Purely Imaginary

Q.

If $z_1$ and $z_2$ are two non-zero complex numbers such that $\left|z_1+z_2\right|=\left|z_1\right|+\left|z_2\right|$, then $\arg \left(\frac{z_1}{z_2}\right)$ is equal to

1. $0$

2. $-\mathrm{\pi }$

3. $-\frac{\mathrm{\pi }}{2}$

4. $\frac{\mathrm{\pi }}{2}$

5. $\mathrm{\pi }$

Q.

if $\frac{z-i}{z+i}$(z ≠ -i) is a purely imaginary number, then z.$\overline{z}$ is equal to

1. 0

2. 1

3. 2

4. None of these

Q. The set of all $\alpha \in R$, for which $w=\frac{1+(1-8\alpha )z}{1-z}$ is a purely imaginary number, for all $z\in C$ satisfying $|z|=1$ and $Rez\ne 1$, is :

1. equal to $R$
2. $\left\{0\right\}$
3. $\{0,\frac{1}{4},-\frac{1}{4}\}$
4. an empty set

Q. Let $a,b,x$ and $y$ be real numbers such that $a-b=1$ and $y\ne 0$. If the complex numbers $z=x+iy$ satisfies Im $\left(\frac{az+b}{z+1}\right)=y$, then which of the following is possible value of $x$?

1. $1-\sqrt{1+y^2}$
2. $-1-\sqrt{1-y^2}$
3. $1+\sqrt{1+y^2}$
4. $-1+\sqrt{1-y^2}$

Q.

The function $\sin x–bx+c$will be increasing in the interval$(–\infty ,\infty )$, if

1. $b\le 1$

2. $b\le 0$

3. $b<–1$

4. $b\ge 0$

Q.

$\text{If}x+iy=\frac{3+5i}{7-6i},\text{then y=}$

1. $\frac{9}{85}$

2. $-\frac{9}{85}$

3. $\frac{53}{85}$

4. none of these

Q. $\frac{3+2isin\theta }{1-2isin\theta }$ will be purely imaginary, if $\theta$ =

[Where n is an integer]

[IIT 1976; Pb. CET 2003]

1. $2n\pi \pm \frac{\pi }{3}$
2. $n\pi +\frac{\pi }{3}$
3. $n\pi \pm \frac{\pi }{3}$
4. None of these

Q. Let $z$ be a complex number satisfying equation $z^n={\left(\overline{z}\right)}^m,$ where $n,m\in N.$ Then

1. If $n=m,$ then number of solutions of the equation will be finite
2. If $n=m,$ then number of solutions of the equation will be infinite
3. If $n\ne m,$ then number of solutions of the equation will be $n+m$
4. If $n\ne m,$ then number of solutions of the equation will be $n+m+1$

Q. If $\left|Z_1\right|+\left|Z_2\right|+\left|Z_3\right|=|Z_1+Z_2+Z_3|$, then $\frac{Z_1{Z}_2}{Z_3^2}+\frac{Z_2{Z}_3}{Z_2^2}+\frac{Z_3{Z}_1}{Z_2^2}$is

1. real number
2. of modulus 1
3. None of these
4. Purely imaginary

Q. $LetA=\{\theta \in (-\frac{\pi }{2},\pi ):\frac{3+2i\sin \theta }{1-2i\sin \theta }\text{is purely imaginary}\}$. The sum of all the elements in $A$ is :

1. $\pi$
2. $\frac{2\pi }{3}$
3. $\frac{3\pi }{4}$
4. $\frac{5\pi }{6}$

Q. If $a_1,a_2,a_3,\cdots ,a_r$ are in $G.P.$ with common ratio $R,$ then the determinant $\left|\begin{array}{ccc}{a}_{r+1}& a_{r+5}& a_{r+9}\\ a_{r+7}& a_{r+11}& a_{r+15}\\ a_{r+11}& a_{r+17}& a_{r+21}\end{array}\right|$ is

1. Propotional to $R$
2. Propotional to $R^2$
3. Independent of $R$
4. Propotional to $R^3$

Q.

If $\mathrm{Re}\left[{(1+\cos \theta +2i\sin \theta )}^{-1}\right]=4$, then the value of $\theta$ is

1. $\frac{\pi }{2}$

2. $\frac{\pi }{3}$

3. $-\frac{\pi }{3}$

4. $\pi$

Q. Let a, b and c are three distinct real positive numbers and n be the number of solution of the equation $a{x}^2-b|x|-c=0$. If $z_1\text{and}{z}_2$ be the complex numbers satisfying $|z-4i|=3$ and having least and greatest amplitudes respectively. Then the value of ${(|z_1{|}^2+|z_2{|}^2)}^{\frac{1}{n}}$

1. $\sqrt{14}$
2. $5\sqrt{2}$
3. $\sqrt{10}$
4. $\frac{5}{\sqrt{2}}$

Q. If $\frac{2z_1}{3z_2}$ is a purely imaginary number, then $\left|\frac{z_1-z_2}{z_1+z_2}\right|$ =
[MP PET 1993]

1. 1

2. 2/3
3. 4/9

4. 3/2

Q. For all complex numbers $z_1,z_2$ satisfying $|z_1|=12$ and $|z_2–3–4i|=5$, then find the minimum value of $|z_1–z_2|$.

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

Q. The value of the expression $(\sin {80}^{\circ }-\cos {80}^{\circ })$ is negative.

1. True
2. False
3. Ambiguous
4. Data insufficient
   

Q. If $y=\sin \left(2{\sin }^{-1}x\right),then\frac{dy}{dx}=$

Q. Let $a,bϵR$ and $a^2+b^2\ne 0$. Suppose $S=\{zϵC:z=\frac{1}{a+ibt},tϵR,t\ne 0\}$, where $i=\sqrt{-1}$. If $z=x+iy$ and $zϵS$, then $(x,y)$ lies on

1. The circle with radius $\frac{1}{2a}$ and centre $(\frac{1}{2a},0)$ for $a>0,b\ne 0$
2. The circle with radius $-\frac{1}{2a}$ and centre $(-\frac{1}{2a},0)$ for $a<0,b\ne 0$
3. The X - axis for $a\ne 0,b=0$
4. The Y - axis for $a=0,b\ne 0$

Q. If $y={\sin }^{-1}(x^2\sqrt{1-x}-\sqrt{x}\sqrt{1-x^4}),$ find $\frac{dy}{dx}.$

1. $\frac{2x}{\sqrt{1-x^4}}-\frac{1}{2\sqrt{x}\sqrt{(1-x)}}$
2. $\frac{2x}{\sqrt{1+x^4}}-\frac{1}{2\sqrt{x}\sqrt{(1-x)}}$
3. $\frac{2x}{\sqrt{1-x^4}}-\frac{1}{2\sqrt{x}\sqrt{(1+x)}}$
4. $\frac{2x}{\sqrt{1+x^4}}-\frac{1}{2\sqrt{x}\sqrt{(1+x)}}$

Q. If $x,y,z$ are all different and not equal to zero and $\left|\begin{array}{ccc}1+x& 1& 1\\ 1& 1+y& 1\\ 1& 1& 1+z\end{array}\right|=0$ then the value of $x^{-1}+y^{-1}+z^{-1}$ is equal to

1. $x^{-1}{y}^{-1}{z}^{-1}$
2. -x-y-z
3. $-1$
4. $xyz$

Q. If $z$ is a complex number satisfying $|z-3-2i|\le 1$, then the maximum value of $|iz+2|$ is

1. $16$
2. $24$
3. $4$
4. $32$

Q. The value of $K$ in order that
$f\left(x\right)=\sin x-\cos x-Kx+b$ decreases for all real values is given by-

1. $K<1$
2. $K\ge \sqrt{2}$
3. $K\ge 1$
4. $K<\sqrt{2}$

Q. Solve: $\underset{x\to \frac{5\pi }{6}}{lim}\frac{cosx-\sqrt{3}sinx}{\pi -6x}$

Q.

If $\frac{z-1}{z+1}$ is purely imaginary number $(z\ne -1)$, find the value of |z|.

Q. Let $a,b,x$ and $y$ be real numbers such that $a–b=1$ and $y\ne 0$. If the complex number $z=x+iy$ satisfies $lm\left(\frac{az+b}{z+1}\right)=y$ , then which of the following is(are) possible value(s) of $x$ ?

1. $1-\sqrt{1+y^2}$
2. $-1-\sqrt{1-y^2}$
3. $-1+\sqrt{1-y^2}$
4. $1+\sqrt{1+y^2}$

Q. What is the additive inverse of $\frac{1}{3-\sqrt{8}}$.

1. $\sqrt{8}+$
2. $-(\sqrt{8}+3)$
3. $\sqrt{3}+8$
4. $-(\sqrt{3}+8)$

Q. If $2a+3b+6c=0,a,b,cϵR$, then the quadratic equation $a{x}^2+bx+c=0$ has

1. at least one root in [2, 3]

2. at least one root in [0, 1]

3. None of the above

4. at least one root in (3, 4]

Q. Find the value of $x$ for which
$\sin h^{-1}\frac{3}{4}+\sin h^{-1}x=\sin h^{-1}\frac{4}{3}$

Q. If $\frac{2z_1}{3z_2}$ is a purely imaginary number, then $\left|\frac{z_1-z_2}{z_1+z_2}\right|$ =
[MP PET 1993]

1. 3/2

2. 1

3. 2/3
4. 4/9

Q.

Find sum of all possible values of $\theta$ in the interval $\left(-\frac{\pi }{2},\pi \right)$ for which $\frac{3+2i\sin \theta }{1-2i\sin \theta }$ is purely imaginary;

1. $\frac{\mathrm{\pi }}{3}$

2. $\mathrm{\pi }$

3. $\frac{2\mathrm{\pi }}{3}$

4. $\frac{\mathrm{\pi }}{2}$