B.Sc. Mathematics Honours C-7 Question Paper 2019 (CBCS)

Unit - I

1. Answer any two questions: $2 \times 2 = 4$
(a) Given $f(x,y,z) = \frac{5xy^{2}}{z^{2}}$, find the relative maximum error in the evaluation of $f(x,y,z)$ at $x = y = z = 1$, if $x, y, z$ have absolute error $\Delta x = \Delta y = \Delta z = 0.1$.
(b) Define the terms:
(i) Computational error, and
(ii) Relative percentage error.
(c) Three approximate values of the number $\frac{1}{3}$ are given as 0.30, 0.33 and 0.34. Which of these three is the best approximation?

Unit - II

2. Answer any one question: $2 \times 1 = 2$
(a) Discuss the condition of convergence of Newton-Raphson method.
(b) Prove that the order of convergence of iteration method is linear.

3. Answer any one question: $5 \times 1 = 5$
(a) Explain the method of Iteration for computing a real root of an equation $f(x) = 0$. Let the iteration function $\phi(x)$ maps the interval [a, b] into itself and is differentiable there. Further there exists a non negative constant $k < 1$ such that $\forall x$ in [a, b], $|\phi'(x)| \le k$, then prove that $\phi(x)$ has exactly one fixed point $\alpha$ on [a, b] and the sequence $\{x_{n}\}$ converges to $\alpha$.
(b) (i) Show that the square root of $N = AB$ is given by $\sqrt{N} \simeq \frac{S}{4} + \frac{N}{S}$ where $S = A + B$.
(ii) Derive the expression for Secant method to find the root of an equation.

Unit - III

4. Answer any one question: $2 \times 1 = 2$
(a) What is called pivoting? Why pivoting is necessary to solve a system of equations using Gaussian elimination method?
(b) State the difference between direct and iterative methods.

5. Answer any one question: $5 \times 1 = 5$
(a) Describe Gauss Jacobi method for solution of a system of linear equation. State the sufficient condition for convergence of this method.
(b) Solve the following equations by Gauss-Jordan elimination method:
$x_{1} + x_{2} + x_{3} = 3$
$2x_{1} + 3x_{2} + x_{3} = 6$
$x_{1} - x_{2} - x_{3} = -3$

Unit - IV

6. Answer any one question: $10 \times 1 = 10$
(a) (i) Given that $f(0) = 2, f(1) = 4, f(2) = 6, f(3) = 10$ and 3rd difference being constant. Find $f(5)$.
(ii) Prove that a divided difference is symmetric function of its arguments. If $f(x) = x^{2}$ then prove that $f[x_{0}, x_{1}, x_{2}]$ is constant and all higher order difference are zero.
(iii) In a country school going children of a certain age group is given for different years as follows:
Year: 1995, 2000, 2005, 2010, 2015
No. of student (in thousand): 304, 329, 357, 387, 421
Estimate the number in the year 2020.
(b) Establish Lagrange's interpolation formula. Show that the Lagrangian functions are invariant under a linear transformation.

Unit - V

7. Answer any one question: $2 \times 1 = 2$
(a) Why does one need to use numerical method instead of analytical method for integration?
(b) What is degree of precision? What is the degree of precision of Weddle's rule?

8. Answer any one question: $5 \times 1 = 5$
(a) Using power method, find the largest eigen value in magnitude and corresponding eigen vector of the matrix $A = \begin{pmatrix} 1 & 3 & 2 \\ -1 & 0 & 2 \\ 3 & 4 & 5 \end{pmatrix}$.
(b) Derive Trapezoidal rule from general quadrature formula and discuss its geometrical significance.

Unit - VI

9. Answer any one question: $5 \times 1 = 5$
(a) Define single step and multistep methods. Use R-K method of order 2 to approximate $y$ when $x = 0.1, 0.2, 0.3$ given that $\frac{dy}{dx} = y - x, y(0) = 2$.
(b) Write down the working rule of modified Euler's method for solving first order differential equation with initial condition. Comment on accuracy of Euler's method in solving a differential equation.