11.

If $\mathrm{\theta } \in \left(\frac{\mathrm{\pi }}{2}, \frac{3\mathrm{\pi }}{2}\right)$, then the value of $\sqrt{4{\cos }^4\left(\mathrm{\theta }\right) + {\sin }^2\left(2\mathrm{\theta }\right)} + 4\cot \left(\mathrm{\theta }\right){\cos }^2\left(\frac{\mathrm{\pi }}{4} - \frac{\mathrm{\theta }}{2}\right)$ is

• $- 2\cot \left(\mathrm{\theta }\right)$

• $2\cot \left(\mathrm{\theta }\right)$

• $2\cos \left(\mathrm{\theta }\right)$

• $2\sin \left(\mathrm{\theta }\right)$

The number ofreal solutions of the equation $\left(\sin \left(\mathrm{x}\right) - \mathrm{x}\right)\left(\cos \left(\mathrm{x}\right) - {\mathrm{x}}^2\right) = 0$ is

• 1

• 2

• 3

• 4

The value of ${\left(\frac{1 + \sqrt{3}\mathrm{i}}{1 - \sqrt{3}\mathrm{i}}\right)}^{64} + {\left(\frac{1 - \sqrt{3}\mathrm{i}}{1 + \sqrt{3}\mathrm{i}}\right)}^{64}$

• 0

• - 1

• 1

• i

Find the maximum value of $\left|z\right|$ when $\left|\mathrm{z} - \frac{3}{\mathrm{z}}\right| = 2$, where z being a complex number.

• 1 + $\sqrt{3}$

• 3

• $1 + \sqrt{2}$

• 1

Given that, x is a real number satisfying $\frac{5{\mathrm{x}}^2 - 26\mathrm{x} + 5}{3{\mathrm{x}}^2 - 10\mathrm{x} + 3} < 0$, then

• $\mathrm{x} < \frac{1}{5}$

• $\frac{1}{5} < \mathrm{x} < 3$

• x > 5

• $\frac{1}{5} < \mathrm{x} < \frac{1}{3} \mathrm{or} 3 < \mathrm{x} < 5$

The value of $\mathrm{\lambda }$ such that the system of equations 2x - y - 2z = 2; x - 2y + z = - 4; x + y + $\mathrm{\lambda }$z = 4, has no solution, is

• 3

• 1

• 0

• - 3

Let ${\mathrm{X}}_{\mathrm{n}} = {\left(1 - \frac{1}{3}\right)}^2{\left(1 - \frac{1}{6}\right)}^2{\left(1 - \frac{1}{10}\right)}^2 ... \left(1 - \frac{1}{{\displaystyle \frac{\mathrm{n}\left(\mathrm{n} + 1\right)}{2}}}\right), \mathrm{n} \ge 2$

Then, the value of $\underset{\mathrm{n}\to \infty }{\lim }{\mathrm{x}}^{\mathrm{n}}$ is

• 1/3

• 1/9

• 1/81

• 0

The variance of first 20 natural numbers is

• 133/4

• 279/12

• 133/2

• 399/4

If the letters of the word 'PROBABILITY' are written down at random in a row, then probability that two B's are together, is

• $\frac{2}{11}$

• $\frac{10}{11}$

• $\frac{3}{11}$

• $\frac{6}{11}$

A fair coin is tossed-at a fixed number of times. If the probability of getting exactly 3 heads equals the probability of getting exactly 5 heads, then the probability of getting exactly one head is

• 1/64

• 1/32

• 1/16

• 1/8