Let $\mathrm{\alpha }, \mathrm{\beta }$ be the roots of x² - x - 1 = 0 and ${\mathrm{S}}_{\mathrm{n}} = {\mathrm{\alpha }}^{\mathrm{n}} + {\mathrm{\beta }}^{\mathrm{n}}$, for all integers $\mathrm{n} \ge 1$. Then, for every integer $\mathrm{n} \ge 2$

• ${\mathrm{S}}_{\mathrm{n}} + {\mathrm{S}}_{\mathrm{n} - 1} = {\mathrm{S}}_{\mathrm{n} +\hspace{0.17em}1}$

• ${\mathrm{S}}_{\mathrm{n}} - {\mathrm{S}}_{\mathrm{n} - 1} = {\mathrm{S}}_{\mathrm{n} +\hspace{0.17em}1}$

• ${\mathrm{S}}_{\mathrm{n} - 1} = {\mathrm{S}}_{\mathrm{n} +\hspace{0.17em}1}$

• ${\mathrm{S}}_{\mathrm{n}} + {\mathrm{S}}_{\mathrm{n} - 1} = 2{\mathrm{S}}_{\mathrm{n} +\hspace{0.17em}1}$

If $\mathrm{\alpha }, \mathrm{\beta }$ are the roots of the quadratic equation x² + px + q = 0, then the values of ${\mathrm{\alpha }}^3 + {\mathrm{\beta }}^3 \mathrm{and} {\mathrm{\alpha }}^4 + {\mathrm{\alpha }}^2{\mathrm{\beta }}^2 + {\mathrm{\beta }}^4$ are respectively

• $3\mathrm{pq} - {\mathrm{p}}^3 \mathrm{and} {\mathrm{p}}^4 - 3{\mathrm{p}}^2\mathrm{q} + 3{\mathrm{q}}^2$

• - p(3q - p²) and (p² - q)(p² + 3q)

• pq - 4 and p⁴ - q⁴

• 3pq - p³ and (p² - q)(p² - 3q)

Let $\mathrm{\alpha }, \mathrm{\beta }$ denote the cube roots of unity other than 1 and $\mathrm{\alpha } \ne \mathrm{\beta }$. Let S = $\sum _{\mathrm{n} = 0}^{302}{\left(- 1\right)}^{\mathrm{n}}{\left(\frac{\mathrm{\alpha }}{\mathrm{\beta }}\right)}^{\mathrm{n}}$. Then, the value of S is

• either - 2w or - 2w²

• either - 2w or 2w²

• either 2w or - 2w²

• either 2w or 2w²

Let p (x) be a quadratic polynomial with constant term 1. Suppose p(x), when divided by x - 1 leaves remainder 2 and when divided by x + 1 leaves remainder 4. Then, the sum of the roots of p(x) = 0 is

• - 1

• 1

• $- \frac{1}{2}$

• $\frac{1}{2}$

If z₁ = 2 + 3i and z₂ = 3 + 4i  be two points on the complex plane. Then, the set of complex number z satisfying ${\left|\mathrm{z} - {\mathrm{z}}_1\right|}^2 + {\left|\mathrm{z} - {\mathrm{z}}_2\right|}^2 = {\left|{\mathrm{z}}_1 - {\mathrm{z}}_2\right|}^2$ represnts

• a straight line

• a point

• a circle

• a pair of straight line

If $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ are roots of x² - x + 1 = 0, then the value of ${\mathrm{\alpha }}^{2013} + {\mathrm{\beta }}^{2013}$ is

• 2

• - 2

• - 1

• 1

If $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$  B are the roots of the quadratic equation is, x² + ax + b = 0, (b $\ne$ 0), then the quadratic equation whose roots are $\mathrm{\alpha } - \frac{1}{\mathrm{\beta }}, \mathrm{\beta } - \frac{1}{\mathrm{\alpha }}$, is

• ax² + a(b - 1)x + (a - 1)² = 0

• bx² + a(b - 1)x + (b - 1)² = 0

• x² + ax + b = 0

• abx² + bx + a = 0

If $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ are the roots of the quadratic equation ax² + bx + c = 0 and 3b² = 16ac, then

• $\mathrm{\alpha } = 4\mathrm{\beta } \mathrm{or} \mathrm{\beta } = 4\mathrm{\alpha }$

• $\mathrm{\alpha } = - 4\mathrm{\beta } \mathrm{or} \mathrm{\beta } = - 4\mathrm{\alpha }$

• $\mathrm{\alpha } = 3\mathrm{\beta } \mathrm{or} \mathrm{\beta } = 3\mathrm{\alpha }$

• $\mathrm{\alpha } = - 3\mathrm{\beta } \mathrm{or} \mathrm{\beta } = - 3\mathrm{\alpha }$

The number of solutions of the equation

$\frac{1}{2}{\log }_{\sqrt{3}}\left(\frac{\mathrm{x} + 1}{\mathrm{x} +\hspace{0.17em}5}\right) + {\log }_{\mathrm{e}}{\left(\mathrm{x} + 5\right)}^2$ = 1

• 0

• 1

• 2

• infinite

80.

If $\sin \left(\mathrm{\alpha }\right), \cos \left(\mathrm{\alpha }\right)$  be the roots of the equation x² - bx + c = 0, Then, which of the following statements is/are correct?

• $\mathrm{c} \le \frac{1}{2}$

• $\mathrm{b} \le \sqrt{2}$

• $\mathrm{c} > \frac{1}{2}$

• b $> \sqrt{2}$