5. If $\mathrm { w } = \alpha + \mathrm { i } \beta$, where $\beta \neq 0$ and $\mathrm { z } \neq 1$, satisfies the condition that $\left( \frac { \mathrm { w } - \overline { \mathrm { w } } \mathrm { z } } { 1 - \mathrm { z } } \right)$ is purely real, then the set of values of z is
(A) $\{ \mathrm { z } : | \mathrm { z } | = 1 \}$
(B) $\{ \mathrm { z } : \mathrm { z } = \overline { \mathrm { z } } \}$
(C) $\{ z : z \neq 1 \}$
(D) $\{ \mathrm { z } : | \mathrm { z } | = 1 , \mathrm { z } \neq 1 \}$
Sol. (D)
$$\begin{aligned} & \frac { \mathrm { w } - \overline { \mathrm { w } } \mathrm { z } } { 1 - \mathrm { z } } = \frac { \overline { \mathrm { w } } - \mathrm { w } \overline { \mathrm { z } } } { 1 - \overline { \mathrm { z } } } \\ & \Rightarrow \quad ( \mathrm { z } \overline { \mathrm { z } } - 1 ) ( \overline { \mathrm { w } } - \mathrm { w } ) = 0 \\ & \Rightarrow \quad \mathrm { z } \overline { \mathrm { z } } = 1 \Rightarrow | \mathrm { z } | ^ { 2 } = 1 \Rightarrow | \mathrm { z } | = 1 \end{aligned}$$

1. Let $\mathrm { a } , \mathrm { b } , \mathrm { c }$ be the sides of a triangle. No two of them are equal and $\lambda \in \mathrm { R }$. If the roots of the equation $\mathrm { x } ^ { 2 } + 2 ( \mathrm { a } + \mathrm { b } + \mathrm { c } ) \mathrm { x } + 3 \lambda ( a b + b c + c a ) = 0$ are real, then
   (A) $\lambda < \frac { 4 } { 3 }$
   (B) $\lambda > \frac { 5 } { 3 }$
   (C) $\lambda \in \left( \frac { 1 } { 3 } , \frac { 5 } { 3 } \right)$
   (D) $\lambda \in \left( \frac { 4 } { 3 } , \frac { 5 } { 3 } \right)$

Sol. (A) $\mathrm { D } \geq 0$ $\Rightarrow \quad 4 ( \mathrm { a } + \mathrm { b } + \mathrm { c } ) ^ { 2 } - 12 \lambda ( \mathrm { ab } + \mathrm { bc } + \mathrm { ca } ) \geq 0$ $\Rightarrow \lambda \leq \frac { \mathrm { a } ^ { 2 } + \mathrm { b } ^ { 2 } + \mathrm { c } ^ { 2 } } { 3 ( \mathrm { ab } + \mathrm { bc } + \mathrm { ca } ) } + \frac { 2 } { 3 }$ Since $| \mathrm { a } - \mathrm { b } | < \mathrm { c } \Rightarrow \mathrm { a } ^ { 2 } + \mathrm { b } ^ { 2 } - 2 \mathrm { ab } < \mathrm { c } ^ { 2 }$
$$\begin{aligned} & | b - c | < a \Rightarrow b ^ { 2 } + c ^ { 2 } - 2 b c < a ^ { 2 } \\ & | c - a | < b \Rightarrow c ^ { 2 } + a ^ { 2 } - 2 a c < b ^ { 2 } \end{aligned}$$
From (1), (2) and (3), we get $\frac { a ^ { 2 } + b ^ { 2 } + c ^ { 2 } } { a b + b c + c a } < 2$. Hence $\lambda < \frac { 2 } { 3 } + \frac { 2 } { 3 } \Rightarrow \lambda < \frac { 4 } { 3 }$.

If $|z| = 1$ and $z \neq \pm 1$, then all the values of $\frac{z}{1-z^2}$ lie on
(A) a line not passing through the origin
(B) $|z| = \sqrt{2}$
(C) the $x$-axis
(D) the $y$-axis

46. If $| z | = 1$ and $z \neq \pm 1$, then all the values of $\frac { z } { 1 - z ^ { 2 } }$ lie on
(A) a line not passing through the origin
(B) $| z | = \sqrt { 2 }$
(C) the $x$-axis
(D) the $y$-axis

Answer

[Figure]
(A)
[Figure]
(B)
[Figure]
(C)
[Figure]
(D)

1. Let $E ^ { c }$ denote the complement of an event $E$. Let $E , F , G$ be pairwise independent events with $P ( G ) > 0$ and $P ( E \cap F \cap G ) = 0$. Then $P \left( E ^ { c } \cap F ^ { c } \mid G \right)$ equals
   (A) $P \left( E ^ { c } \right) + P \left( F ^ { c } \right)$
   (B) $P \left( E ^ { c } \right) - P \left( F ^ { c } \right)$
   (C) $P \left( E ^ { c } \right) - P ( F )$
   (D) $P ( E ) - P \left( F ^ { c } \right)$

Answer ◯
(A)
[Figure]
(B)
[Figure]
(C)
◯
(D)

Let $z _ { 1 }$ and $z _ { 2 }$ be two distinct complex numbers and let $z = ( 1 - t ) z _ { 1 } + t z _ { 2 }$ for some real number $t$ with $0 < t < 1$. If $\operatorname { Arg } ( w )$ denotes the principal argument of a nonzero complex number $w$, then
A) $\left| z - z _ { 1 } \right| + \left| z - z _ { 2 } \right| = \left| z _ { 1 } - z _ { 2 } \right|$
B) $\operatorname { Arg } \left( z - z _ { 1 } \right) = \operatorname { Arg } \left( z - z _ { 2 } \right)$
C) $\left| \begin{array} { c c } \mathrm { z } - \mathrm { z } _ { 1 } & \overline { \mathrm { z } } - \overline { \mathrm { z } } _ { 1 } \\ \mathrm { z } _ { 2 } - \mathrm { z } _ { 1 } & \overline { \mathrm { z } } _ { 2 } - \overline { \mathrm { z } } _ { 1 } \end{array} \right| = 0$
D) $\operatorname { Arg } \left( z - z _ { 1 } \right) = \operatorname { Arg } \left( z _ { 2 } - z _ { 1 } \right)$

Let $z$ be a complex number with non-zero imaginary part. If
$$\frac { 2 + 3 z + 4 z ^ { 2 } } { 2 - 3 z + 4 z ^ { 2 } }$$
is a real number, then the value of $| z | ^ { 2 }$ is $\_\_\_\_$.

Let $A = \left\{ \frac { 1967 + 1686 i \sin \theta } { 7 - 3 i \cos \theta } : \theta \in \mathbb { R } \right\}$. If $A$ contains exactly one positive integer $n$, then the value of $n$ is

If $z \neq 1$ and $\frac{z^{2}}{z-1}$ is real, then the point represented by the complex number $z$ lies
(1) either on the real axis or on a circle passing through the origin
(2) on a circle with centre at the origin
(3) either on the real axis or on a circle not passing through the origin
(4) on the imaginary axis

Let $a = \operatorname { Im } \left( \frac { 1 + z ^ { 2 } } { 2 i z } \right)$, where $z$ is any non-zero complex number. The set $\mathrm { A } = \{ a : | z | = 1$ and $z \neq \pm 1 \}$ is equal to:
(1) $( - 1,1 )$
(2) $[ - 1,1 ]$
(3) $[ 0,1 )$
(4) $( - 1,0 ]$

If $\frac { z - \alpha } { z + \alpha } ( \alpha \in R )$ is a purely imaginary number and $| z | = 2$, then a value of $\alpha$ is :
(1) 1
(2) $\frac { 1 } { 2 }$
(3) $\sqrt { 2 }$
(4) 2

If $\frac { 3 + i \sin \theta } { 4 - i \cos \theta } , \theta \in [ 0,2 \pi ]$, is a real number, then an argument of $\sin \theta + i \cos \theta$ is
(1) $\pi - \tan ^ { - 1 } \left( \frac { 4 } { 3 } \right)$
(2) $\pi - \tan ^ { - 1 } \left( \frac { 3 } { 4 } \right)$
(3) $- \tan ^ { - 1 } \left( \frac { 3 } { 4 } \right)$
(4) $\tan ^ { - 1 } \left( \frac { 4 } { 3 } \right)$

Let $S = \left\{ z = x + iy : \frac { 2z - 3i } { 4z + 2i } \text{ is a real number} \right\}$. Then which of the following is NOT correct?
(1) $y + x ^ { 2 } + y ^ { 2 } \neq - \frac { 1 } { 4 }$
(2) $( x , y ) = \left( 0 , - \frac { 1 } { 2 } \right)$
(3) $x = 0$
(4) $y \in \left( - \infty , - \frac { 1 } { 2 } \right) \cup \left( - \frac { 1 } { 2 } , \infty \right)$

Q61. 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
(1) $3 \pi$
(2) $2 \pi$
(3) $5 \pi$
(4) $4 \pi$