$\mathrm{If} \cos \left(\mathrm{x}\right) = \tan \left(\mathrm{y}\right), \cot \left(\mathrm{y}\right) = \tan \left(\mathrm{z}\right) \mathrm{and} \cot \left(\mathrm{z}\right) = \tan \left(\mathrm{x}\right) \mathrm{then} \sin \left(\mathrm{x}\right) = ?$

• $\frac{\sqrt{5} + 1}{4}$

• $\frac{\sqrt{5} - 1}{4}$

• $\frac{\sqrt{5} + 1}{2}$

• $\frac{\sqrt{5} - 1}{2}$

$\tan \left(81°\right) - \tan \left(63°\right) - \tan \left(27°\right) + \tan \left(9°\right) = ?$

• 6

• 0

• 2

• 4

If x and y are acute angles such that cos(x) + cos(y) = $\frac{3}{2}$sin(x) + sin(y) = $\frac{3}{4}$, then sin(x + y) equals to

• $\frac{2}{5}$

• $\frac{3}{4}$

• $\frac{3}{5}$

• $\frac{4}{5}$

$\mathrm{The} \mathrm{sum} \mathrm{of} \mathrm{the} \mathrm{solutions} \mathrm{in} \left(0, 2\mathrm{\pi }\right) \mathrm{the} \mathrm{equation} \cos \left(\mathrm{x}\right)\cos \left(\frac{\mathrm{\pi }}{3} - \mathrm{x}\right)\cos \left(\frac{\mathrm{\pi }}{3} + \mathrm{x}\right) = \frac{1}{4} \mathrm{is}$

• $4\mathrm{\pi }$

• $\mathrm{\pi }$

• $2\mathrm{\pi }$

• $3\mathrm{\pi }$

$\mathrm{In} \mathrm{any} ∆\mathrm{ABC}, \frac{\left(\mathrm{a} + \mathrm{b} +\hspace{0.17em}\mathrm{c}\right)\left(\mathrm{b} + \mathrm{c} - \mathrm{a}\right)\left(\mathrm{c} +\hspace{0.17em}\mathrm{a} - \mathrm{b}\right)\left(\mathrm{a} +\hspace{0.17em}\mathrm{b} - \mathrm{c}\right)}{4{\mathrm{b}}^2{\mathrm{c}}^2} = ?$

• ${\sin }^2\left(\mathrm{B}\right)$

• ${\cos }^2\left(\mathrm{A}\right)$

• ${\cos }^2\left(\mathrm{B}\right)$

• ${\sin }^2\left(\mathrm{A}\right)$

If the point P(1, 3) undergoes the following transformations successively.

(i) Reflection with respect to the line y = x.

(ii) Translation through 3 units along the positive direction of the X-axis.

(iii) Rotation through an angle of $\frac{\mathrm{\pi }}{6}$ about the origin in the clockwise direction.Then, the final position of the point P is

• $\left(\frac{6\sqrt{3} + 1}{2}, \frac{\sqrt{3}{\displaystyle }{\displaystyle -}{\displaystyle }{\displaystyle 6}}{2}\right)$

• $\left(\frac{\sqrt{7}}{2}, \frac{- \sqrt{5}}{2}\right)$

• $\left(\frac{6 + \sqrt{3}}{2}, \frac{1{\displaystyle }{\displaystyle -} {\displaystyle }{\displaystyle 6\sqrt{3}}}{2}\right)$

• $\left(\frac{6 + \sqrt{3} - 1}{2}, \frac{6 + \sqrt{3}}{2}\right)$

47.

If the mean and variance of a binomial variate X are 8 and 4 respectively, then P(X < 3) equals to 

• $\frac{265}{2^{15}}$

• $\frac{137}{2^{14}}$

• $\frac{137}{2^{16}}$

• $\frac{265}{2^{16}}$

A random variable X has the probability distribution given below.

Its variance is

• $\frac{16}{3}$

• $\frac{4}{3}$

• $\frac{5}{3}$

• $\frac{10}{3}$

A candidate takes three tests in succession and the probability of passing the first test is p. The probability of passing each succeeding test is p or $\frac{\mathrm{p}}{2}$ according as he passes or fails in the preceding one. The candidate is selected, if he passes atleast two tests. The probability that the candidate is selected, is

• p²(2 - p)

• p(2 - p)

• p + p² + p³

• p²(1 - p)

A six-faced unbiased die is thrown twice and the sum of the numbers appearing on the upper face is observed to be 7. The probability that the number 3 has appeared atleast once, is

• $\frac{1}{5}$

• $\frac{1}{2}$

• $\frac{1}{3}$

• $\frac{1}{4}$