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Previous Year Exam Questions of Applied Mathematics-1 of JNTUACEP - M-1 by thamk uoy

  • Applied Mathematics-1 - M-1
  • 2016
  • PYQ
  • Jawaharlal Nehru Technological University Anantapur (JNTU) College of Engineering (CEP), Pulivendula, Pulivendula, Andhra Pradesh, India - JNTUACEP
  • Electronics and Communication Engineering
  • B.Tech
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R15 Code No: 121AB JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech I Year Examinations, August/September - 2016 MATHEMATICS-I (Common to all Branches) Time: 3 hours Max. Marks: 75 Note: This question paper contains two parts A and B. Part A is compulsory which carries 25 marks. Answer all questions in Part A. Part B consists of 5 Units. Answer any one full question from each unit. Each question carries 10 marks and may have a, b, c as sub questions. PART- A (25 Marks) 1 3 4 3 Find the rank of the matrix 3 9 12 9 . 1 3 4 1 [2] d) If A is an n ร— n matrix and A2 = A, then show that each Eigen value of A is 0 or 1. Give an example of a function that is continuous on [-1, 1] and for which mean theorem does not hold with explanation. 1 1 1 Find the maximum and minimum values of ๐‘ฅ + ๐‘ฆ + ๐‘ง subject to ๐‘ฅ + ๐‘ฆ + ๐‘ง = 1. [3] value [2] [3] e) Evaluate [2] f) g) h) Evaluate + ๐‘ฆ 2 ๐‘‘๐‘ฆ๐‘‘๐‘ฅ by changing into polar coordinates. [3] Solve the differential equation ๐‘•๐‘ฅ + ๐‘๐‘ฆ + ๐‘“ ๐‘‘๐‘ฆ + ๐‘Ž๐‘ฅ + ๐‘•๐‘ฆ + ๐‘” ๐‘‘๐‘ฅ = 0. [2] Find the equation of the curve passing through the point (1,1) whose differential equation is ๐‘ฆ โˆ’ ๐‘ฆ๐‘ฅ ๐‘‘๐‘ฅ + ๐‘ฅ + ๐‘ฅ๐‘ฆ ๐‘‘๐‘ฆ = 0. [3] Find Laplace transform of 4 sin ๐‘ก โˆ’ 3 ๐‘ข(๐‘ก โˆ’ 3). [2] 2 ๐‘ก 0 < ๐‘ก < 2 Express ๐‘“(๐‘ก) in terms of Heavisides unit step function ๐‘“ ๐‘ก = [3] 4๐‘ก ๐‘ก>2 1.a) b) c) i) j) โˆž โˆ’๐‘๐‘ฅ 2 ๐‘Ž ๐‘‘๐‘ฅ. 0 ๐‘Ž ๐‘Ž 2 โˆ’๐‘ฆ 2 2 ๐‘ฅ 0 0 PART-B (50 Marks) 2.a) b) 3. 4.a) b) Show that the two matrices A, C-1AC have the same latent roots. 1 2 โˆ’3 For a matrix A = 0 3 2 find the Eigen values of 3A3 + 5A2 โ€“ 6A + 2I. [5+5] 0 0 โˆ’2 OR Reduce the following quadratic form to canonical form and find its rank and signature [10] ๐‘ฅ 2 + 4๐‘ฆ 2 + 9๐‘ง 2 + ๐‘ก 2 โ€“ 12๐‘ฆ๐‘ง + 6๐‘ง๐‘ฅโ€“ 4๐‘ฅ๐‘ฆโ€“ 2๐‘ฅ๐‘กโ€“ 6๐‘ง๐‘ก. Prove that ๐‘ข = ๐‘ฅ + ๐‘ฆ + ๐‘ง, ๐‘ฃ = ๐‘ฅ๐‘ฆ + ๐‘ฆ๐‘ง + ๐‘ง๐‘ฅ, ๐‘ค = ๐‘ฅ2 + ๐‘ฆ 2 + ๐‘ง 2 dependent and find the relation between them. ๐œ•(๐‘ข,๐‘ฃ) ๐œ•(๐‘ฅ,๐‘ฆ) If ๐‘ฅ = ๐‘ข 1 โˆ’ ๐‘ฃ ; ๐‘ฆ = ๐‘ข๐‘ฃ prove that ๐œ•(๐‘ฅ,๐‘ฆ) ร— ๐œ• (๐‘ข,๐‘ฃ) = 1. OR WWW.MANARESULTS.CO.IN are functional [5+5]

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5. Find the volume of the greatest rectangular parallelepiped that can be inscribed in the ๐‘ฅ2 ๐‘ฆ2 ๐‘ง2 ellipsoid ๐‘Ž 2 + ๐‘ 2 + ๐‘ 2 = 1. 6.a) Evaluate b) Evaluate [10] ๐‘ฅ2 โˆž ๐‘ฅ๐‘‘๐‘ฅ 0 1+๐‘ฅ 6 using ฮ“- ฮฒ functions. [5+5] OR Find the volume bounded by the cylinders ๐‘ฅ 2 + ๐‘ฆ 2 = 4 and ๐‘ง = 0. 7. ๐‘ฆ2 ๐‘ฅ ๐‘š โˆ’1 ๐‘ฆ ๐‘›โˆ’1 ๐‘‘๐‘ฅ๐‘‘๐‘ฆ over the positive quadrant of the ellipse ๐‘Ž 2 + ๐‘ 2 = 1. [10] Solve ๐ท2 โˆ’ 4๐ท + 4 ๐‘ฆ = 8๐‘ฅ 2 ๐‘’ 2๐‘ฅ ๐‘ ๐‘–๐‘›2๐‘ฅ. [10] OR A particle is executing simple harmonic motion of period T about a centre O and it passes through the position P(OP = ๐‘) with velocity ๐‘ฃ in the direction OP. Show that the time ๐‘‡ ๐‘ฃ๐‘‡ that elapses before it returns to P is ๐œ‹ ๐‘ก๐‘Ž๐‘›โˆ’1 2๐œ‹๐‘ . [10] 8. 9. ๐‘‘2๐‘ฅ ๐‘‘๐‘ฅ Solve the differential equation ๐‘‘๐‘ก 2 โˆ’ 4 ๐‘‘๐‘ก โˆ’ 12๐‘ฅ = ๐‘’ 3๐‘ก , given that ๐‘ฅ 0 = 1 and ๐‘ฅ โ€ฒ 0 = โˆ’2 using Laplace transform. [10] OR ๐‘‘๐‘ฅ 2 11.a) Using Laplace transform, solve ๐ท + 1 ๐‘ฅ = ๐‘ก cos 2๐‘ก given ๐‘ฅ = 0, ๐‘‘๐‘ก = 0 at ๐‘ก = 0. 10. b) Using Convolution theorem, evaluate ๐ฟโˆ’1 1 ๐‘ (๐‘  2 +2๐‘ +2) . ---ooOoo--- WWW.MANARESULTS.CO.IN [5+5]

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