Purely Real

Q.

What is $\overline{Z}$ in complex numbers?

Q. If z is a complex number , then the minimum value of $|z|+|z-1|$ is

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

Q.

Find the smallest positive integer value of n for which $\frac{(1+i{)}^n}{(1-n{)}^{n-2}}$ is a real number.

Q. If $|z_1|=|z_2|$ and $\arg \left(\frac{z_1}{z_2}\right)=\pi$, then value of $z_1+z_2$ is

Q.

Let $f\left(x\right)=x^n,n$ being a positive integer. The value of $n$ for which the equality $f(a+b)=f\left(a\right)+f\left(b\right).f$ is valid for all $a,b>0$ is

1. $0,2$

2. $1,3$

3. $3,4$

4. None of these

Q. Let $z$ be a unit complex number having the argument $\theta$, $0<\theta <\frac{\pi }{2}$ and satisfying the relation $|z-3i|=3,$ then $\arg (\cot \theta -\frac{6}{z})$

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

Q.

Find the real value of a for which $3i^3-2a{i}^2+(1-a)i+5$ is real.

Q.

Solve the system of equations Re ($z^2$)=0, |z| = 2.

Q. If $z=1+\cos \frac{6\pi }{5}+i\sin \frac{6\pi }{5}$ , then

1. $amp\left(z\right)=\frac{-2\pi }{5}$
2. $amp\left(z\right)=\frac{2\pi }{5}$
3. $|z|=2\cos \frac{2\pi }{5}$
4. $|z|=-2\cos \frac{2\pi }{5}$

Q. If $z=\sin \theta +i(\cos \theta -1),i^2=-1$ is a purely real as well as a purely imaginary number, then the number of value(s) of $\theta \in [0,4\pi ]$ is

Q. The complex number(s) $z$ satisfying $Re\left(z^2\right)=0$ and $|z|=\sqrt{3}$ is (are)

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

Q.

a + ib > c + id can be explained only when

1. b = 0, c = 0

2. b = 0, d = 0

3. a = 0, c = 0

4. a = 0, d = 0

Q. If $a$ and $b$ are positive integers such that $N=(a+ib{)}^3-107i$ is a positive integer, then the value of $N$ is

Q. Let z be a complex number such that the imaginary part of z is non - zero and $a=z^2+z+1$ is real. Then, $a$ cannot take the value

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

Q.

Conjugate of $x+iy$ is

Q.

If $\frac{z-1}{z+1}$ is purely imaginary then what would be the locus of $z$?

Q. If complex numbers $z_1$ and $z_2$ both satisfy $z+\overline{z}=2|z-1|$ and $\arg (z_1-z_2)=\frac{\pi }{3},$ then find the value of $\text{Im}(z_1+z_2)$. (where $\text{Im}\left(z\right)$ denotes the imaginary part of $z$)

1. $\sin \frac{\pi }{3}$
2. $\text{cosec}\frac{\pi }{3}$
3. $\tan \frac{\pi }{3}$
4. $\cot \frac{\pi }{3}$

Q. If $z_1,z_2,z_3$ are vertices of an equilateral triangle $ABC$ such that $|z_1-i|=|z_2-i|=|z_3-i|$, then $|z_1+z_2+z_3|$ equals to

1. $3\sqrt{3}$
2. $\sqrt{3}$
3. $3$
4. $\frac{1}{3\sqrt{3}}$

Q. If $\arg \left(z\right)=\frac{\pi }{3}$ and $\arg (z-1)=\frac{2\pi }{3},$ then $z$ is

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

Q.

Prove that $1+2+3+.....+n=\left[\frac{n(n+1)}{2}\right]$ for $n$ being a natural number.

Q. If $z=1+\cos \frac{6\pi }{5}+i\sin \frac{6\pi }{5}$ , then

1. $amp\left(z\right)=\frac{-2\pi }{5}$
2. $amp\left(z\right)=\frac{2\pi }{5}$
3. $|z|=2\cos \frac{2\pi }{5}$
4. $|z|=-2\cos \frac{2\pi }{5}$

Q. Evaluate the difinite integral: ${\int }_0^1(x{e}^x+\sin \frac{\pi x}{4})dx$

Q.

Find the value off :

$a+2b-5c$ when $a=2$, $b=-3$ and $c=1$

Q. If a and b are real number between 0 and 1 such that the points $z_1=a+i,z_2=b+iand{z}_3=0$
form an equilateral triangle, then

1. 
2. 
3. 
4. None of these

Q. For $a<0$, $\text{arg}\left(ia\right)$ is

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

Q.

When $a=3$, $b=0$, and $c=-2$ , find the value of :

$ab+2bc+3ca+4abc$

Q.

Find the value off :

$3ab+4bc-5ca$ when $a=4$, $b=5$ and $c=-2$

Q. The domain of ${\cos }^{-1}(x^2-4)$ is

1. $[-1,1]$
2. $[-\sqrt{5},-\sqrt{3}]\cap [-\sqrt{5},\sqrt{3}]$
3. $[-\sqrt{5},-\sqrt{3}]\cup [\sqrt{3},\sqrt{5}]$
4. $[3,5]$

Q.

If z is a complex number, then the minimum value of $\left|z\right|+|z-1|$is

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

Q. If z is a complex number such that $z^2=\left(\overline{z}{)}^2\right),$ then

1. None of these

2. z is purely real

3. z is purely imaginary

4. Either z is purely real or purely imaginary