Prove that the equation cos(2x) + asin(x) = 2a - 7 possesses a solution if 2 ≤ a ≤ 6.
Find the values of x. - π < x < π, x ≠ 0 satisfying the equation,
g1 + cosx + cos2x + ... ∞ = 43
Prove that the centre of the smallest circle passing through origin and whose centre lies on y = x + 1 is - 12, 12
Prove by induction that for n ∈ N, n2 + n is an even integer (n ≥ 1)
If N = n!(n n ∈ N, n > 2, then find
limN→∞log2N- 1 + log3N- 1 + ... + lognN- 1
Use the formula limx→0ax - 1x = logea to compute limx→02x - 11 + x - 1
limx→02x - 11 + x - 1= limx→02x - 11 + x - 1 × 1 + x + 11 + x + 1= limx→02x - 1x × limx→01 + x + 1= loge2 × 2 = loge4
If A, B are two square matrices such that AB = A and BA = B, then prove that B2 = B
If dydx + 1 - y21 - x2 = 0 prove that, x1 - y2 + y1 - x2 = A where A is constant.
Evaluate the following integral
∫- 12xsinπxdx
If f(a) = 2, f'(a) = 1, g(a) = - 1 and g'(a) = 2, find the value of limx→agafa - gafxx - a