EXERCISE - A
1) α Is the real cube root of 2 and β, γ are its imaginary cube roots, then (xβ + yγ + zα)/(xγ + yα + zβ) is equal to
a) 2¹⁾³ b) 2¹⁾³ω² c) ω² d) none
2) The modulus-amplitude form of the complex number -1 - I is
a) -√2(cosπ/4 + i sin π/4)
b) √2(cos3π/4 + i sin 3π/4)
c) -√2[cos (-3π/4) + i sin (-3π/4)] d) none
3) The amplitude of √12 + 6{(1- 3i)/(1+ i)} is
a) π/3 b) 2π/3 c) -π/3 d) -2π/3
4) If x√2 = 1+ √-1, then the value of x⁶+ x⁴+ x²+ 1 is
a) 0 b) 4 c) - 4 d) none
5) The value/s of √i + √-i is are
a) √2 b) ±√2 c) ±√2i d) ±√2, ±√2i
6) If y= √(x²+ 6x +8), then (1- y)¹⁾² is equal
a) ±(1/√2) {i√(x+4) - √(x+2)}
b) ±(1/√2) {√(x+4) - √(x+2)}
c) ±(1/√2) {√(x+4) - i√(x+2)} d) none
7) The square root of a²+ 1/a² -4(a - 1/a) - 6 are
a) ±(a - i/a -2) b) ±(a - 1/a -2i) c) ±(a - 1/a +2i) d) none
8) If ³√(x + iy) = a + ib, where a, b, x, y are real, x/a + y/b is equals to
a) 4(b² - a²) b) 4(b² + a²) c) 4(a² - b²) d) none
9) If n is a +ve integer, not a multiple of 3, then {(-1+√3)/2}ⁿ + {(-1- √3)/2}ⁿ is equal to
a) -1 b) 2 c) 0 d) none
10) if x + iy moves on the line 3x + 4y + 5 = 0, then the least value of |x + iy | is
a) 0 unit b) 1/5 unit c) 1 unit d) none
11) If z₁ =√3 i and z₂ = - 1+ √3 i, then amp(z₁ z₂) is equal to
a) 7π/6 b) -5π/6 c) 5π/6 d) none
12) If {(2- i)x + i}/(1+ i)+ {(1+ 2i) y +i}/(1- i) = -1/2+ 5i/2, where x and y are real, then x - y is equals to
a) 1 b) -1 c) 6 d) 8
13) If {(1+ i)x + 2i}/(3+ i)+ {(2- 3i)y +i}/(3- i) = i, where x and y are real, then 4x 9y is equals to
a) 10 b) -10 c) 3 d) -3
14) The modulus of (1 - i)/(3+ i) + 4i/5 is
a) √5 unit b) √11/5 unit c) √5/3 unit d) none
15) The least positive integer n such that {2i/(1+ i)}ⁿ is a positive integer, is
a) 2 b) 4 c) 8 d) 16
16) If (√3+ i)¹⁰⁰ = 2¹⁰¹(a + ib), then a is equal to
a) 4 b) -4 c) 1/4 d) -1/4
17) If (a + i)²/(2a - i)= p + iq, then p²+ q² is equal to
a) (a²+1)²/(4a²-1) b) (a²+1)²/(2a²-1) c) (a²+1)²/(4a²+1) d) none
18) The complex numbers z= x + iy which satisfy the equation |(z - 5i)/(z + 5i)| = 1, lie on
a) x= -5 b) y= 6 c) the x-axis d) y-axis
19) For any complex number z= x + iy, if the imaginary part (2z +1)/(iz +1) is -2, then the locus of z is
a) a straight line b) a circle c) an ellipse d) none
20) If the complex number z= x + it satisfies the condition |(z - k)/(z + ki)| = 1, where k is any real number, then locus of z is
a) a straight line b) a circle c) an ellipse d) none
21) The complex number z= x + iy satisfying the condition amp{(z - i)/(z + i)}= π/4 lies on
a) a straight line b) a circle c) an ellipse d) none
22) The complex number z= x + iy satisfying the condition amp{(z -1)/(z +1)} =π/6 lies on
a) a straight line b) a circle c) an ellipse d) none
23) If the argument of (z - a)(bar z - b) and {√3 + i)(1+ √3 i)}/(1+ i) are equal where z is a complex number and bar z is its conjugate and a, b are real numbers, then the locus of z is
a) a straight line b) a circle c) a parabola d) an ellipse
24) The locus of the complex number z= x + iy satisfying the condition ' real part of 1/z = 1/4 ' , is
a) a straight line at distance of 4 unit from the imaginary Axis.
b) a circle with radius 1 unit.
c) a circle with radius 2 unit.
d) a straight line not passing through the origin.
25) The equation bar b z + b bar z = c, where b is a non zero complex constant and c is a real, represents
a) a circle b) a straight line c) a parabola d) an ellipse
26) if z= x + iy and z . conjugate of z - (2+ 3i)z - (2- 3i). Conjugate of z + 9= 0, then the locus of z in the complex plane is
a) a straight line b) an ellipse c) a circle d) none
27) If z = x + iy and w= (1- iz)/(z - i), then |w| = 1 implies that, in the complex plane
a) z lies on the imaginary axis.
b) z lies on the real axis
c) z lies on the unit circle with centre at origin d) none
28) If x + iy= 1/(2+ cosθ + i sinθ), where x, y, θ are real, then as θ varies , in the complex plane the point z= x + iy moves on
a) a straight line b) an ellipse c) a parabola d) a circle
29) Let i²= -1; then
a) i and - i each has exactly one square root
b) i has two square roots but - i does not have any.
c) neither i nor - i has any square root.
d) i and -i each has exactly two square roots .
30) The expression (1+ i)ⁿ/(1- i)ⁿ⁻² is equal to
a) iⁿ⁺¹ b) - iⁿ⁺¹ c) 1 d) -2iⁿ⁺¹
31) The argument of the complex number z= (1+ i√3)²/4i(1- i√3) is
a) π/6 b) π/4 c) π.2 d) none
32) let z and w be two non-zero complex numbers such that |z| = |w| and amp z + amp w =π, then z is equal to
a) w b) - w c) bar w d) - bar w
33) If z and w be two complex numbers satisfying the equation |(z+w)/(z - w)| = 1, then z/w is a number which is
a) positive real b) negative real c) zero d) none
34) If z₁ and z₂ are two complex numbers such that |z₁ + z₂| = |z₁| + |z₂|, then
a) arg z₁ = arg z₂
b) arg z₁ + arg z₂ = 0
c) arg(z₁z₂)= 0 d) none
35) If z₁ and z₂ are two complex numbers such that |z₁|² + |z₂|² = |z₁ + z₂|², then
a) Re(z₁/z₂)= 0
b) Im(z₁/ z₂) = 0
c) Re(z₁z₂)= 0 d) Im(z₁z₂)= 0
36) If z₁ and z₂ are two complex numbers such that |z₁ + √(z₁²+z₂²)+ |z₁ - √(z₁² -z₂²) is equal to
a) |z₁| b) |z₂| c) |z₁ + z₂| d) |z₁ + z₂|+ |z₁ - z₂|
37) If a²+ b²= 1, then (1+ a + ib)/(1+ a - ib) is equal to
a) a + ib b) a - ib c) b + ia d) b - ia
38) If z is an imagery number and z/(1+ z) is purely imaginary , then z
a) can be neither real purely imaginary
b) is a real c) is purely imaginary
d) satisfies none of these properties
39) In a GP the first term and the common ratio are both (1/2) (√3+ i), then the absolute value of the nth term of the progression is
a) 2ⁿ b) 4ⁿ c) 1 d) none
40) The complex number z satisfying |z - 1| = |z - 3|= |z - i| is
a) 2+ i b) 3/2 + i/2 c) 2 + 2i d) none
41) The value of the expression
1(2- ω)(2- ω²)+ 2(3- ω)(3- ω²)+ ......+ (n -1)(n - ω)(n- ω²), where ω is an imaginary cube root of unity, is
a) [n(n +1)/2]² b) n²(n +1)²/4 c) n²(n +1)²/4 + n d) none
42) The number of solutions of the equation z²= conjugate z, where z is a complex number, is
a) 2 b) 3 c) 4 d) none
43) The solution of the equation |z| - z = 1+ 2i is
a) 2 - 3i/2 b) 3/2 - 2i c) 3/2 + 2i d) -2+ 3i/2
44) The number of solutions of the equation z²+ |z|²= 0, where z is a complex number is,
a) 1 b) 2 c) 3 d) none
45) If x = cosx + i sin k, y= cosm + i sin m, z= cos n + i sin n and x+ y+ z =0, then 1/x + 1/y + 1/z is equal to
a) xyz b) 1 c) Rs x + Re y + Re z d) 0
46) If x + 1/x = 2 cosθ then for any integer n, xⁿ - 1/xⁿ is equal to
a) 2cosnθ b) 2i sin nθ c) - 2i sin nθ d) none
47) For all complex numbers z₁ , z₂ satisfying |z₁| = 12 and |z₂ - 3 - 4i|= 5, the minimum value of |z₁ - z₂| is
a) 0 b) 2 c) 7 d) 17
48) Given that the equation z²+ (p + iq)z + r + is= 0 where p, q, r, s are non zero real numbers, has a real root, then
a) pqr= r²+ p²s
b) pqs = s²+ q²r
c) prs = q²+ r²p
d) qrs = p²+ s²q
49) The equation z²+ conjugate of z² - 2|z|²+ z + conjugate z = 0 represents
a) a straight line b) a circle c) an ellipse d) a parabola
50) The complex numbers sinx + i cos2x and cosx - i sin2x are conjugate to each other for
a) x=0 b) x= nπ c) x = (2n +1)π/2 d) no value of x
51) The greatest value of the modulii of the complex numbers z satisfying the equation|z + 4/z|= 2 is
a) √5 -1 b) √5 c) √5+ 1 d) none
52) For any complex number z, the minimum value of |z|+ |z -1| is
a) 0 b) 1/2 c) 1 d) 3/2
53) Let z₁ and z₂ be two complex numbers such that z₁/z₂ + z₂/z₁ = 1, then
a) z₁, z₂ and the origin are collinear.
b) z₁, z₂ and the origin form a right angled triangle.
c) z₁, z₂ and the origin form an equilateral triangle
d) none of these is true
54) The maximum value of|z| when z satisfies the condition |z - 2/z|= 2, is
a) √3 - 1 b) √3 c) √3+ 1 d) √2+ √3
55) If α is a root of the equation x² + x + 1 =0, then α³ᵐ + α³ⁿ⁺¹ + α³ᵖ⁺² (Where m, n, p are three integers) is equal to
a) -3 b) 3 c) 1 d) 0
56) The area of the triangle in the complex plane formed by the points z, iz and z + iz is
a) |z|² b) (1/4) |z|² c) (1/2) |z|² (1/2) z²
57) If z₁ , z₂ , z₃ , z₄ are the four complex numbers represented by the vertices of a quadrilateral taken in order such that z₁ - z₄ = z₂ - z₃ and amp (z₄ - z₁)/(z₂ - z₁) =π/2, then the quadrilateral is
a) a square b) a rectangle c) a rhombus d) none
58) The complex numbers z₁ , z₂ , z₃ are respectively the vertices A, B, C of a parallelogram ABCD, then the fourth vertex D is
a) (1/2) (z₁ + z₂) b) (1/2) (z₁ + z₂ + z₃) c) (1/2) (z₁ + z₂ + z₃) d) z₁ + z₃ - z₂.
59) Suppose z₁ , z₂ , z₃ are the vertices of an equilateral triangle circumscribing the circle |z|= 2. If z₁ = 1+ √3 i and z₁ , z₂,z₃, are the anticlockwise sense, then z₂ is
a) 1- √3 I b) 2 c) (1 + √3 I d) -2
60) In the complex plane the point -2+ i, 1 + 2i, 4 + 5i and 1+ 4i form
a) a square b) a rectangle c) a parallelogram d) none
61) The statement ' a + ib > c + id where a, b, c, d are real numbers and i²= -1' is
a) correct when a> c and b > d
b) correct when a> c and b = d
c) correct when a> c and b = d=0 d) never true
62) The statement ' mi > ni, where m,n are real numbers and i²= -1' is
a) correct if m > n and m,n are positive rational numbers.
b) correct if m > n and m,n are prime integers
c) correct if m > n>π
d) meaningless for all real numbers m,n.
63) The statement 'i²= -1' is equivalent to
a) i= √-1, taking positive square root.
b) i= √-1, taking negative square root
c) i= ± i.
d) all the statement in above three options.
64) If arg z < 0, then arg(-z) - arg z is equal to
a) π b) -π c) -π/2 d) π/2
65) If z²+ z +1= 0, where z is a complex number, then the value of (z + 1/z)² + (z²+ 1/z²)²+ (z³+ 1/z³)²+......+ (z⁶+ 1/z⁶)² is equal to
a) 6 b) 12 c) 18 d) 54
66) The value of ¹⁰ₖ₌₁∑ (sin(2kπ/11) + i cos(2kπ/11)) is equal to
a) -1 b) - i c) i d) 1
67) The value of ⁶ₖ₌₁∑ (sin(2kπ/7) - i cos(2kπ/7)) is equal to
a) -1 b) - i c) i d) 1
68) Let n (∈N) be a multiple of 5 and x= cos(2π/5) + i sin(2π/5)) then 1+ xⁿ + x²ⁿ + x³ⁿ + x⁴ⁿ is equal to
a) 0 b) 5 c) -5 d) 5i
69) Let x= cos(4π/3) - i sin(4π/3)) then the value of {1+ x)/2)}³ⁿ is equal to
a) (-1)ⁿ b) (i)ⁿ c) (-1)ⁿ/2³ⁿ d) 1/2³ⁿ
70) Let zₙ = cos(2nπ/7) + i sin(2nπ/7), n= 0,1,2,.....6 then z₁, z₂,z₃....z₆ is equal to
a) 0 b) 1 c) -1 d) - i
71) Let zₙ = cos(π/2ⁿ) + i sin(π/2ⁿ) then z₁, z₂, z₃.....to ∞ is equal to
a) 0 b) -1 c) 1 d) i
72) The product of values of (1+ i√3)³⁾⁴ is equal to
a) 80 b) -8i c) -8 d) 8
73) Let α, β are the roots of the equation x² - 2x cosθ +1= 0, then the equation whose roots are αⁿ and βⁿ, is
a) x² + 2x cosnθ +1= 0
b) x² - 2x cosnθ +1= 0
c) x² - 2x cosnθ -1= 0
d) x² - 2x cosnθ -1= 0
74) If |z|= 1 and w= (z -1)/(z +1), z≠ 1 then Re w is equal
a) 0 b) z/|z +1|² c) |z/(z+1)| z/|z +1|² d) √2/|z +1|²
75) If one root ax²+ bx - 2c =0(a, b, c are real) is imaginary and 8a + 2b > c, then
a) a> 0 and c < 0 b) a> 0 and c > 0 c) a< 0, c > 0 d) a< 0, c< 0
76) if x²+ 1 =√3 x, then (x³ - 1/x³)²ⁿ⁺¹ for any n ∈ N is equal to
a) ω b) 0 c) ±1 d) ± i
77) Let z be a complex number and z= (1- t²)+ i √(1+ t²), where t is a real parameter, then the locus of z in the complex plane is
a) a straight line b) a parabola c) a hyperabola d) an ellipse
78) Let z be a complex number and z= 1 - t + i √(t²+ t+2), where t is a real parameter then the locus of z is
a) a straight line b) a parabola c) a hyperabola d) an ellipse
79) If the three complex numbers z₁, z₂ and z₃ are in AP then z₁, z₂, z₃ lie on
a) a straight line b) a circle c) a parabola d) an ellipse
80) Let z₁ and z₂ are two complex numbers such that |z₁| = |z₂|= 1, then|(z₁ - z₂)/(1- z₁ . Conjugate z₂| is equal to
a) 2 b) 1/2 c) 1 d) none
81) The modulus of the complex number {(2+ i√5)/(2- i √5)}¹⁰ + {(2- i√5)/(2+ i √5)}¹⁰ is equal to
a) 2 sin(10 cos⁻¹(2/3))
b) 2 sin(20 cos⁻¹(2/3))
c) 2 cos(20 cos⁻¹(2/3))
d) 2 cos(10 cos⁻¹(2/3))
82) Let C be the set of all complex numbers and A, B be two subsets of C x C defined by A = {(z, w): |z|= |w| and z, w belongs to C}
B= {(z,w): z²= w², z, w belongs to C} then
a) A= B b) ⊂ c) B ⊂ A d) none
83) If |z²-1|= |z|²+ 1, then z lies on
a) the real axis b) the imaginary axis c) a circle d) an ellipse
84) If x + iy= 6i - 3i 1
4 3i -1
20 3 I then
a) x=3, y=1 b) x=1, y=3 c) x=0, y=3 d) x=0, y=0
85) If z and w are two complex numbers such that |zw| =1 and arg z - arg w =π/2, then conjugate of z.w is equal to
a) -1 b) i c) 1 d) - i
86) If z is a complex number such that iz³+ z²+ i =0, then | z |=
a) 2 b) 1 c) √2 d) none
87) (√3/2 + i/2)¹⁷⁷ is equal to
a) I b) - I c) -1 d) √3/2 - i/2
88) Let z and w are two complex numbers such that conjugate of z + i. Conjugate of w =0 and arg zw =π, then arg z is equal to
a) π/4 b) π/2 c) 3π/4 d) 5π/4
89) If ω is an imaginary cube root of 1 then
1 1+ ω² ω²
1- i -1 ω²-1
- i -1+ω -1 is equal to
a) 0 b) 1 c) I d) ω
90) The sum of ¹⁹ₖ₌₁∑ (sin(kπ/5) + i cos(kπ/5)) is
a) purely real and positive b) purely imaginary c) purely real and negative d) none
91) If w is a given complex number outside the circle with centre at origin and radius |a - 1| (a is real), then the points z, satisfying z x conjugate of z - 2 x conjugate of w x z - w x conjugate of w - 2w x conjugate of z+ 5(a -1)²=0, lie on
a) a circle b) a Parabola c) a straight line d) none
92) If Z and W represent diagonally opposite vertices of a square, then the other vertices are given by the complex numbers
a) Z + iW and Z - iW
b) (1/2)(Z + W)± (i/2)(Z - W)
c) (1/2)(Z - W)± (i/2)(Z - W)
d) (1/2)(Z +p- W)± (i/2)(Z - W)
Answer
1c 2c 3c 4a 5d 6c 7b 8c 9a 10c 11b 12b 13c 14c 15c 16d 17c 18c 19a 20a 21b 22b 23b 24c 25b 26c 27b 28d 29d 30d 31c 32d 33c 34a 35a 36d 37a 38a 39c 40c 41b 42c 43b 44d 45d 46b 47b 48c 49d 50d 51c 52c 53c 54c 55d 56c 57b 58d 59d 60c 61c 62d 63c 64a 65b 66b 67c 68b 69c 70 c 71b 72d 73b 74a 75a 76b 77b 78c 79a 80c 81c 82c 83b 84d 85d 86b 87b 88c 89a 90b 91b 92d
EXERCISE - B
1) If a <0, b> 0 then √a. √b is equals to
a) -√(|a|. b) b) √(|a|.b.1) c) √(|a|b) d) none
2) The value of the sum ¹³ₙ₌₁∑(iⁿ + iⁿ⁺¹), where I=-√1, is
a) i b) i -1 c) - i d) 0
3) If n₁, n₂ are two positive integers then
(1+ i)ⁿ₁+ (1+ i³)ⁿ₁+(1+ i⁵)ⁿ₂+(1+ i⁷)ⁿ₂ is a real number if and only if
a) n₁ = n₂ +1 b) n₁ +1= n₂ c) n₁= n₂ d) n₁, n₂ are any two positive integers
4) The complex number 2ⁿ/(1+ 2²ⁿ) + (1+ i)²ⁿ/2ⁿ , n ∈ Z, is equal to
a) 0 b) 2 c) {1+(-1)ⁿ}.iⁿ d) none
5) The smallest positive integral value of n for which {(1- i)/(1+ i)}}ⁿ is purely imaginary with positive imaginary part, is
a) 1 b) 3 c) 5 d) none
6) if (a+ ib)⁵= α + iβ then (b + ia)⁵ is equal to
a) β + iα b) α - iβ c) β - iα d) - α - iβ
7) If I=√-1, the number of values of iⁿ + i⁻ⁿ for different n∈Z is
a) 3 b) 2 c) 4 d) 1
8) Im(z) is equal to
a) (1/2)(z + conjugate of z)i
b) (1/2)(z + conjugate of z)
c) (1/2)(conjugate of z - z)i d) none
9) The values of (1+ i)³+ (1- i)⁶ is
a) I b) 2(-1+ 5i) c) 1 - 5i d) none
10) Taking the value of a square root with positive real part only, the value of √(-3- 4i) + √(3+ 4i) is
a) 1+ I b) 1- 3i c) 1+ 3i d) none
11) sin⁻¹{(z -1)i} , where z is non real, can be the angle of a triangle if
a) Re(z)=1, Im(z)=2 b) Re(z)=1, -1< Im(z)≤1 c) Re(z)+ Im(z)=0 d) none
12) If n is an odd integer, i=√-1 then (1+ i)⁶ⁿ+ (1- i)⁶ⁿ is equal to
a) 0 b) 2 c) -2 d) none
13) If z₁ = 9y⅖- 4 - 10ix, z₂ = 8y²- 20i, where z₁ = conjugate of z₂, then z= x + iy is equal to -2+2i b) -2±2i c) -2±i d) none
14) The complex number sinx - i cos 2x and cosx - i sin 2x are conjugate to each other for
a) x= nπ b) x=0 c) x= (2n +1)π/2 d) no value of x.
15) If z= 1+ i tanα, where π<α < 3π/2, then |z| is equals to
a) secα b) - secα c) cosecα d) none
16) If z is a complex number satisfying the relation |z +1| = z +2(1+ i) then z is
a) (1/2)(1+ 4i) b) (1/2)(3+ 4i) c) (1/2)(1- 4i) d) (1/2)(3 - 4i)
17) If (1+ i)z= (1- i). Conjugate of z then z is
a) t(1- i), t∈ R b) t(1+ i), t∈ R c) t/(1+ i), t∈ R⁺ d) none
18) If z₁, z₂ are two non-zero complex number such that |z₁ + z₂|= |z₁|+ |z₂| then amp(z₁/z₂) is equal to
a)π b) -π c) 0 d) π/2
19) The complex number z is purely imaginary if
a) z. Conj of z is real b) z = conjugate of z c) z + conjugate of z d) none
20) If z= x + iy such that |z +1|= |z -1| and amp(z -1)/(z +1)= π/4 then
a) x=√2+1, y=0 b) x=0 , y= √2+1 c) x=0, y=√2-1 d) x=√2-1, y=0
21) Let z= (cosθ + i sinθ)/(cosθ - i sinθ), π/4< θ < π/2. Then arg z is
a) 2θ b) 2θ -π c) π+2θ d) none
22) If z= (√3+ i)/((√3- i) then the fundamental amplitude of z is
a) -π/3 b) π/3 c) π/6 d) none
23) If (1+2 i)/(2+ i)= r(cosθ + i sinθ) then
a) r=1, θ = tan⁻¹(3/4)
b) r=√5, θ = tan⁻¹(4/3)
c) r=1, θ = tan⁻¹(4/3) d) none
24) If z= x + iy satisfies amp(z -1)= amp(z + 3i) then the value of (x -1): y is equals to
a) 2: 1 b) 1: 3 c) -1:3 d) none
25) Let z be a complex number of constant modulus such that z² is purely imaginary then the number of possible values of z is
a) 2 b) 1 c) 4 d) infinite
26) If ω is an imaginary cube root of unity than (1+ ω - ω²)⁷ equals to
a) 128ω b) - 128ω c) 128ω² d) - 128ω²
27) If ω is a noreal cube root of unity then the expression (1-ω)(1-ω²)(1+ ω⁴)(1+ ω⁸) us equal to
a) 0 b) 3 c) 1 d) 2
28) If 3⁴⁹(x + iy)= (3 0/2+ √3i/2)¹⁰⁰ and x = ky then k is
a) -1/3 b) √3 c) -√3 d) -1/√3
29) x³ᵐ + x³ⁿ⁻¹ + x³ʳ⁻², where m, n r ∈ N is divisible by
a) x²- x +1 b) x²+ x +1 c) x²+ x -1 d) x²- x -1
30) If x²- x +1 then the value of ⁵ₙ₌₁∑(xⁿ - 1/xⁿ)² is equals to
a) 8 b) -10 c) 12 d) none
31) If x² +1= √3 x then the value of ²⁴ₙ₌₁∑(xⁿ - 1/xⁿ)² is equals to
a) 48 b) -48 c) ±48(ω -ω²) d) none
32) The smallest positive integral value of n for which (1+ √3 i)ⁿ⁾² is real
a) 3 b) 6 c) 12 d) 0
33) If I=√-1, ω= non real cube root of unity than
{(1+ i)²ⁿ - (1- i)²ⁿ}/{(1+ ω⁴ - ω²)(1- ω⁴ + ω²)} is equal to
a) 0 if n is even b) 0 for all n ∈ Z c) 2ⁿ⁻¹ . i for all n∈ N d) none
34) If z½ - z +1=0 then zⁿ - z⁻ⁿ, where n is a multiple of 3, is
a) 2(-1)ⁿ b) 0 c) (-1)ⁿ⁺¹ d) none
35) If ω is a non real cube root of unity then
(1+ 2ω+ 3ω²)/(2+ 3ω+ ω²) + (2+ 3ω+ ω²)/(3+ ω+ 2ω²) is equals to
a) -1 b) 2ω c) 0 d) -2ω
36) If (x - 1)⁴ - 16=0 then the sum of non real complex values of x is
a) 2 b) 0 c) 4 d) none
37) If zᵣ= cos(2rπ/5) + i sin(2rπ/5), r= 0,1,2,3,4..... then z₁, z₂, z₃, z₄, z₅ is equals to
a) -1 b) 0 c) 1 d) none
38) If eᶥᶿ = cosθ + i sinθ then for the ∆ABD, eᶥᴬ. eᶥᴮ. eᶥᴰ is
a) - i b) 1 c) -1 d) none
39) If (√3+ i)ⁿ = (√3- i)ⁿ, n ∈ N then the least value of n is
a) 3 b) 4 c) 6 d) none
40) If the 4th roots of unity are z₁, z₂, z₃, z₄ then z₁² + z₂² + ₃² + z₄² is equals to
a) 1 b) 0 c) i d) none
41) If x³-1=0 has the non-real complex roots, α, β then the value of (1+ 2α+β)³ -(3+ 3α +5β)³ is
a) -7 b) 6 c) -5 d) 0
42) If I=√-1 then 4+ 5(-1/2 + i√3/2)³³⁴ - 3(1/2 + i√3/2)³⁶⁵ is equal to
a) 1- i√3 b) -1+ i√3 c) 4√3 i d) i√3
43) If (√3- i)ⁿ = 2ⁿ, n∈ Z the set of integers, then n is a multiple of
a) 6 b) 10 c) 9 d) 12
44) If z(2- i 2√3)²= i(√3+ i)⁴ then amplitude of z is
a) 5π/6 b) - π/6 c) π/6 d) 7π/6
45) If z is a non real root of ⁷√-1 then z⁸⁶ + z¹⁷⁵+ z²⁸⁹ is equals to
a) 0 b) - 1 c) 3 d) 1
46) If α is non real and α = ⁵√1 then the value of ₂|1+α+α²+α⁻²-α⁻¹| is equal to
a) 4 b) 2 c) 1 d) none
47) The value of amp(iω) + amp(iω²), where I=√-1 and ω=³√1= non real, is
a) 0 b) π/2 c) π d) none
48) If α, β be two complex numbers then |α²|+ |β²| is equal to
a) (1/2) (|α+β|² - |α - β|²)
b) (1/2) (|α+β|² + |α - β|²)
c) (|α+β|² - |α - β|²) d) none
49) The set of values of a ∈R for which x²+ i(a -1)x +5=0 will have a pair of conjugate complex roots is
a) R b) {1} c) {a|a²- 2a +21> 0} d) none
50) Nonreal complex numbers z satisfying the equation z³+ 2z²+ 3z +2=0 are
a) (-1± √-7)/2 b) (1+ i√7)/2, (1- i√7)/2 c) - i, (-1+ i√7)/2, (-1- i√7)/2 d) none
51) For a complex number z, the minimum value of |z|+ |z -2| is
a) 1 b) 2 c) 3 d) none
52) If |z|= 1 then (1+ z)/(1+ conjugate of z) is equals to
a) z b) conjugate of z c) z + conjugate of z d) none
53) if α is a non real cube root of unity then |αⁿ|, n∈Z, is equals to
a) 1 b) 3 c) 0 d) none
54) If z be a complex number satisfying z⁴+ z³+ 2z²+ z +1=0 then |z| is equals to
a) 1/2 b) 3/4 c) 1 d) none
55) let z₁ = a + ib, z₂ = p + iq be two unimodular complex such that I'm(z₁ conjugate of z₂)= 1. If ω₁ = a + ip, ω₂ = b + iq then
a) Re(ω₁ω₂)= 1 b) Im(ω₁ω₂)= 1 c) Re(ω₁ω₂)= 0 d) Re(ω₁. Conj of ω₂)= 1
56) Let |z₁ -1|< 1, | z₂ -2|< 2, |z₃ -3|< 3 then |z₁ + z₂+ z₃|
a) is less than 6 b) is more than 3 c) is less than 12 d) lies between 6 and 12
57) If |z - i|≤ 2 and z₀ = 5 + 3i then the maximum value of |iz + z₀| is
a) 2+ √31 b) 7 c) √31 -2 d) none
58) If |z|= max {|z -1|}, |z +1| then
a) |z + conj of z|= 1/2
b) z + conj of z = 1
c) |z + conj of z|= 1 d) none
59) |z - 4|< |z -2| represents the region given by
a) Re(z)> 0 b) Re(z)< 0 c) Re(z)> 2 d) none
60) If log₁/₂ (|z|²+ 2|z|+ 4)/(2|z|²+1)< 0 then region traced by z is
a) |z|< 3 b) 1< |z|< 3 c) |z|> 1 d) |z|< 2
Answer
1b 2b 3 d 4c 5b 6a 7a 8c 9b 10d 11b 12a 13b 14d 15b 16c 17a 18c 19c 20b 21a 22b 23a 24b 25c 26d 27b 28d 29b 30a 31b 32b 33a 34b 35b 36a 37c 38c 39c 40b 41a 42c 43d 44b 45b 46a 47c 48b 49b 50a 51b 52a 53a 54c 55d 56c 57b 58c 59d 60a
EXERCISE -
1) The value of (5- i)- conjugate of (3+ i³) is
a) 2+ 2i b) 2- 2i c) 2i d) - i
2) If k= iⁿ + iⁿ⁺¹ + iⁿ⁺² + iⁿ⁺³, where n belongs to N, the value of k is
a) 0+ 0i b) 1+ 0i c) 1+ i d) 1 - i
3) If z= 1 - i , the reciprocal complex number of z will be
a) 1/2 - i/2 b) - 1/2 - i/2 c) 1/2 - i d) 1/2 + i/2
4) If (x + iy)¹⁾²= p + iq, then the value of x²+ y² will be
a) p+ q b) (p²+ q²)² c) p²- q² d) p - q
5) If (x + iy)³ = u + iv, then the value of u/x+ v/y will be
a) 4(x²+ y²) b) 4(x²- y²) c) x²- y² d) x² + y²
6) If z= (1- i)⁻² + (1+ i)⁻², then the value of Conjugate of z is
a) 0 b) (1+ i)⁻² + (1- i)⁻² c) (1+ i)⁻² - (1- i)⁻² d) none
7) If (z -1)/(z +1) is a purely imaginary quantity, the value of |z| will be
a) 4 b) 2 c) 1 d) 1/4
8) If z= (√3/2 + i/2)⁷+ (√3/2 + i/2)⁷, the value of Im(z) will be
a) (√3 + 1)/2 b) (√3 - 1)/2 c) 1 d) 0
9) If z= (1+ i)²ⁿ⁺¹/(1- i)²ⁿ⁻¹ and n belongs to N the value of |z| is
a) 1 b) 2 c) -1 d) 0
10) The value of (x + yω+ zω²)/(z + xω+ yω²) + (x + yω+ zω²)/(y + zω+ xω²) is
a) -1 b) 1 c) ω d) ω²
11) The value of (3 + 3ω+ 5ω²)⁶ - (2 + 6ω+ 2ω²)³ is
a) 5 b) 4 c) 0 d) 1
12) The value of (1 + 5ω²+ 5ω⁴)(1 + 5ω+ ω²) (5 + ω+ ω²) is
a) 0 b) 4 c) 64 d) 16
13) The value of (1 + ω)(1+ ω²)(1+ ω⁴)(1+ ω⁸).....2n upto 2n number factors is
a) 2²ⁿ b) 2ⁿ c) 1 d) 2
14) If 4√3+ 4i = r(cosθ + i sinθ), the value of r & θ will be
a) 1,π/3 b) 1,π/6 c) 8,π/3 d) 8,π/6
15) If -2 = r(cosθ + i sinθ), the value of θ will be
a) π/2 b) π c) - π/2 d) 0
16) If - 2i = r(cosθ + i sinθ), the value of r & θ will be
a) 2, -π/2 b) 2,π/2 c) 2,π d) 2, - π
17) For (cos70 + i cos 20)= z, the argument z will be
a) 7π/18 b) 7π c) 7π/2 d) 7π/3
18) If z = 1+ i tanθ) is expressed in polars, the value of |z| will be
a) 1 b) 2 c) secθ d) 4
19) Square root of I is
a) ±(1+ i)/√2 b) ±(1- i)/√2 c) ω² d) ω
20) Number of solutions of the equation z²+ |z|²= 0 is
a) 3 b) 1 c) 2 d) 4
21) Equation of locus of z for z= r+ 3 + i√(5- r²) is a/an
a) hyperbola b) circle c) straight line d) ellipse
22) For which value of n. {(1+ i)/(1- i)ⁿ = -1?
a) 3 b) 4 c) 1 d) 2
23) If |z₁|= 12 & |z₂ -3- 4i|= 5, then least value of |z₁ - z₂| is
a) 1 b) 2 c) 3 d) 4
24) If z≠ 0, then the value of arg z + arg of conj of z is
a) 0 b) π c) 2π d) π/4
25) If L, M, N are three positive integers & L/|z₂ - z₃|= M/|z₃ - z₁|= N/|z₁ - z₂| then the value of L²/|z₂ - z₃| + M/(z₃ - z₁) + N/(z₁ - z₂) is
a) abc b) a+ b - c c) 0 d) abc /2
26) If z= (√3+ i)/2, then the value of (z¹⁰¹ + i¹⁰³)¹⁰⁵ is
a) z² b) z c) z³ d) 2z
27) If z is located on a circle having radius 1/2, then the radius of that circle upon which (-1+ 4z) is
a) 1 b) 2 c) √3 d) 4
28) Modulus -amplitude form of 5- 5i is
a) 6(cosπ/2 + i sinπ/2) b) (cosπ/6 + i sinπ/6) c) (cosπ/2 + i sinπ/2) d) 5√2(cos(-π/4) + i sin(-π/4))
29) If (1+ ω+ω²), find the value of ω²⁰⁰⁴ + ω²⁰⁰⁵
a) - ω² b) - ω c) 1 d) 2ω
30) If |z₁|=1, |z₂|= 2, |z₃|= 3 & |z₁ + z₂ + z₃|=1, the value of|9z₁z₂ + 4z₁z₃ + z₃z₂| is
a) 64 b) 6 c) 4 d) 16
31) If 2+ ix & 7/(2+ √3i) are Conjugate, the value of x will be
a) √3 b) 3 c) √2 d) 1/√2
32) If z₁ = 1 - i, z₂ = 2 + 4i, then the value of Iₘ = (z₁z₂/conj of z ₁) is
a) 1 b) 2 c) 3 d) -1
33) If x= 3+ 2i, then the value of x⁴- 4x³+ 4x²+ 8x + 44 is
a) 1 b) 2 c) 5 d) 0
34) If (x -1)³+ 8= 0, the value of x is
a) 1, 1+ 2ω, 1+ 2ω²
b) -1, 1- 2ω, 1+ 2ω²
c) - 1, 1- 2ω, 1- 2ω²
d) - 1, 1+ 2ω, 1- 2ω²
35) If (-1/2 + √3i/2)¹⁰⁰⁰= x + iy, then the value of x and y is
a) x= 1/2, y= √3/2
b) x= 1/2, y= - √3/2
c) x= - 1/2, y= √3/2
d) x= 1, y= 1
36) If tan(α + iβ)= cosθ then the value of α is
a) nπ/2 + π/4
b) nπ + π
c) nπ + (-1)ⁿ π/2
d) nπ ± π/2
37) The polar form of √3+ I is
a) (cos π/3 + i sin π/3)
b) 2(cos π/6 + i sin π/6)
c) 3(cos π/6 + i sin π/6)
d) 2(cos (-π/6) + i sin (-π/6))
38) If z= (4+ 5i)/(5- 6i), |z| is
a) √(41/61) b) √(61/41) c) √2 d) √(1/2)
39) Square root of (5- 12i)/(3+ 4i) is
a) ±(1/2) (√3 i +1)
b) ±(1/•2) ( i +1)
c) ±(1/5) (4 - 7 i)
d) ±(1/5) (2 - 7i)
40) The value of (1 + ω -ω²)⁴ + (1 - ω+ ω²)⁴ is
a) -16 b) 25 c) -8 d) 9
41) If z= x + iy and x <0 , y <0, then arg z is
a) tan⁻¹(y/x) - π b) tan⁻¹(y/x) +π c) tan⁻¹(y/x) d) none
42) If x= 3⁴ - 12x³+ 62x²- 156x + 169 will be
a) 1 b) 3 c) -1 d) 0
43) If 1/(x +ω²) + 1/(y+ω²)+ 1/(z+ ω²) then the value of 1/(x +ω) + 1/(y+ω) + 1/(z +ω) is
a) 2/3 b) 2/ω c) 3ω² d) 1/ω²
44) Cross multiplication of the quantity (1+ √3 i)² is
a) 3/13 - 2i/13 b) -1/8 - √3i/8 c) 1/5 + 2i/5 d) 12/13 + 5i/13
45) If (1- i)ˣ = 2ˣ, the value of x is
a) 1 b) -1 c) 0 d) 2
46) If |z|≥ 3, then the least value of |z + 1/z| is
a) 3/8 b) 8/3 c) 1/8 d) 1/3
47) If z= (1- i √3)/(1+ i √3), then arg(z) is
a) 60° b) 120° c) 240° d) 300°
48) If z²/(z -1) be pure real, then the point z lies on
a) a circle b) a real axis c) either circle or a a real axis d) none
49) The value of 1/(1+ i)²+ 1/(1- i)² is
1) -1 b) 1 c) 0 d) 2
50) The value of {(1+ √-3)/2}⁶+ {(1- √-3)/2}⁹ is
1) 1 b) 0 c) -1d) 1/2
Answer
1b 2a 3d 4b 5b 6a 7c 8d 9b 10a 11c 12 c 13 c 14d 15b 16a 17a 18c 19a 20c 21b 22d 23b 24a 25c 26c 27b 28d 29b 30b 31a 32b 33c 34c 35c 36a 37b 38a 39c 40a 41a 42d 43b 44b 45c 46b 47c 48c 49c 50b
