Purely Imaginary

Q.

The value of the expression ${\sin }^6\left(\theta \right)+{\cos }^6\left(\theta \right)+3{\sin }^2\left(\theta \right)·{\cos }^2\left(\theta \right)$ is

1. $0$

2. $-1$

3. $1$

4. $2$

Q.

Evaluate ${[i^{18}+{\left(\frac{1}{i}\right)}^{25}]}^3$

Q.

For the complex number z, one from z+$\overline{z}$ and z$\overline{z}$ is

1. A real number

2. A imaginary number

3. Both are real numbers

4. Both are imaginary numbers

Q.

If the equation $x^2+2x+3\lambda =0and2x^2+3x+5\lambda =0$ have a non - zero common root, then $\lambda$=

1. 3

2. None

3. 1

4. -1

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

1. 0
2. 1
3. 2
4. None of these

Q.

If Z= $\frac{(13-5i)}{(4-9i)}$ find $z^6$

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Q.

If x is real and satisfies x + 2 > $\sqrt{x+4}$, then

1. 

2. 

3. 

4. 

Q.

Find the multiplicative inverse of each of the complex numbers

$\sqrt{5}+3i$

Q.

If equation $a{x}^2+bx+c$ = 0 has only imaginary roots and c < 0, then a is ______.

1. positive

2. negative

3. Zero

4. Can’t say

Q. Convert the given complex number in polar form: $i$

Q.

$\text{If}\underset{x\to \infty }{lim}(\sqrt{x^2-x+1}-ax-b)=0\text{then the values of a and b are given by}$

1. 

2. 

3. a=-1, b=-1/2

4. 

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.

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/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. Find the multiplicative inverse of the complex number $\sqrt{5}+3i$.

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

Q.

Show that $\left|\frac{z-2}{z-3}\right|=2$ represents a circle. Also, find its centre and radius.

Or

If arg (z - 1) = arg (z + 3 i), find (x - 1) : y, where z = x + iy.

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 $a,b,x$ 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. A value of $\theta$ for which $z=\frac{2+3isin\theta }{1-2isin\theta }$ is purely imaginary, is

1. $\frac{\pi }{3}$
2. $\frac{\pi }{6}$
3. $si{n}^{-1}\left(\frac{1}{\sqrt{3}}\right)$
4. $si{n}^{-1}\left(\frac{\sqrt{3}}{4}\right)$

Q. A value of $\theta$ for which $\frac{2+3isin\theta }{1-2isin\theta }$ is purely imaginary, is:

1. $si{n}^{-1}\left(\frac{1}{\sqrt{3}}\right)$
2. $\frac{\pi }{6}$
3. $\frac{\pi }{3}$
4. $si{n}^{-1}\left(\frac{\sqrt{3}}{4}\right)$

Q. Find the complex number for which |z+1| = z+2(1+i).

Q.

For the equation $x^2$ - 2ax + $a^2$ - 1 = 0, the values of 'a' for which 3 lies in between the roots of given equation is

1. [3, 4]

2. (-, 3) (4, )

3. (2, 4)

4. (-, 2) (4, )

Q.

If Z = $\frac{13-5i}{4-9i}$ can be written in a+ib. Find the value of a+b.

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Q.

In the quadratic equation p(x)=0 with real coefficients has purely imaginary roots. Then, the equation p[p(x)]=0 has

1. Only purely imaginary roots

2. All real roots

3. Two real and two purely imaginary roots

4. Neither real nor purely imaginary roots

Q. Find the general solution for $\cos 4x=\cos 2x$

1. $x=n\pi$ or $\frac{n\pi }{6}$
2. $x=n\pi$ or $\frac{n\pi }{3}$
3. $x=\frac{2n\pi }{3}$
   
4. $x=\pi$
   

Q. Consider a quadratic equation $a{z}^2+bz+c=0$, where a, b, c are complex numbers, then the condition that equation has one purely imaginary root is,

1. $(c\overline{a}-a\overline{c}{)}^2=-(b\overline{c}+c\overline{b}\left)\right(a\overline{b}+\overline{a}b)$
2. $(c\overline{a}+a\overline{c}{)}^2=(b\overline{c}+c\overline{b}\left)\right(a\overline{b}+\overline{a}b)$
3. None of these
4. $(c\overline{a}-a\overline{c}{)}^2=(b\overline{c}-c\overline{b}\left)\right(a\overline{b}-\overline{a}b)$

Q. If $-1+\sqrt{-3}=r{e}^{i\theta }$, then $\theta$ is equal to :

1. $\frac{2\pi }{3}$
2. $-\frac{2\pi }{3}$
3. $\frac{\pi }{3}$
4. $-\frac{\pi }{3}$

Q.

The value of the sum $\sum _{n-1}^{13}(i^n+i^{n+1})$ , where i = $\sqrt{-1}$ equals

1. i-1

2. -i

3. 0

4. i

Q. If ${\sin }^{-1}\left(\sin 5\right)>x^2-4x$, then

1. $x\in (2+\sqrt{9-2\pi },\infty )$
2. $x\in (2-\sqrt{9-2\pi },2+\sqrt{9-2\pi })$
3. $x\in (2-\sqrt{9-2\pi },\infty )$
4. none of these

Q.

If z be a complex number satisfying $z^2$ + 1 = 0 Find the value of $\left|z\right|$.

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