Conjugate of a Complex Number

Q.

The conjugate of $\frac{(2+i{)}^2}{3+i}$ , in the form of a+ib, is

1. 

2. 

3. 

4. 

Q. If $a>0$ and $z=\frac{(1+i{)}^2}{a-i}$, has magnitude $\sqrt{\frac{2}{5}}$ , then $\overline{z}$ is equal to :

1. $-\frac{3}{5}-\frac{1}{5}i$
2. $-\frac{1}{5}-\frac{3}{5}i$
3. $\frac{1}{5}-\frac{3}{5}i$
4. $-\frac{1}{5}+\frac{3}{5}i$

Q. Find the conjugate of $\frac{(3-2i)(2+3i)}{(1+2i)(2-i)}.$

Q. If the sum of roots of the equation $\{x+1\}+2x=4[x+1]-6$ is $p$, then the value of $3p$ is equal to
(where $\left\{x\right\}$ represents fractional part of $x$ and $\left[x\right]$ represents greatest integer function of $x$)

Q. Consider two complex numbers $\alpha$ and $\beta$ as $\alpha ={\left(\frac{a+bi}{a-bi}\right)}^2+{\left(\frac{a-bi}{a+bi}\right)}^2,$ where $a,b\in R$ and $\beta =\frac{z-1}{z+1},$ where $|z|=1.$ Then

1. both $\alpha$ and $\beta$ are purely real
2. both $\alpha$ and $\beta$ are purely imaginary
3. $\alpha$ is purely real and $\beta$ is purely imaginary
4. $\beta$ is purely real and $\alpha$ is purely imaginary

Q. Let n be an odd natural number greater than 1. Then the number of zeros at the end of the sum${99}^n+1$ is

1. 3
2. 4
3. 2
4. None of these

Q. If the matrix $A$ and $B$ are of $3\times 3$ and $(I-AB)$ is invertible, then which of the following statements is/are correct:

1. $I-BA$ is not invertible
2. $I-BA$ is invertible
3. $I-BA$ has for its inverse $I+B(I-AB{)}^{-1}A$
4. $I-BA$ has for its inverse $I+A(I-BA{)}^{-1}B$

Q.

If the difference between the roots of $x^2+ax-b=0$ is equal to the difference between the roots of $x^2-px+q=0$, then $a^2-p^2$ in terms of $b$ and $q$ is equal

1. $-4\left(b+q\right)$

2. $4\left(b+q\right)$

3. $4\left(b-q\right)$

4. $4\left(q-b\right)$

Q. If $a$ is a complex number such that $|a|=1$ and $a{z}^2+z+1=0$ has one purely imaginary root, then $\cos \left(\arg \right(a\left)\right)$ is

1. $\frac{\sqrt{5}-1}{2}$
2. $\frac{\sqrt{2}-1}{2}$
3. $\frac{\sqrt{3}+1}{2}$
4. $\frac{\sqrt{7}-1}{2}$

Q.

If z = 3+5i, then $z^3$ + $\overline{z}$ + 198 =

1. -3-5i

2. -3+5i

3. 3-5i

4. 3+5i

Q. The number of solutions of equation $|z|+|z-1|+|z-2|+|z-3|=4andz\overline{z}-z-\overline{z}=\frac{5}{4}$ where z is a complex number ?

1. 0
2. 2
3. 1
4. Infinite

Q. If $\frac{(a+i{)}^2}{2a-i}=p+iq$, where $a\in R$, then the value of $p^2+q^2$ is

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

Q. If complex numbers $(-3+iy{x}^2)$ and $(x^2+y+4i)$ are conjugates of each other, where $x,y\in R,$ then $(x,y)$ can be

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

Q. Let $Z$ be a complex number and $\overline{Z}$ denotes the conjugate of $Z$. If $2Z-3\overline{Z}=\frac{-27+23i}{1+i}$, then which of the following is/are correct?

1. $Re\left(Z\right)=2$
2. $Im\left(Z\right)=5$
3. $Re\left(Z\right)=5$
4. $Im\left(Z\right)=2$

Q.

The multiplicative inverse of ${10}^{10}$ is

Q. Let $z$ be a complex number such that both real and imaginary parts of $\frac{z}{20}$ and $\frac{20}{\overline{z}}$ lies between $[0,1]$. Then the area of the region in the complex plane that consists of all points $z$ is
Assume $\pi =3.14$

Q. Let $z=x+iy$ be a complex number where $x,y\in R$ and $i=\sqrt{-1}.$ The area bounded by locus of $P\left(z\right)$ satisfying $|z-1|=2Im\left(z\right)$ and coordinate axes on the complex plane, is equal to
$[$ Here, $Im\left(z\right)$ denotes imaginary part of $z$ $]$

1. $\sqrt{3}$ sq. unit
2. $2\sqrt{3}$ sq. unit
3. $\frac{1}{\sqrt{3}}$ sq. unit
4. $\frac{1}{2\sqrt{3}}$ sq. unit

Q. Let $z=x+iy$ be a complex number. If $\frac{1+\overline{z}}{z}$ is a real number, then

1. $x=-\frac{1}{2}\text{or}y=-\frac{1}{2}$
2. $x=-\frac{1}{2}\text{or}y=1$
3. $x=0\text{or}y=-\frac{1}{2}$
4. $x=-\frac{1}{2}\text{or}y=0$

Q. If $z$ is a complex number satisfying $z-12=i(9-2\overline{z}),$ then the value of $z+\overline{z}$ is

1. $10$
2. $4$
3. $6$
4. $8$

Q. Let $z$ is a complex number and $\overline{z}$ is conjugate of $z$. If $(1+i)z=(1-i)\overline{z},$ then the value of $z+i\overline{z}$ is

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

Q. $\text{If}|z_1|=1,|z_2|=2,|z_3|=3$ and $|9z_1{z}_2+4z_1{z}_3+z_2{z}_3|=12$, then the value of $|z_1+z_2+z_3|$ is equal to

1. $2$
2. $3$
3. $4$
4. $6$

Q.

Find the conjuagates of the following complex numbers :

$\left(i\right)4-5i\left(ii\right)\frac{1}{3+5i}\left(iii\right)\frac{1}{1+i}\left(iv\right)\frac{(3-i{)}^2}{2+i}\left(v\right)\frac{(1+i)(2+i)}{3+i}\left(vi\right)\frac{(3-2i)(2+3i)}{(1+2i)(2-i)}$

Q. Let $z_1,z_2,z_3,z_4$ are four points in the Argand plane. If $z$ is a point such that $|z-z_1|=|z-z_2|=|z-z_3|=|z-z_4|$, then the points $z_1,z_2,z_3,z_4$ can lie on which among the following curves?

1. Circle
2. Rectangle
3. Square
4. Triangle

Q. If $z$ is a non zero complex number, where $\frac{z+2}{|z|}=i$, then number of possible value(s) of $z$ is

1. $0$
2. $1$
3. $3$
4. inifinite

Q. If $z=x+iy,|z|=1$ and $\omega =\frac{(1-z{)}^2}{1-z^2}$, then the locus of $\omega$ is equivalent to

1. $|z-2|=|z+2|$
2. $|z-2i|=|z+2i|$
3. $|z-1-i|=|z+1-i|$
4. $|z-i|=|z+i|$

Q. The set of all $\alpha \in R$, for which $w=\frac{1+(1-8\alpha )z}{1-z}$ is a purely imaginary number, for all $z\in C$ satisfying $|z|=1$ and $Rez\ne 1$, is :

1. an empty set
2. $\left\{0\right\}$
3. $\{0,\frac{1}{4},-\frac{1}{4}\}$
4. equal to $R$

Q. If $z$ is a complex number satisfying $\overline{z^2}=1$, where $\overline{z}$ is the conjugate of $z$, then

1. $Re\left(z\right)=0$
2. $z=0$
3. $Im\left(z\right)=0$
4. $Re\left(z\right)=Im\left(z\right)$

Q. Graph of $f(-x)$ will be the mirror image of graph of $f\left(x\right)$ along the $Y-$ axis.

1. True
2. False

Q. The possible number of extremum point(s) for $f\left(x\right)=x^2+2\cos x+2$ on $(0,\frac{\pi }{2})$ is:

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

Q.

The equation$x^2+2(m-1)x+m+5=0$ has an equal and real root. Find the value of $m.$