Modulus of a Complex Number

Q.

What is the value of $e^{-\infty }$?

Q. If the matrix $A=\left(\begin{array}{cc}0& 2\\ K& -1\end{array}\right)$ satisfies $A(A^3+3I)=2I$, then the value of $K$ is

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

Q.

The maximum value of $a\cos \left(x\right)+b\sin \left(x\right)$ is

1. $a+b$

2. $a–b$

3. $\left|a\right|+\left|b\right|$

4. $\sqrt{(a^2+b^2)}$

Q.

If $\tan \left(\theta \right)=-\frac{4}{3}$, then $\sin \left(\theta \right)$ is

1. $-\frac{4}{5}$ but not $\frac{4}{5}$

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

3. $\frac{4}{5}$ but not $-\frac{4}{5}$

4. none of these

Q. Let $z$ be a complex number such that $|z|+z=3+i$ (where $i=\sqrt{-1})$. Then $|z|$ is equal to :

1. $\frac{5}{3}$
2. $\frac{5}{4}$
3. $\frac{\sqrt{34}}{3}$
4. $\frac{\sqrt{41}}{4}$

Q.

How to find the value of $\cos \left(\frac{3\mathrm{\pi }}{2}\right)$?

Q.

$co{t}^{-1}\left(\frac{3}{4}\right)+{\sin }^{-1}\left(\frac{5}{13}\right)=?$

1. ${\sin }^{-1}\frac{63}{65}$

2. ${\sin }^{-1}\frac{12}{13}$

3. ${\sin }^{-1}\frac{65}{68}$

4. ${\sin }^{-1}\frac{5}{12}$

Q. Let $z\in C$ be such that $|z|<1$. If $\omega =\frac{5+3z}{5(1-z)}$, then :

1. $5\text{Re}\left(\omega \right)>4$
2. $5\text{Re}\left(\omega \right)>1$
3. $4\text{Im}\left(\omega \right)>5$
4. $5\text{Im}\left(\omega \right)<1$

Q. If $\frac{z-\alpha }{z+\alpha }(\alpha \in R)$ is a purely imaginary number and $|z|=2$, then a value of $\alpha$ is:

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

Q. The equation $r\cos (\theta -\frac{\pi }{3})=2$ represents

1. a circle
2. a parabola
3. an ellipse
4. a straight line

Q.

Assume X, Y, Z, W and P are matrices of orders $2\times n,3\times k,2\times p,n\times 3andp\times k$ respectivley. Choose the correct answer in Q.21 and Q.22.
If n = p, then the order of the matrix 7X-5Z is
$\left(a\right)p\times 2$
$\left(b\right)2\times 2$
$\left(c\right)n\times 3$
$\left(d\right)p\times n$

Q.

The maximum value of $3\cos \left(\theta \right)+4\sin \left(\theta \right)$ is

1. $3$

2. $4$

3. $5$

4. None of these

Q.

If $z_1$ and $z_2$ be two complex numbers then $\mathrm{Re}\left(z_1{z}_2\right)$ is equal to

1. $\mathrm{Re}\left(z_1\right)\mathrm{Re}\left(z_2\right)$

2. $\mathrm{Re}\left(z_1\right)\mathrm{Im}\left(z_2\right)$

3. $\mathrm{Im}\left(z_1\right)\mathrm{Re}\left(z_2\right)$

4. None of these

Q.

$\text{If}\frac{1-ix}{1+ix}=a+ib,\text{then}{a}^2+b^2=$

1. -1

2. 0

3. none of these

4. 1

Q.

If ${\left|z_1+z_2\right|}^2={\left|z_1\right|}^2+{\left|z_2\right|}^2$, then $\frac{z_1}{z_2}$ is

1. purely real

2. purely imaginary

3. zero

4. neither real nor imaginary

Q.

ABCD is a square, the length of whose side is a. Taking AB and AD as the coordinate axes, the equation of the

circle passing through the vertices of the square, is

1. 

2. 

3. 

4. 

Q.

If $\tan \left(\mathrm{A}\right)=\sqrt{2}-1$ then show that $\sin \left(A\right)·\cos \left(A\right)=\frac{\sqrt{2}}{4}$.

Q. If $|z-3+2i|=4$ then the difference between the greatest value and the least value of $|z|$ is :

1. $2\sqrt{13}$
2. $4+\sqrt{13}$
3. $\sqrt{13}$
4. $8$

Q.

If $a,b,c$ are integers, not all simultaneously equal and $\omega$ is a cube root of unity $(\omega \ne 1)$, then the minimum value of $|a+b\omega +c{\omega }^2|$ is

1. $1$
2. $0$
3. $\frac{\sqrt{3}}{2}$
4. $\frac{1}{2}$

Q. For every integer $n,$ let $a_n$ and $b_n$ be real numbers. Let function $f:R\to R$ be given by
$f\left(x\right)=\{\begin{array}{c}{a}_n+\sin \pi x,\text{for}x\in [2n,2n+1]\\ b_n+\cos \pi x,\text{for}x\in (2n-1,2n)\end{array}$ , for all integers $n.$
If $f$ is continuous, then which of the following hold(s) for all $n?$

1. $a_{n-1}-b_{n-1}=0$
2. $a_n-b_n=1$
3. $a_n-b_{n+1}=1$
4. $a_{n-1}-b_n=-1$

Q. The medians AD and BE of a triangle with vertices A(0, b), B(0, 0) and C(a, 0) are perpendicular to each
other if

1. 
2. 
3. 
4. 

Q.

Evaluate $104x96$ using identities

Q. The inequality $|Z-4|<|Z-2|$ represents the region given by:

1. None of these
2. $\text{Re}\left(z\right)<0$
3. $\text{Re}\left(z\right)\ge 0$
4. $\text{Re}\left(z\right)>0$

Q.

Let $\mathrm{\omega }$ be a complex number such that $2\mathrm{\omega }+1=\mathrm{z}$ where $\mathrm{z}=\sqrt{-3}$. If $\left|\begin{array}{ccc}1& 1& 1\\ 1& -{\mathrm{\omega }}^2-1& {\mathrm{\omega }}^2\\ 1& {\mathrm{\omega }}^2& {\mathrm{\omega }}^7\end{array}\right|=3\mathrm{k}$. Then $\mathrm{k}$ is equal to,

1. $\mathrm{z}$

2. $-1$

3. $1$

4. $-\mathrm{z}$

Q. If the real part of the complex number $z=\frac{3+2i\cos \theta }{1-3i\cos \theta },$ $\theta \in (0,\frac{\pi }{2})$ is zero, then the value of ${\sin }^{2}3\theta +{\cos }^2\theta$ is equal to

Q.

The complex number z which satisfies the condition $\left|\frac{i+z}{i-z}\right|=1$ lies on

1. circle $x^2+y^2=1$

2. the x-axis

3. the y-axis

4. the line x + y =1

Q. If $k+|k+z^2|=|z{|}^2,(k\in R^{-})$, then possible argument of $z$ is

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

Q.

The locus of center of a circle which passes through the origin and cut off a length of $4$ units from the line $x=3$is

1. $y^2+6x=0$

2. $y^2+6x=13$

3. $y^2+6x=10$

4. $x^2+6y=13$

Q. Let $f\left(x\right)=x^2-3x+2.$ Then which of the following is/are true?
(where $Q_1$ denotes first quadrant)

1. area bounded by $y=f\left(|x|\right),$ and $x-$axis, and $y-$axis in $Q_1$ is $\frac{5}{6}$ sq. units
2. area bounded by $y=f\left(|x|\right),$ and $x-$axis is $2$ sq. units
3. area bounded by $y=f\left(|x|\right),$ between $1\le x\le 2$ and $x-$axis is $\frac{1}{6}$sq. units
4. area bounded by $y=f\left(|x|\right),$ and $x-$axis is $1$ sq. units

Q.

The least value of $\left|z\right|$ where $z$ is complex number which satisfies the inequality $\exp \left(\frac{\left(\left|\mathrm{z}\right|+3\right)\left(\left|\mathrm{z}\right|-1\right)}{\left(\left|\mathrm{z}\right|+1\right)}\right)\ge {\log }_{\sqrt{2}}\left(\left|5\sqrt{7}+9\mathrm{i}\right|\right),\mathrm{i}=\sqrt{-1}$, is equal to

1. $2$

2. $3$

3. $8$

4. $\sqrt{5}$