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

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14MAT11

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First Semester B.E. Degree Examination, June/July 2015

Engineering Mathematics - I

Time: 3 hrs.

Max. Marks: 100

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Note: Answer FIVE full questions, selecting at least TWO questions from each part.

MODULE- I

1. a. If \( y = e^{m \sin^{-1} x} \), prove that \( (x^2-1)y_{n+2} + (2n+1)xy_{n+1} + (n^2 - m^2)y_n = 0 \) (07 Marks)
2. b. Find the pedal equation for the curve \( r = a(1 + \cos \theta) \) (06 Marks)
3. c. Derive an expression to find radius of curvature in cartesian form (07 Marks)

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OR

1. a. Find the nth derivative of \( \sin^2 x \cos^3 x \) (07 Marks)
2. b. Show that the curves \( r = a(1+\cos \theta) \) and \( r = b(1-\cos \theta) \) intersect at right angles. (06 Marks)
3. c. Find the radius of curvature when \( x = a \log(\sec t + \tan t) \), \( y = a \sec t \) (07 Marks)

MODULE- II

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1. a. Using Maclaurin's series expand \( \tan x \) upto \( x^5 \) term (07 Marks)
2. b. If \( u = x^3 + y^3 + z^3 - 3xyz \), show that \( x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} + z \frac{\partial u}{\partial z} = 3u \) (06 Marks)
3. c. Find the extreme values of \( f(x, y) = x^3 + y^3 - 3xy \) (07 Marks)

OR

1. a. Evaluate \( \lim_{x \to 0} \frac{e^x \sin x - x - x^2}{x^2 + x \log(1-x)} \) (07 Marks)

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2. b. If \( u = x \log xy \), where \( x^3 + y^3 + 3xy = 1 \), Find \( \frac{du}{dx} \) (06 Marks)
3. c. If \( u = \frac{x}{y-z}, v = \frac{y}{z-x}, w = \frac{z}{x-y} \), find \( \frac{\partial (u, v, w)}{\partial (x, y, z)} \) (07 Marks)

MODULE- III

1. a. Find div F and Curl F where \( \mathbf{F} = \text{grad}(x^3 + y^3 + z^3 - 3xyz) \) (07 Marks)
2. b. Using differentiation under integral sign, Evaluate \( \int_0^\infty \frac{x^{a-1}}{1+x} dx \) (a, 0)
   

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   Hence find \( \int_0^\infty \frac{\log x}{1+x} dx \) (06 Marks)
3. c. Trace the curve \( y^2(a-x) = x^3 \), a > 0, use general rules. (07 Marks)

OR

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1. a. If \( \mathbf{r} = xi + yj + zk \) and \( r = |\mathbf{r}| \), then prove that \( \nabla r^n = n r^{n-2} \mathbf{r} \) (07 Marks)

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2. b. Find the constants a, b, c such that \( \mathbf{F} = (x + y + az)i + (bx + 2y - z)j + (x + cy + 2z)k \) is irrotational. Also find \( \phi \) such that \( \mathbf{F} = \nabla \phi \) (06 Marks)
3. c. Using differentiation under integral sign, Evaluate \( \int_0^\infty \frac{e^{-ax} \sin x}{x} dx \) (07 Marks)

MODULE- IV

1. a. Obtain reduction formula for \( \int \cos^n x dx \) (07 Marks)
2. b. Solve : \( (1 + 2xy \cos x^2 - 2xy)dx + (\sin x^2 - x^2)dy = 0 \) (06 Marks)

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3. c. A body originally at 80°C cools down to 60°C in 20 minutes, the temperature of the air being 40°C. What will be temperature of the body after 40 minutes from the original? (07 Marks)

OR

1. a. Evaluate \( \int_0^{\pi/2} \frac{x^2 \sin^3 x}{\sqrt{1+x^2}} dx \) (07 Marks)
2. b. Solve : \( xy(1+xy^2) \frac{dy}{dx} = 1 \) (06 Marks)
3. c. Find the orthogonal trajectories of the family of confocal conics \( \frac{x^2}{a^2+k} + \frac{y^2}{b^2+k} = 1 \) where k is parameter. (07 Marks)

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MODULE- V

1. a. Solve by Gauss elimination method: \( 5x_1 + x_2 + x_3 + x_4 = 4 \), \( x_1 + 7x_2 + x_3 + x_4 = 12 \), \( x_1 + x_2 + 6x_3 + x_4 = -5 \), \( x_1 + x_2 + x_3 + 4x_4 = -6 \) (07 Marks)
2. b. Diagonalize the matrix \( A = \begin{bmatrix} 8 & -6 & 2 \\ -6 & 7 & -4 \\ 2 & -4 & 3 \end{bmatrix} \) (06 Marks)
3. c. Find the largest eigen value and the corresponding eigen vector of the matrix \( A = \begin{bmatrix} 2 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 2 \end{bmatrix} \) by power method taking the initial eigen vector (1, 1, 1)^T (07 Marks)

OR

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1. a. Solve by L U decomposition method: \( x+5y+z=14, 2x+y+3z=14, 3x+y+4z=17 \) (07 Marks)
2. b. Show that the transformation \( y_1 = 2x_1 - 2x_2 - x_3 \), \( y_2 = 4x_1 + 5x_2 + 3x_3 \), \( y_3 = x_1 - x_2 - x_3 \) is regular and find the inverse transformation. (06 Marks)
3. c. Reduce the quadratic form \( 2x_1^2 + 2x_2^2 + 2x_3^2 + 2x_1x_3 \) into canonical form by orthogonal transformation. (07 Marks)

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