1)(a) Find all the eighth roots of (19 + 7i)
(b) Differentiate tan (5x + 7) w.r.t. cos^{-1} (1-x^{2}/1+x^{2})
(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 x^{2} + y^{2} + z^{2} = 18 , 3x + 3y + 3z = 11 and passed through the origin.
3)(a) Find the area bounded by the x-axis, the curve y = (2x^{2 }+ 7x) and the ordinates x = 5 and x = 7.
(b) Find lim_{x→∞} (1 + x^{2}) / x^{2}
4) (a) Using Simpson’s rule, evaluate the following, taking n = 4
_{0}∫^{π}(1-sin^{2}x)/(1+x)
(b) Find the area bounded by the curves y^{2} = 9x and x^{2} = 9y
5) (a) Evaluate the following
(i) x^{3}-4x/(x^{2}+1)^{2} dx
(ii) ∫ dx/(5+7cosx)
(iii) ∫ x^{1/2}/1+x^{1/4}
(b) Prove the following inequalities:
(i) tan^{–1} x < x for all positive value of x.
(ii) e^{x} – e^{–x} ≥2 x for all x > 0.
(c) Use the Cauchy-Schwartz inequality to solve x3 – 25x^{2} – 4x + 100 = 0
(d) Find the perimeter of the cord r = a (1 + cosθ).