Let X_n = $\left\{\mathrm{z} = \mathrm{x} + \mathrm{iy} : {\left|\mathrm{z}\right|}^2 \le \frac{1}{\mathrm{n}}\right\}$ for all integers $\mathrm{n} \ge 1. \mathrm{Then}, \underset{\mathrm{n} = 1}{\overset{\infty }{\cap }} {\mathrm{X}}_{\mathrm{n}}$ is

• a singleton set

• not a finite set

• an empty set

• a finite set with more than one element

The equation of hyperbola whose coordinates of the foci are (± 8, 0) and the length of latusrectum is 24 units, is

• 3x² - y² = 48

• 4x² - y² = 48

• x² - 3y² = 48

• x² - 4y² = 48

$\cos \left(\frac{2\mathrm{\pi }}{7}\right) + \cos \left(\frac{4\mathrm{\pi }}{7}\right) + \cos \left(\frac{6\mathrm{\pi }}{7}\right)$

• is equal to zero

• lies between 0 and 3

• is a negative number

• lies between 3 and 6

The minimum value of $2^{\sin \left(\mathrm{x}\right)} + 2^{\cos \left(\mathrm{x}\right)}$ is

• $2^{1 - 1/\sqrt{2}}$

• $2^{1 + 1/\sqrt{2}}$

• $2^{\sqrt{2}}$

• 2

45.

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 t_n denotes the nth term of the infinite series $\frac{1}{1!} + \frac{10}{2!} + \frac{21}{3!} + \frac{34}{4!} + \frac{49}{5!} +\hspace{0.17em}...$ . Then, $\underset{\mathrm{n}\to \infty }{\lim }{\mathrm{t}}_{\mathrm{n}}$ is

• e

• 0

• e²

• 1

A poker hand consists of 5 cards drawn at random from a well-shuffled pack of 52 cards. Then, the probability that a poker hand consists of a pair and a triple of equal face values (for example, 2 sevens and 3 kings or 2 aces and 3 queens, etc.) is

• $\frac{6}{4165}$

• $\frac{23}{4165}$

• $\frac{1797}{4165}$

• $\frac{1}{4165}$

If the circle x² + y² + 2gx + 2fy + c = 0 cuts the three circles x² + y² - 5 = 0, x² + y² - 8x - 6y + 10 = 0 and x² + y² - 4x + 2y - 2 = 0 at the extremities of  their diameters, then

• c = - 5

• fg = 147/25

• g + 2f = c + 2

• 4f = 3g

Let f(x) be a differentiable function in [2, 7]. If f(2) = 3 and f'(x) $\le$ 5 for all x in (2, 7), then the maximum possible value of f(x) at x = 7 is

• 7

• 15

• 28

• 14

If ${\sin }^{-1}\left(\frac{\mathrm{x}}{13}\right) + {\csc }^{-1}\left(\frac{13}{12}\right) = \frac{\mathrm{\pi }}{2}$, then the value of x is

• 5

• 4

• 12

• 11