The number of solution(s) of the equation $\sqrt{\mathrm{x} + 1} - \sqrt{\mathrm{x} - 1} = \sqrt{4\mathrm{x} - 1}$ is/are

• 2

• 0

• 3

• 1

The value of $\left|{\mathrm{z}}^2\right| + {\left|\mathrm{z} - 3\right|}^2 + {\left|\mathrm{z} - \mathrm{i}\right|}^2$ is minimum when z equals

• $2 - \frac{2}{3}\mathrm{i}$

• 45 + 3i

• $1 + \frac{\mathrm{i}}{3}$

• $1 - \frac{\mathrm{i}}{3}$

The solution of the equation ${\log }_{101}\left({\log }_7\left(\sqrt{\mathrm{x} + 7} + \sqrt{\mathrm{x}}\right)\right) = 0$ is

• 3

• 7

• 9

• 49

In a $∆\mathrm{ABC}$, $\tan \left(\mathrm{A}\right) \mathrm{and} \tan \left(\mathrm{B}\right)$ are the roots of pq(x² + 1) = r²x. Then, $∆\mathrm{ABC}$ is

• a right angled triangle

• an acute angled triangle

• an obtuse angled triangle

• an equilateral triangle

Let f(x) = 2x² + 5x + 1. If we write f(x) as f(x) = a(x + 1)(x - 2) + b(x - 2)(x - 1) + c(x - 1)(x + 1) for real numbers a, b, c then

• there are infinite number of choices for a, b, c

• only one choice for a but infinite number of choices for b and c

• exactly one choice for each of a, b, c

• more than one but finite number of choices for a, b, c

If $\mathrm{\alpha }, \mathrm{\beta }$ are the roots of ax² + bx + c = 0 ($\mathrm{a} \ne 0$) and $\mathrm{\alpha } + \mathrm{h}, \mathrm{\beta } + \mathrm{h}$ are the roots of px² + qx + r = 0 (p $\ne$ 0), then the ratio of the squares of their discriminants is

• a² : p²

• a : p²

• a² : p

• a : 2p

Suppose that z₁, z₂, z₃ are three vertices of an equilateral triangle in the Argand plane. Let $\mathrm{\alpha } = \frac{1}{2}\left(\sqrt{3} + \mathrm{i}\right) \mathrm{and} \mathrm{\beta }$ be a non-zero complex number. The points ${\mathrm{\alpha z}}_1 + \mathrm{\beta }, {\mathrm{\alpha z}}_2 + \mathrm{\beta }, {\mathrm{\alpha z}}_3 + \mathrm{\beta }$ will be

• the vertices of an equilateral triangle

• the vertices of an isosceles triangle 

• collinear

• the vertices of a scalene triangle

In the Argand plane, the distinct roots of 1 + z + z³ + z⁴ = 0 (z is a complex number) represent vertices of

• a square

• an equilateral triangle

• a rhombus

• a rectangle

149.

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)