1. The locus of a point on the Argand plane represented by the complex number \(z\), which satisfies the condition \(\left|\frac{z - 1 + i}{z + 1 - i}\right| = \left|\mathrm{Re}\left(\frac{z - 1 + i}{z + 1 - i}\right)\right|\) is

1. 1. A straight line that does not contain the point \((-1 + i)\)
2. 2. A circle that does not contain the point \((-1 + i)\)
3. 3. A parabola that does not contain the point \((-1 + i)\)
4. 4. A hyperbola that does not contain the point \((-1 + i)\)

2. Let \(z_1, z_2\) be two complex numbers such that \(\overline{z_1} - i\overline{z_2} = 0\) and \(\arg(z_1 z_2) = \frac{3\pi}{4}\), then \(\arg(z_1) =\)

1. 1. \(\frac{\pi}{4}\)
2. 2. \(\frac{-\pi}{8}\)
3. 3. \(\frac{\pi}{8}\)
4. 4. \(\frac{\pi}{3}\)

3. If \(x + iy = \frac{(3 + 2i)(4 - 7i)(12 + 13i)}{(13 - 12i)(2 - 3i)(11 + 3i)}\), then \(x^2 + y^2 =\)

1. 1. 1
2. 2. 2
3. 3. 1
4. 4. 3

4. What is the modulus of the complex number \((1 + 2i)(-2 + i)\)?

1. 1. \(\sqrt{5}\)
2. 2. 5
3. 3. \(5\sqrt{5}\)
4. 4. \(\sqrt{35}\)

5. Let the complex numbers \(\alpha\) and \(\frac{1}{\alpha}\) lie on circles \((x - x_0)^2 + (y - y_0)^2 = r^2\) and \((x - x_0)^2 + (y - y_0)^2 = 4r^2\) respectively. If \(z_0 = x_0 + iy_0\) satisfies the equation \(2|z_0|^2 = r^2 + 2\), then \(|\alpha| =\)

1. 1. \(\frac{1}{\sqrt{2}}\)
2. 2. \(\frac{1}{2}\)
3. 3. \(\frac{1}{\sqrt{7}}\)
4. 4. \(\frac{1}{3}\)

6. Let \(z = x + yi\), where \(x, y\) are integers and \(i = \sqrt{-1}\). The area of the rectangle whose vertices are the roots of the equation \(z\overline{z}^3 + z(\overline{z})^3 = 350\) is

1. 1. 32
2. 2. 40
3. 3. 48
4. 4. 80

7. Geometrically, the set \(\{z \in C : |z - 2 - 2i| \leq 1\}\) represents

1. 1. A closed circular disc with center at \((-2, -2)\) and with radius 1
2. 2. A closed circular disc with center at \((2, 2)\) and with radius 1
3. 3. A closed circular disc with center at \((1, 1)\) and with radius 0
4. 4. A closed circular disc with center at \((-1, -1)\) and with radius 0.5

8. If \((2 + i)\) is a root of the equation \(x^3 - 5x^2 + 9x - 5 = 0\), then the other roots are

1. 1. 1 and \((2 - i)\)
2. 2. -1 and \((3 + i)\)
3. 3. 0 and 1
4. 4. -1 and \((-2 + i)\)

9. The locus of \(z\) satisfying \(\left|\frac{z - i}{z - 2i}\right| = 2\) is a

1. 1. Hyperbola
2. 2. Circle
3. 3. Straight line
4. 4. Ellipse

10. For how many natural numbers 'n' such that \(1 \leq n \leq 2021\) is \(\left(\frac{1 + i}{1 - i}\right)^n = 1\)?

1. 1. 504
2. 2. 505
3. 3. 506
4. 4. 503

11. If \(x + iy = \frac{3}{2 + \cos\theta + i\sin\theta}\), then \(x^2 + y^2 =\)

1. 1. \(4x - 3\)
2. 2. \(4x + 3\)
3. 3. 0
4. 4. 1

12. Find the conjugate of \(\frac{5i}{7 + i}\)

1. 1. \(\frac{1}{10}(1 - 7i)\)
2. 2. \(\frac{1}{10}(7i - 1)\)
3. 3. \(\frac{1}{10}(7i + 1)\)
4. 4. \(\frac{1}{\sqrt{50}}(1 - 7i)\)

13. If \(\left|\frac{z - 25}{z - 1}\right| = 5\), then \(|z| =\)

1. 1. 5
2. 2. 3
3. 3. 4
4. 4. 10

14. If \(2i\) is a root of \(f(z) = z^4 + z^3 + 2z^2 + 4z - 8 = 0\), then which among the following cannot be a root of \(f(z) = 0\)

1. 1. \(-2i\)
2. 2. 1
3. 3. -2
4. 4. 2

15. Suppose \(z \in C\) has argument \(\theta\) such that \(0 < \theta < \frac{\pi}{2}\) and satisfies the equation \(|z - 3i| = 3\). Then what is the value of \(\cot\theta - \frac{6}{z}\)?

1. 1. \(2i\)
2. 2. \(i\)
3. 3. \(-i\)
4. 4. \(-2i\)

16. If \(A = \left\{z = x + iy \mid \text{real part of } \frac{z - 1}{z - i} = 2\right\}\), then the locus of the point \(P(x, y)\) in the Cartesian plane is

1. 1. A pair of lines passing through \((-1, -1)\)
2. 2. A circle of radius \(\sqrt{2}\) and the centre \(\left(\frac{-1}{2}, \frac{3}{2}\right)\)
3. 3. A pair of lines passing through \((-1, -2)\)
4. 4. A circle of radius \(\frac{1}{2}\)

17. Let \(z \in C\) and \(i = \sqrt{-1}\). If \(a, b, c \in (0, 1)\) be such that \(a^2 + b^2 + c^2 = 1\) and \(b + ic = (1 + a)z\), then \(\frac{1 + iz}{1 - iz} =\)

1. 1. \(\frac{a + ib}{1 + c}\)
2. 2. \(\frac{a - ib}{1 + c}\)
3. 3. \(\frac{a - ib}{1 - c}\)
4. 4. \(\frac{a + ib}{1 - c}\)

18. Let \(a, b \in R\) and the roots \(\alpha, \beta\) of the equation \(z^2 + az + b = 0\) be complex. If the origin, \(\alpha\) and \(\beta\) represent the vertices of an equilateral triangle on the Argand plane, then

1. 1. \(a = b\)
2. 2. \(a^2 = 3b\)
3. 3. \(a^2 = 4b\)
4. 4. \(a = 3b\)

19. Assertion (A): If the arguments of \(\overline{z_1}\) and \(z_2\) are \(\frac{\pi}{5}\) and \(\frac{\pi}{3}\) respectively, then \(\arg(z_1 z_2)\) is \(\frac{2\pi}{15}\). Reason (R): For any complex number \(z\), \(\arg\overline{z} = \frac{\pi}{2} + \arg z\).

1. 1. (A) is true, (R) is true and (R) is the correct explanation for (A)
2. 2. (A) is true, (R) is true but (R) is not the correct explanation for (A)
3. 3. (A) is true but (R) is false
4. 4. (A) is false but (R) is true

20. Let \(z = x + iy\) be a complex number, \(A = \{z : |z| \leq 2\}\) and \(B = \{z : (1 - i)z + (1 + i)\overline{z} \geq 4\}\). Then which one of the following options belongs to \(A \cap B\)?

1. 1. \(\sqrt{3} + \frac{1}{2}i\)
2. 2. \(\frac{1}{2} + \frac{i}{2}\)
3. 3. \(\sqrt{2} + \frac{i}{2}\)
4. 4. \(2 + 2i\)

21. The solutions of the equation \(z^2(1 - z^2) = 16\), \(z \in \mathbb{C}\) lie on the curve

1. 1. \(|z| = 1\)
2. 2. \(|z| = \frac{2}{|z|}\)
3. 3. \(|z|^2 = 3|z| + 2\)
4. 4. \(|z| = 2\)

22. If \(z, \overline{z}, -z, -\overline{z}\) forms a rectangle of area \(2\sqrt{3}\) square units, then one such \(z\) is

1. 1. \(\frac{1}{2} + \sqrt{3}i\)
2. 2. \(\frac{\sqrt{5} + \sqrt{3}i}{4}\)
3. 3. \(\frac{3}{2} + \frac{\sqrt{3}i}{2}\)
4. 4. \(\frac{\sqrt{3} + \sqrt{11}i}{2}\)

23. If \(z_1 = x_1 + iy_1, z_2 = x_2 + iy_2, z_3 = x_1 + \frac{ix_2}{2}, z_4 = 2y_1 + iy_2\) are complex numbers such that \(|z_1| = 1, |z_2| = 2\) and \(\operatorname{Re}(z_1 z_2) = 0\), then

1. 1. \(|z_3| = 1, |z_4| = 2, \operatorname{Im}(z_3 z_4) = 0\)
2. 2. \(|z_3| = 2, |z_4| = 1, \operatorname{Re}(z_3 z_4) = 0\)
3. 3. \(|z_3| = 1, |z_4| = 2, \operatorname{Re}(z_3 z_4) = 0\)
4. 4. \(|z_3| = 2, |z_4| = 1, \operatorname{Re}(z_1 z_2) = \operatorname{Im}(z_2 z_4) = 0\)

24. Assertion (A): If \(z\) is a complex number such that \(|z| \geq 3\), then the least value of \(\left|z + \frac{3}{z}\right|\) is 1. Reason (R): \(|z_1 - z_2| \leq |z_1| + |z_2|\), for any two complex numbers \(z_1, z_2\).

1. 1. (A) is true, (R) is true and (R) is the correct explanation for (A)
2. 2. (A) is true, (R) is true but (R) is not the correct explanation for (A)
3. 3. (A) is true but (R) is false
4. 4. (A) is false but (R) is true

25. \(A(z_1 = 2 + 2i), B(z_2), C(z_3)\) are three points on the Argand plane satisfying \(|z_k - 2| = 2, (k = 1, 2, 3)\). If \(\Delta ABC\) encloses the maximum area, then the sum of imaginary parts of \(z_2\) and \(z_3\) is

1. 1. 1
2. 2. 0
3. 3. 4
4. 4. -4

26. The number of points \(z\) on the Argand plane which satisfy the conditions \(\operatorname{Re}\left(\frac{z - 2}{z - 4i}\right) = 0\) and \(\operatorname{Im}\left(\frac{z - 2}{z - 4i}\right) = 1\) simultaneously is

1. 1. 0
2. 2. 1
3. 3. 2
4. 4. Infinitely many

27. Let \(a = 1 + i\) and \(z = x + iy\). If the curve \(z\overline{z} + az + a\overline{z} - 4 = 0\) is cut by the straight line \((z + \overline{z}) - i(z - \overline{z}) + 2 = 0\) at two points A and B, then the equation of the circle passing through the origin, A and B is

1. 1. \(x^2 + y^2 + 3x - 4y = 0\)
2. 2. \(x^2 + y^2 + x + y = 0\)
3. 3. \(x^2 + y^2 + 6x + 2y = 0\)
4. 4. \(x^2 + y^2 - 7x - 12y = 0\)

28. If \(z\) is a complex number such that \(z^2 + z + 1 = 0\), then \(\left(z + \frac{1}{z}\right)^3 + \left(z^2 + \frac{1}{z^2}\right)^3 + \left(z^3 + \frac{1}{z^3}\right)^3 + \ldots + \left(z^{200} + \frac{1}{z^{200}}\right)^3 =\)

1. 1. 4037
2. 2. -2020
3. 3. 4038
4. 4. 2020 + 673i

29. Let \(z\) be a complex number such that \(|z| - z = 2 + i\), where \(i = \sqrt{-1}\), then \(|z| =\)

1. 1. \(\frac{5}{2}\)
2. 2. \(\frac{\sqrt{41}}{4}\)
3. 3. \(\frac{5}{3}\)
4. 4. \(\frac{5}{4}\)

30. If the amplitude of \(Z - 2 - 3i\) is \(\frac{\pi}{4}\), then the locus of \(Z = x + iy\) is

1. 1. \(x + y - 1 = 0\)
2. 2. \(x - y - 1 = 0\)
3. 3. \(x + y + 1 = 0\)
4. 4. \(x - y + 1 = 0\)

31. Real part of \((\cos 4 + i\sin 4 + 1)^{2020}\) is

1. 1. \(2^{2020}\cos^{2020}2\cos 2020\)
2. 2. \(2^{2020}\cos^{2020}2\cos 4040\)
3. 3. \(2^{1020}\cos^{2020}2\cos 4040\)
4. 4. \(2^{2020}\cos^{2020}1\cos 2020\)

32. If \(|z - 2| = |z - 1|\), where \(z\) is a complex number, then locus of '\(z\)' is a straight line

1. 1. Parallel to x-axis
2. 2. Parallel to y-axis
3. 3. Parallel to \(y = x\)
4. 4. Parallel to \(y = -x\)

33. The radius of the circle represented by \((1 + i)(1 + 3i)(1 + 7i) = x + iy\) is \((i = \sqrt{-1})\)

1. 1. 1000
2. 2. \(10\sqrt{10}\)
3. 3. 10000
4. 4. 100

34. If \(a > 0\) and \(z = x + iy\), then \(\log_{\cos^2\theta}|z - a| > \log_{\cos^2\theta}|z - ai|, (\theta \in R)\) implies

1. 1. \(x > y\)
2. 2. \(x < y\)
3. 3. \(x + y = \cos\theta\)
4. 4. \(x + y < 0\)

35. If \(|z_1 + z_2|^2 = |z_1|^2 + |z_2|^2\), where \(z_1\) & \(z_2\) are two complex numbers, then

1. 1. \(\frac{z_1}{z_2}\) is purely real
2. 2. \(\frac{z_1}{z_2}\) is purely imaginary
3. 3. \(\arg\left(\frac{z_1}{z_2}\right) = \frac{\pi}{4}\)
4. 4. \(\left|\frac{z_1}{z_2}\right| = 1\)

36. A real value of \(x\) will satisfy the equation \(\left(\frac{3 - 4ix}{3 + 4ix}\right) = \alpha - i\beta\) (\(\alpha, \beta\) are real), if

1. 1. \(\alpha^2 - \beta^2 = -1\)
2. 2. \(\alpha^2 - \beta^2 = 1\)
3. 3. \(\alpha^2 + \beta^2 = 1\)
4. 4. \(\alpha^2 - \beta^2 = 2\)

37. If \(z \in C\), then the minimum value of \(|z| + |2z - 3| + |z - 1|\) is

1. 1. 2
2. 2. 1
3. 3. 3
4. 4. 0

38. If \(a, b \in R\) and \(i = \sqrt{-1}\), then the number of ordered pairs of real numbers \((a, b)\) satisfying the condition \((a + bi)^3 = a - bi\) is

1. 1. 3
2. 2. 2
3. 3. 4
4. 4. 5

39. If \(z\) is a complex number, the curves \(|z| = 1, |z - 2| = 1\) and \(|z - 1| = 0\) have a common point at

1. 1. \((0, 1)\)
2. 2. \((2, 0)\)
3. 3. \((1, 0)\)
4. 4. \((0, 2)\)

40. Let \(z = x + iy\) be a complex number \((x, y \in R)\). Let A and B be two sets such that \(A = \{z : |z| \leq 2\}\) and \(B = \{z : (z + 2y) + \overline{z} \geq 4\}\), the area of region \(A \cap B\) is

1. 1. 4
2. 2. \(\pi - 4\)
3. 3. \(\pi\)
4. 4. \(\pi - 2\)

41. \(\left|\frac{1}{i^{2020}} + \frac{2}{i^{2021}} + \frac{3}{i^{2022}} + \frac{4}{i^{2023}}\right| =\)

1. 1. \(3\sqrt{2}\)
2. 2. \(4\sqrt{2}\)
3. 3. \(2\sqrt{2}\)
4. 4. \(\sqrt{2}\)

42. Define \(f : C \to R\) by \(f(z) = |z| \forall z \in C\). Then which of the following is false?

1. 1. \(f(-z) = f(z) \forall z \in C\)
2. 2. \(f(\overline{z}) = f(z) \forall z \in C\)
3. 3. \(f(z^2) = (f(z))^2 \forall z \in C\)
4. 4. \(f(z_1^2 + z_2^2) = f(z_1^2) + f(z_2^2) \forall z_1, z_2 \in C\)

43. The value of \(\left\{i^{22} - \left(\frac{1}{i}\right)^{35}\right\}^2\) is

1. 1. \(2i\)
2. 2. \(i\)
3. 3. \(-i\)
4. 4. \(-2i\)

44. If \(z_1, z_2\) are conjugate complex numbers. Match the items under the following columns?

1. 1. (i-c)(ii-a)(iii-d)(iv-b)
2. 2. (i-c)(ii-a)(iii-b)(iv-d)
3. 3. (i-a)(ii-b)(iii-d)(iv-c)
4. 4. (i-b)(ii-d)(iii-c)(iv-a)

45. If \(x + iy = \frac{1 + 7i}{(2 - i)^2}\), then \(\csc\left(\tan^{-1}\frac{y}{x} - \frac{\pi}{4}\right) =\)

1. 1. 1
2. 2. \(\infty\)
3. 3. -1
4. 4. 0

46. If \((a + ib)^{1/4} = 2 + 3i\), then \(3b - 2a =\)

1. 1. -22
2. 2. -122
3. 3. -598
4. 4. -698

47. If \(z_1 = 1 - 2i, z_2 = 1 + i\) and \(z_3 = 3 + 4i\), then \(\left|\left(\frac{1}{z_1} + \frac{2}{z_2}\right)z_3\right| =\)

1. 1. \(\frac{\sqrt{7}}{2}\)
2. 2. \(\frac{\sqrt{5}}{2}\)
3. 3. \(\frac{\sqrt{45}}{2}\)
4. 4. \(\frac{\sqrt{15}}{2}\)

48. If \(x = \frac{4}{5} + \frac{3}{5}i, y = \frac{\sqrt{3}}{\sqrt{8}} - \frac{\sqrt{5}}{\sqrt{8}}i\), then \(\left(x^2 + \frac{1}{x^2}\right)\left(y^2 - \frac{1}{y^2}\right) =\)

1. 1. \(\frac{-7\sqrt{3}}{5\sqrt{5}}i\)
2. 2. \(\frac{7}{125}i\)
3. 3. \(\frac{7\sqrt{3}}{5\sqrt{5}}i\)
4. 4. \(\frac{\sqrt{15}}{\sqrt{8}}i\)

49. If \((\sqrt{3} + i)^8 - (\sqrt{3} - i)^8 = \alpha + i\beta\), then \(\alpha - \frac{\sqrt{3}}{2}\beta =\)

1. 1. 256
2. 2. \(384\sqrt{3}\)
3. 3. 384
4. 4. \(256\sqrt{3}\)

50. If \(Z = x + iy\) is a complex number and \(\sqrt{x^2 - 2x + 8} + (x + 4)i = y(2 + i)\), then \(Z =\)

1. 1. \(\frac{-28}{9} - \frac{16}{9}i\)
2. 2. \(-2 + 2i\)
3. 3. \(\frac{2}{3} - \frac{2}{3}i\)
4. 4. \(-2 - \frac{2i}{5}\)

51. The locus of \(z = x + iy\) such that \(\operatorname{Im}\left(\frac{z - 3i}{iz + 4}\right) = 0\) is

1. 1. \(x^2 - y^2 + 7y - 12 = 0\)
2. 2. \(x^2 + y^2 - 7y + 12 = 0\)
3. 3. \(x^2 + y^2 - 7y + 12 = 0\) & \((x, y) \neq (0, 4)\)
4. 4. \(x^2 - y^2 + 7y - 12 = 0\) & \((x, y) \neq (0, 4)\)

52. If \(z_1\) and \(z_2\) are the roots of the equation \(x^2 + 2x + 2 = 0\), then \(\frac{-2^{11}(z_1 + 1 + 3i)^{11}}{2^5(z_2 + 1 - 3i)^{11}} =\)

1. 1. 64
2. 2. 32
3. 3. \(16\sqrt{2}\)
4. 4. \(8\sqrt{2}\)

53. Let \(f(x) = ax^2 + bx + c\) and GCD of \(a, b, c\) is 1. If \(\frac{-7 + \sqrt{11}i}{6}\) is a root of \(f(x) = 0\) and \(f\left(\frac{x}{k}\right) - L = (x + 4)(3x - 5)\), then \(k\) and \(L\) are respectively

1. 1. 1, -15
2. 2. 1, 25
3. 3. 7, -15
4. 4. 7, 25

54. \(iz^3 + z^2 - z + i = 0 \Rightarrow |z| =\)

1. 1. \(1/2\)
2. 2. 2
3. 3. 3/2
4. 4. 1

55. If \(\frac{x - 1}{3 + i} + \frac{y - 1}{3 - i} = i\), then the true statement among the following is

1. 1. \(x < 0, y < 0\)
2. 2. \(x < 0, y > 0\)
3. 3. \(x > 0, y < 0\)
4. 4. \(x > 0, y > 0\)

56. The number of integer solutions of the equation \(|1 - i|^x = 2^x\) is

1. 1. 1
2. 2. 0
3. 3. 2
4. 4. 3

57. Multiplicative inverse of the complex number \((\sin\theta, \cos\theta)\)

1. 1. \((+\sin\theta, +\cos\theta)\)
2. 2. \((\sin\theta, -\cos\theta)\)
3. 3. \((\cos\theta, -\sin\theta)\)
4. 4. \((-\cos\theta, \sin\theta)\)

58. \(\sum_{k=0}^{440} i^k = x + iy \Rightarrow x^{100} + x^{99}y + x^{242}y^2 + x^{97}y^3 =\)

1. 1. 0
2. 2. -4
3. 3. 4
4. 4. 1

59. By simplifying \(i^{18} - 3i^7 + i^2(1 + i^4)(i^2)^7\) we get

1. 1. \(-1 + 3i\)
2. 2. \(1 - 3i\)
3. 3. \(1 + 3i\)
4. 4. \(-1 - 3i\)

60. The locus of point \(z\) satisfying \(|z|^2 = \mathrm{Re}(z)\) is a circle with centre

1. 1. \(\left(0, \frac{1}{2}\right)\)
2. 2. \(\left(-\frac{1}{2}, 0\right)\)
3. 3. \(\left(\frac{1}{2}, 0\right)\)
4. 4. \(\left(0, -\frac{1}{2}\right)\)

61. If \((x - iy)^{1/3} = a - ib\), then the value of \(\frac{x}{2a} + \frac{y}{2b}\) is

1. 1. \(2(a^2 - b^2)\)
2. 2. \(4(a^2 - b^2)\)
3. 3. \(a^2 - b^2\)
4. 4. \(\frac{1}{2}(a^2 - b^2)\)

62. If \((x + iy) = \left(\frac{1 + i}{1 - i}\right)^3 - \left(\frac{1 - i}{1 + i}\right)^3\), then the true statement among the following is

1. 1. \(x < y\)
2. 2. \(x > y\)
3. 3. \(x \neq 0\)
4. 4. \(x = y\)

63. The number of complex numbers \(z\) satisfying \(\overline{z} = iz^2\) is

1. 1. 3
2. 2. 4
3. 3. 2
4. 4. 5

64. For any complex number \(z\), the minimum value of \(|z| + |z - 1|\) is

1. 1. 1
2. 2. 0
3. 3. \(1/2\)
4. 4. \(3/2\)

65. If the vertices A, B and C of an isosceles triangle ABC are respectively \(z_1, z_2\) and \(z_3\) and if \(\angle C = 90^\circ\), then

1. 1. \((z_1 - z_2) = (z_1 - z_3)(z_3 - z_2)\)
2. 2. \((z_1 - z_2)^2 = (z_1 - z_3)(z_3 - z_2)\)
3. 3. \((z_1 - z_2)^2 = 2(z_1 - z_3)(z_3 - z_2)\)
4. 4. \(z_1^2 + z_2^2 + z_3^2 = z_1 z_2 z_3 + 2\)

66. Let \(z\) and \(w\) be two complex numbers such that \(\overline{z} + iw = 0\) and \(\mathrm{Arg}(zw) = \pi\). Then \(\mathrm{Arg}(z) =\)

1. 1. \(\frac{3\pi}{4}\)
2. 2. \(\frac{\pi}{2}\)
3. 3. \(\frac{5\pi}{4}\)
4. 4. \(\frac{\pi}{4}\)

67. If the complex numbers \(z_1, z_2, 0\) are vertices of an equilateral triangle, then \(z_1^2 + z_2^2 =\)

1. 1. \(2z_1^2 z_2^2\)
2. 2. \(z_1^2 z_2^2\)
3. 3. \(2z_1 z_2\)
4. 4. \(z_1 z_2\)

68. Let \(z = x + iy\) be a complex number with \(x, y \in \mathbb{Z}\). Then the area (in square units) of the rectangle whose vertices are the roots of the equation \(\overline{z}z^3 + z\overline{z}^3 = 350\) is

1. 1. 48
2. 2. 32
3. 3. 40
4. 4. 44

69. Area of the triangle formed by the complex numbers \(z, iz, z + iz\) in the Argand diagram as vertices is

1. 1. \(\frac{1}{2} \cdot |z|^2\)
2. 2. \(\frac{1}{2} \cdot z^2\)
3. 3. \(z^2\)
4. 4. \(|z|^2\)

70. If \((x - iy)^{1/3} = 2 - i\sqrt{3}\) and the point \(z = (x, y)\) lies on the line \(\frac{x}{2} + \frac{y}{\sqrt{3}} = k\), then \(k =\)

1. 1. 16
2. 2. 2
3. 3. 8
4. 4. 4

71. If \(\left|z + \frac{2}{z}\right| = 2\), then the maximum value of \(|z|\) is

1. 1. \(\sqrt{3} + 1\)
2. 2. \(\sqrt{3} - 1\)
3. 3. \(\sqrt{3}\)
4. 4. infinity

72. A complex number \(z\) among the following which does not satisfy \(z^3 + 27i = 0\) is

1. 1. \(\frac{3\sqrt{3} - 3i}{2}\)
2. 2. \(-3i\)
3. 3. \(\frac{3\sqrt{3} + 3i}{2}\)
4. 4. \(\frac{-3\sqrt{3} + 3i}{2}\)

73. If \(|z - 3i| + |z + 5i| = 4\), then the locus of \(z\) is

1. 1. No such point \(z\) exists
2. 2. Ellipse
3. 3. Parabola
4. 4. Circle

74. \(\sqrt{(-3 + 4i)(8 + 6i)} =\)

1. 1. \(\pm(1 + 2i)\)
2. 2. \(\pm(3 + i)\)
3. 3. \(\pm(1 + 7i)\)
4. 4. \(\pm(7 - i)\)

75. If \(\left(\frac{\sqrt{3} + i}{\sqrt{3} - i}\right)^m = 1\), \(2022 < m < 2029\), then \(m =\)

1. 1. 2022
2. 2. 2024
3. 3. 2028
4. 4. 2026

76. If the point \((x, y)\) satisfies the equation \(\frac{x + i(x - 2)}{3 + i} - i = \frac{2y + i(1 - 3y)}{i - 3}\), then \(x + y =\)

1. 1. 4
2. 2. 2
3. 3. 0
4. 4. -2

77. If \((2x - y + 1) + i(x - 2y - 1) = 2 - 3i\), then the multiplicative inverse of \((x - iy)\) is

1. 1. \(\frac{15}{41} + \frac{12}{41}i\)
2. 2. \(\frac{6}{29} + \frac{15}{29}i\)
3. 3. \(\frac{15}{29} + \frac{6}{29}i\)
4. 4. \(\frac{12}{41} + \frac{15}{41}i\)

78. If \(Z = \alpha + i\beta\) satisfies the equation \(|Z| - Z = 1 + 2i\) and \(|Z| = \sqrt{\alpha^2 + \beta^2}\), then \(Z\overline{Z} =\)

1. 1. \(\frac{5}{2}\)
2. 2. \(\frac{25}{4}\)
3. 3. \(\frac{16}{9}\)
4. 4. \(\frac{36}{25}\)

79. \(\left\{x \in [0, 2\pi] \mid \sin x + i\cos 2x \text{ and } \cos x - i\sin 2x \text{ are conjugate to each other}\right\} =\)

1. 1. \(\left\{\frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{3\pi}{2}, \frac{7\pi}{4}, 2\pi\right\}\)
2. 2. \(\left\{\frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}\right\}\)
3. 3. \(\left\{\frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi\right\}\)
4. 4. \(\phi\)

80. If \(\left|\begin{array}{cc} 2 + 3i & i \\ 1 - 2i & -i \end{array}\right| = x + iy\), then \(x + y =\)

1. 1. -2
2. 2. -4
3. 3. -8
4. 4. 4

81. The values of \(\theta\), for which \(\frac{3 + 2i\sin\theta}{1 - 2i\sin\theta}\) is real are

1. 1. \(\theta = n\pi + \frac{\pi}{3}\) for \(n \in \mathbb{Z}\)
2. 2. \(\theta = n\pi + \frac{\pi}{4}\) for \(n \in \mathbb{Z}\)
3. 3. \(\theta = n\pi + \frac{\pi}{2}\) for \(n \in \mathbb{Z}\)
4. 4. \(\theta = n\pi\) for \(n \in \mathbb{Z}\)

82. If \(z_1 = (2, -1)\) and \(z_2 = (6, 3)\), then \(\mathrm{amp}\left(\frac{z_1 - z_2}{z_1 + z_2}\right) =\)

1. 1. \(\frac{3\pi}{4} - \tan^{-1}\left(\frac{1}{4}\right)\)
2. 2. \(\frac{\pi}{4} - \tan^{-1}\left(\frac{1}{4}\right)\)
3. 3. \(\frac{3\pi}{4} + \tan^{-1}\left(\frac{1}{4}\right)\)
4. 4. \(\frac{\pi}{4} + \tan^{-1}\left(\frac{1}{4}\right)\)

83. The number of all possible solutions of the equation \(z^3 + \overline{z} = 0\) is

1. 1. 4
2. 2. 5
3. 3. 3
4. 4. 6

84. \(S = \{z \in C \mid |z - 1 + i| = 1\}\) represents

1. 1. A circle with centre \((-1, 1)\) and radius 1 unit
2. 2. A circle with centre \((1, 2)\) and radius 5 units
3. 3. A circle with centre \((1, -1)\) and radius 1 unit
4. 4. An ellipse with centre \((1, -1)\)

85. If \(\left|z - \frac{2}{z}\right| = 2\), then the greatest value of \(|z|\) is

1. 1. \(\sqrt{3} - 1\)
2. 2. \(\sqrt{3}\)
3. 3. \(\sqrt{3} + 1\)
4. 4. \(\sqrt{3} + 2\)

86. If \(\sqrt{-3 - 4i} = re^{i\theta}\), then \(r^2 \tan\theta =\)

1. 1. -5
2. 2. 5
3. 3. 10
4. 4. -10

87. If \(Z_1 = 2 - 3i\) and the roots of the equation \(z^3 + bz^2 + cz + d = 0\) are \(i, z_1\) and \(\overline{z_1}\), then \(b + c + d =\)

1. 1. 13
2. 2. -13
3. 3. 9 - 10i
4. 4. 10 - 10i

88. Let the two values of \(z = \sqrt{\frac{1 - i}{1 + i}}\) be \(z_1\) and \(z_2\). If \(-\frac{\pi}{2} < \mathrm{Arg}(z_1) < \mathrm{Arg}(z_2) < \pi\), then \(\arg(z_1) + \arg(z_2) =\)

1. 1. \(\frac{\pi}{4}\)
2. 2. \(\frac{3\pi}{2}\)
3. 3. \(\frac{\pi}{3}\)
4. 4. \(\frac{\pi}{2}\)

89. If C is a point on the straight line joining the points A(-2 + i) and B(3 - 4i) in the Argand plane and \(\frac{AC}{CB} = \frac{1}{2}\), then the argument of C is

1. 1. \(\tan^{-1}3\)
2. 2. \(\tan^{-1}2 - \pi\)
3. 3. \(\tan^{-1}2\)
4. 4. \(\pi - \tan^{-1}3\)

90. If \(\alpha\) is the modulus of \(z_1 = 4 + 3i\), then a point that does not lie in the region represented by \(|z - \overline{z_1}| \leq \alpha\) is

1. 1. \(z_1 - 2i\)
2. 2. \(z_1\)
3. 3. \(2z_1 - 7i\)
4. 4. \(3z_1 - (10 + 8i)\)

91. If \(z_1, z_2, z_3\) are the vertices of an equilateral triangle and \(z\) is its circum centre, then

1. 1. \(z_1^2 + z_2^2 + z_3^2 = 3z^2\)
2. 2. \(z_1^2 + z_2^2 + z_3^2 = z^2\)
3. 3. \(z_1^2 + z_2^2 + z_3^2 = 2z^2\)
4. 4. \(z_1^2 + z_2^2 + z_3^2 = 4z^2\)

92. For real numbers \(a\) and \(b\), if \(4a + i(3a - b) = b - 6i\) and \(z = a + \frac{b}{4}i\), then \(\left|\frac{z}{a}\right| =\)

1. 1. \(2\sqrt{2}\)
2. 2. \(6\sqrt{2}\)
3. 3. \(\sqrt{2}\)
4. 4. 2

93. If \(z = (1 - i)^3(x + i)\) is a purely imaginary number for \(x = x_1\) and \(z\) is a purely real number for \(x = x_2\), then \(x_1 x_2 =\)

1. 1. -1
2. 2. 0
3. 3. 1
4. 4. 2

94. The modulus of the conjugate of \(Z = \frac{-2 + i}{(1 - 2i)^2}\) is

1. 1. \(\frac{1}{5}\)
2. 2. \(\frac{1}{\sqrt{5}}\)
3. 3. \(\frac{1}{25}\)
4. 4. \(\sqrt{5}\)

95. If \(z_1 = 2 + 5i, z_2 = -1 + 4i\) and \(z_3 = i\), then \(\left|\frac{z_1 - z_3}{z_3 - z_2}\right| =\)

1. 1. \(\sqrt{2}\)
2. 2. \(2\sqrt{2}\)
3. 3. \(5\sqrt{2}\)
4. 4. \(4\sqrt{2}\)

96. The locus of the variable point \(z = x + iy\) whose amplitude is always equal to \(\theta\), is

1. 1. \(x^2 + y^2 = \tan^2\theta\)
2. 2. \(y = x\tan\theta\)
3. 3. \(\frac{x^2}{\sin^2\theta} + \frac{y^2}{\cos^2\theta} = 1\)
4. 4. \(\frac{x^2}{\sin^2\theta} - \frac{y^2}{\cos^2\theta} = 1\)

97. If \(z = x + iy\) represents a point in the Argand plane, then a point which is not in the region represented by \(|z - 1 + i| \leq 2\) is

1. 1. \(\frac{1 - i}{2}\)
2. 2. 1
3. 3. \(\frac{1 - i}{4}\)
4. 4. \(i\)

98. Let the locus of a point \(z\) in the Argand plane satisfying the condition \(\operatorname{Re}(z^2) = 4\) be \(C_1\) and the locus of \(z\) satisfying the condition \(\operatorname{Im}(z^2) = 4\) be \(C_2\). Then the number of common points of the two curves \(C_1\) and \(C_2\) are

1. 1. 0
2. 2. 3
3. 3. 4
4. 4. 2

99. If \(z\) is a point on the circle \(|z| = 1\) with \(\mathrm{Arg}(z) = \frac{\pi}{6}\), then \(\frac{z^{12} + 1 - z^6}{z^{12} + iz^6 - 1} =\)

1. 1. \(2 + 3i\)
2. 2. \(3i\)
3. 3. \(3 + 2i\)
4. 4. \(4 + 3i\)

100. If \(-3 + ix^2y\) and \(x^2 + y + 4i\) are complex conjugates, then \(x =\)

1. 1. 0
2. 2. \(\pm 1\)
3. 3. \(\pm 3\)
4. 4. \(\pm 4\)

101. If \(Z = 1 + \cos\theta - i\sin\theta\), \(0 < \theta < \pi\), then \(\left||z - 1|^2 - \left|\frac{z}{4}\right|^2\right|^{1/2} =\)

1. 1. \(\sqrt{2}\cos\theta\)
2. 2. \(\sqrt{2}\sin\theta\)
3. 3. \(\cos\left(\frac{\theta}{2}\right)\)
4. 4. \(\sin\left(\frac{\theta}{2}\right)\)

102. In the Argand plane, the values of \(Z\) satisfying the equation \(|z - 1| = |i(z + 1)|\) lie on

1. 1. The Y-axis
2. 2. A Parabola
3. 3. A Hyperbola
4. 4. The X-axis

103. \(\operatorname{Arg}\left(\frac{4 + 2i}{1 - 2i} + \frac{3 + 4i}{2 + 3i}\right)\) lies in the interval

1. 1. \(\left(0, \frac{\pi}{2}\right)\)
2. 2. \(\left(\frac{\pi}{2}, \pi\right)\)
3. 3. \(\left(\pi, \frac{3\pi}{2}\right)\)
4. 4. \(\left(\frac{3\pi}{2}, 2\pi\right)\)

104. The multiplicative inverse of \(z\) is

1. 1. \(\frac{1}{z}\)
2. 2. \(\overline{z}\)
3. 3. \(\frac{\overline{z}}{|z|^2}\)
4. 4. \(\frac{z}{|z|^2}\)

105. If \(z_1 = 2 + 3i\), \(z_2 = 4 - 5i\) and \(z_3\) are three points in the Argand plane such that \(5z_1 + xz_2 + yz_3 = 0\) \((x, y \in R)\) and \(z_3\) is the midpoint of the line segment joining the points \(z_1\) and \(z_2\), then \(x + y =\)

1. 1. -5
2. 2. 0
3. 3. 4
4. 4. -1

106. If \(z_1\) and \(z_2\) are complex numbers such that \(|z_1 + z_2| = |z_1| + |z_2|\), then the difference in the amplitudes of \(z_1\) and \(z_2\) is

1. 1. \(\frac{\pi}{4}\)
2. 2. \(\frac{\pi}{3}\)
3. 3. \(\frac{\pi}{2}\)
4. 4. 0

107. If \(i = \sqrt{-1}\), then \(1 + i^2 + i^4 + i^6 + \ldots + i^{2024} =\)

1. 1. \(i\)
2. 2. \(-i\)
3. 3. 1
4. 4. -1

108. If \(\frac{1 + i\cos\theta}{1 - 2i\cos\theta}\) is purely real, then \(\cos^3\theta + \sin^2\theta + \cos\theta + 1 =\)

1. 1. 0
2. 2. 1
3. 3. 2
4. 4. \(\frac{3}{4}(2 + \sqrt{2})\)

109. If \(\alpha, \beta\) are non-zero integers and \(z = (\alpha + i\beta)(2 + 7i)\) is a purely imaginary number, then minimum value of \(|z|^2\) is

1. 1. 0
2. 2. 2809
3. 3. 2808
4. 4. 1

110. \(\operatorname{Arg}\left(\sin\frac{6\pi}{5} + i\left(1 + \cos\frac{6\pi}{5}\right)\right) =\)

1. 1. \(\frac{5\pi}{6}\)
2. 2. \(\frac{6\pi}{5}\)
3. 3. \(\frac{2\pi}{5}\)
4. 4. \(\frac{9\pi}{10}\)

111. If \(x + iy = \sqrt{\frac{3 + i}{1 + 3i}}\), then \((x^2 + y^2)^2 =\)

1. 1. 0
2. 2. 1
3. 3. 2
4. 4. 3

112. If the imaginary part of \(\frac{2z + 1}{iz + 1}\) is -2, then the locus of the point representing \(z\) in the Argand plane is

1. 1. a circle
2. 2. a straight line
3. 3. a parabola
4. 4. an ellipse

113. If the value of \(\sqrt{-5 - 12i} + \sqrt{7 + 24i}\) is a negative real number \(k\), then \(k =\)

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

114. Let \(z = x + iy\) be a point in the Argand plane. If the amplitude of \(\left(\frac{z - 3}{z + 2i}\right)\) is \(\frac{\pi}{2}\), then the locus of \(z\) is

1. 1. A circle
2. 2. A straight line
3. 3. A semicircular arc not containing the origin
4. 4. A semicircular arc containing the origin

115. If a point P denotes the complex number \(z = x + iy\) in the Argand plane and if \(\frac{z - (2 + i)}{z + (1 - 2i)}\) is purely real, then the locus of P is

1. 1. The line \(x + 3y - 5 = 0\) excluding the point \((-1, 2)\)
2. 2. The circle \(x^2 + y^2 - x - 3y = 0\) excluding the point \((-1, 2)\)
3. 3. The line \(x + 3y - 5 = 0\) and the circle \(x^2 + y^2 - x - 3y = 0\) excluding the point \((-1, 2)\)
4. 4. The circle \(x^2 + y^2 - 2x - 6y + 5 = 0\) excluding the point \((-1, 2)\)

116. If \(i = \sqrt{-1}\), then \(\sum_{n=0}^{\infty}\left(\frac{i}{3}\right)^n =\)

1. 1. \(\frac{9 - 3i}{10}\)
2. 2. \(9 - 3i\)
3. 3. \(9 + 3i\)
4. 4. \(\frac{9 + 3i}{10}\)

117. If \(i = \sqrt{-1}\), then \(\operatorname{Arg}\left(\frac{(1 + i)^{2025}}{(1 - i)^{2022}}\right) =\)

1. 1. \(-\pi\)
2. 2. \(\frac{\pi}{4}\)
3. 3. \(\frac{3\pi}{4}\)
4. 4. \(\frac{-3\pi}{4}\)

118. The locus of \(z\) such that \(\left|\frac{z - i}{z + i}\right| = 2\), where \(z = x + iy\), is

1. 1. \(3x^2 + 3y^2 + 10y + 3 = 0\)
2. 2. \(3x^2 - 3y^2 - 10y - 3 = 0\)
3. 3. \(3x^2 + 3y^2 + 10y - 3 = 0\)
4. 4. \(x^2 + y^2 - 5y + 3 = 0\)

119. If the roots of the equation \(z^2 - i = 0\) are \(\alpha\) and \(\beta\), then \(|\operatorname{Arg}\beta - \operatorname{Arg}\alpha| =\)

1. 1. \(2\pi\)
2. 2. \(\frac{\pi}{2}\)
3. 3. \(\pi\)
4. 4. \(\frac{\pi}{4}\)

120. If \(i^2 = -1\), then \((1 + \sqrt{3}i)^{2022} - (\sqrt{3} - i)^{2022} =\)

1. 1. \(2^{2023}\)
2. 2. 0
3. 3. \(2^{2022}\)
4. 4. \(3^{1011}\)

121. If \(\left(\frac{\sqrt{3} + i}{\sqrt{3} - i}\right)^4 + \left(\frac{\sqrt{3} - i}{\sqrt{3} + i}\right)^4 = r\,\mathrm{cis}\,\theta\), then one of the values of \(\sqrt{r\,\mathrm{cis}\,\theta}\) is

1. 1. \(\mathrm{cis}\left(\frac{3\pi}{4}\right)\)
2. 2. \(\mathrm{cis}\left(\frac{3\pi}{2}\right)\)
3. 3. \(\mathrm{cis}\left(\frac{\pi}{3}\right)\)
4. 4. \(\mathrm{cis}\,\pi\)

122. If \(z = x + iy\) and the point P in the Argand plane represents \(z\), then the locus of \(z\) satisfying the equation \(|z - 2| + |z - 2i| = 4\) is

1. 1. \(4x^2 + 3xy + 4y^2 - 6x - 6y + 8 = 0\)
2. 2. \(3x^2 + 2xy + 3y^2 - 8x - 8y + 6 = 0\)
3. 3. \(3x^2 + 2xy + 3y^2 - 8x - 8y = 0\)
4. 4. \(4x^2 + 3xy + 4y^2 - 6x - 6y = 0\)