Cube Root of a Complex Number

Q. If $\alpha$ and $\beta$ are the distinct roots of the equation $x^2+(3{)}^{1/4}x+3^{1/2}=0,$ then the value of ${\alpha }^{96}({\alpha }^{12}-1)+{\beta }^{96}({\beta }^{12}-1)$ is equal to

1. $28\times 3^{25}$
2. $56\times 3^{24}$
3. $52\times 3^{24}$
4. $56\times 3^{25}$

Q. Find the matrix $A$ satisfying the matrix equation:

$\left[\begin{array}{cc}2& 1\\ 3& 2\end{array}\right]A\left[\begin{array}{cc}-3& 2\\ 5& -3\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

Q. Using binomial theorem, prove that $6^n-5n$ always leaves remainder $1$ when divided by $25$.

Q. If $z=-1$, then principal value of the $\arg \left(z^{2/3}\right)$ is/are

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

Q. If $\alpha$ and $\beta$ are imaginary cube roots of unity, then ${\alpha }^4+{\beta }^4+\frac{1}{\alpha \beta }$
[IIT 1977]

1. 0

2. 3

3. 1
4. 2

Q. The value of ${\left(\frac{-1+i\sqrt{3}}{1-i}\right)}^{30}$ is:

1. $2^{15}i$
2. $-2^{15}$
3. $-2^{15}i$
4. $6^5$

Q.

The value of $x$, when ${\left(2\right)}^{x+4}·{\left(3\right)}^{x+1}=288$ is

1. 1

2. $-1$

3. 0

4. None

Q.

The value of $\frac{\left(a+b\omega +c{\omega }^2\right)}{\left(b+c\omega +a{\omega }^2\right)}+\frac{\left(a+b\omega +c{\omega }^2\right)}{\left(c+a\omega +b{\omega }^2\right)}$ will be

1. $1$

2. $-1$

3. $2$

4. $-2$

Q. If $x^2+y^2=a^2$, then ${\int }_0^a\sqrt{1+{\left(\frac{dy}{dx}\right)}^2}dx=$

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

Q. The least positive integer $n$ for which ${\left(\frac{1+i\sqrt{3}}{1-i\sqrt{3}}\right)}^n=1,\text{is}$

1. $2$
2. $5$
3. $6$
4. $3$

Q. In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor axis is $10$ and one of the foci is at $(0,5\sqrt{3})$, then the length of its latus rectum is :

1. $10$
2. $5$
3. $6$
4. $8$

Q. Let $z=\frac{\sqrt{3}}{2}-\frac{i}{2}.$ Then the smallest positive integer $n$ such that $(z^{95}+i^{67}{)}^{94}=z^n$ is

Q. Let $z_0$ be a root of the quadratic equation, $x^2+x+1=0$. If $z=3+6i{z}_0^{81}-3i{z}_0^{93}$, then $\arg z$ is equal to :

1. $\frac{\pi }{4}$
2. $\frac{\pi }{6}$
3. $\frac{\pi }{3}$
4. $0$

Q. Let $\omega \ne 1$ be a cube root of unity. Then the minimum of the set
$\{|a+b\omega +c{\omega }^2{|}^2:a,b,c$ distinct non-zero integers} equals

Q. If $x=\omega -{\omega }^2-2,$ then the value of $x^4+3x^3+2x^2-11x-6$ is

Q. If $\frac{1}{a+\omega }+\frac{1}{b+\omega }+\frac{1}{c+\omega }+\frac{1}{d+\omega }=\frac{2}{\omega },$ where $a,b,c$ are real and $\omega$ is non real cube root of unity, then:

1. $\frac{1}{a+{\omega }^2}+\frac{1}{b+{\omega }^2}+\frac{1}{c+{\omega }^2}+\frac{1}{d+{\omega }^2}=\frac{2}{{\omega }^2}$
2. $abc+bcd+abd+acd=2$
3. $a+b+c+d=2abcd$
4. $\frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c}+\frac{1}{1+d}=2$

Q. The value of the expression $1(2-\omega )(2-{\omega }^2)+2(3-\omega )(3-{\omega }^2)+\cdots \cdots +(n-1)(n-\omega )(n-{\omega }^2)$ is where $\omega$ is an imaginary cube root of unity is

1. ${\left(\frac{n(n+1)}{2}\right)}^2-n$
2. ${\left(\frac{n(n+1)}{2}\right)}^2$
3. ${\left(\frac{n(n+1)}{2}\right)}^2+n$
4. ${\left(\frac{n(n+1)}{2}\right)}^2+2n$

Q. If $i=\sqrt{-1}$ and $\omega$ is non real cube root of unity, then $\frac{(1+i{)}^{2n}-(1-i{)}^{2n}}{(1+{\omega }^4-{\omega }^2)(1-{\omega }^4+{\omega }^2)}$ is equal to

1. $0,$ when $n$ is even
2. $2^{n-1}\cdot i^n,$ when $n$ is odd
3. $0\forall n\in I$
4. $2^{n-1}\cdot i^n,$ when $n$ is even

Q. If $\omega$ is a non-real cube root of unity, then the value of $1\cdot (2-\omega )(2-{\omega }^2)+2\cdot (3-\omega )(3-{\omega }^2)+\cdots$ sum upto $19$ terms is

1. $56680$
2. $44080$
3. $84390$
4. $64580$

Q. Let $1,\omega$ and ${\omega }^2$ be the cube roots of unity. The least possible degree of a polynomial with real coefficients, having $2{\omega }^2,3+4\omega ,3+4{\omega }^2$ and $5-\omega -{\omega }^2$ as roots is

Q. If $x+\frac{1}{x}=1$ and $p=x^{4000}+\frac{1}{x^{4000}}$ and $q$ be the digit at unit place in the number $2^{2^n}+1,n\in N$ and $n>1$, then the value of $p+q=$

Q. If $1,\omega ,{\omega }^2$ be the three cube roots of unity, then
$(1+\omega )\prod _{n=1}^{2n-1}(1+{\omega }^{2^n})=$

Q. If $5^{40}$ is divided by $11,$ then remainder is $\alpha$ and if $2^{2003}$ is divided by $17,$ then remainder is $\beta .$ Then the value of $(\beta -\alpha )$ is

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

Q. The total number of distinct $x\in R$ for which $\left|\begin{array}{ccc}x& x^2& 1+x^3\\ 2x& 4x^2& 1+8x^3\\ 3x& 9x^2& 1+27x^3\end{array}\right|=10$ is

Q. Let $z=1+ai$ be a complex number, $a>0$, such that $z^3$ is a real number. Then the sum $1+z+z^2+....+z^{11}$ is equal to :

1. $1365\sqrt{3}i$
2. $1250\sqrt{3}i$
3. $-1250\sqrt{3}i$
4. $-1365\sqrt{3}i$

Q. If $\omega$ is a complex cube root of unity, then the value of $(a+b{)}^2+(a\omega +b{\omega }^2{)}^2+(a{\omega }^2+b\omega {)}^2$ is

1. $6ab$
2. $3ab$
3. $12ab$
4. $ab$

Q. Let D be the domain of the
real valued function $f$
defined by $f\left(x\right)=\sqrt{25-x^2}$

Then, write D.

Q. If the polynomial $5x^3+Mx+N$ is divisible by $x^2+x+1$, then $|M+N|=$

Q. Let $\omega$ is an imaginary cube roots of unity then the value of $2(\omega +1)({\omega }^2+1)+3(2\omega +1)(2{\omega }^2+1)+........+(n+1)(n\omega +1)(n{\omega }^2+1)$is

1. 
2. 
3. 
4. 

Q.

When $4^{101}+6^{101}$ is divided by $25$, the remainder is:

1. $20$

2. $10$

3. $5$

4. $0$