SPPU Engineering Maths 3 - December 2015 Exam Question Paper

SPPU Computer Engineering (Semester 4)
Engineering Maths 3
December 2015

Total marks: --
Total time: --

INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

Solve any one question from Q1 and Q2

1 (a) (i) (D²-2D-3) y=3e^-3x sin e^-3x + cos (e^-3x).

4 M

1 (a) (ii) (D²-2D+2) y=e^x tan x. (By variation of parameters).

4 M

1 (a) (iii)

$x^2 \dfrac {d^2 y}{dx^2}-2x \dfrac {dy}{dx} - 4y=x^2$

4 M

1 (b) Find the Fourier transform of e^-|x| and hence show that:

$\int^{\infty}_{-\infty} \dfrac {e^{i\lambda x}}{1+\lambda^2} d \lambda = \pi e^{-|x|}$

4 M

2 (a) An unchanged condenser of capacity C charged by applying an e.m.f. of value

$\dfrac {t}{\sqrt{LC}}$ through the LEDs of inductance L and of negligible resistance. The charge Q on the place of condenser satisfied the differential equation:

$\dfrac {d^2Q}{dt^2} + \dfrac {Q}{LC} = \dfrac {E}{L}\sin \dfrac {t}{\sqrt{LC}}$ Prove that the charge at any time t is given by:

$Q= \dfrac {EC}{2} \left [ \sin \dfrac {t}{\sqrt{LC}} - \dfrac {t}{\sqrt{LC}} \cos \dfrac {t}{\sqrt{LC}} \right ]$

4 M

2 (b) Find the Inverse Z-transform (any one):

$i) \ F(z) = \dfrac {z+2}{z^2 - 2z+1} \ \text {for }|z|>1. \\ ii) \ F(z) = \dfrac {10z}{(z-1)(z-2)} \ \text {(Use inversion integral method).}$

4 M

2 (c) Solve the following difference equation to find {f(k)}:

$f(k+1)+ \dfrac {1}{4} f(k) = \left ( \dfrac {1}{4} \right )^k, \ k\ge 0, \ f(0)=0$

4 M

Solve any one question from Q3 and Q4

3 (a) The first four moments of distribution about the value 4 are -1.5, 17, -30 and 108. Obtain the first four central moments, mean, standard deviation and coefficient of skewness and kurtosis.

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3 (b) If the probability that a concrete cube fails is 0.001. Determine the probability that out of 1000 cubes:
i) exactly two
ii) more than one cubes will fail.

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3 (c) Show that:

$\overline F= ( y \sin z - \sin x) \overline i (x \sin z +2 yz)\overline j + (xy\cos z+y^2)\overline k$ is irrotational and hence find scalar function ϕ s.t. F=∇ϕ.

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4 (a) Find the directional derivative of ϕ=4xz³-3x²y²z at (2, -1, 2) along a line equally inclined with co-ordinate axes.

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4 (b) For a solenoidal vector field F, show that:
curl curl curl curl F=∇⁴F.

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4 (c) The regression equations are:
8x+10y+66=0 and 40x-18y=214.
The value of variance of x is 9. Find.
i) The mean values of x and y
ii) The correlation coefficient between x and y
iii) The standard deviation of y.

4 M

Solve any one question from Q5 and Q6

5 (a) Find the work done in moving a particle once round the ellipse:

$\dfrac {x^2}{16}+ \dfrac {y^2}{4}=1, \ z=0$ under the field of force given by:

$\overline{F} = (2x-y+z)\overline i + (x+y-z^2) \overline j + (3x-2y + 4z) \overline{k}$

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5 (b) Evaluate:

$\iint_s (\nabla \times \overline{F} ) \cdot \widehat{n} \ dS$ where

$\overline F = (x^3 - y^3) \overline {i} - xyz \overline j + y^2 \overline {k}$ and S is the surface x²+4y²+z²-2x=4 above the plane x=0.

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5 (c) Evaluate:

$\iint_s \overline {F} \cdot \overline {dS}$ using divergence theorem, where

$\overline F= x^3 \overline i + y^3 \overline j+z^3 \overline k$ and S is the surface of sphere x²+y²+z²=a².

5 M

6 (a) If

$\overline F = x^2 \overline i + (x-y) \overline j + (y+z) \overline k$ displaces a particle from A(1, 0, 1) to B(2, 1, 2) along the straight line AB, find work done.

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6 (b) Evaluate:

$\int_C (e^x dx + 2ydy - dx)$ where C is the curve x²+y²=4, z=2.

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6 (c) Evaluate:

$\int_s \overline F \cdot \overline {dS}$ using Gauss divergence theorem, where:

$\overline F = 2xy\overline i + yz^2 \overline j + xz\overline k$ and S is the region bounded by:
x=0, y=0, z=0, y=3, x+2z=6.

5 M

Solve any one question from Q7 and Q8

7 (a) Show that u=y³-3x²y is harmonic function. Find its harmonic conjugate and the corresponding analytic function f(z) in terms of z.

5 M

7 (b) Using Cauchy's integral formula, evaluate:

$\int_C \dfrac {2z^2 + z +5}{(z-3 /2)^2}dz$ where C is

$\dfrac {x^2}{4} + \dfrac {y^2}{9} = 1.$

4 M

7 (c) Find the bilinear transformation which maps the points z=1, i, -1, onto the points w=0, 1, ∞.

4 M

8 (a) If f(z) is an analytic function v²=u, then show that f(z) is constant function.

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8 (b) Using residue theorem evaluate:

$\int_C \dfrac {z}{z^4 +13z^2 + 36} dz$ where 'C' is the circle

$z| = \dfrac {5}{2}.$

5 M

8 (c) Find the map of the circle |z=i|=1 under the transformation

$w=\dfrac {1}{w}$ into w-plane.

4 M

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