B.Sc. Mathematics Honours GE-4 Question Paper 2022 (CBCS)

The figures in the margin indicate full marks.
Candidates are required to give their answers in their own words as far as practicable.

Group A

1. Answer any four questions: 5 x 4 = 20
(a) (i) Deduce Newton-Cotes quadrature formula.
(ii) Evaluate: $(\frac{\Delta^{2}}{E})x^{3}$
(b) Given $(n+1)$ distinct points $x_{0},x_{1},x_{2},.....,x_{n}$ and $(n+1)$ ordinates $y_{0},y_{1},.....,y_{n}$, there is a polynomial $p(x)$ of degree n that interpolates to $y_{i}$ at $x_{i}, i=0,1,....,n$. Prove that this polynomial is unique.
(c) Describe the Regula-Falsi method for finding the root of the equation $f(x)=0$. What are the advantages and disadvantages of this method.
(d) Let f(x) be a function. Describe least square method to approximate a polynomial.
(e) Describe Gauss-elimination method for numerical solution of a system of linear equations.
(f) Evaluate y(1.0) from the differential equation $\frac{dy}{dx}=y+x^{2}$ with $y(0)=1$ taking $h=0.2$, by Euler's method correct upto two decimal places.

Group B

2. Answer any two questions: 10 x 2 = 20
(a) (i) Derive the Simpson integration formula in the form $$\int_{a}^{b}f(x)dx=\frac{b-a}{3}[f(a)+4f(\frac{a+b}{2})+f(b)]-\frac{(b-a)^{5}}{2880}f^{(4)}(\xi)$$ where $a < \xi < b$. What is the error if $f(x)$ is a polynomial of degree 3.
(ii) Find the value of $\int_{0}^{1}\frac{1}{1+x}dx$ using Simpson's $1/3$ rule and mid-point formula using $h=0.5$.
(b) (i) Derive the convergence criteria for Newton-Raphson method. Also determine the order of convergence of this method.
(ii) Describe power method to find the largest magnitude eigen value of a square matrix.
(c) Solve the following system of equations by Gauss-Seidal iteration method correct upto three significant figures:
$3x+y+z=3$
$x+4y+z=2$
$2x+y+5z=5$
(d) Compute the percentage error in the time period $T=2\pi\sqrt{\frac{l}{g}}$ for $l=1m$ if the error in the measurement of l is 0.01.

Group C

[PARTIAL DIFFERENTIAL EQUATIONS AND APPLICATIONS]
Full Marks: 60
Time: 3 Hours
1. Answer any five questions: 2 x 5 = 10
(i) Find the order and degree of the following PDE: $$(\frac{\partial u}{\partial x})^{2}+(\frac{\partial u}{\partial y})^{2}=1$$
(ii) Form a PDE by the elimination of the arbitrary constants a, b from $z=ax+by$.
(iii) Determine whether the equation $\frac{\partial^{2}u}{\partial x^{2}}+2\frac{\partial^{2}u}{\partial y^{2}}=0$ is hyperbolic, parabolic or elliptic.
(iv) Write and classify Laplace's equation.
(v) Give an example of a homogeneous linear second order PDE.
(vi) State Kepler's second law.
(vii) Write the Lagrange's auxiliary equations for the PDE $zxp+zyq=xy$.
(viii) A particle describes the curve $p^{2}=ar$ under a force F to the pole. Find the law of force.

2. Answer any four questions: 5 x 4 = 20
(i) Form a PDE by eliminating the function f from $z=f(x^{2}-y^{2})$.
(ii) Using Lagrange's method solve the PDE $(y+z)p+(z+x)q=x+y$.
(iii) Show that the characteristics equation of the PDE $x^{2}\frac{\partial^{2}u}{\partial x^{2}}+2xy\frac{\partial^{2}u}{\partial x\partial y}+y^{2}\frac{\partial^{2}u}{\partial y^{2}}=0$ represents a family of straight lines passing through the origin.
(iv) Find the complete integral of $x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}+z\frac{\partial u}{\partial z}=au+\frac{xy}{z}$
(v) A particle describes a curve whose equation is $r=a\sec^{2}\frac{\theta}{2}$ under a force to the pole. Find the law of force.
(vi) A particle describes the path $r=a\tan\theta$ under a force to the origin. Find its acceleration in terms of r.

3. Answer any three questions: 10 x 3 = 30
(i) Transform the partial differential equation $\frac{\partial^{2}u}{\partial x^{2}}-4\frac{\partial^{2}u}{\partial x\partial y}+4\frac{\partial^{2}u}{\partial y^{2}}=0$ to canonical form and hence solve it.
(ii) Apply the method of separation of variables to obtain a formal solution u(x, y) of the problem which consists of $\frac{\partial^{2}u}{\partial x^{2}}-\frac{\partial^{2}u}{\partial y^{2}}=0$ with the conditions: $u(0,y)=u(\pi,y)=0, y \ge 0$, $u(x,0)=\sin^{3}x$, $\frac{\partial u(x,0)}{\partial y}=0, 0 \le x \le \pi$
(iii) Find the solution of the initial boundary value problem: $u_{tt}=u_{xx}$, $0 < x < 2, t > 0$, $u(x,0)=\sin(\frac{\pi x}{2})$, $0 \le x \le 2$, $u_{t}(x,0)=0, 0 \le x \le 2$, $u(0,t)=0, u(2,t)=0, t \ge 0$
(iv) Find the solution of the cauchy problem for the first order PDE $x\frac{\partial z}{\partial x}+y\frac{\partial z}{\partial y}=z$ on $D=\{(x,y,z):x^{2}+y^{2} \ne 0, z > 0\}$ with the initial condition $x^{2}+y^{2}=1, z=1$.
(v) Show that the path described under the inverse square law of distance will be an ellipse, a parabola or a hyperbola according as $v^{2} <, = \text{ or } > \frac{2\mu}{r}$.