Download VTU B-Tech/B.E 2019 June-July 1st And 2nd Semester 2015 Jan 2014 Dec 14MAT11 Engineering Mathematics I Question Paper

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14MAT11

First Semester B.E. Degree Examination, Dec.2014/Jan.2015

Engineering Mathematics - I

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Time: 3 hrs.

Max. Marks:100

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Note: Answer any FIVE full questions, selecting ONE full question from each part,

PART – 1

1. a. If Y = cos(m log x), prove that x ²y?2 + (2n +1)xy ?1 + (m² +n²)y? =0. (07 Marks)
2. b. Find the angle of intersection between the curves r = a log ? and r = a / log ?. (06 Marks)
3. c. Derive an expression to find radius of curvature in Cartesian form. (07 Marks)

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1. a. If sin?¹ y = 2 log(x +1) prove that (x² +1)Y?2 + (2n +1)(x +1)y?1 + (n² + 4)y? = 0 . (07 Marks)
2. b. Find the pedal equation rn = sec h n? . (06 Marks)
3. c. Evaluate lim (sin x)^(1/x). (07 Marks) x?0



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PART -2

1. a. Find the first four non zero terms in the expansion of f(x) = e? sin x (07 Marks)
2. b. If cosu = (x + y) / (v(x+vy)) show that x (?u/?x) + y (?u/?y) = 1/2 (06 Marks)
3. c. If u = x²+ y², v = xy + yz + zx and w=x+y+z. Find ?(u, v, w) / ?(x, y, z) Hence interpret the result. (07 Marks)



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1. a. If w = f(x, y), x = rcos? , y = r sin ? show that (?w/?x)² + (?w/?y)² = (?w/?r)² + (1/r²) (?w/?)² (07 Marks)
2. b. Evaluate lim ((sin x) / x)^(1/x²) (06 Marks) x?0
3. c. Examine the function f(x, y) = 1+sin(x²+y²) for extremum. (07 Marks)

PART - 3

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1. a. A particle moves along the curve x = 2t², y = t² -4t, z = 3t -5 . Find the components of velocity and acceleration at t = 1 in the direction i -2j+2k . (07 Marks)
2. b. Using differentiation under integral sign, evaluate ?0^8 (e? sin x) dx (06 Marks)
3. c. Use general rules to trace the curve y² (a- x) = x² (x +a), a>0 (07 Marks)

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1. a. If v = ? x r, prove that curl v = 2? where ? is a constant vector. (07 Marks)
2. b. Show that div(curlA)=0. (06 Marks)
3. c. If r = xi + yj + zk and r= |r|. Find grad div(rn) (07 Marks)

PART 4

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1. a. Obtain the reduction formula for ? cosn x dx . (07 Marks)
2. b. Solve (xy³ + y)dx + 2(x²y² + x + y4 )dy = 0 . (06 Marks)
3. c. Show that the orthogonal trajectories of the family of cardioids r = a cos²(?/2) is another family of cardioids r= b sin²(?/2) (07 Marks)

1. a. Evaluate ?0^(p/2) x sin 2x cos x dx. (07 Marks)

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2. b. Solve dy/dx + y tan x = y² sec x . (06 Marks)
3. c. If the temperature of the air is 30°C and a substance cools from 100°C to 70°C in 15 minutes, find when the temperature will be 40°C. (07 Marks)

PART-5

1. a. Solve 5x + 2y + z =12, x +4y +2z =11, 2x -3y -5z = 2 by Gauss elimination method. (06 Marks)

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2. b. Diagonalize the matrix, A = | 1 -1 2 | (07 Marks) | -1 1 2 | | 2 2 -3 |
3. c. Determine the largest eigen value and the corresponding eigen vector of A= | 2 1 1 | | 1 3 -1 | | 1 -1 2 | Starting with [0, 0, 1]? as the initial eigenvector. Perform 5 iterations. (07 Marks)

1. a. Show that the transformation y1 = x1 + 2x2 + 5x3, y2 = 2x1 + 4x2 +11x3, y3 = -x1 + 2x3 is regular and find the inverse transformation. (06 Marks)
2. b. Solve by LU decomposition method 2x + y + z =12, 8x-3y + 2z = 20, 4x +11y -z =33. (07 Marks)

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3. c. Reduce the quadratic form 2x² + 2y² - 2xy -2yz-2zx into canonical form. Hence indicate its nature, rank, index and signature. (07 Marks)

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