The point (3, 2) undergoes the following three transformations in the order given

(i) Reflection about the line y = x.

(ii) Translation by the distance 1 unit in the positive direction of x-axis.

(iii) Rotation by an angle $\frac{\mathrm{\pi }}{4}$ about the origin in  anti-clockwise direction.

Then, the final position of the point is

• $\left(- \sqrt{18}, \sqrt{18}\right)$

• (- 2, 3)

• $\left(0, \sqrt{18}\right)$

• (0, 3)

If X is a poisson variate such that a = P(X = 1) = P(X = 2), then P(X = 4) is equal to

• $2\mathrm{\alpha }$

• $\frac{\mathrm{\alpha }}{3}$

• ${\mathrm{\alpha e}}^{ - 2}$

• ${\mathrm{\alpha e}}^2$

Suppose X follows a binomial distribution with parameters n and p, where 0 < p < 1. If $\frac{\mathrm{P}\left(\mathrm{X} = \mathrm{r}\right)}{\mathrm{P}\left(\mathrm{X} = \mathrm{n} - \mathrm{r}\right)}$ is independent of n for every r, then p is equal to

• $\frac{1}{2}$

• $\frac{1}{3}$

• $\frac{1}{4}$

• $\frac{1}{8}$

In an entrance test there are multiple choice questions. There are four possible answers to each question, of which one is correct. The probability that a student know the answer to a question is 9/10. If he gets the correct answer to a question, then the probability that he was guessing is 

• $\frac{37}{40}$

• $\frac{1}{37}$

• $\frac{36}{37}$

• $\frac{1}{9}$

There are four machines and it is known that exactly two of them are faulty. They are teste done by one, in a random order till both the faulty machines are identified. Then, the probability that only two tests are need is

• $\frac{1}{3}$

• $\frac{1}{6}$

• $\frac{1}{2}$

• $\frac{1}{4}$

A fair coin is tossed 100 times. The probability of getting tails an odd number of times is

• $\frac{1}{2}$

• $\frac{1}{4}$

• $\frac{1}{8}$

• $\frac{3}{8}$

47.

If pth, qth, rth terms of a geometric progression are the positive numbers a, b and c respectively,then the angle· between the vectors (log(a))²i + (log(b))²j + (log(c))²k and (q - r)i + (r - p)j + (p - q)k is

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

• ${\sin }^{-1}\left(\frac{1}{\sqrt{{\mathrm{a}}^2 + {\mathrm{b}}^2 + {\mathrm{c}}^2}}\right)$

• $\frac{\mathrm{\pi }}{4}$

$$\begin{gathered} \mathrm{If} \mathrm{\alpha }, \mathrm{\beta }, \mathrm{\gamma } \mathrm{are} \mathrm{length} \mathrm{of} \mathrm{the} \mathrm{altitudes} \mathrm{of} \mathrm{a} ∆\mathrm{ABC} \mathrm{with} \mathrm{area} ∆, \mathrm{then} \\ \frac{{∆}^2}{{\mathrm{R}}^2}\left(\frac{1}{{\mathrm{\alpha }}^2} + \frac{1}{{\mathrm{\beta }}^2} + \frac{1}{{\mathrm{\gamma }}^2}\right) = ? \end{gathered}$$

• ${\sin }^2\left(\mathrm{A}\right) + {\sin }^2\left(\mathrm{B}\right) + {\sin }^2\left(\mathrm{C}\right)$

• ${\cos }^2\left(\mathrm{A}\right) + {\cos }^2\left(\mathrm{B}\right) + {\cos }^2\left(\mathrm{C}\right)$

• ${\tan }^2\left(\mathrm{A}\right) + {\tan }^2\left(\mathrm{B}\right) + {\tan }^2\left(\mathrm{C}\right)$

• ${\cot }^2\left(\mathrm{A}\right) + {\cot }^2\left(\mathrm{B}\right) + {\cot }^2\left(\mathrm{C}\right)$

In an acute angled triangle, cot(B)cot(C) + cot(A)cot(C) + cot(A)cot(B) = ?

• - 1

• 0

• 1

• 2

$\mathrm{x} = \log \left(\frac{1}{\mathrm{y}} + \sqrt{1 + \frac{1}{{\mathrm{y}}^2}}\right) \Rightarrow \mathrm{y} = ?$

• $\tanh \left(\mathrm{x}\right)$

• $\coth \left(\mathrm{x}\right)$

• $\mathrm{sech}\left(\mathrm{x}\right)$

• $\mathrm{csch}\left(\mathrm{x}\right)$