Finding equation of the line in two-dimensional space

Question

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  lim_x→∞    (1 + x²) / x²

4) (a) Using Simpson’s rule, evaluate the following, taking n = 4

₀∫^π(1-sin²x)/(1+x)

(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) ∫  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 – 25x² – 4x + 100 = 0

(d) Find the perimeter of the cord r = a (1 + cosθ).