Tan Inverse

Q. The smallest positive integral value of $n$ for which $(1+i{)}^{2n}=(1-i{)}^{2n}$ is

1. $4$
2. $8$
3. $2$
4. $12$

Q. If $Z$ is a non zero complex number such that $Re\left(Z\right)=0,$ then

1. $Re\left(Z^2\right)=0$
2. $Im\left(Z^2\right)\ne 0$
3. $Im\left(Z^2\right)=0$
4. $Re\left(Z^2\right)+Im\left(Z^2\right)=0$

Q.

The least positive integer $k$ for which $k\left(n^2\right)\left(n^2-1^2\right)\left(n^2-2^2\right)\left(n^2-3^2\right)\dots \left[n^2-{\left(n-1\right)}^2\right]=r!$ for some positive integers $r$ is?

1. $2002$

2. $2004$

3. $1$

4. $2$

Q. Which of the following is a complex number?

1. $(\sqrt{-5},i^4)$
2. $(i^{2020},i^3)$
3. $(i^{2022},\pi )$
4. $(\sec \frac{\pi }{2},i^4)$

Q. The least positive integers $n$ such that $\frac{(2i{)}^n}{(1-i{)}^{n-2}},i=\sqrt{-1},$ is a positive integer, is

Q.

The number of terms in the expansion of${(a+b+c)}^n$, where $n\in N$ .

Q. Which of the following is a complex number?

1. $(\sec \frac{\pi }{2},i^4)$
2. $(\sqrt{-5},i^4)$
3. $(i^{2020},i^3)$
4. $(i^{2022},\pi )$

Q. The value of $(1-i{)}^n{(1-\frac{1}{i})}^n$ is equal to

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

Q. The value of $(1-i{)}^n{(1-\frac{1}{i})}^n$ is equal to

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

Q. If $Z$ is a non zero complex number such that $Re\left(Z\right)=Im\left(Z\right),$ then

1. $Re\left(Z^2\right)=Im\left(Z^2\right)$
2. $Im\left(Z^2\right)=0$
3. $Re\left(Z^2\right)=0$
4. $Re\left(Z^2\right)+Im\left(Z^2\right)=0$

Q. The value of $\sum _{n=0}^{100}{i}^{n!},$ where $i=\sqrt{-1}$ is

1. $97+2i$
2. $90+2i$
3. $95+2i$
4. $95+i$

Q. The value of ${\left(\frac{1+i}{1-i}\right)}^{496}$ is