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- Advanced Engineering Mathematics - AEM
- 2015
- PYQ
**I K G Punjab Technical University - IKGPTU**- Automobile Engineering
- B.Tech
**1 Views**- Uploaded 3 months ago

Roll No. Total No. of Questions : 09 B.Tech. (2011 Onwards) Total No. of Pages : 02 (Sem.–1) ENGINEERING MATHEMATICS – I Subject Code : BTAM-101 Paper ID : [A1101] Time : 3 Hrs. Max. Marks : 60 INSTRUCTIONS TO CANDIDATES : 1. 2. 3. 4. SECTION-A is COMPULSORY consisting of TEN questions carrying T WO marks each. SECTION - B & C. have FOUR questions each. Attempt any FIVE questions from SECTION B & C carrying EIGHT marks each. Select atleast T WO questions from SECT ION - B & C. SECTION-A l. Solve the following : a) Find the length of any arc of the curve r = a sin2 . 2 b) If z = f (x, y) and x = eu + e–v, y = e–u ev, prove that z z z z – x –y . u v x y c) In polar co-ordinates x = r cos , y = r sin , show that d) Using Euler’s theorem, prove that if tan u = ( x, y ) r. (r , ) u u x3 y 3 y sin 2u . , then x x y x–y e) Write Taylor’s series for a function of two variables. f) Find the value of ‘a’ for which the vectors 3iˆ 2 ˆj 9kˆ and iˆ ajˆ 3kˆ are perpendicular. g) If r = xiˆ yjˆ zkˆ and r r , show that f (r ) f (r ) r . h) State Stoke’s theorem. i) Find the volume common to the two cylinders x2 + y2 =a2 and x2 + z2 = a2. 1 j) Evaluate ( x 2 yz ) dS, where C is the curve defined by x = 4y, z = 3 from 2, ,3 2 C to (4, 1, 3). 1 | M-54091 (S1)-10

SECTION-B 2. Find the radius of curvature at any point of following curves : a) x = a (cos t + t sin t), y = a(sin t – t cos t) (4) b) S = a log (sec + tan ) + a sec tan (4) 3. The cardiod r = a (1 + cos ) revolves about the initial line. Find the volume of the solid generated. (8) 4. a) Find the minimum value x2 + y2 + z2 of subject to the condition that xyz = a3. (4) b) Find the maximum and minimum values of 2(x2 – y2) – x4 + y4. (4) 5. a) If f (x, y) = tan–1 (xy), find an approximate value of f (1.1,0.8) using the Taylor’s series linear approximation. (3) x3 2 y 3 , ( x, y ) (0, 0) b) Show that the function f (x, y) = x 2 y 2 0 ( x, y ) (0, 0) (i) is continuous at (0, 0) (ii) possesses partial derivatives at (0, 0) . (5) SECTION-C 6. Find the centre of gravity of a plate whose density (x, y) is constant and is bounded by the curves y = x2 and y = x + 2. Also, find the moment of inertia about the axis. (8) 7. a) If a sin iˆ cos ˆj kˆ, b cos iˆ – sin ˆj – 3kˆ and c 2iˆ 3 ˆj – kˆ, d (a (b c )) at = 0. find d (4) b) A particle moves along the curve x = 3t2, y = t2 – 2t and z = t3. Find its velocity and acceleration at t = 1 in the direction of iˆ ˆj kˆ . (4) 8. Verify Gauss divergence theorem for f = 4x iˆ – 2y2 ĵ + z2 k̂ , taken over the region bounded by the cylinder x2 + y2 = 4, z = 0 and z = 3. (8) 2 9. a) Evaluate the integral y2 2 0 0 y 2 x y2 1 dxdy . b) Prove that div ( f v ) = f (div v ) + (grad f ). v , where f is scalar function. 2 | M-54091 (4) (4) (S1)-10

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