GTU BE 2019 Summer Question Papers || Gujarat Technological University
Enrolment No.
GUJARAT TECHNOLOGICAL UNIVERSITY
--- Content provided by FirstRanker.com ---
BE - SEMESTER-I &II (NEW) EXAMINATION — SUMMER-2019
Subject Code: 3110014 Date: 06/06/2019
Subject Name: Mathematics — I
Time: 10:30 AM TO 01:30 PM Total Marks: 70
Instructions:
--- Content provided by FirstRanker.com ---
- Attempt all questions.
- Make suitable assumptions wherever necessary.
- Figures to the right indicate full marks.
| Marks | |
|---|---|
| Q.1 (a) Use L’Hospital’s rule to find the limit of lim (1/(x-1) - 1/lnx) | 03 |
| (b) Define Gamma function and evaluate ?08 e-x dx. | 04 |
| (c) Evaluate ?03 ?01 xey2dydx. | 07 |
| Q.2 (a) Define the convergence of a sequence (an) and verify whether the sequence whose nth term is an = ((-1)n(n+1)/n) converges or not. | 03 |
| (b) Sketch the region of integration and evaluate the integral ?R (x2 - 2x2)dA where R is the region inside the square |x| + |y| = 1. | 04 |
| (c) (i) Find the sum of the series ?n=18 1/(n(n+1)) and ?n=18 n/(n+1)! | 07 |
| (ii) Use Taylor’s series to estimate sin38°. | |
| OR | |
| (c) Evaluate the integrals ?01 ?01 (x2+y2) dxdy and ?0p ?01 r2sin(?) drd?. | 07 |
| Q.3 (a) If an electrostatic field E acts on a liquid or a gaseous polar dielectric, the net dipole moment P per unit volume is P(E) = e(eE - e-E)/ (eE + e-E). Show that limE?0 P(E) = 0. | 03 |
| (b) For what values of the constant k does the second derivative test guarantee that f(x,y) = x3 + kxy +y2 will have a saddle point at (0,0)? A local minimum at (0,0)? | 04 |
| (c) Find the series radius and interval of convergence for ?n=18 (3x-2)n/(n24n). For what values of x does the series converge absolutely? | 07 |
| OR | |
| Q.3 (a) Determine whether the integral ?08 x/(1+x3) converges or diverges. | 03 |
| (b) Find the volume of the solid generated by revolving the region bounded by y = vx and the lines y = 1, x = 4 about the line y=1. | 04 |
| (c) Test the convergence of the series ?n=18 (n2+1)/(n3+1) and ?n=18 (1/v(n2+n)). | 07 |
| Q.4 (a) Show that the function f(x,y) = xy/(x2+y2) has no limit as (x,y) approaches to (0,0). | 03 |
| (b) Suppose f is a differentiable function of x and y and g(u,v) = f(eu + sinv, eu + cosv). Use the following table to calculate gu(0,0), gv(0,0), gu(1,2) and gv(1,2). | 04 |
| f | g | fx | fy | |
|---|---|---|---|---|
| (0,0) | 3 | 6 | 4 | 2 |
| (1,2) | 6 | 3 | 5 | 8 |
| (c) Find the Fourier series of 2p-periodic function f(x) = x2, 0 < x < 2p and hence deduce that p2/6 = ?n=18 1/n2. | 07 |
| OR | |
| Q.4 (a) Verify that the function u = e-k2t . sinkx is a solution of the heat conduction equation ut = k2uxx. | 03 |
| (b) Find the half-range cosine series of the function f(x) = {-2, -2<x<0; 0, 0<x<2}. | 04 |
| (c) Find the points on the sphere x2 + y2 + z2 =4 that are closest to and farthest from the point (3,1, -1). | 07 |
| Q.5 (a) Find the directional derivative Du f(x,y) if f(x,y) =x3-3xy + 4y2 and u is the unit vector given by angle ? = p/6. What is Du f(1,2)? | 03 |
| (b) Find the area of the region bounded by the curves y = sinx, y = cosx and the lines x = 0 and x = p/2. | 04 |
| (c) Prove that A = [[2, 0, 0], [1, 2, 1], [1, 0, 3]] is diagonalizable and use it to find A13. | 07 |
| OR | |
| Q.5 (a) Define the rank of a matrix and find the rank of the matrix A = [[2, -1, 0], [4, 5, -3], [1, -4, 7]]. | 03 |
| (b) Use Gauss-Jordan algorithm to solve the system of linear equations 2x1 + 2x2 -x3 + x5 =0, -x1- x2 +2x3-3x4+x5=0, x1 +x2 -2x3-x5=0, x3 + x4 + x5 = 0 | 04 |
| (c) State Cayley-Hamilton theorem and verify if for the matrix A=[[4, 0, 1], [1, -2, 1], [-2, 0, 1]]. | 07 |
--- Content provided by FirstRanker.com ---
This download link is referred from the post: GTU BE 2019 Summer Question Papers || Gujarat Technological University