Multiples and Powers

Q.

Write 102 as the sum of the powers of 2

1. $2^6+2^5+2^2+2^1$

2. $2^6+2^5+2^2+2^5$

3. $2^7+2^5+2^2+2^1$

4. $2^6+2^5+2^2+2^1+2^0$

Q.

which of the following indicates the product of 5 taken 6 times ?

1. $6^{\prime }5$

2. $5^6$

3. $6\times 5$

4. $6^5$

Q.

Write 16 as the sum of the powers of 2.

1. $2+2^2+2^3+2$

2. $2^2+2^2+2^3$

3. $2^3+2^2+2^3$

4. $2^2+2+2^3$

Q.

which of the following indicates sum of 5 taken 6 times ?

1. $6\times 5$

2. $6^5$

3. $6^{\prime }5$

4. $5^6$

Q. Let $z$ be a complex number satisfying equation $z^p={\overline{z}}^q$, where $p,qϵN$, then

1. If $p=q$, then number of solutions of equation will be infinite
2. If $p\ne q$, then number of solutions of equation will be $p+q$
3. If $p\ne q$, then number of solutions of equation will be $p+q+1$
4. If $p=q$, then number of solutions of equation will be finite

Q.

If -1, $\alpha$, ${\alpha }^3$, ${\alpha }^5$, $\overline{\alpha }$, $\overline{{\alpha }^3}$, $\overline{{\alpha }^5}$ are roots of the equation $z^7$ + 1 = 0.Find the value of $\alpha$$\overline{\alpha }$, ${\alpha }^3$$\overline{{\alpha }^3}$, ${\alpha }^5$$\overline{{\alpha }^5}$, $\alpha$$\overline{\alpha }$ + ${\alpha }^3$$\overline{{\alpha }^3}$ + ${\alpha }^5$$\overline{{\alpha }^5}$.

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Q.

Find the value of 1 + $\alpha$ + ${\alpha }^2$ + ${\alpha }^3$ +.............. ${\alpha }^{n-1}$. If ${\alpha }^k$ = $cos\frac{2\pi k}{n}+isin\frac{2\pi k}{n}$,

K = 0, 1, 2, ..........n-1

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Q. If $\alpha =\cos \left(\frac{8\pi }{11}\right)+i\sin \left(\frac{8\pi }{11}\right),$ then $Re(\alpha +{\alpha }^2+{\alpha }^3+{\alpha }^4+{\alpha }^5)$ is equal to

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

Q.

If 1, $\alpha$, ${\alpha }^2$, ${\alpha }^3$, .................${\alpha }^6$ are the roots of the equation x = $(-1{)}^{\frac{1}{7}}$.Find the value of

${\alpha }^4$ + ${\alpha }^6$ + ${\alpha }^7$ + ${\alpha }^5$ + ${\alpha }^7$ + ${\alpha }^8$ + ${\alpha }^7$ + ${\alpha }^9$ + ${\alpha }^{10}$

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Q.

Fill all the roots of the equation.

$x^6$ - $x^5$ + $x^4$ - $x^3$ + $x^2$ - x + 1 = 0 1 + x $\ne 0$

1. $cos\frac{(2n+1)\pi }{7}$ + $isin\frac{(2n+1)\pi }{7}$; n = 0, 1, 2, 3, 4, 5, 6

2. $cos\frac{2n\pi }{7}$ + $isin\frac{2n\pi }{7}$; n = 0, 1, 2, 3, 4, 5, 6

3. $cos\frac{(n+1)\pi }{7}$ + $isin\frac{(n+1)\pi }{7}$; n = 0, 1, 2, 3, 4, 5, 6

4. $cos\frac{(n+1)\pi }{7}$ + $isin\frac{(n+1)\pi }{7}$; n = 0, 1, 2, 3, 4, 5, 6