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