Equality of 2 Complex Numbers

Q. Let $z\in C$ with $\text{Im}\left(z\right)=10$ and it satisfies
$\frac{2z-n}{2z+n}=2i–1$ for some natural number $n$. Then

1. $n=20$ and $\text{Re}\left(z\right)=10$
2. $n=20$ and $\text{Re}\left(z\right)=-10$
3. $n=40$ and $\text{Re}\left(z\right)=10$
4. $n=40$ and $\text{Re}\left(z\right)=-10$

Q. Let $z=x+iy$ be a complex number where $x$ and $y$ are intergers. The area of the rectangle whose vertices are the roots of the equation $\overline{z}{z}^3+z{\overline{z}}^3=350$ is

1. $48\text{sq. units}$
2. $32\text{sq. units}$
3. $40\text{sq. units}$
4. $80\text{sq. units}$

Q.

If z is a complex number, then z.$\overline{z}$ = 0 if and only if

1. z=0

2. Re(z)=0

3. Im(z)=0

4. None of these

Q.

If $x^3+y^3–3axy=0$, then prove that $\frac{d^{2}y}{d{x}^2}=\frac{2a^{2}xy}{{(ax-y^2)}^3}$.

Q.

For every real number c$\ge 0$, find all the complex number z which satisfy the equation

$|z{|}^2$ - 2iz + 2c(1 + i) = 0

1. c + i( -1 Where 0

2. c + i( -1) Where 0 - 1

3. c + i( -2) Where c > -1

4. c + i( -2) Where 0

Q. The equation $(x^2-5x+1)(x^2+x+1)+8x^2=0$ has

1. four real and distinct roots
2. three real and distinct roots
3. two real and distinct roots
4. only one real root

Q. If $z$ is a complex number and $\overline{z}$ is it's conjugate, then the number of solution(s) of $z^3+\overline{z}=0$ is

Q.

The imaginary part of $\frac{{\left(1+i\right)}^2}{i\left(2i-1\right)}$is

1. $\frac{4}{5}$

2. $0$

3. $\frac{2}{5}$

4. $\frac{-4}{5}$

Q.

If $8f\left(x\right)+6f(1/x)=x+5$and $y=x^{2}f\left(x\right),$ then $\frac{dy}{dx}$ at $x=–1$ is equal to

1. $0$

2. $\frac{1}{14}$

3. $-\frac{1}{14}$

4. $1$

Q. If |z|=1 and arg $\left(z\right)=\frac{\pi }{4}$ then which of the following is /are true?

1. Modulus of $(z^5-iz)$ is $\sqrt{2}$
2. Modulus of $(z^4-iz)$ is 1
   
3. Principal argument of $(z^5-iz)$ is $\frac{\pi }{4}$
   
4. Principal argument is $(z^5-iz)$ is $\frac{\pi }{2}$

Q.

The sum of distinct values of $\lambda$ for which the systems of equations
$(\lambda -1)x+(3\lambda +1)y+2\lambda z=0$
$(\lambda -1)x+(4\lambda -2)y+(\lambda +3)z=0$
$2x+(3\lambda +1)y+3(\lambda -1)z=0$
has non-zero solutions, is

Q. The area bounded by $|\arg \left(z\right)|\le \frac{\pi }{4}$ and $|z-1|<|z-3|$ is

1. $4$ sq. units
2. $8$ sq. units
3. $2$ sq. units
4. $1$ sq. unit

Q. $\int \frac{xdx}{2-x^2+\sqrt{2-x^2}}$ equals to
(where $C$ is constant of integration)

1. $\ln |1+\sqrt{2+x^2}|+C$
2. $-\ln |1+\sqrt{2-x^2}|+C$
3. $-x\ln |1-\sqrt{2-x^2}|+C$
4. $x\ln |1-\sqrt{2+x^2}|+C$

Q. If $(x+iy)(2-3i)=4+i$, then values of $x$ and $y$ are

1. $x=\frac{5}{11},y=\frac{7}{11}$
2. $x=\frac{5}{13},y=\frac{7}{13}$
3. $x=\frac{5}{11},y=\frac{7}{13}$
4. $x=\frac{5}{13},y=\frac{14}{13}$

Q. The number of complex number(s) $z$ satisfying $|z-3-i|=|z-9-i|$ and $|z-3+3i|=3$ is

1. one
2. infinitely many
3. two
4. four

Q. If the equation $z^4+a_1{z}^3+a_2{z}^2+a_{3}z+a_4=0$ where $a_1,a_2,a_3,a_4$ are real coefficients different from zero, has a pure imaginary root, then the expression $\frac{a_3}{a_1{a}_2}+\frac{a_1{a}_4}{a_2{a}_3}$ has the value equal to

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

Q.

If $\alpha ,\beta$ be the roots of $x^2$+px+q=0 and $\alpha +h,\beta +h$ are the roots of $x^2$+rx+s=0 , then

1. 

2. 

3. 

4. 

Q. Complex number $z$ satisfying the equation $z^2\times (1+i)=-7+17i,$ is/are

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

Q.

If point$\mathrm{P}(\mathrm{a},\mathrm{b})$ lies on the straight line $3x+2y=13$ and the point $Q(b,a)$ lies on the straight line $4x-y=5$, then the equation of the line PQ is

1. $x–y=5$

2. $x+y=5$

3. $x+y=-5$

4. $x–y=-5$

Q. $z_1=a+ib$ and $z_2=c+id$ are two complex numbers then $z_1>z_2$ is meaningful if

1. $a>c$ and $b>d$
2. $a>c$ and $b=d=0$
3. $a+c>b+d$
4. $a=c=0$ and $b>d$

Q.

Find the real values of x and y, if

$\left(i\right)(x+iy)(2-3i)=4+i\left(ii\right)(3x-2iy)(2+i{)}^2=10(1+i\left)\right(iii)\frac{(1+i)x-2i}{3+i}+\frac{(2-3i)y+i}{3-i}=i(iv\left)\right(1+i\left)\right(x+iy)=2-5i$

Q. If $z$ is a complex number, then the number of solution(s) for the equation $z^2=\overline{z}$ is

1. $1$
2. $2$
3. $4$
4. $5$

Q. Find the complex number z satisfying the equations $\left|\frac{z-12}{z-8i}\right|=\frac{5}{3},\left|\frac{z-4}{z-8}\right|=1$

1. 
2. 
3. 
4. 

Q. If $a+bi=\frac{c+i}{c-i},a,b,c\in R$, show that $a^2+b^2=1$ and $\frac{b}{a}=\frac{2c}{c^2-1}$

Q. If $(\frac{1}{1+2i}+\frac{3}{1-i})\left(\frac{3-2i}{1+3i}\right)$ is reducible to $a+ib$, then values of $a$ and $b$ are

1. $a=\frac{7}{10},b=\frac{-13}{5}$
2. $a=\frac{7}{10},b=\frac{-11}{5}$
3. $a=\frac{7}{11},b=\frac{-13}{10}$
4. $a=\frac{7}{11},b=\frac{-13}{5}$

Q. Let $f\left(x\right)$ be a polynomial of degree three, if the curve $y=f\left(x\right)$ has relative extrema at $x=\pm \frac{2}{\sqrt{3}}=4x^2+y^2=4$ in two parts. Then find the integral part of areas of these two parts.

Q.

If $x<0$ is a real number, then $arg\left(x\right)=$

Q. If $\sqrt{5-12i}+\sqrt{-5-12i}=z$, then principal value of $\arg z$ can be

1. $-\frac{\pi }{4}$
2. $\frac{\pi }{4}$
   
3. $\frac{3\pi }{4}$
4. $\frac{3\pi }{4}$

Q. Match the colums:

$\begin{array}{cc}Column1& Column2\\ \text{a. The set of points}z\text{satisfying}|z-i|z||=|z+i|z||& \text{p. an ellipse with eccentricity}\frac{4}{5}\\ \text{is contained in or equal to}& \\ \text{b. The set of points}z\text{satisfying}|z+4|+|z-4|=10& \text{q. the set of points}z\text{satisfying}\\ \text{is contained in or equal to}& \text{Img}z=0\\ \text{c. If}|w|=2,\text{then the set of points}z=w-\frac{1}{w}& \text{r. the set of points}z\text{satisfying}\\ \text{is contained in or equal to}& |\text{Img}z|\le 10\\ \text{d. If}|w|=1,\text{then the set of points}z=w+\frac{1}{w}& \text{s. the set of points}z\text{satisfying}\\ \text{is contained in or equal to}& |\text{Re}z|<2\\ & \text{t. the set of points}z\text{satisfying}\\ & |z|\le 3\end{array}$

1. $a-q,r;b-p;c-p,s,t;d-q,r,t$
2. $a-p,q,r;b-p;c-s,t;d-q,r,t$
3. $a-q,r;b-p,t;c-p,t;d-q,r$
4. $a-p,r;b-p,q;c-s,t;d-p,r$

Q. $\text{If}u+iv=(x+iy{)}^3\text{then,}\frac{u}{x}+\frac{v}{y}=$

1. $4(x^2-y^2)$
2. $4(x^2+y^2)$
3. $2(x^2-y^2)$
4. $2(x^2+y^2)$