Calculate Karl Pearson's coefficient of of correlation between x and y for the following data and interpret the result.

(1, 6), (2, 5), (3, 7), (4, 9), (5, 8), (6, 10), (7, 11), (8, 13), (9, 12)

The marks obtained by 10 candidates in English and Mathematics are given below.

Estimate the probable score for Mathematics if the marks obtained in English are 24.

23.

A committee of 4 persons has to be chosen from 8 boys and 6 girls, consisting of at least one girl. Find the probability that the committee consists of more girls than boys.

An urn contains 10 white and 3 black balls while another urn contains 3 white and 5 black balls. Two balls are drawn from the second urn. Find the probability that the ball drawn from the second urn is a white ball.

Find the locus of a complex number, z = x + iy, satisfying the relation $\frac{\left|\mathrm{z} - 3\mathrm{i}\right|}{\left|\mathrm{z} + 3\mathrm{i}\right|} \le \sqrt{2}$. Illustrate the locus of z in the Argand plane.

Solve the following differential equation:

${\mathrm{x}}^{2}d\mathrm{y} + (\mathrm{xy} + {\mathrm{y}}^2)\mathrm{dx} = 0 \mathrm{when} \mathrm{x} = 1 \mathrm{and} \mathrm{y} = 1$.

For any three vectors, $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$, show that $\overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{b}} - \overrightarrow{\mathrm{c}}, \overrightarrow{\mathrm{c}} - \overrightarrow{\mathrm{a}}$ are coplanar.

Find a unit vector perpendicular to each of the vectors $\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{b}} \mathrm{where} \overrightarrow{\mathrm{a}} = 3\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 2\hat{\mathrm{k}} \mathrm{and} \hat{\mathrm{i}} + 2\hat{\mathrm{j}} - 2\hat{\mathrm{k}}$.

In an automobile factory, certain parts are to be fixed into the chassis in a section before it moves into another section. On a given day, one of the three persons A, B, C carries out this task. A has 45 % chance, B has 35 % and C has 20 % chance of doing the task. The probability that A, B and C will take more than the allotted time is $\frac{1}{6}, \frac{1}{10} \mathrm{and} \frac{1}{20}$ respectively. I it is found that the time taken is more than the allotted time, what is the probability that A has done the mask?

The difference between mean and varianve of a bionomial distribution is 1 and the difference of their squares is 11. Find the distribution.