The angle of elevation of the top the tower from the foot of the building is 45 �. If the tower is 30 m high, and the angle of elevation of top of building from the foot of the tower is 30°. Find the height of the building.

Find the area of the triangle ABC with A(1,-4) and mid-points of sides through A being (2,-1) and (0,1).

Find that non-zero value of k, for which the quadratic equation kx² + 1 – 2(k-1)x + x² = 0 has equal roots. Hence find the roots of the equation.

Two difference dice are rolled together. Find the probability of getting:

(i) the sum of numbers on two dice to be 5.

(ii) even numbers on both dice.

If S_n, denotes the sum of first n terms of an A.P., prove that S₁₂ = 3(S₈-S₄)