The least positive value of t, so that the lines $\mathrm{x} = \mathrm{t} + \mathrm{\alpha }, \mathrm{y} + 16 = 0 \mathrm{and} \mathrm{y} = \mathrm{\alpha x}$ are concurrent is

• 2

• 4

• 16

• 8

In $∆\mathrm{ABC}, \mathrm{if} {\mathrm{a}}^2{\cos }^2\left(\mathrm{A}\right) - {\mathrm{b}}^2 - {\mathrm{c}}^2 = 0$ then

• $\frac{\mathrm{\pi }}{4} < \mathrm{A} < \frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{2} < \mathrm{A} < \mathrm{\pi }$

• $\mathrm{A} = \frac{\mathrm{\pi }}{2}$

• $\mathrm{A} < \frac{\mathrm{\pi }}{4}$

$\left\{\mathrm{x} \in \mathrm{R}: \left|\cos \left(\mathrm{x}\right)\right| \ge \sin \left(\mathrm{x}\right)\right\} \cap \left[0, \frac{3\mathrm{\pi }}{2}\right]$ is equal to

• $\left[0, \frac{\mathrm{\pi }}{4}\right] \cup \left[\frac{3\mathrm{\pi }}{4}, \frac{3\mathrm{\pi }}{2}\right]$

• $\left[0, \frac{\mathrm{\pi }}{4}\right] \cup \left[\frac{\mathrm{\pi }}{2}, \frac{3\mathrm{\pi }}{2}\right]$

• $\left[0, \frac{\mathrm{\pi }}{4}\right] \cup \left[\frac{5\mathrm{\pi }}{4}, \frac{3\mathrm{\pi }}{2}\right]$

• $\left[0, \frac{3\mathrm{\pi }}{2}\right]$

If ${\log }_{0.2}\left(\mathrm{x} - 1\right) > {\log }_{0.04}\left(\mathrm{x} + 5\right)$, then

• - 1 < x < 4

• 2 < x < 3

• 1 < x < 4

• 1 < x < 3

The number of real roots of equation log_e(x) + ex = 0 is

• 0

• 1

• 2

• 3

26.

Let x₁, x₂, ..., x₁₅ be 15 distinct numbers chosen from 1, 2, 3, ..., 15. Then, the value of (x₁ - 1)(x₂ - 1)(x₃ - 1)...(x₁₅ - 1) is

• always $\le$ 0

• 0

• always even

• always odd

A particle starts moving from rest from a fixed point in a fixed direction. The distance s from the fixed point at a time t is given by s = t² + at - b + 17, where a and b are real numbers. If the particle comes to rest after 5 s at a distance of s = 25 units from the fixed point, then values of a and b are, respectively

• 10, - 33

• - 10, - 33

• - 8, 33

• - 10, 33

$\underset{\mathrm{n}\to \infty }{\lim }\frac{\sqrt{1} + \sqrt{2} + ... + \sqrt{\mathrm{n} - 1}}{\mathrm{n}\sqrt{\mathrm{n}}} \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\frac{1}{2}$

• $\frac{1}{3}$

• $\frac{2}{3}$

• 0

If the vertex of the conic y - 4y = 4x - 4a always lies between the straight lines x + y = 3 and 2x + 2y - 1 = 0, then

• 2 < a < 4

• $- \frac{1}{2} < \mathrm{a} < 2$

• 0 < a < 2

• $- \frac{1}{2} < \mathrm{a} < \frac{3}{2}$

Number of intersecting points of the conics 4x² + 9y² = 1 and 4x² + y² = 4 is

• 1

• 2

• 3

• 0