Euler's Representation

Q.

The value of ${\left(\frac{1+\sin {\displaystyle \frac{2\mathrm{\pi }}{9}}+i\cos {\displaystyle \frac{2\mathrm{\pi }}{9}}}{1+\sin {\displaystyle \frac{2\mathrm{\pi }}{9}}-i\cos {\displaystyle \frac{2\mathrm{\pi }}{9}}}\right)}^3$ is

1. $\frac{-1}{2}\left(1-i\sqrt{3}\right)$

2. $\frac{1}{2}\left(1-i\sqrt{3}\right)$

3. $\frac{-1}{2}\left(\sqrt{3}-i\right)$

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

Q.

If $ɑ,\beta \ne 0,f\left(n\right)={ɑ}^n+{\beta }^n$ and $\left|\begin{array}{ccc}3& 1+f\left(1\right)& 1+f\left(2\right)\\ 1+f\left(1\right)& 1+f\left(2\right)& 1+f\left(3\right)\\ 1+f\left(2\right)+1+f\left(3\right)& 1+f\left(3\right)& 1+f\left(4\right)\end{array}\right|$ $=k{(1-ɑ)}^2{(1-\beta )}^2{(ɑ-\beta )}^2$, then $k=$

1. $\alpha \beta$

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

3. $1$

4. $-1$

Q. Let z be a complex number and a be a real parameter, such that $z^2+az+a^2=0$. Then

1. Locus of z is a pair of straight lines
2. $Arg\left(z\right)=\pm \frac{2\pi }{3}$
3. Locus of z is a circle
4. $|z|=|a|$

Q. If $z_1,z_2$ are complex numbers such that $Re\left(z_1\right)=|z_1-1|$, $Re\left(z_2\right)=|z_2-1|$ and $arg(z_1-z_2)=\frac{\pi }{6}$, then $Im(z_1+z_2)$ is equal to:

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

Q.

What is the domain of ${\tan }^{-1}\left(x\right)$?

Q.

If $z=r{e}^{i\theta }$, then $\left|e^{iz}\right|$ is equal to

1. $e^{r\sin \theta }$

2. $e^{-r\sin \theta }$

3. $e^{-r\cos \theta }$

4. $e^{r\cos \theta }$

Q. If $z$ and $\omega$ are two complex numbers such that $|z\omega |=1$ and $\arg \left(z\right)-\arg \left(\omega \right)=\frac{3\pi }{2},$ then $\arg \left(\frac{1-2\overline{z}\omega }{1+3\overline{z}\omega }\right)$ is
(Here $\arg \left(z\right)$ denotes the principal argument of complex number $z$ )

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

Q. If $f:R\to [\frac{-\pi }{4},\frac{\pi }{2})$ defined by $f\left(x\right)={\tan }^{-1}(x^4-x^2-\frac{7}{4}+{\tan }^{-1}\alpha )$ is surjective, then

1. ${\cos }^{-1}\left(\frac{1-{\alpha }^2}{1+{\alpha }^2}\right)=2$
2. $\alpha +\frac{1}{\alpha }=2\text{cosec}2$
3. ${\sin }^{-1}\left(\frac{2\alpha }{{\alpha }^2+1}\right)=\pi -2$
4. ${\tan }^{-1}\left(\frac{2\alpha }{{\alpha }^2-1}\right)=2-\pi$

Q.

If $(\sec \alpha +\tan \alpha )(\sec \beta +\tan \beta )(\sec \gamma +\tan \gamma )=\tan \alpha \tan \beta \tan \gamma$, then$(\sec \alpha -\tan \alpha )(\sec \beta -\tan \beta )(\sec \gamma -\tan \gamma )=$

1. $cot\alpha cot\beta cot\gamma$

2. $\tan \alpha \tan \beta \tan \gamma$

3. $cot\alpha +cot\beta +cot\gamma$

4. $\tan \alpha +\tan \beta +\tan \gamma$

Q.

Prove that $\left(1+cota-\cos eca\right)\left(1+\tan a+seca\right)=2$

Q.

If $y={\sin }^{-1}\left(\frac{x}{2}\right)+{\cos }^{-1}\left(\frac{x}{2}\right)$, then the value of $\frac{dy}{dx}$is

1. $1$

2. $-1$

3. $0$

4. $2$

Q. If $|z-1|=1$, then the value of $\tan \left(\frac{\text{arg}(z-1)}{2}\right)-\frac{2i}{z}$ is

1. $1$
2. $-1$
3. $i$
4. $-i$

Q.

The direction cosines of the line x = y = z are
[MP PET 1989]

1. 
   

2. 
   

3. 
   

4. None of these

Q.

Explain Eulers formula.

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 z is a non - real root of $\sqrt[7]{7-1}$, then $z^{86}+z^{175}+z^{289}$ is equal to:

1. -1

2. 0

3. 1

4. 3

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

Q. If $z_n=\cos \frac{\pi }{(2n+1)(2n+3)}+i\sin \frac{\pi }{(2n+1)(2n+3)},n\in N,$ then the value of $\underset{k\to \infty }{lim}(z_1\cdot z_2\cdot z_3\cdots z_k)$ is

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

Q. Let a complex number $\alpha (\alpha \ne 1)$ be a root of the equation $z^{12}-z^7-z^5+1=0$ and $S_n=\sum _{k=0}^{n-1}{\alpha }^k.$ Then which of the following is FALSE?

1. $S_7=0$ and $S_5=0$
2. $S_7=0$ but $S_5\ne 0$
3. $S_5=0$ and $S_7\ne 0$
4. $|\alpha |=1$

Q.

Express $sin\frac{\pi }{5}+i(1-cos\frac{\pi }{5})$ in polar form.

Q. Consider the function $f\left(x\right)=\sin \left({\sin }^{-1}2x\right)+\sec \left({\sec }^{-1}3x\right)+\tan \left({\tan }^{-1}4x\right)$ and $g\left(x\right)=9x$ , then which of the following is correct?

1. $f\left(x\right)$ and $g\left(x\right)$ are equal functions
2. $f\left(x\right)$ is an odd function
3. Number of integers in range of $f\left(x\right)$ are $4$
4. Number of integers in domain of $f\left(x\right)$ are $2.$

Q. If $\frac{1}{{\alpha }_k+i}({\alpha }_k\in R)$ are $8$ vertices of a regular octagon for $k=1,2,3,\dots ,8$, where $i=\sqrt{-1}$, then the area of the octagon is

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

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.

Write the general expression of Euler’s formula.

Q. The value of $e^{2+i}$ is

1. $e^2[\cos 1+i\sin 1]$
2. $\cos 1+i\sin 1$
3. $e^{-2}[\cos 1+i\sin 1]$
4. $e[\cos 1+i\sin 1]$

Q.

Convert the given complex number into polar form: $1-i$

Q. Number of points of discontiuity of the function $f\left(x\right)=\frac{1}{2\sin x-1}$ in $[0,2021\pi ]$ is

1. $2020$
2. $2021$
3. $2022$
4. $2023$

Q. If $|z-1|=1$, then the value of $\tan \left(\frac{\text{arg}(z-1)}{2}\right)-\frac{2i}{z}$ is

1. $1$
2. $-1$
3. $i$
4. $-i$

Q.

If $z$ = $x+iy$, $|z+1|$ = $|z-1|$ and arg $⟮\frac{z-1}{z+1}⟯$ = $\frac{\pi }{4}$, then z =

1. 

2. 

3. 

4. 

Q. If $\sqrt{1-c^2}=nc-1$ for all permissible values of $c$ and $n,$ and $z=e^{i\theta },$ 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. $1+2c\cos \theta$