nth Root of a Complex Number

Q. The set of values of $a$ for which the function $f:R\to R$ given by $f\left(x\right)=x^3+(a+2)x^2+3ax+5$ is one-one, is

1. $[1,4]$
2. $(-\infty ,4)$
3. $(-\infty ,1)\cup (4,\infty )$
4. $(1,\infty )$

Q. The values of $x$ which satisfies the equation $(x-1{)}^3+8=0$ are

1. $-1,1-2\omega \text{and}1-2{\omega }^2$
2. $1,1-2\omega \text{and}1-2{\omega }^2$
3. $-1,1+2\omega \text{and}1+2{\omega }^2$
4. $1,1+2\omega \text{and}1+2{\omega }^2$

Q. If $1,\omega ,{\omega }^2,\dots ,{\omega }^{n-1}$ are the $n^{\text{th}}$ roots of unity of $x^n=1.$ Then the value of $(5-\omega )(5-{\omega }^2)\dots (5-{\omega }^{n-1})$ is equal to

1. $\frac{5^n+1}{4}$
2. $4^{n-1}$
3. $1$
4. $\frac{5^n-1}{4}$

Q. The complex number $\omega$ which satisfies the equation $z^3=8i$ and lying in the second quadrant on the complex plane.

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

Q.

Evaluate the following :

$\left(i\right)2x^3+2x^2-7x+72,whenx=\frac{3-5i}{2}\left(ii\right)x^4-4x^3+4x^2+8x+44,whenx=3+2i\left(iii\right)x^4+4x^3+6x^2+4x+9,whenx=-1+i\sqrt{2}\left(iv\right)x^6+x^4+x^2+1,whenx=\frac{1+i}{\sqrt{2}}\left(v\right)2x^4+5x^3+7x^2-x+41,whenx=-2-\sqrt{3}i$

Q.

If ${\alpha }_0,{\alpha }_1,{\alpha }_2\dots ,{\alpha }_{n-1}$ be the n, nth roots of the unity, then the value of ${\sum }_{i=0}^{n-1}\frac{{\alpha }_i}{(3-{\alpha }_i)}$ is equal to

1. $\frac{n-1}{3^n-1}$

2. $\frac{n+1}{3^n-1}$

3. $\frac{n}{3^n-1}$

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

Q. If $1,\omega ,{\omega }^2$ are three cube roots of unity, then $(1-\omega +{\omega }^2)(1+\omega -{\omega }^2)$ is

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

Q. The sum of roots of the equation $x^8=1$ whose real part is positive is

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

Q. If $z=\cos \theta +i\sin \theta$ be a root of the equation $a_0{z}^n+a_1{z}^{n-1}+a_2{z}^{n-2}+\dots \dots +a_{n-1}z+a_n=0$, then

1. $a_1\sin \theta +a_2\sin 2\theta +\dots \dots +a_n\sin n\theta =0$
2. $a_0+a_1\cos \theta +a_2\cos 2\theta +...+a_n\cos n\theta =-1$
3. $a_0+a_1\cos \theta +a_2\cos 2\theta +...+a_n\cos n\theta =0$
4. $a_0+a_1\cos \theta +a_2\cos 2\theta +...+a_n\cos n\theta =1$

Q. The root(s) of the equation $x^4=16$ is/are :

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

Q. If $z_1,z_2,z_3,z_4,z_5$ are roots of the equation $z^5+z^4+z^3+z^2+z+1=0$, then the value of $\left|\sum _{i=1}^5{z}_i^4\right|$ is

Q. $f\left(\pi /2\right)$ equals

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

Q. If $z_1$ and $z_2$ are two $n^{\text{th}}$ roots of unity, then $\arg \left(\frac{z_1}{z_2}\right)$ is a multiple of

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

Q. If the lines $\frac{x-4}{2}=\frac{y+5}{4}=\frac{z-1}{-3}$ and $\frac{x-2}{1}=\frac{y+1}{3}=\frac{z}{2}$ intersects, find their point of intersection.

Q. Simplified form of $\frac{(\sqrt{3}-i{)}^6}{(1+i{)}^8}$ is

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

Q. Consider a function $f:C\to C$ defined as $f\left(z\right)=z^{12}+2z^{11}+3z^{10}+...+12z+13$ and $S$ be the set defined as $\left\{z|Re\left[f\right(z)-z\left(\frac{z^{13}-1}{(z-1{)}^2}\right)]=\frac{13}{4}\right\},$ where $Re\left(z\right)$ denotes the real part of complex number $z.$ If $\alpha =\cos \frac{2\pi }{13}+i\sin \frac{2\pi }{13},$ where $i=\sqrt{-1},$ then

1. $f\left(\alpha \right)\cdot f\left({\alpha }^2\right)\cdots f\left({\alpha }^{12}\right)=(13{)}^{11}$
2. $f\left(\alpha \right)\cdot f\left({\alpha }^2\right)\cdots f\left({\alpha }^{12}\right)=(13{)}^{12}$
3. the maximum area of the quadrilateral formed by joining four points lying in $S$ is $8.$
4. the maximum area of the quadrilateral formed by joining four points lying in $S$ is $16.$

Q. The complex number $z$ in the second quadrant which satisfies the equation $z^3=8i$ is

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

Q. If ${\left(\frac{1+i}{1-i}\right)}^x=1$, then

1. $x=4n+1,$ where $n\in N$
2. $x=2n+1$
3. $x=4n$
4. $x=2n$

Q. If $x=2+2^{1/3}+2^{2/3}$ then value of $x^3-6x^2+6x-2$ is

1. 0
2. 3
3. -1
4. 1

Q. If $|2z-1|=|z-2|$ and $z_1,z_2,z_3$ are complex numbers such that $|z_1-\alpha |<\alpha ,|z_2-\beta |<\beta$, then $|\frac{z_1+z_2}{\alpha +\beta }|$

1. $<|z|$
2. $<2|z|$
3. $>|z|$
4. $>2|z|$

Q. If $sec\alpha =\frac{2}{\sqrt{3}}$ then find the value of $\frac{1-cosec\alpha }{1+cosec\alpha }$ where $\alpha$ is in IV quadrant.

Q. The value of ${\sin }^{-1}\left(\cos \frac{53\pi }{5}\right)$ is

1. $\frac{3\pi }{5}$
2. $\frac{3\pi }{5}$
3. $\frac{\pi }{10}$
4. $-\frac{\pi }{10}$

Q. If $1,\omega ,{\omega }^2,{\omega }^3.....,{\omega }^{n-1}$ are the n, $n^{th}$ roots of unity, then $(1-\omega )(1-{\omega }^2).....(1-{\omega }^{n-1})$ equals

1. 1
2. $n^2$
3. n
4. 0

Q. Let $f:[0,\frac{\pi }{2}]\to [0,1]$ be a differentiable function such that $f\left(0\right)=0,f\left(\frac{\pi }{2}\right)=1$, then

1. $f^{{}^{\prime }}\left(\alpha \right)=\sqrt{1-\left(f\right(\alpha ){)}^2}$ for all $\alpha ϵ(0,\frac{\pi }{2})$
2. $f^{{}^{\prime }}\left(\alpha \right)=\frac{2}{\pi }$ for all $\alpha ϵ(0,\frac{\pi }{2})$
3. $f\left(\alpha \right)f^{{}^{\prime }}\left(\alpha \right)=\frac{1}{\pi }$ for at least one $\alpha ϵ(0,\frac{\pi }{2})$
4. $f^{{}^{\prime }}\left(\alpha \right)=\frac{8\alpha }{{\pi }^2}$ for at least one $\alpha ϵ(0,\frac{\pi }{2})$

Q. $If1,\omega ,{\omega }^2,{\omega }^3.....,{\omega }^{n-1}arethen,n^{th}rootsofunity,then(1-\omega )(1-{\omega }^2).....(1-{\omega }^{n-1})equals$

1. 
2. 
3. 
4. 

Q. Solve: ${\tan }^{-1}\frac{1-x}{1+x}=\frac{1}{2}{\tan }^{-1}x,(x>0)$

Q. lf $\alpha ,\beta ,\gamma$ are the roots of the equation $x^3-7x+7=0$, then the value of ${\alpha }^{-4}+{\beta }^{-4}+{\gamma }^{-4}$ is

1. $\frac{11}{7}$
2. $\frac{3}{7}$
3. $-\frac{11}{7}$
4. $-\frac{3}{7}$

Q. If $\alpha ,\beta$ are the roots of the equation $a{x}^2+bx+c=0$, show that $\log (a-bx+c{x}^2)=\log a+(\alpha +\beta )x-\frac{{\alpha }^2+{\beta }^2}{2}{x}^2+\frac{{\alpha }^3+{\beta }^3}{3}{x}^3-...$

Q. $\frac{{}^{\alpha }{C}_2}{{}^{\alpha +1}{C}_4}$ =

1. $\frac{12}{(\alpha +1)(\alpha -2)}$
2. $\frac{4}{(\alpha +1)(\alpha -3)}$
3. $\frac{12}{(\alpha +1)(\alpha -2)(\alpha -3)}$
4. $\frac{1}{3}$

Q. lf ${\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4$ are the roots of the equation $3x^4-(l+m)x^3+2x+5l=0$ and sum of all the roots is equal to $3$ and ${\alpha }_1{\alpha }_2{\alpha }_3{\alpha }_4=10$, then $(l,m)$ is equal to

1. $(6,3)$
2. $(-6,-3)$
3. $(-6,15)$
4. $(6,15)$