The sum of the fourth powers of the roots of the equation

x³ + x + 1 = 0 is

• - 2

• - 1

• 1

• 2

let $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ be the roots of the quadratic equation ax² + bx + c = 0. Observe the lists given below

	List-I		List-II
(i)	$\mathrm{\alpha } = \mathrm{\beta }$	(A)	(ac2)1/3 + (a2c)1/3 + b = 0
(ii)	$\mathrm{\alpha } = 2\mathrm{\beta }$	(B)	2b2 = 9ac
(iii)	$\mathrm{\alpha } = 3\mathrm{\beta }$	(C)	b2 = 6ac
(iv)	$\mathrm{\alpha } = {\mathrm{\beta }}^2$	(D)	3b2 = 16ac
		(E)	b2 = 4ac
		(F)	(ac2)1/3 + (a2c)1/3 = b

The correct match of List-I from List-II is

The roots (x - a) (x - a - 1) + (x - a - 1) (x - a - 2) + (x - a) (x - a - 2) = 0, a $\in$ R are always

• equal

• imaginary

• real and distinct

• rational and equal

Let f(x) = x + ax + b, where a, b $\in$ R. If f(x) = 0 has all-its roots imaginary, then the roots of f(x) + f'(x) + f"(x) = 0 are

• real and distinct

• imaginary

• equal

• rational and equal

If $\mathrm{\alpha }, \mathrm{\beta }, \mathrm{\gamma }$ are the roots of x³ + 4x + 1 = 0, then the equation whose roots are $\frac{{\mathrm{\alpha }}^3}{\mathrm{\beta } + \mathrm{\gamma }}, \frac{{\mathrm{\beta }}^2}{\mathrm{\gamma } + \mathrm{\alpha }}, \frac{{\mathrm{\gamma }}^2}{\mathrm{\alpha } + \mathrm{\beta }}$ is

• x³ - 4x - 1 = 0

• x³ - 4x + 1 = 0

• x³ + 4x - 1 = 0

• x³ + 4x + 1 = 0

If $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ are the roots of x² - 2x + 4 = 0, then the value of ${\mathrm{\alpha }}^6 + {\mathrm{\beta }}^6$ is

• 32

• 64

• 128

• 256

If n is an integer which leaves remainder one when divided by three, then ${\left(1 + \sqrt{3}\mathrm{i}\right)}^{\mathrm{n}} + {\left(1 - \sqrt{3}\mathrm{i}\right)}^{\mathrm{n}}$ equals

• - 2^{n + 1}

• 2^{n + 1}

• - (- 2)ⁿ

• - 2ⁿ

If ∝, ß, y are the roots of the equation x³ - 6x² + 11x - 6 = 0 and if a = ∝² + ß² + γ², b = ∝ß + ßγ + γ∝ and  c = (∝ + ß)(ß + γ)(γ + ∝), then the correct inequality among the following is

• $\mathrm{a} < \mathrm{b} < \mathrm{c}$

• $\mathrm{b} < \mathrm{a} < \mathrm{c}$

• $\mathrm{b} < \mathrm{c} < \mathrm{a}$

• $\mathrm{c} < \mathrm{a} < \mathrm{b}$

319.

Let $\mathrm{\alpha } \ne 1$ be a real root of the equation x³ - ax² + ax - 1 = 0, where $\mathrm{a} \ne - 1$ is a real number. Then, a root of this equation, among the following, is

• ${\mathrm{\alpha }}^2$

• $- \frac{1}{\mathrm{\alpha }}$

• $\frac{1}{\mathrm{\alpha }}$

• $- \frac{1}{{\mathrm{\alpha }}^2}$

$\mathrm{If} \mathrm{f}\left(\mathrm{x}\right) = \left|\begin{array}{ccc}2\cos \left(\mathrm{x}\right)& 1& 0\\ \mathrm{x} - \frac{\mathrm{\pi }}{2}& 2\cos \left(\mathrm{x}\right)& 1\\ 0& 1& 2\cos \left(\mathrm{x}\right)\end{array}\right|, \mathrm{then} \mathrm{f}\text{'}\left(\mathrm{\pi }\right) = ?$

• 0

• 2

• $\frac{\mathrm{\pi }}{2}$

• $\mathrm{\pi } - 6$