Geometrical Representation of a Complex Number

Q. Let $z_1$ and $z_2$ be complex numbers such that $z{{}_1}^2-4z_2=16+20i.$ Suppose the roots $\alpha \&\beta \text{of}{x}^2+z_{1}x+z_2+m=0$ for some complex number $m$ satisfy $|\alpha -\beta |=2\sqrt{7}.$

The complex number $m$ lies on

1. A square with side $7$ and centre $(–4,5)$
2. A circle with radius $7$ and centre $(4,5)$
3. A square with side $7$ and centre $(4,5)$
4. A circle with radius $7$ and centre $(–4,5)$

Q. Let $S=S_1\cap S_2\cap S_3,$ where
$S_1=\{z\in C:|z|<4\},$ $S_2=\{z\in C:\text{Im}\left(\frac{z-1+\sqrt{3}i}{1-\sqrt{3}i}\right)>0\}$ and $S_3=\{z\in C:\text{Re}z>0\}$

$\underset{z\in S}{min}|1-3i-z|=$

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

Q. All the possible points in the set $S=\{\frac{\alpha +i}{\alpha -i}:\alpha \in R,i=\sqrt{-1}\}$ lies on

1. a straight line whose slope is $1.$
2. a straight line whose slope is $-1.$
3. a circle whose radius is $\sqrt{2}$ units.
4. a circle whose radius is $1$ unit.

Q.

Region represented by x ≥ 0, y ≥ 0 is

1. Third Quadrant

2. First Quadrant

3. Second Quadrant

4. Fourth Quadrant

Q.

How do you find the conjugate of a complex number in polar form $?$

Q. The complex number $z=-1+i\sqrt{3}$ can be represented in polar form as $(\text{where}cis\theta =\cos \theta +i\sin \theta )$

1. $\sqrt{2}cis\left(\frac{-2\pi }{3}\right)$
2. $2cis\left(\frac{\pi }{3}\right)$
3. $2cis\left(\frac{2\pi }{3}\right)$
4. $\sqrt{2}cis\left(\frac{\pi }{3}\right)$

Q. Let $S$ be the set of all complex numbers $z$ satisfying $|z-2+i|$ $\ge \sqrt{5}$. If the complex number $z_0$ is such that $\frac{1}{|z_0-1|}$ is the maximum of the set $\{\frac{1}{|z-1|}:z\in S\}$, then the principal argument of $\frac{4-z_0-\overline{z_0}}{z-\overline{z_0}+2i}$ is

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

Q.

The feasible solution of a L.P.P. belongs to

1. First and third quadrant

2. Second quadrant

3. Only first quadrant

4. First and second quadrant

Q.

The principal value of ${\cos }^{-1}\left\{\left(\frac{1}{\sqrt{2}}\right)\left[\cos \left(\frac{9\mathrm{\pi }}{10}\right)-\sin \left(\frac{9\mathrm{\pi }}{10}\right)\right]\right\}$is

1. $\frac{3\mathrm{\pi }}{10}$

2. $\frac{17\mathrm{\pi }}{20}$

3. $\frac{7\mathrm{\pi }}{10}$

4. None of these

Q.

Write the argument of -i.

Q. If $z=x+iy$ is a complex number satisfying the equation $|z-(1+i){|}^2=2$ and $p=\frac{2}{z}$, then the locus of $p$ in argand plane is

1. $x+y=1$
2. $x-y+1=0$
3. $x-y=1$
4. $x+y+1=0$

Q. If the real part of the complex number ${(1-\cos \theta +2i\sin \theta )}^{-1}$ is $\frac{1}{5}$ for $\theta \in (0,\pi ),$ then the value of the integral $\underset{0}{\overset{\theta }{\int }}\sin xdx$ is equal to :

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

Q. The equation $|z-i|=|z-1|,i=\sqrt{-1},$ represents:

1. the line through the origin with slope $-1.$
2. the line through the origin with slope $1.$
3. a circle of radius $1\text{units}.$
4. a circle of radius $\frac{1}{2}\text{units}.$

Q. If the centre of a regular hexagon is at origin and one of the vertices on Argand plane is at $1+2i$ and its perimeter is $k\text{units}$, then the numerical value of $k^2$ is

Q. Let $P\left(z\right)=z^3+a{z}^2+bz+c,$ where $a,b,c\in R$. If there exists a complex number $w$ such that the three roots of $P\left(z\right)$ are $w+3i,w+9i$ and $2w-4$, where $i^2=-1$, then the value of $a+b+c$ is
​​​​​​​(correct answer + 1, wrong answer - 0.25)

1. $-280$
2. $136$
3. $280$
4. $-136$

Q.

A point $z$ moves on Argand diagram in such a way that $|z-3i|=2$, then its locus will be

1. $y$ axis

2. Straight line

3. Parabola

4. None of these

Q.

The product of perpendiculars drawn from the origin to the lines represented by the equation $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0,$ will be

1. 

2. 

3. 

4. 

Q. If $|z-2-i|=|z|\left|\sin (\frac{\pi }{4}-\arg (z\left)\right)\right|$, where $z$ is in first quadrant, then the locus of $z$ is

1. a ray.
2. a part of a circle.
3. a part of an ellipse.
4. a part of a parabola.

Q. If $\frac{\pi }{3}$ and $\frac{\pi }{4}$ are the arguments of $z_1$ and $\overline{z_2}$, then the value of $\arg \left(z_1{z}_2\right)$ is

1. $\frac{5\pi }{12}$
2. $\frac{\pi }{12}$
3. $-\frac{5\pi }{12}$
4. $-\frac{\pi }{12}$

Q. Consider two complex number $z$ and $\omega$ satisfying $|z|=1$ and $|\omega -2|+|\omega -4|=2$. Then which of the following statement(s) is/are correct ?

1. $\text{Re}\left(z\omega \right)$ can never exceed $4$.
2. If $\arg z=\arg \omega$, then $|z+2|$ is equal to $3$.
3. The minimum value of $|z-\omega |$ is equal to $2$.
4. The maximum value of $|z-w|$ is equal to $6$.

Q. If $z_1,z_2,z_3$ are the vertices of a triangle in argand plane such that $|z_1-z_2|=|z_1-z_3|,$ then $\arg (\frac{2z_1-z_2-z_3}{z_3-z_2})$ is

1. $\pm \frac{\pi }{3}$
2. $0$
3. $\pm \frac{\pi }{2}$
4. $\pm \frac{\pi }{6}$

Q. If $z=\frac{\sqrt{3}}{2}+\frac{i}{2},$ Then $\arg (-z)$ will be

1. $\frac{5\pi }{6}$
2. $-\frac{\pi }{6}$
   
3. $-\frac{5\pi }{6}$
4. $\frac{\pi }{6}$

Q. Let $\arg \left(z\right)$ represent the principal argument of the complex number $Z$.
Then, $|z|=3$ and $\arg (z-1)-\arg (z+1)=\frac{\pi }{4}$ intersect

1. at infinitely many points
2. nowhere
3. exactly at one point
4. exactly at two points

Q. If $X=\{8^n-7n-1|n\in N\}$ and

$Y=\{49n-49|n\in N\}.$

Then

1. $X=Y$
2. $Y\subset X$
3. $X\subset Y$
4. $X\cap Y=\varphi$

Q. If the points $A\left(z\right),B(-z)$ and $C(1-z)$ are the vertices of an equilateral triangle $ABC$, then

1. sum of possible $z$ is $\frac{1}{2}$
2. Sum of possible $z$ is $1$
3. Product of possible $z$ is $\frac{1}{4}$
4. Product of possible $z$ is $\frac{1}{2}$

Q. Converse of the statement"If two triangles are congruent, then they have the same measure".

1. If two triangles have the same measure, then they are congruent
2. If two triangles have the same measure, then they are not congruent
3. If two triangles are not congruent, then they do not have the same measure
4. If two triangles do not have the same measure, then they are not congruent

Q. For any integer $n,$ the argument of $z=\frac{(\sqrt{3}+i{)}^{4n+1}}{(1-i\sqrt{3}{)}^{4n}}$ is

1. $\frac{\pi }{6}$
2. $\frac{\pi }{3}$
3. $\frac{\pi }{2}$
4. $\frac{2\pi }{3}$

Q. If $(3+x^{2020}+x^{2021}{)}^{2022}=a_0+a_{1}x+a_2{x}^2+\cdots +a_{10}{x}^n$, then the value of $a_0-\frac{1}{2}{a}_1-\frac{1}{2}{a}_2+a_3-\frac{1}{2}{a}_4-\frac{1}{2}{a}_5+a_6\cdots$ is
(correct answer + 2, wrong answer - 0.50)

1. $2^{2021}$
2. $1$
3. $2^{2022}$
4. $0$

Q. If $A\left(z_1\right),B\left(z_2\right),C\left(z_3\right)$ are the vertices of the triangle ABC such that $\frac{(z_1-z_2)}{(z_3-z_2)}=(\frac{1}{\sqrt{2}}+\frac{i}{\sqrt{2}})$, then triangle $ABC$ is

1. Isosceles
2. Equilateral
3. Right angled
4. Obtuse angled

Q.

The equation of the locus of $z$ such that $\left|\frac{(z-i)}{(z+i)}\right|=2$, where $z=x+iy$is a complex number is