Download JNTUA B.Tech 1-1 R13 2014 Dec Supply 13A54102 Mathematics II Question Paper

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Code: 13A54102

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B.Tech I Year (R13) Supplementary Examinations December/January 2014/2015

Time: 3 hours Max. Marks: 70

MATHEMATICS – II

(Common to EEE, ECE, EIE, CSE and IT)

PART - A

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(Compulsory Question)

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1 Answer the following: (10 X 02 = 20 Marks)

1. Find the sine series of f(x) = k in (0,p).
2. If f(x) = x + x² in -p < x < p then find a_n.

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3. Obtain the complete solution for p + q = sin x + sin y.
4. Find a₀, f (x) = |cos x|, (-p,p).
5. Find P.I of (D²-2DD') z = x³ y.
6. State one dimensional heat equation.
7. Find the Eigen values for the matrix
   

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   2	-2	1
   -5	3	2
   -2	4	1

8. Write condition for the system AX = B is consistent.
9. Find the rank of

   1	-9	6
   4	8	5
   7	9	4

10. Using Euler's method find the solution of the initial problem dy/dx = log(x + y), y(0) = 2 at x = 0.2 by assuming h = 0.2.

PART - B

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(Answer all five units, 5 × 10 = 50 Marks)

UNIT - I

2 Reduce the quadratic form 3x² + 5y² + 3z² – 2yz + 2zx - 2xy to the canonical form. Also specify the matrix of transformation.

OR

3 State and prove Cayley-Hamilton theorem.

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UNIT - II

4 Find the root of x log₁₀ x - 1.2 = 0 by Newton Raphson method corrected to three decimal places.

OR

5 Evaluate ?₀¹ x e^x dx taking 4 intervals. Using (i) Trapezoidal rule. (ii) Simpson's 1/3 rd rule.

UNIT - III

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6 Use fourth order Runge-Kutta method to compute y for x = 0.1, given dy/dx = 1 + x², y(0) = 1 take h = 0.1.

OR

7 Find the Half range Fourier sine series f(x) = x(p - x) 0 = x = p and hence deduce that: (i) S 1/(2n-1)⁶ = p⁶/960, (ii) S 1/n⁴ = p⁴/90

UNIT - IV

8 Find the Fourier cosine transform of f(x) = e^-x².

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OR

9 Solve Z-transform Y_k+1 + Y_k = k, (k = 0), y(0) = 0.

UNIT - V

10 Solve the equation ?u/?t = ?²u/?x² with boundary conditions u(x, 0) = 3 sin(npx), u(x, t) = 0, u(a, t) = 0, where 0 < x < 1, t > 0.

OR

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11 A tightly stretched string with fixed end points x = 0 and x = l is initially in a position given by y = y₀ sin³ (px/l). if it is selected from rest from this position, find the displacement y(x, t).

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