B.Sc. Mathematics Honours C-7 Question Paper 2021 (CBCS)

Theory

Group - A

1. Answer any three of the following questions: $12 \times 3 = 36$
(a) (i) Factorize the matrix $\begin{pmatrix} 2 & -2 & 1 \\ 5 & 1 & -3 \\ 3 & 4 & 1 \end{pmatrix}$ into the form LU, where L and U are lower and upper triangular matrices and hence solve the system of equations
$2x - 2y + z = 2$
$5x + y - 3z = 0$
$3x + 4y + z = 9$
(ii) Fit a parabola to the following data by taking 'x' as independent variable.
x: 1, 2, 3, 4, 5, 6, 7, 8, 9
y: 2, 6, 7, 8, 10, 11, 11, 10, 9
(b) (i) Given $\frac{dy}{dx} = y^{2} - x^{2}$, where $y(0) = 2$. Find $y(0.1)$ and $y(0.2)$ as a solution of this equation by fourth order Runge-Kutta method.
(ii) Solve the following system of equations by Gauss-Seidal method correct up to four significant figures:
$2x + y + 5z = 5$
$3x + y + z = 3$
$x + 4y + z = 2$
(c) (i) Consider the equation $5x^{3} - 20x + 3 = 0$. Find the root, using iteration method, lying on the interval [0, 1] correct up to 5 decimal places.
(ii) Find the value of $\int_{0}^{1} \frac{dx}{1+x^{2}}$ taking 5 sub-intervals, by trapezoidal rule, correct to five significant figures. Also find the error by comparing with the exact value.
(d) (i) Find the method of iteration for numerical integration.
(ii) If $x = \alpha$ be a root of the equation $f(x) = 0$ which is rewritten as $x = \phi(x)$. If $\phi(x)$ is continuous and $|\phi'(x)| \le l$ where $0 < l < 1$, in an interval $I$ containing $\alpha$, then prove that the sequence $(x_{n})$ of iterations determined from $x_{n+1} = \phi(x_{n})$, $(n = 0, 1, 2, \dots)$ converges to the root $\alpha$.
(iii) Let $y = 5x^{7} - 4x$. Find the percentage error in $y$ at $x = 1$, if the error in $x$ is $\Delta x = 0.04$.
(e) Find the basic principle for Newton-Raphson method with its geometrical meaning. Find advantages and disadvantages of Newton-Raphson method. How can you use this method for an assigned root of a positive real number.
(f) Establish the Gauss Legendre Quadrature formula for numerical integration $\int_{a}^{b} f(x) dx$ and then establish composite Simpson's $\frac{1}{3}$rd rule from it. Evaluate $\int_{0}^{1} x^{3} dx$, by Simpson's $\frac{1}{3}$rd rule with $n = 5$.

Group - B

2. Answer any two of the following questions: $2 \times 2 = 4$
(a) Find $f(x)$, when its first difference is $x^{3} + 4x^{2} + 2x + 7$.
(b) Define Round off error and Truncation error.
(c) Show that the maximum error in linear interpolation is given by $\frac{h^{2}M_{2}}{8}$ where $M_{2} = \max_{0 \le x \le 1} |f''(x)|$.
(d) Compare between Newton-Cote's quadrature and Gaussian quadrature.

Practical

Group - A

1. Answer any one of the following questions: $15 \times 1 = 15$
(a) Write a program to find a root of the equation $x^{3} - 3x + 1 = 0$ by Newton-Raphson method.
(b) Write a program to solve an ordinary differential equation by modified Euler's method, $\frac{dy}{dx} = x^{2} + y^{2}$ with $y(0) = 1$ at $y(0.2)$ and $y(0.4)$.
(c) Write a program on Lagrange's interpolation polynomial to find the value of a certain point from the given set of data. Find the value of 1.75 from the set of data:
x: 1, 1.5, 2, 3.2, 4.5
y: 5, 8.2, 9.2, 11, 16

Group - B

2. Answer any one of the following questions: $5 \times 1 = 5$
(a) Write a program to find the sum of the following series: $1/1 + 1/2 + 1/3 + 1/4 + \dots + 1/N$.
(b) Write a program to enter 100 integers into an array and sort them in an ascending order.
(c) Write a program to find the value of the integration by Trapezoidal rule, $\int_{0}^{5} e^{-x} dx$ by taking 6 intervals.