B.Sc. Mathematics Honours C-7 Question Paper 2018 (CBCS)

The figures in the right-hand margin indicate full marks.
Candidates are required to give their answers in their own words as far as practicable.
Illustrate the answers wherever necessary.
[Calculator is allowed in examination Hall]

UNIT-I

1. Answer any two questions: 2 × 2
(a) If a number 0.05418 is approximated to 0.05411, find the number of significant digits for such approximation.
(b) Define the terms:
(i) Truncation error
(ii) Round off error
(c) Let, $u = 4x^{6} + 3x - 9$. Find the percentage error in computing u at $x = 1.1$, if the error in x is 0.05.

UNIT-II

2. Answer any one question: 1 × 2
(a) Write down the equation $x^{3} + 2x - 10 = 0$ in the form $x = \phi(x)$ such that the iterative scheme about $x = 2$ converges.
(b) What do you mean by the term as iterative method has the rate of convergence $p(\ge 1)$?

3. Answer any one question: 1 × 5
(a) Find the iterative formula for finding $\sqrt[K]{N}$, where N is a real number, using Newton-Raphson formula. Hence evaluate $\sqrt[3]{2}$ correct upto four significant figure. State the condition of convergence of this method.
(b) Describe the method of false position for finding a real root of an equation $f(x) = 0$ and obtain the corresponding iteration formula. Discuss its advantages and disadvantages in comparison to Newton-Raphson Method.

UNIT-III

4. Answer any one question: 1 × 2
(a) State the conditions for convergence of Gauss-Seidel method for solving a system of linear equations. Are they necessary and sufficient?
(b) Define ill-conditioned and well-conditioned system of Linear equation.

5. Answer any one question: 1 × 5
(a) Consider a system of equations
$x + y - z = 2$
$2x + 3y + 5z = -3$
$3x + 2y - 3z = 6$
Solve the system of equations by LU decomposition method.
(b) Describe Gauss elimination method with pivoting for solution of a system of linear equation. What is the total number of operations required for this method?

UNIT-IV

6. Answer any one question: 1 × 10
(a) (i) Prove that $f(x + Kh) = \sum_{i=0}^{K} ({^{K}C_{i}}) \Delta^{i} f(x)$
(ii) Find the missing term of the following table:
$x$: 0, 1, 2, 3, 4, 5
$f(x)$: 0, ?, 8, 15, ?, 35
(iii) Obtain the Error in the Lagrange Interpolating Polynomial. Also show that the maximum error in linear interpolation is given by $\frac{(x_{0} - x_{1})^{2}}{8} M$ where $M = max |f''( \xi )|$, $x_{0} \le \xi \le x_{1}$.
(b) What is the nth order forward differences of a polynomial of degree n? If h is very small prove that $\Delta^{n+1} f(x) \approx h^{n+1} f^{n+1}(x)$. Find the value of Sec $31.5^{\circ}$ using the following table:
$\theta$ (in degree): $31^{\circ}$, $32^{\circ}$, $33^{\circ}$, $34^{\circ}$
$\tan \theta$: 0.6008, 0.6249, 0.6494, 0.6747

UNIT-V

7. Answer any one question: 1 × 2
(a) Show that Simpson's $\frac{1}{3}$ rule is exact for integrating a polynomial of degree 3.
(b) If $f(x)$ is a quadratic polynomial, deduce that $\int_{1}^{3} f(x) dx \cong \frac{1}{12} [f(0) + 22f(2) + f(4)]$

8. Answer any one question: 1 × 5
(a) Derive Simpson's one-third Rule from Newton cotes formula.
(b) Describe the method of least squares to fit a straight line $y = ax + b$. In some determinations of the value v of carbon dioxide dissolved in a given volume of water at different temperature $\theta$, the following pairs of values were obtained:
$\theta$: 0, 5, 10, 15
$v$: 1.80, 1.45, 1.18, 1.00
Obtain by the method of least square a relation of form $v = a + b \theta$ which best fit to this data.

UNIT-VI

9. Answer any one question: 1 × 5
(a) Describe Euler's method for solving first order differential equation with initial condition. Compute $y(1.2)$ for the problem $\frac{dy}{dx} = 1 + xy$, $y(1) = 1$ by modified Euler's method taking $h = 0.1$.
(b) Find the values of $y(0.1)$ and $y(0.2)$ using Runge Kutta Method of 4th order taking $h = 0.1$. Given that $\frac{dy}{dx} = xy + y^{2}$, $y(0) = 1$.