the lesson for reference when necessary

9


First test

General tests

(Algebra and trigonometry)

First : choose the correct answer from the given answers : 1 The point which belongs to the solution set of the following inequalities: x > 2 , y > 1 , x + y H 3 is: A (2, 1) B (1, 2) C (3, 2) D (1, 3) 2 If A is a matrix of order 1 * 3, B t is a matrix of order 1 * 3 then it is possible to carryout the following operation: A A+B B Bt + At C A Bt D AB 3 The solution set of the equations 2x - 3y = 1 , 3x + 2y = 7 is: A (1, 2) B (2, 1) C (2, 3) D (3, 2) 4 The perimeter of a circular sector is 10cm, and the length of its arc equals 2cm, then its area in square centimetres equals: A

B 8 C 10 D 20 4 5 The solution set of the equation sin X + cos X = 0 where 180c < X < 360c equals: A {210c} B {225c} C {240c} D {315c}

Second : Answer the following questions: 6 A Solve the system of the following linear equations using matrices . 2x - 3y = 4 , 3x + 4y = 23 B Prove that the identity: sini sin(90c - i) tani = 1 - cos2 i 7

A

Find the area of the triangle in which its vertices are (-4, 2), (3, 1), (-2, 5) using matrices. ........................................................................................................................................................................................................................ B Find the solution set of the equation 2sin X + 1 =0 where X = ]0, 2r[ ........................................ x 0 0

8

A

Find the values of x which satisfy the equation

B

A boat was observed from the top of the lighthouse of height 50 metres , it was found that its depression angle 35c. Find the distance between the boat and the top of the lighthouse.

1 x x 5 2 x

= 3x

................................................

......................................................................................................................................................................................................................................

9

A AB

is a chord of length 8cm, is opposite to the central angle of measure 60c. Find to the nearest tenth the area of the minor circular segment in which its chord A B . ......................................................................................................................................................................................................................................

B

Determine the solution set of the following inequalities graphically: x H 0 , y H 0 , x + 3y G 7 , 3x + 4y G 14 then find from the solution set the values of x, y which make the value of the function: P = 30x + 50 y is greatest as possible. ...................................................................................................................................... Work book - Second term

65


General tests Second test

(Algebra and trigonometry)

First choose the correct answer from the given answers: 1 If A is a matrix of order 2 * 3, Bt is a matrix of order 1 * 3, then AB is a matrix of order: A 3*3 B 3*1 C 2*1 D 1*2 2 The point which belongs to the solution set of the inequalities : x H 0 , y H 0 , 2x + y < 4 , x + 3y < 6 is: A (1, -3) B (3, 0) C (2, 3) 3 If A

2x 2 4 3

2

D

(1, 1)

D

5

D

csc2 i

= 10 then x equals: B

3

C

4

4 1 + cot2 i equals ........... in the simplest form: A sin2 i B cos2 i C sec2 i

Second : Answer the following questions: 5 A Solve the system of the following linear equations using Cramar's rule: 2x - 3y = 3 , x + 2y = 5

6

cos X * tan X csc X

B

Prove that the identity

A

-2 2 l b 12 23 l Find the matrix A which satisfies the relation : A * b = 8 13 4 3

= 1 - cos2 X

.......................................................................................................................................................................................................................................

B

r Find the general solution of the equation: cos ( 2 - i) = 1 2

.......................................................................................................................................................................................................................................

7

2 -4 l If A t = b , Prove that A2 - 5A + 2I = 4 3 B The measure of the central angle of a circular segment is 90c and the area of its surface is 56 cm2. Find the length of its radius. A

.......................................................................................................................................................................................................................................

8

A

From a point on the ground 50 metres distance to the base of a vertical pole. It is found that the measure of the elevation angle of the top of the pole to be 19c 24'. find to the nearest metre , the height of the pole from the ground. .......................................................................................................................................................................................................................................

B

Find the maximum value of the objective function P= 2x + y given that: xH0 , yH0 2x + 3y G 18 , -4x + y H -8 .......................................................................................................................................................................................................................................

66

Mathematics - first form secondary


General tests Third test

(Algebra and trigonometry)

First : Choose the correct answer from the given answers: 1 If the matrix A of order 3 * 3, then number of the elements of A equals: A 3 B 6 C 9 D 12 2 The point which belongs to the solution set of the following inequalities : 2x + y < 4 , x + 3y < 6 is: A (1, -4) B (2, 1) C (1, 2) D (3, -1) 3 The point which belongs to the solution set of the following inequalities: x > 2 , y > 1 , x + y H 3 is: A (3, 1) B (1, 2) C (3, 2) D (1, 3) 4 The general solution of the equation cos i = 1 is: A nr B 2n r C r +nr D r + 2n r 2 2 5 The area of an equilateral triangle of side length 6cm equals: A 6 3 cm2 B 9 3 cm2 C 12 3 cm2 D 18 3 cm2 Second : Answer the following questions: 6

7

A

-2 3 l b a b l b 1 0 l Find a , b, c and d if: b + = 4 -1 c d 2 3

B

Prove that the identity:

A

Find the area of the triangle in which its vertices are (2, 4), (5, 2), (-3, -2) using determinants.

cot C 1 + cot2 C

...............................................................................................

= sin C * cos C

......................................................................................................................................................................................................................................

B

Find the general solution of the equation cos 2i = sini ......................................................................................................................................................................................................................................

8

-1 2 If Y = b 0 l , prove that Y2 - 2 Y - 3 I = 3 B Find the area of the regular pentagon of side length 10cm to the nearest square centimetres. A

......................................................................................................................................................................................................................................

9

A

The length of an arc of a circular sector is 7cm, and its perimeter equals 25cm. Find its area. ............................................................................................................................................................................................................... B One of the factories produces two types of blowing music instruments, the first type needs 25 units of copper, 4 units of nickel , the second types needs 15 units of copper, 8 units of nickel, if the available quantity in the factory from copper is 95 units and 32 units of nickel. The profit of an instrument of the first type is 60 pounds and the profit of an instrument of the second type is 48 pounds. What is the number of instruments should the factory produced from each type such that it get the maximum profit. Work book - Second term

67


General tests Fourth Test

(Geometry)

First Complete each of the following: 1 If A = 2 i + 3 j , B = 3 i - j then 2 A - B = .................................. 2 If A = (-2, 1), C = (-3, K) are parallel then K = .................................. 3 If A = (-4, 4), B = (5, -8), C â&#x2C6;&#x2C6; A B where AC : CB = 2 : 1, then C = (..................., ..................) 4 If the two lines L1: 3x - 2y + 7 = 0, L2: ax + 3y + 5 = 0 are perpendicular, then a = ...................... 5 The vector equation of the line which passes through the point (2, -3) and its directions vector (3, 4) is .................................. Second : Answer the following questions: 6

A

If ||-8 A || = 5 ||K A || Find the value of K. ......................................................................................................................................................................................................................................

B

Find the length of perpendicular drawn from the point (1, 2) on the line whose equation is 5x - 12y - 7 = 0 ......................................................................................................................................................................................................................................

7

ABCD is a quadrilateral , E is the mid-point of AB , F is the mid-point CD . Prove that: BC + AD = 2 EF B Find the equation of the line which passes through the point of intersection of the two lines whose equations are 2x + y = 5, and r = (1, 0) + t (1, 1) and passes through the point (5, 3).

8

A

If the point C (2, 5) divides AB by the ratio 4 : 1, where A (8, 3). Find the coordinates of the point B.

B

Prove that the triangle whose vertices are the points Y (4, 2), X(3, 5), Z(-5, -1) is a right angled triangle at Y, then calculate the area of the circle which passes through its vertices.

A

......................................................................................................................................................................................................................................

9 If L1 : 3x + 2y - 7 = 0, L2: 2x - 3y + 4 =0, find : A Measure of the acute angle between L , L . 1 2 ......................................................................................................................................................................................................................................

B

The vector equation of the line which passes through the point of intersection of the two lines L1, L2 and the point (3, 4). ......................................................................................................................................................................................................................................

68

Mathematics - first form secondary


General tests

Fifth test

(Geometry)

First : Complete each of the following: 1 If A = (2, 3), B = (-1, 2) then AB = .................................. 2 If A = (4, 2), B = (1, -2) then || A - B || = .................................. 3 If A (-3, 4), B (6, -8), then the x-axis divides AB by the ratio ............... : ................ 4 Measure of the acute angle between the two lines whose slopes 1 , -2 equals ............................... 2

5 Length of the perpendicular from (1, 1) to the line whose equation X + Y = 0 equals ........................ Second : Answer the following questions: 6

A

If K ||4 A || = ||-3 A ||. find the value of K. ......................................................................................................................................................................................................................................

B

Find the equation of the line which passes through the point (-1, 0), and the point of intersection of the two lines whose equations 2x - y + 4 = 0 , x + y + 5 = 0 ......................................................................................................................................................................................................................................

7

If A = (3, 4), B = (5, -1), C = (2, -2) are three vertices of the parallelogram ABCD find the coordinates of the fourth vertex D. .............................................................................................................. B Prove that the two lines r = (0, 4) + t (1, -2), 2x + y + 2 = 0 are parallel then find the shortest distance between them.

8

A

If A = (-1, 4), B = (5, -1), find the coordinates of the point C which divides AB internally by the ratio 2 : 1.

B

Acircle of centre the origin point, prove that the two chords drawn in the circle and whose equations: 3x + 4y + 10 = 0 and 5x - 12y + 26 = 0 are equal in length.

A

9 ABCD is a trapezium in which AD \\ BC , if A(7, -1), B(3, -1), C(2, 1), D(5, y) : A Find the value of y. ......................................................................................................................................................................................................................................

B

Find the area of the trapezium ABCD. ......................................................................................................................................................................................................................................

Work book - Second term

69


General tests Sixth test

(Geometry)

First : complete each of the following: 1 If A = (-1, 5), B = (2, 1) then || AB || = .................................. 2 If ||3K A || = ||-15 A ||, then K = .................................. 3 If the point (3, 6) is the midpoint of AB where A(-3, 7), then the coordinates of the point B is (................ , ................) 4 If (6, 4), (3, M) are two direction vectors of perpendicular straight lines, then M = ............... 5 The equation of the straight line passing through the two points (2, 0), (0, -3) is ..................... Second : Answer the following questions: 6 A If A (2, 2), B (4, -2), C (-2, 0), D (1, K) and DA , CB are perpendicular vectors, then find the value of K. ......................................................................................................................................................................................................................................

B

If A(2, 3), B(-2, 1), find the ratio by which AB is divided by the x-axis , then find the coordinates of the division point. ......................................................................................................................................................................................................................................

7

A

ABCD is a parallelogram , its diagonals are intersecting at N, M is a point belongs to its plane. Prove that: MA + MB + MC + MD = 4 MN .

B

Find the equation of the straight line passing through the point of intersection of the two lines whose equation are 2x - 7y + 9 = 0 , 3x + 2y - 4 = 0 and makes an angle of measure r with the positive direction of the x-axis. 4

......................................................................................................................................................................................................................................

8

A

If A(3, 5), B (-1, M), || AB || = 4, find the value of M. ......................................................................................................................................................................................................................................

B AB

is a diameter of circle M , If B (-7, 11), M (-2, 3), Find the equation of the tangent to the circle at the point A. ......................................................................................................................................................................................................................................

9 If the straight line whose equation 4x - 3y = 12 intersects the two axes at the two points A, B Find: A Area of the triangle OAB where O is the origin point. ......................................................................................................................................................................................................................................

B

The shortest distance from the origin point to the straight line AB . ......................................................................................................................................................................................................................................

70

Mathematics - first form secondary


Model Answer Unit (1) Matrices Lesson (1-1) 1

1- 56, the distance between the two cities B, C

B

2- Zero, the distance between the city C and itself , 3- The value is itself x11 = 0, x22 = 0, x33 = 0

E

1- 3 * 3 2- ∀i, j ∈ {1, 2, 3} a = 0, b = 2, c = 2, d = 6 a = 5, b = 5, c = 0 , d= 10 B a=2,b=2 d=1,e=1 6 F

3 5 8

A

A= f

0

1

2

-1

0

1

-2

-1

0

Lesson 1-2 Impossible A 2 possible, b B 5

C

p

, At = f

-1

0

-1

8

14

0

from sections A, B) 4x + 5y G 60 (Time of test is at most 60 minutes) x and y must be positive and each point shown in the region of solution represents different permutation between number of questions of section A and section B, and there exists 23. Answers may vary, for ex. y H x , y G x 5 A Because the points of the boundary line B 2x - y = 2 are not solutions for this system Lesson (2 - 3) 3

0

-1

-2

1

0

-1

2

1

0

p

A

The minimum value : 7

B

The maximum value : 14

General Exercises 1

A B

C

l

a = 16, b = 6 , c = 4 , d = 8

D

Exercise 1 - 3 3

0 0

AB = b

0 0

A D E

-1

2

2

A B

C

15

2 3

(x + 1)(y2 + 1)-(y + 1)(x2 + 1) = xy2 + x + y2 + yx

F

-7

B

+y-x

-1

2

- 84

G

H

636

I

-1

x = 1 , y = -1, the s.s. = {(1, -2)} x = 3 , y = 2 , the s.s. = {(3, 2)}

x = -1 , y = 2, the s.s. = {(-1, 2)} 12 square units 22 12 square units 4 C

3

Exercise 1 - 5 a=2 A 2 C

6

A B

B

x = 48, y = -8, the s.s = {(48, -8)} Substituting the points in the equation, then: 5 + a = c , 1 + 3 a = c we get a = 2, c =7

2 3

6 7 11

A

A= A A C

x = 4 , y = -1

B

a=3,b=1

4

6

the s.s {(3, -1)}

6

Unit 2 : linear programing Lesson (2 - 1) (3, 9) A 2

A

OM , MC

(0, 1)

Lesson (2 - 2) Let x be the number of questions in section (A), y be the 4 number of questions in section (B). Write the system of linear inequalities which represent this situation: x H 4 (you should answer at least 4 questions from section (A)) y H3 (you should answer at least 3 questions from section (B)) x + y H 10 (you should answer at least 10 questions

E

5 6

F

3 2

B

AM

C

AB AD

is equivalent to

,

BC

,

OE

BM

,

BA

is equivalent to

,

DA

is equivalent to CB ,

BC

is equivalent to MC

, ,

CD

,

MB is equivalent to DM , MA

is equivalent to CM

D = (-3 , -8) , M (0, -7) , N (0, 7) || AB || = 5 , || CD || = 5 A B

M(-2, -1) , N (9, 6) , R (-5, 1)

Lesson (3-2) 1

A

= (3, -4), ||

C

= (-3, -6), ||

2

A

M

B

M

C D

F A

3

A

M

5

A

A

|| = 5 C

B

= (-12, 5), ||

B

|| = 13

|| = 3 5

= -4 i - 3 j , || M || = 5 = 8 i - 6 j , || N || = 10 = -5 i - 12 j , || F || = 13 = 2 2 j , || A || = 2 2 r = (16 , 6r ) B N = (6 , 4 )

7

A

= -60 i F = 10 2 i a = -6

8

A

2M

B B

3 2

B

, ED , ME , AO is equivalent to DC

AM

a = 1 , b = 3, c = 3, d = -6 B

1 1

A

BM is equivalent to MD

b 1 1 l -3 9

the s.s = {(4, -1)} the s.s = {( 13 , 10 )}

D

C

General Exercises 4

The minimum value in zero, the maximum value is 20 45 minutes for running, 15 minutes for walking

Unit 3: Vectors Lesson (3-1)

a=±6

a = -2 , a = 4 D a =1 or a = 2 x = 2, y = 6, the s.s. = {(2, 6)} x = 5, y = 1, the s.s = {(5, 1)}

C

7

A

a for ex.-2 < 2 ‫ ؛‬then 12 > - 12 ]3, ∞[ ]1, 2] B

Unit test

B (a + x) - A(b + y) = bx - ay 2

- 5 > -8 then 25 < 64 True ‫ ؛‬for ex. : 5 > 3 then 15 < 13 -9 > -12 then - 15 < - 121 False : the inequality sign does not change in this case, i.e. the correction 1 > 1 b

l but either A or B is the zero matrix

Exercise 1 - 4 1

True‫ ؛‬for ex. 5 > 3 then 25 > 9 False‫ ؛‬the correction, then a2 > b2 for ex.

Z

B

- 10

j B

b = -3

4N

C

-2 N

AD

C

D

-M

D

AD

Lesson (3-3) 1

A

DC

9

A

2 i 4 i

B

Work book - Second term

B

-6 -2

AC

j j

C

2 10

71


Model Answer 10 11

C ( -5 , 3) K=4 A

Lesson (4-4): B

30 unit of area

1

2 6

-

=3( M 13 cm BC

2 6

)

N

8

(4, -1)

10 12 13

General Exercises 1

A

2

A

3

A

6

= (2 17 , 76c) B C F = 20 j (2, -3) , (0, 13) , (5, 12) D

|| AB || = 5 = - 70 j

Z

(0, 3)

4

First: A (3, -1), B (1, 2), C (7, 6) Second: 13 square unit

1

A

|| AB || = 5 , || CD || = 5

2

A

-2

-6

i

=

B

C

A

a=5

18 25

E (7 , 4)

B

47 25 B

+

A

-1

B

5

L = -4 , M = 3

6

First: K = 5 C

6

A

2

Second: D (-3, 2)

C

10 12

3

K= 2 M=-

B

3

4 C B

5 2

D

(2, 5) , (5, -2) 3 5 square unit

11

L = 35

N = 25

(8, r3 )

B

5 6

A

9

10

C

2

4

B

(0 ,5)

D

(0, 13)

(5 , -2) , (2 ,0)

4

C = (1 , 13 )

5

C = (-1 , 1)

6

(17 , -13)

7

(1 , -1)

8

2:1,3

2

5

2

r

= (0 , 5)

r

= K (1, 2)

4

5 7

y=x+5 3

6 8

3x + 2y – 6 = 0 4 , zero , 2

9

2 3

10

3

11

r

= (5 , 0) + k(3 , 1)

12

x = 3 + k , y = -5 + k

13

r

= (5 , 1) + k(3 , 4)

14

x – 5 y - 35 = 0

17

x+2y+7=0

16

8 x + 3 y – 22 = 0

Lesson (4 - 3): 90c 1 45c 6

16 17

90c 90c r = (-6, 0) + K(1, 1) 36 square unit B C 14 45c 18c 26' A B -28 12,5 18

21

10 , 45c, 45c 22

9 11 13

72

2 7

7

3 8

45c

10 12 15

y=6 B C

-2 4 r = (6, 0) + K(1, -1)

C

19

3

2, 9 2

1 3 5

4 7

x+2=0

2

3

B

-1

C

1

D

1

C

-1

D

G

tan2 i

H

cos i cot2 i

A F

cos i 2

7

D

4

D

C

8

A

A

r + 2rn, 56 r + 2rn where n ∈ Z 6 r r + rn, where n ∈ Z B 2rn ! , where n ∈ Z C 6 6 7 r 5 r 5 , 4 rn B 0, ,6r r A 0, 2r, C 6 4 6 A

A

A B

6

3

, - 9 , -2 , 7

cos2 i -2

A

A

x = 5cm , y = 9.4cm y = 9.4cm , x = 10.7cm a = 45c , b = 45c a =55c 91' , b = 34c 50' 59'' A C = 7cm, m(dA) = 56c, m(dC) = 34c

37 m 2 54 m 735 m 4 1.85m, 4.23 m The horizontal distance = 15 m, height of the building = 30 + 8 = 38 m The height of the minaret = 71 m, 33c 50'

Lesson (5 - 5) 14cm 1

Lesson (4 - 2): -2 3

-1

2

Lesson (5 - 4)

3

1

A

B

(0 , 7 ) 2 Externally by the ration 3 : 7 internally by the ratio 2 : 1 ( 17 , 0)

B

35x + 10y – 28 = 0

Lesson ( 5 - 3) 1

Lesson (4-1): A

5

Lesson (5 - 2)

5 D

A

A

x–3=0

x+y–2=0

1

Unit 4: the straight line 1 2

3

Unit 5 : trigonometry

6

B

= (1, 3) + K (-2, 1)

x+y–4=0

E

Accumulative test: 1

r

Lesson (5 - 1)

,9 j

j

2

2

5

Lesson (4 - 5): r = K (3, 4) 1 4 6

Unit test:

3

y=0 5 6 units B C 71c 33' 54'' 15 E B D B C A 3 4 5 3 3 3 2 5 A B C D x+y–3=0,4 2 (6, 0) , 17 11 Length of each chord equals two units K = 2, y = -3 length of perpendicular = 3 , Area = 9 A

D

Lesson (3-4)

B

2

9.6 cm

2 5

24cm2

3 20cm 6 A

A

B

Exercise 5 - 6 3cm A B 9cm 1 irad = 11 , Area = 56cm2 3 7 7 9

43.34cm2 61.42cm2

8

10

B

4 3 unit of area 42 unit of area

1

4 r + 2 nr 3

C

- tan2i

20

1 3

(2 , 4) , 12 ‫ ﹰ‬52 ‫ ﹶ‬36 ‫ ﹾ‬, 9 unit of area

C

cot2 i 210c C 2 The measures of two given angles

B

8cm

B

]-2, 2[

A

sin2 i

1 3 4

D

6 8

Mathematics - first form secondary

240c

B

B

Accumulative test 30c

D

8.588cm2 11.18cm2

Exercise 5 - 7 8 unit of area A 1 24 unit of area A 2 General Exercises -1 A 1 r + 2 nr or A 4 3

120c

C

2 4 9.06 cm

5

C

12cm

B

cot i

C

1


Egym wb10 v2 eng