Purely Real

Q. Let $A=\{x:x=2n+1,n\in W\}$ and $B=\{y:y=2n,n\in N\}.$ If a relation $R$ is defined from $A$ to $B,$ then which of the following is void relation?

1. $R=\left\{\right(x,y):x+y=11\}$
2. $R=\left\{\right(x,y):x+2y=21\}$
3. $R=\left\{\right(x,y):x+y=8\}$
4. $R=\left\{\right(x,y):2x+y=14\}$

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 $z_1$ and $z_2$ are two complex numbers satisfying the equation $\left|\frac{z_1+z_2}{z_1-z_2}\right|=1,$ then $\frac{z_1}{z_2}$ is a number which is

1. Positive real

2. Negative real

3. Zero or purely imaginary
4. None of these

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 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. If $w=\frac{-9+3i}{1-2i}$, then

1. $|w|=3\sqrt{2}$
2. $\arg w=\frac{\pi }{4}$
3. $|w|=\sqrt{2}$
4. $\arg w=-\frac{3\pi }{4}$

Q. If $w=\frac{-9+3i}{1-2i}$, then

1. $|w|=3\sqrt{2}$
2. $\arg w=\frac{\pi }{4}$
3. $|w|=\sqrt{2}$
4. $\arg w=-\frac{3\pi }{4}$

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. $\frac{3+2isin\theta }{1-2isin\theta }$ will be real, if $\theta$ =
[IIT 1976; EAMCET 2002]

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

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

1. z is purely real

2. z is purely imaginary

3. Either z is purely real or purely imaginary

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