Logarithm of a Complex Number

Q.

If $\tan \left(\theta \right)+sec\left(\theta \right)=e^x$, then $\cos \left(\theta \right)$ equals

1. $\frac{e^x+e^{-x}}{2}$

2. $\frac{2}{e^x+e^{-x}}$

3. $\frac{e^x-e^{-x}}{2}$

4. $\frac{e^x-e^{-x}}{e^x+e^{-x}}$

Q.

Find the general value of ${log}_e$(i).

1. n I

2. n I

3. n I

4. n I

Q. Let $S=\{1,2,3\}$. Determine whether the functions $f:S\to S$ defined as below has inverse. Find $f^{-1},$ if it exists.
(i) $f=\left\{\right(1,1),(2,2),(3,3\left)\right\}$
(ii) $f=\left\{\right(1,2),(2,1),(3,1\left)\right\}$
(iii) $f=\left\{\right(1,3),(3,2),(2,1\left)\right\}$

Q.

If $\tan \left(\theta \right)=\sqrt{\frac{3}{2}}$ the sum of the infinite series $1+2(1-\cos (\theta \left)\right)+3(1-\cos {\left(\theta \right))}^2+4(1-\cos {\left(\theta \right))}^3+..............\infty$ is

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

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

3. $\frac{5}{2\sqrt{2}}$

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

Q. The general value of $\log (1+i)+\log (1-i)$ is

1. $\log 2+i2k\pi ,k\in Z$
2. $\log \sqrt{2}+i2k\pi ,k\in Z$
3. $2\log 2+i2k\pi ,k\in Z$
4. $\log 3+i2k\pi ,k\in Z$

Q. $\log (-1+\sqrt{3}i)$ can be expressed in cartesian form as

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

Q. If $|z+4|\le 3,$ then the maximum value of $|z+1|$ is

Q. If $z=z\left(x\right)$ and $(2+\cos x)\frac{dz}{dx}+\left(\sin x\right)\cdot z=\sin x$, $z\left(x\right)>0$ and $z\left(\frac{\pi }{2}\right)=3$, then $z\left(\frac{\pi }{3}\right)$ is

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

Q. Detemine if $f$ defined by $f\left(x\right)=\{\begin{array}{cc}{x}^2\sin \frac{1}{x},& \text{if}x\ne 0\\ 0,& \text{if}x=0\end{array}$ is a continuous function?

Q. If $x_0$ is the solution of the equation $2^{1+{\left({\log }_{2}x\right)}^2}+(x^{{\log }_{2}x}{)}^2=3,$ then the value of ${\sin }^{-1}\left(x_0\right)+{\tan }^{-1}\left(\frac{2x_0}{2-(x_0{)}^2}\right)+{\cot }^{-1}\left(2\right)$ equals

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

Q. ${\log }_{\frac{\pi }{4}}(-1+\sqrt{3}i)$ can be expressed in cartesian form as

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

Q. The expression $\tan \left(i{\log }_e\left(\frac{2-3i}{2+3i}\right)\right)$ is equal to

1. $-\frac{12}{3}$
2. $-\frac{3}{2}$
3. $-\frac{12}{5}$
4. $0$

Q.

If z = $i^i$ . Express ${log}_e$z in A+iB form.Find the value of A and B.

1. A = , B =

2. A = 0 , B =

3. A = , B = 0

4. A = , B =

Q.

If $\sec \left(\theta \right)=4$, how do you use the reciprocal identity to find $\cos \left(\theta \right)$?

Q. If $x=-9$ is a root of

$\left|\begin{array}{ccc}x& 3& 7\\ 2& x& 2\\ 7& 6& x\end{array}\right|=0$,
then other two roots are

Q. If $z=(1-i{)}^{-i}$, then the correct option(s) is/are

1. $Re\left(z\right)=e^{-\frac{\pi }{4}}\cos \left(\log 2\right)$
2. $Im\left(z\right)=-e^{-\frac{\pi }{4}}\sin \left(\frac{1}{2}\log 2\right)$
3. $Im\left(z\right)=-e^{-\frac{\pi }{4}}\sin \left(\log 2\right)$
4. $Re\left(z\right)=e^{-\frac{\pi }{4}}\cos \left(\frac{1}{2}\log 2\right)$

Q. $\log (-1+\sqrt{3}i)$ can be expressed in cartesian form as

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

Q. The value of ${\log }_2(-3)$ is

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

Q.

What is the value of sin(${log}_e{i}^i)$ ?

1. 0

2. 1

3. -1

4. 2

Q.

Find the value of i.$i^i$ is _______.

1. 

2. 

3. 

4. 

Q.

Find the value of i.$i^i$ is _______.

1. i.$e^{\frac{\pi }{2}}$

2. i.$e^{-\frac{\pi }{2}}$

3. i.$e^{\frac{\pi }{4}}$

4. i.$e^{-\frac{\pi }{4}}$

Q. The value of $\log \left(i^i\right)$ is

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

Q. Given that $\sin hx=\frac{5}{12}$, find the values of $\cos hx$
Determine the value of $x$ as a natural logarithm

Q. The value of $z$ satisfying the equation $\log z+\log z^2+...+\log z^n=0$

1. $\cos \frac{4m\pi }{\left(n\right(n+1)}-i\sin \frac{4m\pi }{n(n+1)},m=1,2,....$
2. $\cos \frac{4m\pi }{n(n+1)}+i\sin \frac{4m\pi }{n(n+1)},m=1,2,...$
3. $0$
4. $\sin \frac{4m\pi }{n(n+1)}+i\cos \frac{4m\pi }{n(n+1)},m=1,2,....$

Q. $\underset{n\to \infty }{lim}\left\{lo{g}_{n-1}\right(n\left)lo{g}_n\right(n+1\left)lo{g}_{n+1}\right(n+2)\dots \dots lo{g}_{(n^k-1)}(n^k\left)\right\}$ is equal to

1. n
2. k
3. 
4. None of these

Q.

The locus of z satisfying the inequality $lo{g}_{\frac{1}{3}}$|z+1| > $lo{g}_{\frac{1}{3}}$|z-1| is

1. None of these

2. R(z)<0

3. R(z)>0

4. I(z)<0

Q. If $A_1{A}_2{A}_3...A_n$ be a regular polygon of $n$ sides and
$\frac{1}{A_1{A}_2}=\frac{1}{A_1{A}_3}+\frac{1}{A_1{A}_4},$then

1. $n=5$
2. $n=6$
3. $n=7$
4. none of these.

Q. $\log (-1+\sqrt{3}i)$ can be expressed in cartesian form as

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

Q. ${\log }_{\frac{\pi }{4}}(-1+\sqrt{3}i)$ can be expressed in cartesian form as

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

Q. If $f^{\prime \prime }\left(x\right)=\frac{\cos \left(\log x\right)}{x},f^{\prime }\left(1\right)=0$ and $y=f\left(\frac{2x+3}{3-2x}\right)$ then $\frac{dy}{dx}$ is equal to

1. $\left(\sin \left(\log x\right)\right)\frac{1}{\cos x}$
2. $\sin \left(\log \left(\frac{2x+3}{3-2x}\right)\right)$
3. $\frac{12}{{(3-2x)}^2}\sin \left(\log \left(\frac{2x+3}{3-2x}\right)\right)$
4. None of these