Euler's Representation

Q.

What is the Derivative of $y={\sec }^2\left(x\right)$?

Q.

If $S_n={\cos }^n\theta +{\sin }^n\theta$ then the value of $3S_4-2S_6$ is given by :

1. $4$

2. $0$

3. $1$

4. $7$

Q. The domain and range of the function ${\text{cosec}}^{-1}\sqrt{{\log }_{\left(\frac{3-4\sec x}{1-2\sec x}\right)}2}$ are respectively

1. $R;(-\frac{\pi }{2},\frac{\pi }{2})$
2. $R^{+};(0,\frac{\pi }{2})$
3. $(2n\pi -\frac{\pi }{2},2n\pi +\frac{\pi }{2})-\left\{2n\pi \right\},n\in Z;(-\frac{\pi }{2},\frac{\pi }{2})-\left\{0\right\}$
4. $(2n\pi -\frac{\pi }{2},2n\pi +\frac{\pi }{2})-\left\{2n\pi \right\},n\in Z;(0,\frac{\pi }{2})$

Q. If $z=e^{\frac{i\pi }{13}}$ then $\frac{1}{1-z}$ is equal to $(i=\sqrt{-1})$

1. $1-z^2+z^4-z^6+z^8-z^{10}+z^{12}$
2. $1+z^2+z^4+z^6+z^8+z^{10}+z^{12}$
3. $z-z^3+z^5-z^7+z^3-z^{11}$
4. $z+z^3+z^5+z^7+z^5+z^{11}$

Q. Let $z={(\frac{\sqrt{3}}{2}+\frac{i}{2})}^5+{(\frac{\sqrt{3}}{2}-\frac{i}{2})}^5.$
If $R\left(z\right)$ and $I\left(z\right)$ respectively denote the real and imaginary parts of $z,$ then :

1. $R\left(z\right)>0$ and $I\left(z\right)>0$
2. $R\left(z\right)<0$ and $I\left(z\right)>0$
3. $R\left(z\right)=-3$
4. $I\left(z\right)=0$

Q. Find the value of $tan\frac{1}{2}[si{n}^{-1}\frac{2x}{1+x^2}+co{s}^{-1}\frac{1-y^2}{1+y^2}]$
$|x|<1,y>0andxy<1$

1. $\pi +\frac{x+y}{1-xy}$
2. $\frac{1-xy}{x+y}$
3. $\frac{x+y}{1-xy}$
4. $\pi +\frac{1-xy}{x+y}$

Q. The value of $\sum _{k=1}^{13}\frac{1}{\sin (\frac{\pi }{4}+\frac{(k-1)\pi }{6})\sin (\frac{\pi }{4}+\frac{k\pi }{6})}$ is equal to

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

Q. If $z=e^{\frac{i\pi }{13}}$ then $\frac{1}{1-z}$ is equal to $(i=\sqrt{-1})$

1. $1-z^2+z^4-z^6+z^8-z^{10}+z^{12}$
2. $1+z^2+z^4+z^6+z^8+z^{10}+z^{12}$
3. $z-z^3+z^5-z^7+z^3-z^{11}$
4. $z+z^3+z^5+z^7+z^5+z^{11}$

Q. If $2ta{n}^{-1}x=si{n}^{-1}\frac{2x}{1+x^2}$, then:

1. $x\in R$
2. $-1\le x\le 1$
3. $x\le -1$
4. $x\ge 1$

Q.

If $|u|<1,|v|<1$, and $z=\frac{u-v}{1+\overline{u}v}$, then least value of $|z|$ is

1. $\frac{|u|-|v|}{1+|u||v|}$

2. $\frac{|u|+|v|}{1+|u||v|}$

3. $\frac{||u|-|v||}{1-|u||v|}$

4. None of these

Q. If $\cos A+\cos B=\cos C,\sin A+\sin B=\sin C$
then the value of expression $\frac{\sin (A+B)}{\sin 2C}$ is

Q.

If z=r$e^{i\theta }$, then |$e^{iz}$|=

1. $e^{rsin\theta }$

2. $e^{-rsin\theta }$

3. $e^{-rcos\theta }$

4. $e^{rcos\theta }$

Q. If $\prod _{p=1}^r{e}^{ip\theta }=1$ where $\prod$ denotes the continued product, then the most general value of $\theta$ is

1. $\frac{2n\pi }{r(r-1)},n\in Z$
2. $\frac{2n\pi }{r(r+1)}n\in Z$
3. $\frac{4n\pi }{r(r-1)},n\in Z$
4. $\frac{4n\pi }{r(r+1)},n\in Z$

Q. If $a=cis2\alpha ,b=cis2\beta$, then $\cos (\alpha -\beta )$ is
where $cis\theta =\cos \theta +i\sin \theta$

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

Q. If $z=r{e}^{i\theta }$, then find the value of $|e^{iz}|$

1. $e^{i\theta }$
2. $e^{-i\sin \theta }$
3. $e^{-r\sin \theta }$
4. $e^{-r\cos \theta }$

Q. If $|z_1+z_2|=|z_1-z_2|$ then $\arg z_1-\arg z_2=$

1. $(2n+1)\frac{\pi }{2},n\in Z$
2. $n\pi +\frac{\pi }{4},n\in Z$
3. $2n\pi ,n\in Z$
4. $(2n+1)\pi ,n\in Z$

Q. If $\sqrt{1-c^2}=nc-1$ for all permissible values of $c$ and $n$, where $z=e^{i\theta },$ then $\frac{c}{2n}(1+nz)(1+\frac{n}{z})$ is equal to

1. $1-c\cos \theta$
2. $1+2c\cos \theta$
3. $1+c\cos \theta$
4. $1-2c\cos \theta$

Q. Let $z=cos\theta +isin\theta$. The value of ${\sum }_{m=1}^{15}Im\left(z^{2m-1}\right)$ at $\theta =2^{\circ }$

1. $\frac{1}{sin{2}^{\circ }}$
2. $\frac{1}{3sin{2}^{\circ }}$
3. $\frac{1}{2sin{2}^{\circ }}$
4. $\frac{1}{4sin{2}^{\circ }}$

Q. The value of $cos\left(\frac{1}{2}co{s}^{-1}\left[cos\right(-\frac{14\pi }{5}\left)\right]\right)$

1. $sin\left(\frac{-\pi }{5}\right)$
2. $cos\left(\frac{2\pi }{5}\right)$
3. $-cos\left(\frac{3\pi }{5}\right)$
4. $cos\left(\frac{9\pi }{5}\right)$

Q. The value of ${\sum }_{k=1}^6(sin\frac{2k\pi }{7}-icos\frac{2k\pi }{7})$ is

1. -1
2. \N
3. -i
4. i

Q.

If $|u|<1,|v|<1$, and $z=\frac{u-v}{1+\overline{u}v}$, then least value of $|z|$ is

1. $\frac{|u|-|v|}{1+|u||v|}$

2. $\frac{|u|+|v|}{1+|u||v|}$

3. $\frac{||u|-|v||}{1-|u||v|}$

4. None of these

Q. If $\tan \theta -i,$ where $\theta \in (-\frac{\pi }{2},\frac{\pi }{2})$
can be written as $r{e}^{i\beta }$, then $(r,\beta )$ is

1. $(\sec \theta ,\theta -\frac{\pi }{2})$
2. $(\sec \theta ,\theta +\frac{\pi }{2})$
3. $(\text{cosec}\theta ,\theta -\frac{\pi }{2})$
4. $(\text{cosec}\theta ,\frac{\pi }{2}-\theta )$