5. Complex Numbers(11th)

SECTION A: Multiple Choice Questions

1. Defining the set $A = \{ \theta \in [0, 2\pi] :$ $1 + 10 Re \left( \frac{\cos \theta + i \sin \theta}{ \cos \theta - i \sin \theta} \right) = 0$ , then the sum $\sum_{\theta \in A} \theta^2$ is equal to:

(A) $\frac{21}{4} \pi^2$

(B) $6 \pi^2$

(D) $8 \pi^2$

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2. If the locus of $z \in \mathbb{C}$ such that $Re \left( \frac{z-1}{2z+i} \right)$ $+$ $Re \left( \frac{\overline{z}-1}{2\overline{z}-i} \right) = 2$ , is a circle of radius $r$ and center $(a, b)$ , then $\frac{15ab}{r^2}$ is equal to:

(A) 16

(B) 24

(D) 18

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3. Among the statements:

(S1): The set $\{ z \in \mathbb{C} - \{-i\} : |z| = 1 \text{ and } \frac{z-i}{z+i} \text{ is purely real} \}$ contains exactly two elements, and
(S2): The set $\{ z \in \mathbb{C} - \{-1\} : |z| = 1 \text{ and } \frac{z-1}{z+1} \text{ is purely imaginary} \}$ contains infinitely many elements.

(A) both are incorrect

(B) both are correct

(D) only (S1) is correct

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4. Let the product of $\omega_1 = (8 + i) \sin \theta + (7 + 4i) \cos \theta$ and $\omega_2 = (1 + 8i) \sin \theta + (4 + 7i) \cos \theta$ be $\alpha + i\beta, i = \sqrt{-1}$ . Let $p$ and $q$ be the maximum and the minimum values of $\alpha + \beta$ respectively. Then $p + q$ is equal to:

(A) 130

(B) 150

(D) 140

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5. If $z_1, z_2, z_3 \in \mathbb{C}$ are the vertices of an equilateral triangle, whose centroid is $z_0$ , then $\sum_{k=1}^{3} (z_k - z_0)^2$ is equal to:

(A) 0

(B) 1

(D) -i

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6. Let $z \in \mathbb{C}$ be such that $\frac{z^2 + 3i}{z - 2 + i} = 2 + 3i$ . Then the sum of all possible values of $z^2$ is :

(A) $-19 + 2i$

(B) $-19 - 2i$

(D) $19 + 2i$

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7. Let $z$ be a complex number such that $|z| = 1$ . If $\frac{2 + kz}{1 + z^2} = kz$ , $k \in \mathbb{R}$ , then the maximum distance of $k + ik^2$ from the circle $|z - (1 + 2i)| = 1$ is :

(A) $\sqrt{5} + 1$

(B) 3

(D) 2

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8. Let $|z_1 - 8 - 2i| \leq 1$ and $|z_2 - 2 + 6i| \leq 2, z_1, z_2 \in \mathbb{C}$ . Then the minimum value of $|z_1 - z_2|$ is :

(A) 3

(B) 10

(D) 13

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9. If $\alpha + i\beta$ and $\gamma + i\delta$ are the roots of $x^2 - (3 - 2i)x - (2i - 2) = 0, i = \sqrt{-1}$ , then $\alpha\gamma + \beta\delta$ is equal to:

(A) 2

(B) -6

(D) -2

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10. Let $O$ be the origin, the point $A$ be $z_1 = \sqrt{3} + 2\sqrt{2}i$ , the point $B (z_2)$ be such that $\sqrt{3} |z_2| = |z_1|$ and $\arg(z_2) = \arg(z_1) + \frac{\pi}{6}$ . Then

(A) area of triangle $ABO$ is $\frac{11}{4}$

(B) area of triangle $ABO$ is $\frac{11}{\sqrt{3}}$

(D) $ABO$ is an obtuse angled isosceles triangle

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11. If $\alpha$ and $\beta$ are the roots of the equation $2z^2 - 3z - 2i = 0$ , where $i = \sqrt{-1}$ , then
$16 \cdot Re \left( \frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{15} + \beta^{15}} \right) \cdot Im \left( \frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{15} + \beta^{15}} \right)$ is equal to:

(A) 441

(B) 312

(D) 398

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12. The number of complex numbers $z$ satisfying $|z| = 1$ and $\left| \frac{z}{z} + \frac{\overline{z}}{z} \right| = 1$ , is :

(A) 8

(B) 10

(D) 6

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13. Let $\left| \frac{z - i}{2z + i} \right| = \frac{1}{3}, z \in \mathbb{C}$ , be the equation of a circle with center at $C$ . If the area of the triangle, whose vertices are at the points $(0, 0), C$ and $(\alpha, 0)$ is 11 square units, then $\alpha^2$ equals:

(A) $\frac{121}{25}$

(B) 100

(D) 50

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14. Let the curve $z(1 + i) + \overline{z}(1 - i) = 4, z \in \mathbb{C}$ , divide the region $|z - 3| \leq 1$ into two parts of areas $\alpha$ and $\beta$ . Then $|\alpha - \beta|$ equals :

(A) $1 + \frac{\pi}{3}$

(B) $1 + \frac{\pi}{6}$

(D) $1 + \frac{\pi}{4}$

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15. Let $z_1, z_2$ and $z_3$ be three complex numbers on the circle $|z| = 1$ with $\arg(z_1) = -\frac{\pi}{4}$ , $\arg(z_2) = 0$ and $\arg(z_3) = \frac{\pi}{4}$ . If $|z_1 \overline{z_2} + z_2 \overline{z_3} + z_3 \overline{z_1}| = \alpha + \beta \sqrt{2}, \alpha, \beta \in \mathbb{Z}$ , then the value of $\alpha^2 + \beta^2$ is :

(A) 41

(B) 29

(D) 31

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16. Let $z$ be a complex number such that the real part of $\frac{z-2 i}{z+2 i}$ is zero. Then, the maximum value of $|z-(6+8 i)|$ is equal to

(A) 8

(B) 12

(D) $\infty$

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17. The sum of all possible values of $\theta \in[-\pi, 2 \pi]$ , for which $\frac{1+i \cos \theta}{1-2 i \cos \theta}$ is purely imaginary, is equal to :

(A) $4 \pi$

(B) $3 \pi$

(D) $5 \pi$

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18. Let $z$ be a complex number such that $|z+2|=1$ and $lm \left(\frac{z+1}{z+2}\right)=\frac{1}{5}$ . Then the value of $|Re(\overline{z+2})|$ is

(A) $\frac{2 \sqrt{6}}{5}$

(B) $\frac{24}{5}$

(D) $\frac{1+\sqrt{6}}{5}$

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19. If the set $R=\{(a, b): a+5 b=42, a, b \in \mathbb{N}\}$ has $m$ elements and $\sum_{n=1}^{m}\left(1-i^{n!}\right)=x+i y$ , where $i=\sqrt{-1}$ , then the value of $m+x+y$ is

(A) 12

(B) 4

(D) 5

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