Complex Numbers

Q.

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

1. a = 0, c = 0

2. b = 0, c = 0

3. b = 0, d = 0

4. a = 0, d = 0

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

1. 16

2. -32
3. 32
4. -16

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

1. 0

2. 2⁷

3. 2⁶

4. None of these

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. $|\beta |=1$
4. $\beta ϵ(-1,0)$

Q. Let $z=x+iy$ be a complex number such that $|z|=1$, where $i=\sqrt{-1}$. Match List - I with List - II.

$\begin{array}{cccc}\text{List-I}& & \text{List - II}& \\ \text{(I)}& \text{Re}\left(\frac{iz}{1+z^2}\right)\text{is equal to}& \text{(P)}& 0\\ \text{(II)}& \text{Im}\left(\frac{iz}{1+z^2}\right)\text{can be equal to}& \text{(Q)}& 1\\ \text{(III)}& \text{Number of integers NOT in the}& \text{(R)}& \frac{1}{2}\\ & \text{range of}\text{Im}\left(\frac{iz}{1+z^2}\right)\text{is equal to}& & \\ \text{(IV)}& \frac{1}{2\pi }\arg \left(\frac{iz}{1+z^2}\right)\text{is equal to}& \text{(S)}& -\frac{1}{2}\\ & (\text{where}-\pi <\arg (z)\le \pi )& & \\ & & \text{(T)}& -\frac{1}{4}\\ & & \text{(U)}& \frac{1}{4}\end{array}$

Which of the following is only CORRECT combination?

1. $\text{I}\to \text{P},\text{Q}$
2. $\text{I}\to \text{P},\text{Q},\text{R}$
3. $\text{II}\to \text{Q, R, S}$
4. $\text{III}\to \text{Q, R}$

Q. Let $z=x+iy$ be a complex number such that $|z|=1$, where $i=\sqrt{-1}$. Match List - I with List - II.

$\begin{array}{cccc}\text{List-I}& & \text{List - II}& \\ \text{(I)}& \text{Re}\left(\frac{iz}{1+z^2}\right)\text{is equal to}& \text{(P)}& 0\\ \text{(II)}& \text{Im}\left(\frac{iz}{1+z^2}\right)\text{can be equal to}& \text{(Q)}& 1\\ \text{(III)}& \text{Number of integers NOT in the}& \text{(R)}& \frac{1}{2}\\ & \text{range of}\text{Im}\left(\frac{iz}{1+z^2}\right)\text{is equal to}& & \\ \text{(IV)}& \frac{1}{2\pi }\arg \left(\frac{iz}{1+z^2}\right)\text{is equal to}& \text{(S)}& -\frac{1}{2}\\ & (\text{where}-\pi <\arg (z)\le \pi )& & \\ & & \text{(T)}& -\frac{1}{4}\\ & & \text{(U)}& \frac{1}{4}\end{array}$

Which of the following is only CORRECT combination?

1. $\text{III}\to \text{P, Q}$
2. $\text{IV}\to \text{T, U}$
3. $\text{III}\to \text{P, Q, T}$
4. $\text{III}\to \text{S, T, U}$

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 complex number $\frac{2^n}{(1+i{)}^{2n}}+\frac{(1+i{)}^{2n}}{2^n},n\in I$ is equal to

1. $2$
2. $\{1+(-1{)}^n\}\cdot i^n$
3. $0$
4. $\{1-(-1{)}^n\}\cdot i^n$

Q. Quadratic equation $x^2+(a-1)ix+5=0(a\in R)$ will have a pair of conjugate complex roots, if

1. $a=1$
2. $a^2-2a+21>0$
3. $a^2-2a+21<0$
4. $a^2+2a-21>0$

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

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