Complex Numbers

Q. If ${(i^{23}+{\left(\frac{1}{i}\right)}^{29})}^2=k,$ then the value of $k^2$ is

Q. The value of $i^{57}+\frac{1}{i^{125}}$ is :

1. $0$
2. $-2i$
3. $2i$
4. $2$

Q. Let $\alpha ,\beta$ be real and z be a complex number. If $z^2+\alpha z+\beta =0$ has two distinct roots on the line $Re\left(z\right)=1$, then it is necessary that

1. $\beta ϵ(1,\infty )$
2. $\beta ϵ(0,1)$
3. $\beta ϵ(-1,0)$
4. $|\beta |=1$

Q. Let $z$ be a complex number satisfying the equation $z^2-(3+i)z+m+2i=0,$ where $m\in R.$ Suppose the equation has a real root. The additive inverse of non-real root, is

1. $1-i$
2. $1+i$
3. $-1-i$
4. $-2$

Q. The value of $(1+i{)}^8+(1-i{)}^8$ is
[RPET 2001; KCET 2001]

1. 16

2. -16
3. 32
4. -32

Q. The value of $\sqrt{-3}\cdot \sqrt{-75}$ is

1. $15$
2. $15i$
3. $-15$
4. $-15i$

Q. Which among the following expressions are equivalent $(\forall n\in I)$.

1. $\frac{2^n}{(1-i{)}^{2n}}+\frac{(1+i{)}^{2n}}{2^n}$
2. $\frac{(1+i{)}^{2n}}{2^n}+\frac{(1-i{)}^{2n}}{2^n}$
3. $\frac{(1-i{)}^{2n}}{2^n}+\frac{2^n}{(1+i{)}^{2n}}$
4. $\frac{2^n}{(1+i{)}^{2n}}+\frac{2^n}{(1-i{)}^{2n}}$

Q. Let $f:C\to C$ be a function defined as $f\left(z\right)=\frac{z+i}{z-i}$ for all complex numbers $z\ne i$ and $z_n=f\left(z_{n-1}\right)$ for all $n\in N$. If $z_0=K+i$ and $z_{2020}=1+2020i$, then the value of $\left(\frac{1}{K}\right)$ is

Q. If a, b are non-zero real numbers of opposite signs, then which of the following is/are true?

1. $ab>0$
2. $ab<0$
3. $\frac{a}{b}>0$
4. $\frac{a}{b}<0$

Q. Which among the following expressions are equivalent $(\forall n\in I)$.

1. $\frac{2^n}{(1-i{)}^{2n}}+\frac{(1+i{)}^{2n}}{2^n}$
2. $\frac{(1+i{)}^{2n}}{2^n}+\frac{(1-i{)}^{2n}}{2^n}$
3. $\frac{(1-i{)}^{2n}}{2^n}+\frac{2^n}{(1+i{)}^{2n}}$
4. $\frac{2^n}{(1+i{)}^{2n}}+\frac{2^n}{(1-i{)}^{2n}}$

Q. The value of $(1+i{)}^6+(1-i{)}^6$ is
[RPET 2002]

1. 0

2. 2⁷

3. 2⁶

4. None of these

Q. Value of $i^n+i^{n+1}+i^{n+2}+i^{n+3}$, when $n\in I$ is equal to

1. $1$
2. $-1$
3. $0$
4. $2$

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

Q. The value of $(1+i{)}^6+(1-i{)}^6$ is
[RPET 2002]

1. 0

2. 2⁷

3. 2⁶

4. None of these

Q. The value of $(1+i{)}^8+(1-i{)}^8$ is
[RPET 2001; KCET 2001]

1. 16

2. -16
3. 32
4. -32

Q. Let $\alpha ,\beta$ be real and z be a complex number. If $z^2+\alpha z+\beta =0$ has two distinct roots on the line $Re\left(z\right)=1$, then it is necessary that

1. $\beta ϵ(1,\infty )$
2. $\beta ϵ(0,1)$
3. $\beta ϵ(-1,0)$
4. $|\beta |=1$

Q. Value of $i^n+i^{n+1}+i^{n+2}+i^{n+3}$, when $n\in I$ is equal to

1. $1$
2. $-1$
3. $0$
4. $2$

Q.

If a, b, c are in G.P, then

1. $a(b^2+c^2)=c(b^2+c^2)$

2. $a(b^2+c^2)=c(a^2+b^2)$

3. $a^2(b+c)=c^2(a+b)$

4. none of these

Q. If $a<0,b>0,$ then $\sqrt{a}\cdot \sqrt{b}$ is equal to

1. $-\sqrt{|a|\cdot b}$
2. $\sqrt{|a|\cdot b}\cdot i$
3. $\sqrt{|a|b}$
4. None of the above

Q. The value of $\frac{i^{592}+i^{590}+i^{588}+i^{586}+i^{584}}{i^{582}+i^{580}+i^{578}+i^{576}+i^{574}}+2=$