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- Advanced Engineering Mathematics - AEM
- 2012
- PYQ
**I K G Punjab Technical University - IKGPTU**- Automobile Engineering
- B.Tech
**0 Views**- Uploaded 1 month ago

Roll No. Total No. of Pages : 03 Total No. of Questions : 09 B.Tech. (Sem.–1 st ) ENGINEERING MATHEMATICS-I Subject Code : BTAM-101 (2011 & 2012 Batch) Paper ID : [A1101] Time : 3 Hrs. Max. Marks : 60 INSTRUCTION TO CANDIDATES : 1. 2. 3. 4. SECTION-A is COMPULSORY consisting of TEN questions carrying TWO marks each. SECTION - B & C. have FOUR questions each. Attem p t any FIVE questions from SECTI ON B & C ca rrying EIGHT marks each. Select atleast TWO questions from SECTION - B & C. SECTION-A l. a) Find asymptotes, parallel to axes, of the curve : x2 y 2 – xy 2 – x2 y + x + y + 1 = 0. b) Write a formula to find the volume of the solid generated by the revolution, about y-axis, of the area bounded by the curve x = f(y), the y-axis and the abscissae y = a and y = b. c) What is the value of (u , v ) ( x, y ) ? ( x , y ) (u , v ) d) If an error of 1% is made in measuring the length and breadth of a rectangle, what is the percentage error in its area? e) Find the equation of the tangent plane to the surface z2 = 4(1 + x 2 + y2) at (2, 2,6). 1 2 x f) What is the value of 0 [N- 1- 1288 ] x2 xydxdy

1 1 x g) Give geometrical interpretation of 0 dxdy . 0 ^ ^ h) Show that the vector field F = (x 2 - y 2 + x) i – (2xy + y) j is irrotational. i) What is the value of ( xy i^ yz j^ zxk^ ) ? j) State Stoke’s theorem. SECTION-B 2. Trace the following curves by giving their salient feature: a) x3 + y 3 = 3axy. b) r = a(l + cos) (4,4) 3. a) Find the perimeter of the cardioid r = a (1–cos ). b) Find the area bounded by two parabolas y2 = 4x and x2 = 4y. (4,4) 4. a) If u u u u y z y z 0. , show that x x y z z x b) State Euler’s theorem for homogeneous functions and apply it to show that x u u y = 3 tan u x y x2 y 2 where sin u = x y (4,4) 5. a) Find points on the surface z2 = xy + 1 nearest to the origin. b) Find percentage error in the area of an ellipse if one percent error is made in measuring its major and minor axes. (4,4) [N- 1- 1288 ]

SECTION-C 6. a) Evaluate the following integral by changing the order of integration : 3 4 x 0 ( x y) dxdy 1 2 2 2 b) Find the volume of the ellipsoid x y z = 1. a 2 b2 c2 (4,4) 7. a) Find a unit vector normal to the surface x3 + y 3 + 3xyz = 3 at the point (1, 2, –1). ^ b) If F = (x + y + 1) ^i + j – (x + y) k^ , show that F . curl F = 0. (4,4) 8. a) Compute the line integral ( y 2 dx x 2 dy ) , where C is the boundary C of the triangle whose vertices are (1,0), (0,1) and (-1, 0). b) Compute ^ F Nd s , where F = 6 zi^ 4 ^j yk^ and S is the portion S of the plane 2x + 3y + 6z = 12 in the first octant. (4,4) 9. State Gauss Divergence theorem and verify it for ^ ^ ^ 2 2 2 F = ( x yz )i ( y zx) j ( z xy )k taken over the rectangular parallelopiped 0 x a, 0 y b, 0 z c. (8) [N- 1- 1288 ]

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