Q.1 If z = 3 + 4i, what is |z| (the modulus of z)?
5
7
1
25
Explanation - The modulus of z = a + bi is |z| = sqrt(a^2 + b^2). So |3 + 4i| = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5.
Correct answer is: 5
Q.2 If z1 = 1 + i and z2 = 2 - i, what is z1 + z2?
3
3 + 2i
3 + 0i
1 + i
Explanation - Add the real and imaginary parts separately: (1 + 2) + (1 - 1)i = 3 + 0i.
Correct answer is: 3 + 0i
Q.3 If z = 2 - 3i, find the conjugate of z.
2 + 3i
-2 + 3i
-2 - 3i
3 - 2i
Explanation - The conjugate of z = a + bi is a - bi. So conjugate of 2 - 3i is 2 + 3i.
Correct answer is: 2 + 3i
Q.4 If z = i, what is z^2?
-1
1
i
-i
Explanation - By definition, i^2 = -1.
Correct answer is: -1
Q.5 Which of the following is a purely imaginary number?
5i
3 + 4i
7
-2 + 0i
Explanation - A purely imaginary number has the form 0 + bi. Here 5i fits the form.
Correct answer is: 5i
Q.6 If z = 1 + i√3, find arg(z) (principal argument).
π/3
π/6
2π/3
π/2
Explanation - arg(z) = arctan(b/a) = arctan(√3/1) = π/3.
Correct answer is: π/3
Q.7 If z = 1 - i, what is 1/z?
(1 + i)/2
(1 - i)/2
1 + i
i/2
Explanation - 1/z = conjugate(z)/|z|^2 = (1 + i)/[(1)^2 + (-1)^2] = (1 + i)/2.
Correct answer is: (1 + i)/2
Q.8 If z = 3(cos π/4 + i sin π/4), find z^2 in polar form.
9(cos π/2 + i sin π/2)
9(cos π/4 + i sin π/4)
6(cos π/2 + i sin π/2)
9(cos π + i sin π)
Explanation - z^2 = r^2 [cos(2θ) + i sin(2θ)] = 3^2[cos(π/2) + i sin(π/2)] = 9(cos π/2 + i sin π/2).
Correct answer is: 9(cos π/2 + i sin π/2)
Q.9 Which of the following represents the imaginary unit i on the Argand plane?
Point (0,1)
Point (1,0)
Point (0,-1)
Point (-1,0)
Explanation - i is represented as 0 + 1i, which is the point (0,1) on the Argand plane.
Correct answer is: Point (0,1)
Q.10 If z = 2 + 2i, what is |z|^2?
4
8
2√2
16
Explanation - |z|^2 = (Re(z))^2 + (Im(z))^2 = 2^2 + 2^2 = 4 + 4 = 8.
Correct answer is: 8
Q.11 If z = -1 + i√3, which quadrant does it lie in the Argand plane?
II
I
III
IV
Explanation - Re(z) < 0 and Im(z) > 0 → Quadrant II.
Correct answer is: II
Q.12 If z = 4(cos π/3 + i sin π/3), find z^3 in polar form.
64(cos π + i sin π)
64(cos π/9 + i sin π/9)
12(cos π + i sin π)
64(cos π/3 + i sin π/3)
Explanation - z^3 = r^3 [cos(3θ) + i sin(3θ)] = 4^3[cos π + i sin π] = 64(cos π + i sin π).
Correct answer is: 64(cos π + i sin π)
Q.13 Find the square roots of z = 4 (principal root).
2
-2
2i
-2i
Explanation - The principal square root of 4 is 2.
Correct answer is: 2
Q.14 If z = x + iy, what is the real part of z^2?
x^2 - y^2
2xy
x^2 + y^2
xy
Explanation - z^2 = (x + iy)^2 = x^2 - y^2 + 2xyi, so real part = x^2 - y^2.
Correct answer is: x^2 - y^2
Q.15 If z = cos θ + i sin θ, then z̅ (conjugate of z) equals?
cos θ - i sin θ
sin θ + i cos θ
cos θ + i sin θ
-cos θ + i sin θ
Explanation - Conjugate of a complex number a + bi is a - bi.
Correct answer is: cos θ - i sin θ
Q.16 If z1 = 1 + i and z2 = 1 - i, compute z1 * z2.
2
0
1 + i
1 - i
Explanation - z1 * z2 = (1 + i)(1 - i) = 1 - i^2 = 1 + 1 = 2.
Correct answer is: 2
Q.17 If z = 5i, find the modulus of z.
5
0
i
-5
Explanation - Modulus |z| = sqrt(0^2 + 5^2) = sqrt(25) = 5.
Correct answer is: 5
Q.18 Which of the following is the polar form of z = 1 + i?
√2(cos π/4 + i sin π/4)
√2(cos π/2 + i sin π/2)
2(cos π/4 + i sin π/4)
1(cos π/2 + i sin π/2)
Explanation - r = |z| = √(1^2 + 1^2) = √2, θ = arctan(1/1) = π/4 → z = √2(cos π/4 + i sin π/4).
Correct answer is: √2(cos π/4 + i sin π/4)
Q.19 If z = -i, find z^4.
1
-1
i
-i
Explanation - z^4 = (-i)^4 = (i^2)^2 = (-1)^2 = 1.
Correct answer is: 1
Q.20 If z = cos π/6 + i sin π/6, then 1/z = ?
cos(-π/6) + i sin(-π/6)
cos π/6 + i sin π/6
-cos π/6 - i sin π/6
cos π/3 + i sin π/3
Explanation - 1/z = z̅ / |z|^2. |z| = 1, z̅ = cos θ - i sin θ = cos(-π/6) + i sin(-π/6).
Correct answer is: cos(-π/6) + i sin(-π/6)
Q.21 If z = 2 + 2√3 i, find arg(z).
π/3
π/6
2π/3
π/4
Explanation - arg(z) = arctan(Im/Re) = arctan(2√3/2) = arctan(√3) = π/3.
Correct answer is: π/3
Q.22 If z = 1 + i, express z^2 in a + bi form.
2i
0 + 2i
1 + 2i
0 - 2i
Explanation - z^2 = (1 + i)^2 = 1 + 2i + i^2 = 1 + 2i - 1 = 2i.
Correct answer is: 2i
Q.23 If z = 1 - i, what is |z|?
√2
1
2
0
Explanation - |z| = sqrt(1^2 + (-1)^2) = sqrt(2).
Correct answer is: √2
Q.24 Which of the following complex numbers lies on the negative real axis?
-3 + 0i
-3 + i
3 + 0i
i
Explanation - A point lies on the negative real axis if its imaginary part is 0 and real part is negative.
Correct answer is: -3 + 0i