Complex Numbers

Q.

Find the multiplicative inverse of $(1+i)$.

Q.

Find the principal and general solutions of $\mathrm{cosec}x=-2$

Q.

If $z=\frac{7-i}{3-4i}$, then $z^{14}$ is equal to

1. $2^7$

2. $2^{7}i$

3. $2^{14}i$

4. $-2^{7}i$

5. $-2^{14}$

Q.

If $x^2+y^2=t+\frac{1}{t}$ and $x^4+y^4=t^2+\frac{1}{t^2}$, then $\frac{dy}{dx}$ is equal to:

1. $\frac{y}{x}$

2. $-\frac{y}{x}$

3. $\frac{x}{y}$

4. $-\frac{1}{x^{3}y}$

Q.

If $S_n=\sum _{k=1}^{4n}{\left(-1\right)}^{\frac{k\left(k+1\right)}{2}}{k}^2$, then $S_n$ can take values

1. $1056$

2. $1088$

3. $1120$

4. $1332$

Q.

If $\tan \left(\theta \right)=t$, then $\tan \left(2\theta \right)+sec\left(2\theta \right)=$

1. $\frac{1+t}{1-t}$

2. $\frac{1-t}{1+t}$

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

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

Q.

${\cos }^2\left(\frac{\pi }{6}+\theta \right)-{\sin }^2\left(\frac{\pi }{6}-\theta \right)=?$

1. $\left(\frac{1}{2}\right)\cos 2\theta$

2. $0$

3. $\left(-\frac{1}{2}\right)\cos 2\theta$

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

Q. Let $\alpha$ and $\beta$ are complex numbers satisfying $|\alpha +1+i|=1$ and $|\beta -2-3i|=6$ such that $6|\alpha {|}_{max}-|\beta {|}_{max}=\sqrt{a}-\sqrt{b};a,b\in R^{+}$ then the value of $\sqrt{b^2-2a}$ is

1. $3$
2. $5$
3. $7$
4. $8$

Q.

If $\tan \left(A-B\right)=1$ and $\sec \left(A+B\right)=\frac{2}{\sqrt{3}}$, then the smallest positive value of $B$ is

1. $\frac{25}{24}\mathrm{\pi }$

2. $\frac{19}{24}\mathrm{\pi }$

3. $\frac{13}{24}\mathrm{\pi }$

4. $\frac{11}{24}\mathrm{\pi }$

Q. If $\frac{1}{6}\sin \theta ,\cos \theta$ and $\tan \theta$ are in GP, then the general value of $\theta$ is

1. 
2. 
3. 
4. 

Q.

Find The Cube Root Of $15625$ Number By Prime Factorization Method

Q.

$\frac{d}{dx}\left\{{\sin }^{-1}\left[2ax\sqrt{(1–a^2{x}^2)}\right]\right\}=$

1. $\frac{2a}{\sqrt{(a^2–x^2)}}$

2. $\frac{a}{\sqrt{(a^2–x^2)}}$

3. $\frac{2a}{\sqrt{(1–a^2{x}^2)}}$

4. $\frac{a}{\sqrt{(1–a^2{x}^2)}}$

Q.

If $A+B=\frac{\pi }{2}$, the maximum values of $\cos A\cos B$ is

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

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

3. $1$

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

Q. The complex number $\frac{2^n}{(1+i{)}^{2n}}+\frac{(1+i{)}^{2n}}{2^n},n\in I$ is equal to

1. $2$
2. $\{1+(-1{)}^n\}\cdot i^n$
3. $0$
4. $\{1-(-1{)}^n\}\cdot i^n$

Q. Let $z$ be a complex number satisfying the equation $z^2-(3+i)z+m+2i=0,$ where $m\in R.$ Suppose the equation has a real root. The additive inverse of non-real root, is

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

Q. The value of $\frac{i^{592}+i^{590}+i^{588}+i^{586}+i^{584}}{i^{582}+i^{580}+i^{578}+i^{576}+i^{574}}+2=$

Q. If $\left[\begin{array}{cc}1/25& 0\\ x& 1/25\end{array}\right]={\left[\begin{array}{cc}5& 0\\ -a& 5\end{array}\right]}^{-2}$, then the value of $x$ is:

1. $\frac{1}{125}$
2. $\frac{2a}{125}$
3. $\frac{2a}{25}$
4. $\frac{2a}{625}$

Q.

If $a,b,c$ are non-coplanar vectors and $\lambda$ is a real number, then $\left[\lambda \right(a+b){\lambda }^{2}b\lambda c]=[ab+cb]$ for?

Q.

If the lines joining the foci of the ellipse $\left(\frac{x^2}{a^2}\right)+\left(\frac{y^2}{b^2}\right)=1$, where $a>b$ and an extremity of its minor axis are inclined at an angle of $60°$, then the eccentricity of the ellipse is:

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

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

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

4. $\frac{\sqrt{7}}{3}$

5. $\sqrt{3}$

Q.

A value of $\theta$ satisfying $\sin \left(5\theta \right)-\sin \left(3\theta \right)+\sin \left(\theta \right)=0,$ such that $0<\theta <\frac{\pi }{2}$ is

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

2. $\frac{\pi }{6}$

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

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

Q.

The value of $2^n\left\{1.3.5......\left(2n-3\right)\left(2n-1\right)\right\}$ is $?$

1. $\frac{\left(2n\right)!}{n!}$

2. $\frac{\left(2n\right)!}{2^n}$

3. $\frac{n!}{\left(2n\right)!}$

4. None of these

Q.

A real value of x satisfies the equation $\frac{3-4ix}{3+4ix}=a-b(a,bϵR),\text{if}{a}^2+b^2$

1. 1

2. -1

3. 2

4. -2

Q. The value of $i^{57}+\frac{1}{i^{125}}$ is :

1. $0$
2. $2i$
3. $-2i$
4. $2$

Q.

The modulus and amplitude of $\frac{\left(1+2i\right)}{1-{\left(1-i\right)}^2}$ are

1. $\sqrt{2}$ and $\frac{\mathrm{\pi }}{6}$

2. $1$ and $0$

3. $1$ and $\frac{\mathrm{\pi }}{3}$

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

Q.

The line $\frac{x-2}{3}=\frac{y-3}{4}=\frac{z-4}{0}$ is parallel to

1. xy-plane

2. yz-plane

3. zx-plane

4. None of these

Q. If $a<0,b>0,$ then $\sqrt{a}\cdot \sqrt{b}$ is equal to

1. $-\sqrt{|a|\cdot b}$
2. $\sqrt{|a|b}$
3. $\sqrt{|a|\cdot b}\cdot i$
4. None of the above

Q. Let $p$ be the sum of all possible determinants of order $2$ having $0,1,2$ and $3$ as their four entries. Let $\alpha$ be the common root of the equations
$x^2+ax+[m+1]=0$
$x^2+bx+[m+4]=0$
$x^2-cx+[m+15]=0$
such that $\alpha >p$ and $a+b+c=0$.
If $m=\underset{n\to \infty }{lim}\frac{1}{n}\sum _{r=1}^{2n}\frac{r}{\sqrt{n^2+r^2}}$, then the value of $\alpha +p$ is
( $[.]$ denotes the greatest integer function)

Q. Quadratic equation $x^2+(a-1)ix+5=0(a\in R)$ will have a pair of conjugate complex roots, if

1. $a=1$
2. $a^2-2a+21>0$
3. $a^2-2a+21<0$
4. $a^2+2a-21>0$

Q.

How do you write the expression $6^{\frac{3}{2}}$ in radical form?

Q.

Find the maximum and minimum values of x+sin2x on $[0,2\pi ]$