GTU BE 2020 Summer Question Papers || Gujarat Technological University
Subject Code: 3110015 Date:09/11/2020
Subject Name: Mathematics II
Time: 10:30 AM TO 01:30 PM Total Marks: 70
Seat No.: Enrolment No.
GUJARAT TECHNOLOGICAL UNIVERSITY
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BE - SEMESTER 1&2 EXAMINATION — SUMMER 2020Subject Code: 3110015 Date:09/11/2020
Subject Name: Mathematics II
Time: 10:30 AM TO 01:30 PM Total Marks: 70
Instructions:
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- Attempt all questions.
- Make suitable assumptions wherever necessary.
- Figures to the right indicate full marks.
| Marks | |
|---|---|
| Q1 (a) Evaluate ? F · dr along the parabola y2 = x between the points (0, 0) and (1, 1) where F=x2i+xy j | 03 |
| (b) Find the work done in moving particle from A (1, 0, 1) to B (2,1,2) along the straight-line AB in the force field F=xi+(x-y)j+(y+z)k | 04 |
| (c) Verify Green's theorem for ?c(xydx— y2dy) where C is the boundary of the region bounded by the ellipse 3x2 +4y2 =12 | 07 |
| Q.2 (a) Find the Laplace transform of te-t sin 3t | 03 |
| (b) Find the inverse Laplace transform of 5/(s-1)(s2 +2s+5) | 04 |
| (c) Show that the vector field F =(ysinz—sinx)i+(xsinz+2yz)j+(xycosz+y2)k is conservative and find the corresponding scalar potential. | 07 |
| OR | |
| (c) Show that F = 2xyz i +(x2z +2y)j +x2 y k is irrotational and find a scalar function f such that F = grad f. | 07 |
| Q3 (a) Find the directional derivative of f(x,y)=xy+xey +cos(xy) at the point P(1,0) in the direction of v = 3i —4j. | 03 |
| (b) Find the inverse Laplace transform of log| (s+1)/s |. | 04 |
| (c) Find the singular solution and general solution of y+ px = x4p2 | 07 |
| OR | |
| Q3 (a) Find the Laplace transform of (cosat —cosbt)/t . | 03 |
| (b) Show that ?08 (dx)/(x2 +4)2 = p/32. | 04 |
| (c) Find the power series solution of y''—2xy =0; y(0)=1 near x=0. | 07 |
| Q4 (a) Solve (d3x)/(dt3) -2(d2x)/(dt2) +(dx)/(dt) +x=et with x=2, dx/dt=-1 at t=0. | 03 |
| (b) Solve (D2 — 1)y =xe-x sinx | 04 |
| (c) Solve (D2 — 1)y =xe-x sinx | 07 |
| OR | |
| Q4 (a) Solve dy/dx +xsin2y=x3cos2 y | 03 |
| (b) Using method of variation of parameter, solve (d2y)/(dx2) +4y=tan2x. | 04 |
| (c) Using method of undetermined coefficients solve (d2y)/(dx2) -2(dy)/(dx) +y=xex. | 07 |
| Q.5 (a) Classify the singular points of x2y"+xy' -2y =0. | 03 |
| (b) Solve (d2y)/(dx2) +9y =sin2xsin x. | 04 |
| (c) Solve (i) (x2 +3x2 y2 )dx+(3x2y+ y3 )dy =0. (ii) dy/dx + ycotx =2cosx. | 07 |
| OR | |
| Q5 (a) Solve dy/dx = (x2 - y2)/x2 | 03 |
| (b) Solve x2 (d2y)/(dx2) -3x(dy)/(dx) +y = cos(lnx). | 04 |
| (c) Using Frobenius method solve 2x2y"+xy'=(x+1)y =0. | 07 |
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