Given two matrices A and B

$\left|\begin{array}{ccc}1& - 2& 3\\ 1& 4& 1\\ 1& - 3& 2\end{array}\right|$ and B = $\left|\begin{array}{ccc}11& - 5& - 14\\ - 1& - 1& 2\\ - 7& 1& 6\end{array}\right|$,

Find Ab and use this result to solve the following system of equation:

x - 2y + 3z = 6, x + 4y + z = 12, x - 3y + 2z = 1

Using properties of determinants, prove that:

$\left|\begin{array}{ccc}1 + {\mathrm{a}}^2 +\hspace{0.17em}{\mathrm{b}}^2& 2\mathrm{ab}& - 2\mathrm{b}\\ 2\mathrm{ab}& 1 - {\mathrm{a}}^2 +\hspace{0.17em}{\mathrm{b}}^2& 2\mathrm{a}\\ 2\mathrm{b}& - 2\mathrm{a}& 1 - {\mathrm{a}}^2 -\hspace{0.17em}{\mathrm{b}}^2\end{array}\right| = {\left(1 + {\mathrm{a}}^2 +\hspace{0.17em}{\mathrm{b}}^2\right)}^3$

Solve the equation for x: ${\sin }^{-1}\left(\frac{5}{\mathrm{x}}\right) + {\sin }^{-1}\left(\frac{12}{\mathrm{x}}\right) = \frac{\mathrm{\pi }}{2}, \mathrm{x} \ne 0$

If A, B and C represent switches in 'on' position and A', B' and C' represent them in 'off' position. Construct a switching circuit representing the polynomial ABC + AB'C + A'B'C. Using Boolean algebra, prove that the given polynomial can be simplified to C(A + B'). Construct an equivalent switching circuit.

If y = ${\mathrm{e}}^{{\mathrm{mcos}}^{-1}\left(\mathrm{x}\right)}$, prove that:
$\left(1 - {\mathrm{x}}^2\right)\frac{d^2\mathrm{y}}{d{\mathrm{x}}^2} - \mathrm{x}\frac{d\mathrm{y}}{d\mathrm{x}} = {\mathrm{m}}^2\mathrm{y}$

Find the smaller area enclosed by the circle x² + y² and the line x + y = 2.

Given that the observations are:

(9, - 4), (10, - 3), (11, - 1), (12, 0), (13, 1), (14, 3), (15, 5), (16, 8).

Find the two lines of regression and estimate the value of y when x = 13.5.

In a contest the competitions are awarded marks out of 20 by two judges. the scores of the 10 competitors are given below. Calculate Spearman's rank correlation.

Competitors	A	B	C	D	E	F	G	H	I	J
Judge A	2	11	11	18	6	5	8	16	13	15
Judge B	6	11	16	9	14	20	4	3	13	17

19.

An urn contains 2 white and 2 black balls. A ball is drawn at random. If it is white, it is not replaced into the urn. Otherwise it is replaced with another ball of the same colour. The process is repeated. Find the probability that the third ball drawn is black.

Three persons A, B and C shoot to hit a target. If A hits the target four times in five trials, B hits it three times in four trials and C hits it two times in three trials, find the probability that:

(i) Exactly two perons

(ii) At least two persons hit the target

(iii) None hit the target