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- Advanced Engineering Mathematics - AEM
- 2013
- PYQ
**I K G Punjab Technical University - IKGPTU**- Automobile Engineering
- B.Tech
**1 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. Answer briefly : (a) Identify the symmetry of the polar curve r = sin (b) If u = F(x – y, y – z, z – x), then show that (c) If J = . 2 u u u = 0. x y z (u , v ) ( x, y ) , J = , then show that JJ = 1, where J stands ( x, y ) (u , v ) for Jacobian. (d) Evaluate e y dydx . y 0 x (e) Find the polar equation of the curve x 2 + (y – 3) 2 = 9 given in Cartesian form . (f) State Gauss Divergence Theorem. (g) If F = grad(x3 + y3 + z3 – 3xyz), then find div F . [N-1- 1512 ]

^ ^ (h) Find the work done by the force field F = (y – x2) i + (z – y 2) j ^ ^ ^ ^ + (x – z 2 ) k over the curve r (t) = t i + t 2 j + t 3 k , 0 t 1, from (0,0,0) to (1,1,1). (i) Obtain the local extreme values of the function f(x, y) = xy. (j) The period of a simple pendulum is T = 2 l / g , find the maximum error in T due to possible error up to 1% in l and 2.5% in g. SECTION-B 2. (a) Trace the curve y2(a – x) = x2(a + x) by giving all salient features in detail. (b) If 1 and 2 be the radii of curvature at the extremities of two conjugate diameters of an ellipse x2 a2 y2 b2 = 1, then prove that ( 1) 2/3 + (2 )2/3 (ab)2/3 = a2 + b 2 . (4, 4) 3. (a) Find the entire length of the Cardiode r = a( 1 + cos ). Also show that upper half is bisected by the ray = /3. (b) The area bounded by an arc of the curve x = a( – sin ), y = a(1 – cos ), 0 2 and the x-axis is revolved around x-axis. Find the volume of the solid generated. (4, 4) 4. (a) Transform the equation 1 (b) If u = sin 2u x 2 2u y 2 = 0 into polar co-ordinates. x y u u 1 , then prove that x y tan u. (5, 3) x y x y 2 5. (a) A rectangular box open at the top is to have a volume of 32 cubic feet. Find the dimensions of the box requiring the least material for its construction. (b) Expand f (x, y) = sin xy in ascending powers of (x –1) and (y – (/2)) up to second degree terms. (4, 4) [N-1- 1512 ]

SECTION-C 6. (a) Find the area lying inside the curve r = a(l + cos ) and outside the curve r = a. 1 2 x (b) Evaluate: xy dxdy by changing the order of integration. (4, 4) 0 x2 7. (a) Prove the identity ( F G) = F ( G) G ( F ) + (G ) F – ( F ) G . ^ ^ ^ (b) If F 4 x z i y2 j y zk , then evaluate ^ F N ds , where S is the S surface of the cube bounded by x = 0, x = 1, y = 0, y = 1, z = 1, z =1. (4, 4) ^ ^ ^ 8. (a) Verify Stoke’s theorem for the field F = (2 x y ) i yz 2j y 2 z k , over the upper half surface of x 2 + y 2 + z 2 = 1 bounded by its projection on the xy-plane. (b) Compute the line integral ( y 2dx x 2dy) about the triangle whose C vertices are (1, 0), (0, 1), ( – 1, 0). 9. (a) Verify Green’s theorem for (5, 3) [(3x 2 8 y 2 ) dx (4 y 6 xy )dy] , where C C is the boundary of the region by x = 0, y = 0, x + y = 1. 1 1 x 1 x y (b) Evaluate the triple integral 0 0 [N-1- 1512 ] 0 xyz dx dy dz . (5, 3)

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