Q.1(a): Find a real root of x3 – x – 1 = 0 between 1 and 2 by bisection method. Q.1(b):Find by Newton Raphson’s method, the real root of the equation 3x=cos x + 1 Q.2(a): Given that : sin 45° = 0.7071, sin50° = 0.7660, sin55° = 0.8192, sin 60° = 0.8660, find the value of sin 52°. Q.2(b): Compute the value of f(x) for x = 2.5 from the following table x : 1,2,3,4 f (x) : 1, 8, 27, 64 Q.3(a): Find dy/dx of x = 0.1 from the following table: x : 0.1, 0.2, 0.3, 0.4 f (x) : 0.9975, 0.9900, 0.9776, 0.9604 Q.3(b): Evaluate ∫6,0 dx/(1+x^2) by by using Simpson’s 3/8 rule and Simpson’s 3/8 rule Q.4(a): Solve the following equations by Gauss-Seidel method:10x-2y-2z=6,−x+10y−2z=7,-x-y+10z=8 Q.4(b): Solve the following system of equations by Crout’s method. 2x+y+4z=12, 8x-3y+2z=20, 4x+11y-z=33 Q.5(a): Solve dy/ dx =1−y, y(0)=0 in the range 0≤x≤0.3 by taking h = 0.1 by modified Euler’s method. Q.5(b): Use Runge-Kutta method to find y, when x = 1.2 in steps of 0.1, given that dy/dx=x^2 + y^2 and y (1)=1.5. Q.6(a): Given that dy/dx=x^2(1+y) and y(1)=1, y(1.1)=1.233, y(1.2)=1.548, y(1.3)=1.979. Find y(1.4) by Milan’s predictor character method. Q.6(b): Solve the Elliptic equation Uxx +Uyy=0 for the following square mesh with boundary values as shown. Q.7(a): Find L ((cos at- cos bt) / t) Q.7(b): Find L^-1 (1 / (s^3(s^2+a^2)) Q.8(a): Find the mean and variance of the Poisson's distribution. Q.8(b)7: Ten percent of Screws produced in a certain factory turnout to be defective. Find the probability that in a sample of 10 screws chosen at random, exactly two will be defective.