Polar Representation of a Complex Number

Q.

$\cos \frac{2\pi }{15}\cos \frac{4\pi }{15}\cos \frac{8\pi }{15}\cos \frac{16\pi }{15}=$

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

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

3. $\frac{1}{8}$

4. $\frac{1}{16}$

Q. Number of complex numbers $z$ such that $|z|=1$ and $|\frac{z}{\overline{z}}+\frac{\overline{z}}{z}|=1$ is
$(\arg z\in [0,2\pi \left)\right)$

1. $4$
2. $6$
3. $8$
4. more than $8$

Q.

If $y=sec\left({\tan }^{-1}x\right),$ then $\frac{dy}{dx}$ at $x=1$.

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

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

3. $1$

4. $\sqrt{2}$

Q. If $\left|\frac{z_1}{z_2}\right|=1$ and $\arg \left(z_1{z}_2\right)=0,$ then

1. $z_1=z_2$
2. $|z_2{|}^2=z_1{z}_2$
3. $z_1{z}_2=1$
4. $|z_1+z_2|=0$

Q. If $|z|=1$ and $z\ne \pm 1$, then all the values of $\frac{z}{1-z^2}$ lie on

1. The X - axis
2. The Y - axis
3. $a$ line not passing through the origin
4. $|z|=\sqrt{2}$

Q.

$\text{Find z, ~if}|z|=4andarg\left(z\right)=\frac{5\pi }{6}$

Q. Let $y=f\left(x\right)$ be a thrice differentiable function defined on R such that f(x) = 0 has atleast 7 distinct zeros, then minimum number of zeros of the equation $f\left(x\right)+9f^{{}^{\prime }}\left(x\right)+27f^{{}^{\prime \prime }}\left(x\right)+27f^{{}^{\prime \prime \prime }}\left(x\right)=0$ is

Q. Suppose the limit
$L=\underset{n\to \infty }{lim}\sqrt{n}\underset{0}{\overset{1}{\int }}\frac{1}{(1+x^2{)}^n}dx$
exists and is larger than $\frac{1}{2}$. Then

1. $\frac{1}{2}<L<2$
2. $2<L<3$
3. $3<L<4$
4. $L\ge 4$

Q. In a $\Delta$ ABC if b = 3, c = 5 and cos(B - C) = $\frac{7}{25}$, then find the value of $tan\frac{A}{2}$

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

Q.

If $x=\cos \theta ,y=\sin 5\theta$, then $(1-x^2)d^{2}y/d{x}^2-xdy/dx$ is equal to:

1. $-5y$

2. $5y$

3. $25y$

4. $-25y$

Q. Let $z_1,z_2,z_3$​ be three distinct complex numbers lying on a circle whose centre is at the origin. If $z_i+z_j{z}_k,$ ​where $i,j,k\in \{1,2,3\}$ and $i\ne j\ne k$ are real numbers, then the value of $4(z_1\times z_2\times z_3)$ is

Q.

The principle value of the amplitude of (1 + i) is

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

2. $\frac{\pi }{12}$

3. $\frac{3\pi }{4}$

4. $\pi$

Q.

If $x_r=cos\left(\frac{\pi }{3^r}\right)+isin\left(\frac{\pi }{3^r}\right)$, then $x_1{x}_2{x}_3.......to\infty is$:

1. -1

2. -i

3. 1

4. i

Q. If $\alpha \le 2{\sin }^{-1}x+{\cos }^{-1}x\le \beta ,$ then

1. $\alpha =\frac{-\pi }{2},\beta =\frac{\pi }{2}$
2. $\alpha =\frac{-\pi }{2},\beta =\frac{3\pi }{2}$
3. $\alpha =0,\beta =\pi$
4. $\alpha =0,\beta =2\pi$

Q. Let $z_1,z_2,z_3$​ be three distinct complex numbers lying on a circle whose centre is at the origin. If $z_i+z_j{z}_k,$ ​where $i,j,k\in \{1,2,3\}$ and $i\ne j\ne k$ are real numbers, then the value of $4(z_1\times z_2\times z_3)$ is

Q. If the lines $a\overline{z}+\overline{a}z+b=0$ and $c\overline{z}+\overline{c}z+d=0$ are mutually perpendicular, where $a$ and $c$ are non zero complex numbers, while $b$ and $d$ are real numbers, then

1. $a\overline{a}+c\overline{c}=0$
2. $a\overline{c}$ is purely imaginary
3. $\arg \left(\frac{a}{c}\right)=\pm \frac{\pi }{2}$
4. $\frac{a}{\overline{a}}=\frac{c}{\overline{c}}$

Q.

Express the following complex numbers in the form $r(cos\theta +isin\theta )$.

$\left(i\right)1+itan\theta \left(ii\right)tan\alpha -i\left(iii\right)1-sin\alpha +icos\alpha \left(iv\right)\frac{1-i}{cos\frac{\pi }{3}+isin\frac{\pi }{3}}$

Q. Number of non negative integral solutions of $A+B+C+D=12$
where $A,B>0$ and $0<D<5$ is

Q. If $\alpha ,\beta ϵ(\frac{\pi }{2},\pi )$ and $\alpha <\beta$, then which one of the following is true?

1. $e^{cos\alpha -cos\beta }<\frac{\alpha }{\beta }$
2. $e^{cos\beta -cos\alpha }<\frac{\beta }{\alpha }$
3. $e^{cos\alpha -cos\beta }<\frac{\beta }{\alpha }$
4. $e^{cos\beta -cos\alpha }<\frac{\alpha }{\beta }$

Q. If $(1-i\sqrt{3}{)}^2(z\left)\right(4i)=(1+i\sqrt{3})$, then $Ampz$ is

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

Q. Let $\alpha$ and $\beta$ be the roots of $x^2+x+1=0$. If $n$ be positive integer, then ${\alpha }^n+{\beta }^n$ is

1. $2\cos \frac{2n\pi }{3}$
2. $2\sin \frac{2n\pi }{3}$
3. $2\cos \frac{n\pi }{3}$
4. $2\sin \frac{n\pi }{3}$

Q.

Write $\left(i^{25}\right)$ in polar form.

Q.

If a=cos$\alpha$+isin$\alpha$, b=cos$\beta$+isin$\beta$,

c=cos$\gamma$+isin$\gamma$ and $\frac{b}{c}$+$\frac{c}{a}$+$\frac{a}{b}$=1, Then

cos($\beta$-$\gamma$)+cos($\gamma$-$\alpha$)+cos($\alpha$-$\beta$) is equal to

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

Q. The principal amplitude of ${\left(\sin {40}^{\circ }+i\cos {40}^{\circ }\right)}^5$ is

1. ${70}^{\circ }$
2. ${-110}^{\circ }$
3. ${110}^{\circ }$
4. ${-70}^{\circ }$

Q. If $\alpha +\beta ={90}^0$ and $\alpha =2\beta ,$ then ${\cos }^2\alpha +{\sin }^2\beta$ equals to

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

Q. $\text{If}{z}_1=a+ib\text{and}{z}_2=c+id$ are complex numbers such that $|z_1|=|z_2|=1\text{and}Re\left(z_1\overline{z_2}\right)=0$, then the pair of complex numbers $w_1=a+ic\text{and}{w}_2=b+id$ satisfies

1. $|w_1|=1$
2. $|w_2|=1$
3. $Re\left(w_1\overline{w_2}\right)=0$
4. All of these

Q. If $Z=\frac{1-\sqrt{3}i}{1+\sqrt{3}i}$ then find $arg\left(z\right).$

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

Q. Change to polar coordinates the equation: $x^2+y^2=2ax$

Q. If $\cos \alpha +2\cos \beta +3\cos \gamma =\sin \alpha +2\sin \beta +3\sin \gamma =0$, then the value of $\sin 3\alpha +8\sin 3\beta +27\sin 3\gamma$ is

1. $\sin (\alpha +\beta +\gamma )$
2. $3\sin (\alpha +\beta +\gamma )$
3. $18\sin (\alpha +\beta +\gamma )$
4. $\sin (\alpha +2\beta +3\gamma )$

Q. If $\cos \theta =\frac{9}{15}$ then find $\tan \theta$