Find the numbers a and b such that A² + aA + bI = 0 for the matrix A =\(\begin{bmatrix} 3 & 2 \\ 1 & 1 \end{bmatrix}\)

Find the value of tan \(\left( {{\pi}\over{8}} \right)\)

Find the locus of a point such that the sum of its distances from the points (0, 2) and (0, -2) is 6.

Show that \(\tan\left(\frac{\pi}{3}+x\right)\tan\left(\frac{\pi}{3}-x\right)=\frac{2\cos2x+1}{2\cos2x-1}\)

Find \(\frac{dy}{dx}\) if x = 15(t - sin t); y = 18(1 - cos t).

Separate the intervals in which the function x³ + 8x² + 5x - 2 is increasing or decreasing.

Find the future value of an ordinary annuity of Rs.1000 a year for 5 years for 5 years at 7% p.a compounded annually.

A card from pack 52 cards is lost. From the remaining cards of the pack, two cards are drawn and are found to be hearts. Find the probability of the missing card to be a heart? 

Two phychologist ranked 12 candidates in the selection list as below:

X	1	2	3	4	5	6	7	8	9	10	11	12
Y	12	9	6	10	3	5	4	7	8	2	11	1

Find the rank correlation co-efficient.

Solve the following LPP graphically. Maximize \(Z={ x }_{ 1 }+{ x }_{ 2 }\)
Subject to the constraints \({ x }_{ 1 }-{ x }_{ 2 }\le -1,{ -x }_{ 1 }+{ x }_{ 2 }\le 0\quad and\quad { x }_{ 1 }+{ x }_{ 2 }\ge 0\)