Deemed University 2009 M.Tech Mechanical Engineering University: Lingayas University Term: I Title of the : Numerical Techniques - Question Paper

Roll No. ..

 

Lingayas University, Faridabad

M.Tech. (Part Time) (ECE, EL, CE, ME, IT)

Examination October, 2009

Numerical Techniques

Paper: MA -501

[Time: 3 Hours] [Max. Marks: 100]

 

Before answering the question, candidate should ensure that they have been supplied the correct and complete question paper. No complaint in this regard, will be entertained after examination.

 

Note: There are eight questions in three sections (A, B, C)

• Question No. one is compulsory
• Out of remaining seven questions answer any four selecting two questions from each section that is from B and C.

 

Section A

Q No. 1. Select the correct answer of the following multiple choice questions

(i) If the equation f(x) = 0 has a root of order m, than f ^m-1() is

(a) 1 (b) 2 (c) (d) 0

(ii) The convergence of Newton-Raphson method is of order

(a) 1 (b) 2 (c) 1.618 (d) 1.84

(iii) the real roots of an equation f(x)=0 are the abscissa of points where the graph crosses the

(a) y axis (b) x-axis (c) y=x (d) y= -x)

(iv) The equation f(x) =0 has a simple root in the interval. (1,2). This root is to be determined correct to two decimal places using Bi section method. The required number of iterations are

(a) 5 (b) 6 (c) 7 (d) 8

(v) the nth divided difference can be expressed as the ratio of two determinants each of order

(a) n (b) n+1 (c) n-1 (d) n²

(vi) If f(x) = -6 x³ + 11 x² 6x+1, then ∆³ f(x) is equal to

(a) 36 (b) -36 (c) 32 (d) 1

(vii) Following are the normal equations

y = na + b x +cx²

xy = ax + bx² +cx³

x²y = ax² + b x³ +cx⁴ of

(a) Straight line (b) Ellipse (c) Parabola (d) None of these

(viii) is equal to

(a) (b) (c) (d) none of these

(ix) - h⁴ y^iv () is the error in

(a) trapezoidal rule (b) Simpsons rule (c) Simpsons rule
(d) Booles rule

(x) In Simpsons rule, function y = f(x) is assumed a polynomial of degree

(a) three (b) four (c) two (d) one

(xi) If is an eigen value of multiplicity 3 of a square matrix A of order 3 and the rank of A-I is 3, then the number of linearly independent eigen vectors are

(a) 3 (b) 2 (c) 1 (d) 0

(xii) eigen value problems arising from

(a) Markov process (b) Stretching of an elastic membrane

(c) Both (a) ∆ (b) (d) none of these

(xiii) If the rank of the product of a m x n matrix A of rank , and n x p matrix B of rank , then

(a) +-n ≤ ≤ min. (,) (b) +-n ≤ min. (,)

(c) +-n ≤ min. (, ) (d) +-n = ≤ min. (, )

(xiv) When the step size is reduced by times, then by Adam Moulton method, how many times the error reduces

(a) (b) (c) (d)

(xv) The global error in modified Eulers method is

(a) O (h) (b) O(h²) (c) O(h³) (d) O(h⁴)

(xvi) h⁵ y^v is the truncation error in

(a) Milines Predictor method (b) Mline corrector method

(c) Adams Moultion Predictor method (d) Adams Moulton corrector method

(xvii) Crank-Nicolson method is

(a) Explicit method (b) Implicit method

(c) Both (a) & (b) (d) None of these

(xviii) A homogenous system always has a trivial solution. Non-trivial solution exist if and only if

(a) = no. of unknowns (b) > No. of unknowns

(c) < no. of unknowns (d) none of these

(xix) The partial differential equation Au_xx + Bu_xy +Cu_yy +f (x,y,z,u_x,u_y)=0 is called hyperbolic if

(a) B² 4Ac < 0 (b) B² 4 Ac =0

(c) B² 4AC > 0 (d) none of these

(xx) The accuracy by finite difference method increases if we use

(a)   forward differences (b) background differences

(c) Central difference (d) any of these

 

 

Section B

 

Q-2. (a) What is the rate of convergence of Newton-Raphson method in case of multiple roots? What modification can be done in the method so that it has quadratic convergence in case of multiple roots also. Explain your answer.

(10)

(b) Explain Muller method to find the roots of equation. (10)

Q-3. (a) Derive the Milnes predictor corrector formula? (10)

(b) Apply modified Eulers method to find y (0.1) and y(0.2) if

=xy and y(0) =1 (10)

Q-4. (a) Discuss finite difference method for solving (10) (b) Why A.D.I. method is often preferred in two and three dimensional problems? Discuss in details. (10)

Section - C

Q-5. (a) Find the eigen values and eigen vectors of the matrix (10)

 

 

(b) Solve the following equations by Gauss Jordan method

X₁ + 2x₂ + x₃ = 8;

2x₁ + 3x₂ + 4x₃ = 20;

4x₁ + 3x₂ + 2x₃ = 16; (10)

 

Q-6. (a) By Newtons divided difference formula, find the values of f(8) and f(15) from the following data. (10)

(b) Obtain cubic spline for every subinterval from the given data

With the end conditions m₀ = m₃ = 0. Hence find an estimate of f (2.5) (10)

Q-7. (a) Derive the shooting method for second order linear differential equations. (10)

(b) A river is 80 m. wide. The depth y of the river at a distance x from one bank is given by the following table (10)

Find approximately the area of the cross section.

Q-8. (a) Explain the Galerkin method to solve a boundary value problem. (10)

(b) Using Simpsons rule evaluate

dx dy, taking h=k = 0.5 (10)