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

• $1 + \frac{\mathrm{i}}{3}$

• $1 - \frac{\mathrm{i}}{3}$

If $\underset{\mathrm{x}\to 0}{\lim }\frac{2\mathrm{asin}\left(\mathrm{x}\right) - \sin \left(2\mathrm{x}\right)}{{\tan }^3\left(\mathrm{x}\right)}$ exists and is equal to 1, then the value of $\mathrm{\alpha }$ is

• 2

• 1

• 0

• - 1

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

The number of digits in 20³⁰¹ $\left(\mathrm{given}, {\log }_{10}\left(2\right) = 0.3010\right)$ is

• 602

• 301

• 392

• 391

6.

If R be the set of all real numbers and f : R ➔ R is given by f(x) = 3x² + 1. Then, the set f^-1([1, 6]) is

• $\left\{- \sqrt{\frac{5}{3}}, 0, \sqrt{\frac{5}{3}}\right\}$

• $\left[- \sqrt{\frac{5}{3}}, \sqrt{\frac{5}{3}}\right]$

• $\left[- \sqrt{\frac{1}{3}}, \sqrt{\frac{1}{3}}\right]$

• $\left(- \sqrt{\frac{5}{3}}, \sqrt{\frac{5}{3}}\right)$

The value of $\tan \left(\frac{\mathrm{\pi }}{2}\right) + 2\tan \left(\frac{2\mathrm{\pi }}{5}\right) + 4\cot \left(\frac{4\mathrm{\pi }}{5}\right)$ is

• $\cot \left(\frac{\mathrm{\pi }}{5}\right)$

• $\cot \left(\frac{2\mathrm{\pi }}{5}\right)$

• $\cot \left(\frac{4\mathrm{\pi }}{5}\right)$

• $\cot \left(\frac{3\mathrm{\pi }}{5}\right)$

Let the number of elements of the sets A and B be p and q, respectively. Then, the number of relations from the set A to the set B is

• 2^{p + q}

• 2^pq

• p + q

• pq

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

If y = 4x + 3 is parallel to a tangent to the parabola y² = 12x, then its distance from the normal parallel to the given line is

• $\frac{213}{\sqrt{17}}$

• $\frac{219}{\sqrt{17}}$

• $\frac{211}{\sqrt{17}}$

• $\frac{210}{\sqrt{17}}$