Properties of Cube Root of a Complex Number

Q. Write ${\cot }^{-1}\left(\frac{1}{\sqrt{x^2-1}}\right),x>1$ in the simplest form.

Q.

If $A=\left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right)$, find $f\left(A\right)$ where $f\left(x\right)=2x^2-3x+5$.

Q. If $i=\sqrt{-1}$, then $4+5{(-\frac{1}{2}+\frac{\sqrt{3}}{2}i)}^{334}+3{(-\frac{1}{2}+\frac{\sqrt{3}}{2}i)}^{365}$ is equal to

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

Q. $\sin \left({\tan }^{-1}x\right),|x|<1$ is equal to

1. $\frac{x}{\sqrt{1+x^2}}$
2. $\frac{1}{\sqrt{1+x^2}}$
3. $\frac{x}{\sqrt{1-x^2}}$
4. $\frac{1}{\sqrt{1-x^2}}$

Q.

If $1,\omega and{\omega }^2$are the cube roots of unity, ${\left(3+{\omega }^2+{\omega }^4\right)}^6$ is equal to

1. $64$

2. $729$

3. $2$

4. $0$

Q.

If $1,\omega and{\omega }^2$are the cube roots of unity, then $\left(1-\omega +{\omega }^2\right)\left(1-{\omega }^2+{\omega }^4\right)\left(1-{\omega }^4+{\omega }^8\right)\left(1-{\omega }^8+{\omega }^{16}\right)......upto2n$ factors is

1. $2n$

2. $2^{2n}$

3. $1$

4. $-2^{2n}$

Q.

If $x^2+x+1$ is a factor of $a{x}^3+b{x}^2+cx+d$, then the real root of $a{x}^3+b{x}^2+cx+d=0$ is:

1. $-\frac{d}{a}$

2. $\frac{c}{a}$

3. $\frac{b}{a}$

4. None of these

Q.

${\left(\frac{-1+\sqrt{-3}}{2}\right)}^{100}+{\left(\frac{-1-\sqrt{-3})}{2}\right)}^{100}$is equal to

1. $2$

2. $0$

3. $-1$

4. $1$

Q. If $\alpha$ and $\beta$ are the roots of the equation $x^2-4x+1=0$ $(\alpha >\beta ),$ then the value of $f(\alpha ,\beta )=\frac{{\beta }^3}{2}{\text{cosec}}^2\left(\frac{1}{2}{\tan }^{-1}\frac{\beta }{\alpha }\right)+\frac{{\alpha }^3}{2}{\sec }^2\left(\frac{1}{2}{\tan }^{-1}\frac{\alpha }{\beta }\right)$ is

Q. If the equation $\sec \theta +\text{cosec}\theta =c$ has two real roots between $0$ and $2\pi ,$ then the least integer which $c^2$ cannot exceed must be

Q.

If $A=\left(\begin{array}{cc}3& -5\\ -4& 2\end{array}\right)$, show that $A^2-5A-14I=0$.

Q.

If a, b, c belongs to Q then roots of the equation (b+c-2a)x²+(c+a-2b)x+(a+b-2c)=0 are

a)rational b) non real c) irrational d) equal

Q. The sum of ${162}^{\text{th}}$ power of the roots of the equation $x^3-2x^2+2x-1=0$ is

Q. If the four roots of the equation $z^4+z^3+2z^2+z+1=0$ form a quadrilateral on the Argand plane, then the area of the quadrilateral is

1. $\frac{\sqrt{3}+2}{8}$
2. $\frac{\sqrt{3}+2}{4}$
3. $\frac{\sqrt{3}+2}{2}$
4. $\frac{\sqrt{3}+2}{3}$

Q.

If $A=\left[\begin{array}{cc}2& -3\\ p& q\end{array}\right]$, find $p$and $q$ so that $A^2=I$.

Q.

If $A=\left(\begin{array}{cc}6& 5\\ 7& 6\end{array}\right)$, show that $A^2-12A+I=0$.

Q. If w is a complex cube root of unity, show that
$\left(\left[\begin{array}{ccc}1& w& w^2\\ w& w^2& 1\\ w^2& 1& w\end{array}\right]+\left[\begin{array}{ccc}w& w^2& 1\\ w^2& 1& w\\ w& w^2& 1\end{array}\right]\right)\left[\begin{array}{c}1\\ w\\ w^2\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$

Q. The value of ${\sin }^{-1}\frac{3}{5}-{\sin }^{-1}\frac{8}{17}$ is:

1. ${\cos }^{-1}\frac{60}{85}$
2. ${\cos }^{-1}\frac{24}{85}$
3. ${\cos }^{-1}\frac{84}{85}$
4. ${\cos }^{-1}\frac{15}{17}$

Q. If $\alpha ,\beta$ be the roots of the equation $u^2-2u+2=0$ and if $\cot \theta =x+1$, then $\frac{\left[\right(x+\alpha {)}^n-(x+\beta {)}^n]}{[\alpha -\beta ]}$ is equal to

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

Q. The value of $(1+\cos \theta +i\sin \theta {)}^n+(1+\cos \theta -i\sin \theta {)}^n$ is

1. $2^{n+1}.{\cos }^n\frac{\theta }{2}.\cos \frac{n\theta }{2}$
2. $2^n.{\cos }^n\frac{\theta }{2}.\cos \frac{\theta }{2}$
3. $2^n.{\cos }^n\frac{\theta }{2}.\cos \frac{n\theta }{2}$
4. $2^{n+1}.{\cos }^n\frac{\theta }{2}.\cos \frac{\theta }{2}$

Q. Let $z$ be a complex number satisfying $z+z^{-1}=1$. A possible value of $n$ when $z^n+z^{-n}$ is minimum, is

1. $12$
2. $15$
3. $10$
4. $11$

Q. Let $\omega \ne 1$ be a cube root of unity. For distinct non-zero integers $a,b,$ and $c,$ the minimum value of $z=|a+b\omega +c{\omega }^2{|}^2$ is

1. $0$
2. $1$
3. $3$
4. $4$

Q. Let $z_1,z_2,z_3,z_4$ are distinct complex numbers representing the vertices of a quadrilateral $ABCD$ taken in order. If $z_1-z_3=z_2-z_4$ and $\arg \left(\frac{z_4-z_1}{z_2-z_1}\right)=\frac{\pi }{2},$ then the quadrilateral is

1. rhombus
2. square
3. rectangle
4. trapezium

Q. If $\int \frac{\cos \theta }{5+7\sin \theta -2{\cos }^2\theta }d\theta =A{\log }_e|B\left(\theta \right)|+C,$ where $C$ is a constant of integration, then $\frac{B\left(\theta \right)}{A}$ can be:

1. $\frac{5(2\sin \theta +1)}{\sin \theta +3}$
2. $\frac{5(\sin \theta +3)}{2\sin \theta +1}$
3. $\frac{2\sin \theta +1}{\sin \theta +3}$
4. $\frac{2\sin \theta +1}{5(\sin \theta +3)}$

Q. $A=[a_{ij}{]}_{3\times 3}$ is a diagonal matrix such that $|A|=27000$ and $x_1,x_2,x_3\in N$ are elements along principal diagonal, then

1. Minimum value of $x_1+x_2+x_3$ is $60$
2. Number of such matrix is ${}^{\prime }{N}^{\prime }$ then ${}^{\prime }{N}^{\prime }$ is perfect square
3. Number of such matrix is ${}^{\prime }{N}^{\prime }$ then ${}^{\prime }{N}^{\prime }$ is perfect cube
4. Minimum value of $x_1+x_2+x_3$ is $90$

Q. If $z^2+z+1=0$ where z is a complex number, then the value of
${(z+\frac{1}{z})}^2+{(z^2+\frac{1}{z^2})}^2+{(z^3+\frac{1}{z^3})}^2+......+{(z^6+\frac{1}{z^6})}^2$is

1. 18
2. 54
3. 6
4. 12

Q. If $\int \frac{{\sec }^2\theta \sin 2\theta d\theta }{\cos 3\theta }=f\left(\theta \right)+C$ for some arbitrary constant $C,$ then $f\left(\theta \right)$ is equal to

1. $\frac{2}{3}\ln \left|{\left(\frac{\cos \theta +\cos \frac{\pi }{6}}{\cos \theta -\cos \frac{\pi }{6}}\right)}^{\tan \frac{\pi }{6}}{e}^{\sec \theta }\right|$
2. $\frac{2}{3}\ln \left|{\left(\frac{\cos \theta -\cos \frac{\pi }{6}}{\cos \theta +\cos \frac{\pi }{6}}\right)}^{\tan \frac{\pi }{6}}{e}^{\cos \theta }\right|$
3. $\frac{2}{3}\ln \left|{\left(\frac{\cos \theta +\cos \frac{\pi }{6}}{\cos \theta -\cos \frac{\pi }{6}}\right)}^{\tan \frac{\pi }{6}}{e}^{\sec (\pi -\theta )}\right|$
4. $\frac{2}{3}\ln \left|{\left(\frac{\cos \theta +\cos \frac{\pi }{6}}{\cos \theta -\cos \frac{\pi }{6}}\right)}^{\tan \frac{\pi }{6}}{e}^{\cos (\pi -\theta )}\right|$

Q. If $f\left(x\right)={\tan }^{-1}(\text{cosec}\left({\tan }^{-1}x\right)-\tan \left({\cot }^{-1}x\right));x>0,$ then the value of $8f^{\prime }\left(1\right)$ is

Q. If $\alpha ,\beta$ are non real numbers satisfying $x^3-1=0$ then the value of $\left|\begin{array}{ccc}\lambda +1& \alpha & \beta \\ \alpha & \lambda +\beta & 1\\ \beta & 1& \lambda +\alpha \end{array}\right|$ is equal to

1. ${\lambda }^3$
2. 0
3. ${\lambda }^3+1$
4. ${\lambda }^3-1$

Q. If $a_r=\cos \frac{2r\pi }{9}+i\sin \frac{2r\pi }{9},r=1,2,3,...,i=\sqrt{-1},$ then the determinant $\left|\begin{array}{ccc}{a}_1& a_2& a_3\\ a_4& a_5& a_6\\ a_7& a_8& a_9\end{array}\right|$ is equal to

1. $a_1{a}_9-a_3{a}_7$
2. $a_2{a}_6-a_4{a}_8$
3. $a_9$
4. $a_5$