$\mathrm{In} ∆\mathrm{ABC}, \mathrm{if} \mathrm{bcos}\left(\mathrm{\theta }\right) = \mathrm{c} - \mathrm{a}, (\mathrm{where} \mathrm{\theta } \mathrm{is} \mathrm{an} \mathrm{acute} \mathrm{angle}), \mathrm{then} (\mathrm{c} - \mathrm{a}) \tan \left(\mathrm{\theta }\right) = ?$

• $2\sqrt{\mathrm{ca}}\cos \left(\frac{\mathrm{B}}{2}\right)$

• $2\sqrt{\mathrm{ac}}\sin \left(\frac{\mathrm{B}}{2}\right)$

• $2\mathrm{cacos}\left(\frac{\mathrm{B}}{2}\right)$

• $\text{2casin}\left(\frac{\mathrm{B}}{2}\right)$

$\mathrm{Given} \mathrm{below} \mathrm{is} \mathrm{the} \mathrm{distribution} \mathrm{of} \mathrm{a} \mathrm{random} \mathrm{variable} \mathrm{X}$

X = x	1	2	3	4
P(X = x)	$\mathrm{\lambda }$	$2\mathrm{\lambda }$	3$\mathrm{\lambda }$	4$\mathrm{\lambda }$

If $\mathrm{\alpha } = \mathrm{P}\left(\mathrm{X} < 3\right) \mathrm{and} \mathrm{\beta } = \mathrm{P}\left(\mathrm{X} > 2\right), \mathrm{then} \mathrm{\alpha } : \mathrm{\beta } = ?$

• 2 5

• 3 4

• 4 5

• 3 7

3.

If the pair of lines x - 16pxy - y² = 0 and x² - 16qxy - y = 0 are such that each pair bisects the angle between the other pair, then pq = 

• $\frac{- 1}{64}$

• $\frac{1}{64}$

• $\frac{ - 1}{8}$

• $\frac{1}{8}$

$\underset{\mathrm{n}\to \infty }{\lim }\left[\frac{1^{\mathrm{k}} + 2^{\mathrm{k}} + 3^{\mathrm{k}} + ... + {\mathrm{n}}^{\mathrm{k}}}{{\mathrm{n}}^{\mathrm{k} + 1}}\right] = ?$

• $\frac{1}{\mathrm{k}}$

• $\frac{2}{\mathrm{k} \hspace{0.17em}+ 1}$

• $\frac{1}{\mathrm{k} + 1}$

• $\frac{2}{\mathrm{k}}$

$$\begin{gathered} \mathrm{The} \mathrm{coefficient} \mathrm{of} {\mathrm{x}}^5 \mathrm{in} \mathrm{the} \mathrm{expansion} \mathrm{of} \\ (1 + \mathrm{x}{)}^{21} + (1 +\mathrm{x}{)}^{22} + ... + (1 + \mathrm{x}{)}^{30} \mathrm{is} \end{gathered}$$

• ${}_{31}\mathrm{C}_6 - {}_{21}\mathrm{C}_6$

• ${}_{51}\mathrm{C}_5$

• ${}_9\mathrm{C}_5$

• ${}_{30}\mathrm{C}_5 + {}_{20}\mathrm{C}_5$

Let A = {- 4, - 2, - 1, 0, 3, 5} and f : A $\to$ IR be defined by

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) = \left\{3\mathrm{x} - 1 \mathrm{for} \mathrm{x} > 3 \\ {\mathrm{x}}^2 + 1 \mathrm{for} - 3 \le \mathrm{x} \le 3 \\ 2\mathrm{x} - 3 \mathrm{for} \mathrm{x} < - 3\right\} \\ \mathrm{Then} \mathrm{the} \mathrm{range} \mathrm{of} \mathrm{f} \mathrm{is} \end{gathered}$$

• $\left\{- 11, 5, 2, 1, 10, 14\right\}$

• $\left\{- 11, - 7, 2, 1, 8, 14\right\}$

• $\left\{- 11, 5, 2, 1, 8, 14\right\}$

• $\left\{- 11, - 7, - 5, 1, 10, 14\right\}$

If a circle with radius 2.5 units passes through the points (2, 3) and (5, 7), then its centre is

• (1 5, 2)

• (7, 10)

• (3, 4)

• (3 5, 5)

The circumcentre of the triangle formed by the points (1, 2, 3) (3, - 1, 5), (4, 0, - 3) is

• (1, 1, 1)

• (2, 2, 2)

• (3, 3, 3)

• $\left(\frac{7}{2}, \frac{ - 1}{2}, 1\right)$

$$\begin{gathered} \mathrm{If} \mathrm{f} : \mathrm{IR} \to \mathrm{IR} \mathrm{is} \mathrm{defined} \mathrm{by} \\ \mathrm{f}\left(\mathrm{x}\right) = \left\{\mathrm{x} - 1, \mathrm{for} \mathrm{x} \le 1 \\ 2 - {\mathrm{x}}^2, \mathrm{for} 1 < \mathrm{x} \le 3 \\ \mathrm{x} - 10, \mathrm{for} 3 < \mathrm{x} < 5 \\ 2\mathrm{x}, \mathrm{for} \mathrm{x} \ge 5\right\} \\ \mathrm{then} \mathrm{the} \mathrm{set} \mathrm{of} \mathrm{points} \mathrm{of} \mathrm{discontinuity} \mathrm{of} \mathrm{f} \mathrm{is} \end{gathered}$$

• $\mathrm{IR} - \left\{1, 5\right\}$

• $\left\{1, 3, 5\right\}$

• $\left\{1, 5\right\}$

• $\mathrm{IR} - \left\{1, 3, 5\right\}$

A bag P contains 5 white marbles and 3 black marbles. Four marbles are drawn at random from P and are put in an empty bag Q. If a marble drawn at random from Q is found to be black then the probability that all the three black marbles in P are transfered to the bag Q

• $\frac{1}{7}$

• $\frac{6}{7}$

• $\frac{1}{8}$

• $\frac{7}{8}$