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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
• 1184 Views
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#### Previous Year Exam Questions of Applied Mathematics-1 of JNTUACEP - M-1 by thamk uoy / 2

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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  d) If A is an n × 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 𝑥 + 𝑦 + 𝑧 subject to 𝑥 + 𝑦 + 𝑧 = 1.  value   e) Evaluate  f) g) h) Evaluate + 𝑦 2 𝑑𝑦𝑑𝑥 by changing into polar coordinates.  Solve the differential equation 𝑕𝑥 + 𝑏𝑦 + 𝑓 𝑑𝑦 + 𝑎𝑥 + 𝑕𝑦 + 𝑔 𝑑𝑥 = 0.  Find the equation of the curve passing through the point (1,1) whose differential equation is 𝑦 − 𝑦𝑥 𝑑𝑥 + 𝑥 + 𝑥𝑦 𝑑𝑦 = 0.  Find Laplace transform of 4 sin 𝑡 − 3 𝑢(𝑡 − 3).  2 𝑡 0 < 𝑡 < 2 Express 𝑓(𝑡) in terms of Heavisides unit step function 𝑓 𝑡 =  4𝑡 𝑡>2 1.a) b) c) i) j) ∞ −𝑏𝑥 2 𝑎 𝑑𝑥. 0 𝑎 𝑎 2 −𝑦 2 2 𝑥 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 −3 For a matrix A = 0 3 2 find the Eigen values of 3A3 + 5A2 – 6A + 2I. [5+5] 0 0 −2 OR Reduce the following quadratic form to canonical form and find its rank and signature  𝑥 2 + 4𝑦 2 + 9𝑧 2 + 𝑡 2 – 12𝑦𝑧 + 6𝑧𝑥– 4𝑥𝑦– 2𝑥𝑡– 6𝑧𝑡. Prove that 𝑢 = 𝑥 + 𝑦 + 𝑧, 𝑣 = 𝑥𝑦 + 𝑦𝑧 + 𝑧𝑥, 𝑤 = 𝑥2 + 𝑦 2 + 𝑧 2 dependent and find the relation between them. 𝜕(𝑢,𝑣) 𝜕(𝑥,𝑦) If 𝑥 = 𝑢 1 − 𝑣 ; 𝑦 = 𝑢𝑣 prove that 𝜕(𝑥,𝑦) × 𝜕 (𝑢,𝑣) = 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 𝑥2 𝑦2 𝑧2 ellipsoid 𝑎 2 + 𝑏 2 + 𝑐 2 = 1. 6.a) Evaluate b) Evaluate  𝑥2 ∞ 𝑥𝑑𝑥 0 1+𝑥 6 using Γ- β functions. [5+5] OR Find the volume bounded by the cylinders 𝑥 2 + 𝑦 2 = 4 and 𝑧 = 0. 7. 𝑦2 𝑥 𝑚 −1 𝑦 𝑛−1 𝑑𝑥𝑑𝑦 over the positive quadrant of the ellipse 𝑎 2 + 𝑏 2 = 1.  Solve 𝐷2 − 4𝐷 + 4 𝑦 = 8𝑥 2 𝑒 2𝑥 𝑠𝑖𝑛2𝑥.  OR A particle is executing simple harmonic motion of period T about a centre O and it passes through the position P(OP = 𝑏) with velocity 𝑣 in the direction OP. Show that the time 𝑇 𝑣𝑇 that elapses before it returns to P is 𝜋 𝑡𝑎𝑛−1 2𝜋𝑏 .  8. 9. 𝑑2𝑥 𝑑𝑥 Solve the differential equation 𝑑𝑡 2 − 4 𝑑𝑡 − 12𝑥 = 𝑒 3𝑡 , given that 𝑥 0 = 1 and 𝑥 ′ 0 = −2 using Laplace transform.  OR 𝑑𝑥 2 11.a) Using Laplace transform, solve 𝐷 + 1 𝑥 = 𝑡 cos 2𝑡 given 𝑥 = 0, 𝑑𝑡 = 0 at 𝑡 = 0. 10. b) Using Convolution theorem, evaluate 𝐿−1 1 𝑠(𝑠 2 +2𝑠+2) . ---ooOoo--- WWW.MANARESULTS.CO.IN [5+5]