أقوى مراجعات ليلة الامتحان للثانوية العامة 2020 فى مادة الجبر والهندسة الفراغية لغات

Algebra and solid Exam 2017 model

(1)

page(2)

Q(5) In the opposite figure: If Z1 and Z2 and Z1 Z2 are complex numbers then Z2 =

c e

d f

-2i i

-i

Y

Z1Z2

Z1

X

2i

1 πi Z 1Z 2 re (2 π +θ )i 2 = = e = cos π + i sin π = i Z2 = θi 2 2 Z1 re

()

1

Q(6)

c e

()

If the point (-2,4,m) lies on the sphere

(X + 2)2 + (Y − 1)2 + (Z − 3 )2 = 25

then one value of m=….

d f

6 7

3 + (m − 3 ) = 25 2

2

∴m - 3 = 4

8 9

∴m = 7

Q(7) If ω is an imaginary cube root of unity then (1 + ω − ω2 ) = ……

7

c e

128 ω -128 ω

(− 2ω )

2 7

d f

128 ω2 -128 ω2

= −128ω 4 = −128ω 2

Q(8) If A = (1, 2, -4), B = (1, 1, k - 1) and || A + B || = 7 unit of length then k = ............................

c e

-1,11 1,11

{-1,11}

d f

-11,-1 1,12

Algebra and solid Exam 2017 model

(1)

page(3)

Q(9) If A = (4, -k, 6) , B = (2, 2, m) and A // B , then k + m =.........................

c e

12 -1

d

2

f

3

4 = − K = 6 ∴ m = 3 , K = -4 ∴ K + m = -1 2 2 m

Q(10) 4 non collinear and coplanar points. Find the number of line segments joining each two of them?

c e 4

5 6

d f

7

d f

-1

8

C2 = 6

Q(11) 1 + 3ω + 3ω2 = ……

c e

-2 0

ω

1 + 3 × −1 = −2

Q(12) If

c e

10 20

X+ Y

P4 = 360 , 2 X + Y = 5040 then

d

C2 X = ……

30

f

X + Y = 6 , 2X + Y = 7 ∴ x = 1 , y = 5

Y

40 5

c 2 = 10 ∴

Algebra and solid Exam 2017 model

Q(19) Find Z = − 8

1 + 3i

(1)

page(6)

where i2 = −1 in the trigonometric form then find the

two square roots of the number Z in the exponential form

(

)

(

)

(

)

1 − 3i − 8 1 + 3i − 8 1 − 3i −8 = = = −2 1 − 3i = −2 + 2 3i × 2 + 1 3 1 + 3i 1 − 3i 1 − 3i 2 Y X = −2 , Y = 2 3 , r = X2 + Y2 = 22 + 2 3 = 4 tanθ = = 3 (-,+ ) X nd o o o θ in the 2 ∴ θ = 180 − 60 = 120 Z = 4 cos 120o + i sin 120o Z=

(

( )

( )

)

⎛ 120o + 2Kπ 120o + 2Kπ ⎞ + i sin Z = 2⎜ cos ⎟ 2 2 ⎝ ⎠ πi ⎛ 120o 120o ⎞ + i sin When K = 0 ∴ Z1 = 2 ⎜ cos ⎟ = 2e 3 2 2 ⎠ ⎝ o o ⎛ When K=-1 2 cos− 120 + i sin− 120

(

)

⎞

4

Q(20) The second , third and fourth terms in the expansion of (X + a )n

according to the descending power of X are : 16,112,448 find the value of X,a,n T3 112 = = 7 ∴ n − 2 + 1 × a = 7 ∴ n − 1 × a = 7 → (1) T2 16 2 X 2 X T4 448 = = 4 ∴ n − 3 + 1 × a = 4 ∴ n − 2 × a = 4 → (2 ) 3 X 3 X T3 112 n − 1 × 3 = 7 ∴ 12(n − 1) = 14(n − 2 ) ∴ 2n = 16 ∴ n = 8 2 n−2 4 7 × a = 7 ∴ a = 2 ∴ a = 2X , T = 16 = 8 C (a )(X )7 2 1 X 2 X 16 = 8 × 2X × X 7 ∴ X 8 = 1 ∴ X = ±1

Algebra and solid Exam 2017 model

(2)

page(1)

Answer the following questions 20 questions From 1to 12 choose the correct answer

Q(1) If the X axis cut the sphere which center (3,-4,12) and its radius length 13cm at the two points A and B then AB equals

c e

6units 24units

(X − 3)2 + 16 + 144 = 132

d f

8units

d f

- 2ω

26units

∴ (X - 3 ) = 9 ∴ X = 0 or X = 6 ∴the two points A(0,0,0) and B(6,0,0) ∴ AB = 6 2

2⎛ 2⎞ Q(2) ω ⎜ 1 − 12 + ω ⎟ = ………

c e

⎝

⎠

ω

-2 2

2ω

ω2 (1 − ω + ω2 ) = ω2 × −2ω = −2ω3 = −2 Q(3) If Z1 = 2 + 2 3i

c

− 120 o

e

120

,

Z2 = −3 − 3 3 i then arg Z1 + Z2 =…….

o

Z1 + Z2 = −1 −

3i

3rd quad

d

135 o

f

60

o

θ = tan−1 3 − π = −120 o

Q(4) If P and Q are the coefficients of X in the expansions of (1 + X )2n and (1 + X )2n − 1 then n

c e

d f

P = Q

2P = Q P = 2 n Cn and Q =

Cn ∴ P = 2 Q

2n−1

P = 2Q

P+Q = 0

Algebra and solid Exam 2017

page(3)

Q(9) If the vector A makes angles of measure α , β , θ with X , Y and Z axes then sin2 α + sin2 β + sin2 θ = ...

c

1

d

2

e

-1

f

-2

1 − cos α + 1 − cos β + 1 − cos θ = 3 − (cos α + cos β + cos θ ) = 2 2

2

2

2

2

Q(10) How many ways we can put(distribute) 10 identical balls into 6 distinct bins

c e

d f

151200 210

6 + 10 −1

n+ r −1

C10 = 3003

(

Q(11) a + bω + aω

c e

2

Cr

3003 3000

n = 6 , r = 10

)(a + bω

2

+ aω

4

)

d b2 − a 2 f (a − b)2

1 a-b

(bω − aω )(bω2 − aω2 ) = (b − a )2 Q(12) The least positive integer n which makes n −1 C5 + n − 1C6 < n C7 Is ……………

c e ∵ ∴

d

13

f

14 n −1

C5 +

n −1

1 < 1 7 n-6

n

C6 < C7

n

n

∴ C6 < C7

∴ 7 < n-6

∴

∴ n > 13

10 15 n

6 n−6 ∴ n = 14

<

n 7 n−7

2

Algebra and solid Exam 2017 model

Q(19)

(2)

page(6)

Find in the trigonometric and exponential forms the roots of the

equation Z

4

(

)

= 81 −

3 i then write the solution set

X = 8 , Y = -8 3

∴r =

(

82 + − 8 3

)

2

= 16

3 in the 4th quadrant ∴ θ = − π 3 4 ∴ Z = 16⎛⎜ cos⎛⎜ − π ⎞⎟ + i sin⎛⎜ − π ⎞⎟ ⎞⎟ ⎝ 3⎠ ⎝ 3 ⎠⎠ ⎝ ⎛ − π + 2πr − π + 2πr ⎞⎟ ⎜ 3 + i sin 3 ∴ Z = 2⎜ cos ⎟ 4 4 ⎜ ⎟ ⎝ ⎠ −πi − − π π + i sin = 2e 12 When r=0 then Z1 = 2 cos 12 12 5π i π π 5 5 ⎛ ⎞ + i sin When r=1 then Z2 = 2⎜ cos ⎟ = 2e 12 12 12 ⎠ ⎝ −7 π i When r=-1 then Z3 = 2 cos − 7 π + i sin − 7 π = 2e 12 12 12 11π i π π 11 11 When r=2 then Z4 = 2 cos + i sin = 2e 12 12 12 , tan θ =

)

(

(

S.S={ 2e

−πi 12

, 2e

5π i 12

, 2e

(

−7 π i 12

)

, 2e

11π i 12

)

}

9

Q(20) In the expansion: ⎛⎜ X 2 − 1 ⎞⎟ Find ngeneral term ⎝ X ⎠ oThe term free of X

To get the General term:

Tr +1 = 9Cr (- X-1) × (X2 ) r

9 −r

= 9Cr (− 1) X −r × X18− 2r = 9 Cr (− 1) × X18− 3r

To get the term free of X

18 − 3r = 0 ∴ 3r = 18 ∴ r = 6

r

r

∴ T7 = 9 C6 (− 1) × X18− 3×6 = 84 6

Algebra and solid Exam 2017 model

(3)

page(1)

Answer the following questions 20 questions From 1to 12 choose the correct answer Z

Q(1) In the opposite figure, A B C D A B C D is a cube of / side length 2 units , then find AB • BD 1 2

c

− 1 2 / AB = (2,2,2 ) − (2,0,0 ) = (0,2,2 )

e

(0,2,2) • (− 2,−2,0)

d

1

f

-1

D/

C/

B

A/

/

D C

X

A

B

BD = (0,0,0 ) − (2,2,0 ) = (− 2,−2,0 )

= −4 =−1 8 2 0+4+4× 4+4+4 Q(2) If Z is a complex number of unite modulus and argument θ then the argument of 1 + Z is…………………………

1+ Z c π −θ d −θ 2

e

f

θ

θ−π

1 + Z = 1 + Z = Z ∴ argument is θ 1+ Z 1+ 1 Z

( (

)

⎛ 3 + i 4n + 1 ⎞ ⎟ = ....

Q(3) The amplitude of ⎜⎜ 4n ⎟ ⎝ 1− i 3 ⎠

c e

π 3 π 6

⎛ 3 +i ⎞ ⎜ ⎟× ⎜ ⎟ ⎝ 1− i 3 ⎠

)

d f

( 3 + i) = (i) × ( 3 + i) = 4n

2π 3 5π 6

⎛ ⎞ ∴ amplitude tan − 1 ⎜⎜ 1 ⎟⎟ = π 6 ⎝ 3⎠

3 +i

2 3 Q(4) The co-efficient of X in (1 − X + X ) is 5

c e

d f

-30 -10

-20 30

= (1 + X(X − 1))5 = C0 + C1X(X − 1) + C2 X (X − 1)2 + C3 X (X − 1)3 5

5

5

5

= −2 C2 − C3 = −20 − 10 = −30

5

2

5

3

Y

Algebra and solid Exam 2017 model

(3)

c e

d f

page(2)

Q(5) If θ is the measure of the angle included between A = (2, 0, 2) , B = (0, 0, 4), then θ = ..........

90o 45o

cos θ =

60o 30o

(2,0,2) • (0,0,4 )

⎛ ⎞ = 8 = ∴ θ = cos − 1 ⎜⎜ 1 ⎟⎟ = 45 o ⎝ 2⎠ 22 + 0 2 + 22 × 0 2 + 0 2 + 4 2 8 2

Q(6) If the point C(2,2,6) is the mid point of AB where A(1,−4,0 ) Then the point B

c e

d f

(1,10,11) (3,12,12)

(3,8,12) (2,10,12)

B= (2 × 2 − 1,2 × 2 + 4,2 × 6 − 0 ) = (3,8,12 ) Q(7) If the vector A makes angles of measure α , β , θ with X , Y and Z axes then cos 2α + cos 2β + cos 2θ = ...

c e

d f

1 -1 2

2

2

(

2 -2 2

2

2

)

2 cos α − 1 + 2 cos β − 1 + 2 cos θ − 1 = 2 cos α + 2 cos β + 2 cos θ − 3 = = 2 × 1 − 3 = −1

Q(8) If L1 : X + 2 = Y + 3 = Z + 5 is perpendicular to the line −1 3 2 L 2 : X = Y − 5 = Z − 6 then the value of 3K+2m=….

2 K m

c

-1

d

2

e

0

f

4

(− 1,3,2) • (2, K, m) = 0 ∴ 3K + 2m = 2

Algebra and solid Exam 2017 model

(3)

page(3)

Q(9) If A = (-1, 3, 4) , B (0, -2, 5) , then ||AB ||=…………….

c

d

3 3

e

4 3

f

3 2

AB = B − A = (0,−2,5 ) − (− 1,3,4 )

∴ AB

=

5 3

(− 1)2

+ (5 )2 + (− 1)2 = 3 3

Q(10) How many triangles can be formed by joining the vertices of an octagon

c e 8

d f

10 8

336 56

C3 = 56

⎛ ⎞⎛ ⎞ 5 Q(11) ⎜ 1 + 2ω + 12 ⎟⎜ 1 + 2ω + 14 ⎟ = ⎝ ω ⎠⎝ ω ⎠

c e

d f

0 1

(1 + 2ω

2

)(

+ ω 1 + 2ω + ω

2

)= ω

2

×ω = ω

3

-1 2

= 1

Q(12) If Cr = Cr − 1 and Pr = Pr + 1 then the value of n is n

n

n

n

c

3

d

e

4

f

r+r−1= n

∴r = n+1 2

∵ n Pr = n Pr + 1 ∴

5 2 n n−r

∴ n − n + 1 = 1 ∴ 2n - n - 1 = 2 ∴ n - 1 = 2 ∴ n = 3 2

=

n n−r−1

∴ n-r = 1