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

  • Advanced Engineering Mathematics - AEM
  • 2016
  • PYQ
  • I K G Punjab Technical University - IKGPTU
  • Automobile Engineering
  • B.Tech
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16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 Roll No. Total No. of Questions : 09 B.Tech. (2011 Onwards) 16058 16058 16058 16058 : 03 Total16058 No. of Pages 16058 16058 16058 (Sem.–1) 16058 16058 16058 ENGINEERING MATHEMATICS – 16058 I 16058 16058 16058 16058 Subject Code : BTAM-101 Paper ID : [A1101] 16058 Time : 3 Hrs. 16058 16058 16058 16058 16058 Max. Marks 16058 : 60 16058 16058 INSTRUCTIONS TO CANDIDATES : 16058 16058 16058 1. 16058 2. 3. 4. 16058 5. 16058 1. 16058 SECTION-A is COMPULSORY consisting of TEN questions16058 carrying T WO marks 16058 16058 16058 16058 16058 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 -16058 B & C. 16058 16058 16058 16058 16058 Symbols used have their usual meanings. Statistical tables, if demanded, may be provided. 16058 16058 16058 16058 SECTION-A 16058 16058 16058 16058 16058 16058 16058 16058 Solve the following :  y – x z 16058 –x 2 u 16058 2 u 2 u16058 16058 a) If u16058 = f ,  , then show that x x  y y  z z  0 . xz   xy b) Find the stationary points of the function f(x, y) = x3 + y3 – 63(x + y) + 12xy. 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 c) If u = x2 – y2 and v = 2xy and x = r cos , y = r sin , then find the value of 16058 (u, v) .  (r, ) d) For what values of a, b, and c the vector function 16058 16058 16058  16058  16058 16058  F  ( x  y  az )i  (bx  3 y – z ) j  (3x  cy  z )k is irrotational.  e) Calculate the circulation of the field F  ( x – y)iˆ  xjˆ around the circle x2 + y2 = 1. 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 = x2 from (–1,1) to (2,4). 16058 16058 f) Find the length of one arc of the cycloid x = a( – sin ), y = a (1 – cos ) . 2 ln 3 ln 3 16058 16058 g) Evaluate 16058   0 y /2 2 e x dxdy. 16058 16058 16058 16058 h) State Stoke’s theorem. 16058 16058 16058 i) Evaluate  xydx  ( x  y )dy, along the curve C : y 16058 16058 16058 x y  z e16058 dz dy dx16058 . 16058 16058 16058 16058 C a x x y 16058 16058 16058 j) Evaluate :   00 0 16058 1 | M-54091 16058 16058 16058 16058 16058 16058 (S1)-358 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058

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16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 SECTION-B 2. 16058 16058 a) Find the radius of curvature at any point of the curve r = a(l – cos ) and prove that 2 /r is constant. 16058 16058 16058 16058 16058 16058 16058 16058 b) Trace the curve y2(x2 + y2) + a2(x2 – y2) = 0 by giving all its features in detail. 16058 16058 3. 16058 16058 4. 16058 16058 b) Find the centre of gravity of the arc of the curve x = a(t + sin t), y = a(l – cos t) in the first16058 quadrant. 16058 16058 16058 16058 16058 a) If u = f (r), where r2 = x2 + y2, then show that 16058 16058 5. 16058 a) Find the volume of the solid formed by revolving the curve y2 (2a – x) = x3 about its 16058 16058 16058 16058 16058 16058 asymptote. 16058 16058 16058  2u   2u 16058 x 2 y 2  f (r )  16058 1 f (r ) r 16058 b) Use Lagrange’s method of undetermined coefficients to show that the rectangular solid of maximum16058 volume that can be inscribed16058 in a sphere is16058 a cube. 16058 16058 16058 16058 16058 16058 16058 a) Find the first three terms of the Taylor’s series expansion of ex log(l + y) in the neighbourhood of (0,0) 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 b) Use Euler’s theorem to prove that 16058 16058 x2  2u 16058 x 16058 16058 2  2 xy  x y 16058 16058 16058  2u16058 2  2u sin u cos 2u y  , whenever u  sin –1  2 3  x y xy y 4 cos u  16058 16058 16058    16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 SECTION-C 16058 16058 16058 16058 16058 2 2 x – x2 6. a) Evaluate   0 16058 16058 16058 ( x 2  y 2 ) dy dx by changing into polar coordinates. 0 16058 16058 16058 16058 b) Find the volume bounded by the cylinder x2 + y2 = 4 and the planes y + z = 4 and z = 0. 16058 16058 7. 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 a) Prove the identity :        16058 a  r  a n16058 (a  r ) r 16058 16058   – , where is a constant vector. a  n n n 2 r r r   16058 16058 16058 2 | M-54091 16058 16058 16058 16058 16058 16058 (S1)-358 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058

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16058 16058 16058 16058 16058 16058 16058 16058 16058 16058  b) A 16058 vector field is16058 given by F16058  18 ziˆ –12 j  3 ykˆ . Evaluate the surface16058 integral 16058 16058 16058 16058    F ·nds, where S is the part of the plane 2x + 3y + 6z = 12 in the first octant. S 16058 16058 8. 16058 16058 9. 16058 16058 16058 16058 16058 16058 16058 16058 Verify the Gauss Divergence theorem for a vector field defined by  F = ( x 2 – yz )iˆ  ( y 2 – xz ) ˆj  ( z 2 – xy)kˆ taken around the rectangular parallelepiped 16058 16058 16058 16058 16058 0 x a, 0 y b, 16058 0 z  c. a) Find the directional derivative of f (x, y.z) = x y2 + y z3 at (2,–1,1) in the direction of normal to the surface x log z – y216058 = –4 at (–1,2,1). 16058 16058 16058 16058 16058 16058 16058 16058 b) State Green’s theorem in plane and use it to evaluate 16058 16058 16058 16058  (3x 2 16058 16058 – 8 y 2 )dx  (4 y – 6 xy )dy, 16058 16058 16058 C where C is the boundary of the region defined by x = 0, y = 0, x + y = 1 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 3 | M-54091 16058 16058 16058 16058 16058 16058 (S1)-358 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058 16058

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