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Previous Year Exam Questions of Advanced Engineering Mathematics of IKGPTU - AEM by Ravichandran Rao

  • Advanced Engineering Mathematics - AEM
  • 2015
  • PYQ
  • I K G Punjab Technical University - IKGPTU
  • Automobile Engineering
  • B.Tech
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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

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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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