1)(a) Find all the eighth roots of (19 + 7i)
(b) Differentiate tan (5x + 7) w.r.t. cos-1 (1-x2/1+x2)
(c) Find complex conjugate of ( 7 + 6i) /(2 + 3i)
2)(a) Find the equation of the line in two-dimensional space that passes through the point (2, 3) and is parallel to the line 2x + 3y = 5.
(b) Find the equation of the sphere, which contains the circle x2 + y2 + z2 = 18 , 3x + 3y + 3z = 11 and passed through the origin.
3)(a) Find the area bounded by the x-axis, the curve y = (2x2 + 7x) and the ordinates x = 5 and x = 7.
(b) Find limx→∞ (1 + x2) / x2
4) (a) Using Simpson’s rule, evaluate the following, taking n = 4
(b) Find the area bounded by the curves y2 = 9x and x2 = 9y
5) (a) Evaluate the following
(i) x3-4x/(x2+1)2 dx
(ii) ∫ dx/(5+7cosx)
(iii) ∫ x1/2/1+x1/4
(b) Prove the following inequalities:
(i) tan–1 x < x for all positive value of x.
(ii) ex – e–x ≥2 x for all x > 0.
(c) Use the Cauchy-Schwartz inequality to solve x3 – 25x2 – 4x + 100 = 0
(d) Find the perimeter of the cord r = a (1 + cosθ).