A psychologist selected a random sample of 22 students. He grouped them in 11 pairs so that the students in each pair have nearly equal scores in an intelligence test. In each pair, one student was taught by method A and the other by method B and examined after the course. The marks obtained by them after the course are as follows:

Calculate Sperman's Rank correlation.

In a college, 70% students pass in physics, 75% pass in Mathematics and 105 students fail in both. One student is chosen at random. What is the probability that:

(i) He passes in Physics and Mathematics.

(ii) He passes in Mathematics given that he passes in Physics.

(iii) He passes in Physics given  that he passes in Mathematics.

23.

A bag contains 5 white and 4 black balls and another bag contains 7 white and 9 black balls. A ball is drawn from the first bag and two balls are drawn from the second bag. What is the probability of drawing one white and two black balls ?

Using De Moivre's theorem, find the least positive integer n such that ${\left(\frac{2\mathrm{i}}{1 + \mathrm{i}}\right)}^{\mathrm{n}}$ is a positive integer.

Prove that: $\overrightarrow{\mathrm{a}}.(\overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}) \times \left(\overrightarrow{\mathrm{a}} + 2\overrightarrow{\mathrm{b}} + 3\overrightarrow{\mathrm{c}}\right) = \left[\overrightarrow{\mathrm{a}}\overrightarrow{\mathrm{b}}\overrightarrow{\mathrm{c}}\right]$

Find the equation of a line passing through the points P(- 1, 3, 2) and Q(- 4, 2, - 2). Also, if the points R(5, 5, $\mathrm{\lambda }$) is collinear with the points P and Q, then find the value of $\mathrm{\lambda }$.

Find the equation of the plane passing through the points (2, - 3, 1) and (- 1, 1, - 7) and perpendicular to the plane x - 2y + 5z + 1 = 0

In a bolt factory, three machines A, B and C manufacturere 25 %, 35 %, and 40 % of the total production respectively. Of their respective outputs, 5 %, 4 % and 2 % are defective. A bolt is drawn at random from the total production and it is found to be defective. Find the probability that it was manufactured by machine C.

On dialing certain telephone numbers, assume that on an average, one telephone number out of five is busy. ten telephone numbers are randomly selected and dialed. Find the probability that at least three of them will be busy.

The coefficient of correlation between between the values denoted by X and Y is 0.5. The mean of of X is 3 and that of Y is 5. Their standard deviations are 5 and 4 respectively. Find:

(i) the two lines of regression

(ii) the expected value of Y, when X is given 14.

(iii) the expected value of X, when Y is given 9.