B.Sc. Mathematics Honours C-7 Question Paper 2020 (CBCS)

Theory

Answer any two from the following questions: $2 \times 20$
1. (a) Explain Newton-Raphson method to solve the equation $g(x)=0$.
(b) Find the rate of convergence of Newton-Raphson method.
(c) Find a real root of the equation $f(x) \equiv x^{3} - 2x - 5 = 0$ lies between 2 and 3 by Regula-Falsi method.
2. (a) Discuss Gauss-elimination method to solve the system of linear equation.
(b) Solve the following equation by Gauss-elimination method.
$2x_{1} + x_{2} + x_{3} = 4$
$x_{1} - x_{2} + 2x_{3} = 2$
$2x_{1} + 2x_{2} - x_{3} = 3$
(c) State the differences between direct and iterative methods.
3. (a) Find an LU-decomposition of the matrix and use it to solve the system $Ax = \begin{pmatrix} -3 \\ -12 \\ 6 \end{pmatrix}$ where $x = \begin{pmatrix} x_{1} \\ x_{2} \\ x_{3} \end{pmatrix}$ and $A = \begin{pmatrix} 2 & 7 & 5 \\ 6 & 20 & 10 \\ 4 & 3 & 0 \end{pmatrix}$.
(b) Deduce Lagrange interpolation method.
4. (a) Describe Euler's method and modified Euler's method to solve the following differential equation $\frac{dy}{dx} = f(x,y)$, given $y(x_{0}) = y_{0}$.
(b) Given $\frac{dy}{dx} = x^{2} + y^{2}$ and when $x=0, y=1$. Find the values of $y(0.1)$ by fourth order Runge-Kutta method.

Practical

Answer any one from the following questions: $1 \times 10$
1. Write a program to evaluate $\int_{1.2}^{3} (x \log 2x + \sin 2x) dx$ by trapezoidal rule taking 140 subintervals.
2. Write a program to find the value of $y(0.1)$ from the differential equation $\frac{dy}{dx} = x^{2} + y, y(0) = 1$.
3. Write a program to find the sum of the following series: $1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{10000}$.
4. Write a program to find a real root of the equation $x^{3} - 2x - 1 = 0$ by bisection method.
5. Write a program to solve the equation $2x - \sin x - 1 = 0$ using fixed point iteration method.
6. Write a program to find a real root of $x^{5} + 3x^{2} - 1 = 0$ by Newton-Raphson method.
7. Write a program to compute $\int_{0}^{\pi/2} \sin x dx$ by using Simpson's $\frac{1}{3}$ rule with 200 sub intervals.
8. Evaluate the integral $\int_{0.4}^{1.6} \frac{x}{\sin x} dx$ by weddle's rule by taking 120 sub-intervals.
9. Given $y' = 3x + y^{2}, y(1) = 1.2, h = 0.1$. Find $y(1.8)$ R-K method of four order.
10. Write a program to find a root of the equation $x \sin x - 1 = 0$ by secant method.
11. Using iterative formula to compute $\sqrt[7]{125}$ correct to five significant digits.
12. Find a real root of the equation $\log x = \cos x$ using Regula-falsi method correct to three significant figures.
13. Fit a linear curve to the data:
x: 4, 6, 8, 10, 12
y: 13.72, 12.90, 12.01, 11.14, 10.31
14. If the prescribed curve be $f(x) = \alpha + \beta x + \gamma x^{2}$, estimate $\alpha, \beta$ and $\gamma$ by least square method from the following data:
x: 2, 4, 6
y: 3.97, 12.85, 91.29
15. Write a program to compute $\int_{1}^{2} \sqrt{\frac{x^{2}-1}{x}} dx$ by using Simpson's $\frac{1}{3}$ rule using 1000 sub-intervals.
16. Evaluate the integral $\int_{0}^{0.5} e^{x} dx$ by five-point Gaussian quadrature.
17. Solve the following system of linear equations by LU decomposition method:
$x+y+z=1$
$4x+3y-z=6$
$3x+5y+3z=4$
18. Apply Newton's backward difference formula to obtain the value of y at $x=1.2$ using the following table:
x: 0, 1, 2, 3, 4
f(x): 1, 1.5, 2.2, 3.1, 4.3
19. Use Lagrange's interpolation formula to find f(x) when $x=0$ from the following table:
x: -1, -2, 2, 4
f(x): -1, -9, 11, 69
20. Solve the following system of equations by Gaussian elimination method:
$3x+2y+z=10$
$2x+3y+2z=14$
$x+2y+3z=14$
21. Solve the following by Euler's modified method: $\frac{dy}{dx} = \log(x+y), y(0)=2$, at $x=1.4$ with $h=0.2$.
22. Solve the following system by Gauss Seidel method:
$20x+5y-2z=14$
$3x+10y+z=17$
$x-4y+10z=23$
23. Solve the following systems of equation by Gauss-Jacobi's iteration method.
24. Find by power method, the numerically largest eigen value and the corresponding eigen vector of the following matrix: $\begin{pmatrix} 1 & 3 & 2 \\ -1 & 0 & 2 \\ 3 & 4 & 5 \end{pmatrix}$.
25. Find the value of $e^{x}$ when $x=0.612$ using Newton's forward difference method:
x: 0.61, 0.62, 0.63, 0.64, 0.65
f(x): 1.840431, 1.858928, 1.877610, 1.896481, 1.915541
26. The distance (d) that a car has travelled at time (t) is given below:
Time (t): 0, 2, 4, 6, 8
Distance (d): 0, 40, 160, 300, 380
27. Evaluate $y(0.02)$ given $y' = x^{2} + y, y(0)=1$ by modified Euler's method.
28. Write a program to find the value of $y(0.1)$ from the differential equation $\frac{dy}{dx} = x + y + 100, y(0) = 1.2$ by fourth order Runge Kutta method.
29. If $f(0)=1, f(0.1)=0.9975, f(0.2)=0.9900, f(0.3)=0.9800$ and hence find $f(0.05)$ using Newton's forward formula.
30. Given $\log_{10} 654 = 2.8156, \log_{10} 658 = 2.8182, \log_{10} 659 = 2.8189, \log_{10} 661 = 2.8202$, find $\log_{10} 656$ using Newton's forward formula.