Square Root of a Complex Number

Q.

The square root of 3 - 4i is

1. $\pm$(2+i).

2. $\pm$(2-i).

3. $\pm$(1-2i).

4. $\pm$(1+2i).

Q.

If $\omega$ is an imaginary cube root of unity, then $(1+\omega -{\omega }^2{)}^7$ equals

1. $128\omega$

2. $-128\omega$

3. $128{\omega }^2$

4. $-128{\omega }^2$

Q.

If $\omega$ is an imaginary cube root of unity, then $(1+\omega -{\omega }^2{)}^7$ equals

1. $128\omega$

2. $-128\omega$

3. $128{\omega }^2$

4. $-128{\omega }^2$

Q. $\text{If}(x^2-4)\sqrt{x^2-1}<0$ then $x$ will lie in the interval

1. $(1,2)$
2. $(-2,-1)$
3. $[-1,1]$
4. $(-2,2)$

Q. If $a=\cos \frac{4\pi }{3}+i\sin \frac{4\pi }{3}$, then the value of $(1+a{)}^{3n}$ is

1. $-1$
2. $0$
3. $1$
4. $(-1{)}^n$

Q.

$2\tan h^{-1}\left(\frac{1}{\sqrt{2}}\right)=$

1. $\cos h^{-1}\left(\frac{1}{3}\right)$

2. $\cos h^{-1}\sqrt{3}$

3. $\cos h^{-1}3$

4. $\cos h^{-1}\frac{(\sqrt{2}+1)}{(\sqrt{2}–1)}$

Q. If the inequality $\sqrt{\frac{-3}{x^2+2x+10}}\ge 0$ holds good, then $x\in$

1. $[0,\infty )$
2. $[2,14]$
3. $(-\infty ,0]$
4. $\varphi$

Q. If $\omega$ is a cube root of unity, and the value of
$(1-\omega )(1-{\omega }^2)(1-{\omega }^4)(1-{\omega }^8)=\alpha ,$ and
$\frac{a+b\omega +c{\omega }^2}{b+c\omega +a{\omega }^2}+\frac{a+b\omega +c{\omega }^2}{c+a\omega +b{\omega }^2}=\beta$ then

1. $\alpha +\beta =8$
2. $\alpha \beta =9$
3. $\alpha \beta =-9$
4. $\alpha +\beta =-9$

Q.

If a=$\sqrt[2]{22i}$ then which of the following is correct?

1. a=1+i

2. $a=\sqrt{2}\times (1+i)$

3. 1

4. None of these