SPPU Engineering Maths 3 - December 2014 Exam Question Paper

SPPU Computer Engineering (Semester 4)
Engineering Maths 3
December 2014

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

Answer any one question from Q1 and Q2

1 (a) Solve any two:
i) (D²+6D+9) y=x^-3 e^-3x
ii) (D²-2D+2) y =e^x tan x ( by variation of parameters method)

i i i) x^{2} \frac{d^{2} y}{d x^{2}} - 3 x \frac{d y}{d x} + 5 y = x^{2} \sin (\log x)

$iii) \ \ x^2 \dfrac {d^2 y}{dx^2}-3x \dfrac {dy}{dx} + 5y =x^2 \sin (\log x)$

8 M

1 (b) Find the Fourier sine cosine transforms of e^-mx m > 0.

4 M

2 (a) The currents x and y in the coupled circuits are given by:
(LD + 2R)x - Ry=E
(LD + 2R)y- Rx =0.
Find the general values of x and y in terms of t.

4 M

2 (b) Find the inverse z-transform (any one):

i) F (z) = \frac{10 z}{(z - 1) (z - 2)} (b y i n v e r s i o n i n t e g r a l m e t h o d) i i) F (z) = \frac{z}{(z - \frac{1}{4}) (z - \frac{1}{5})}, | z | > \frac{1}{4} .

$i) \ F(z) = \dfrac {10z}{(z-1)(z-2)} \ (by \ inversion \ integral \ method) \\ ii) \ F(z) = \dfrac{z}{\left ( z- \frac {1}{4}\right )\left ( z- \frac {1}{5} \right )}, \ |z|> \dfrac {1}{4}.$

4 M

2 (c) Solve the difference equation:
f(K+1)- f(K)=1, K≥0, f(0)=0.

4 M

Answer any one question from Q3 and Q4

3 (a) The first four moments about 44.5 of a distribution are -0.4, 2.99, -0.08 and 27.63. Calculate moments about means, coefficients of Skewness and Kurtosis.

4 M

3 (b) The incidence of certain disease is such that on the average 20% of workers suffer from it. If 10 workers are selected at random, find the probability that:
i) exactly 2 workers suffer from disease.
ii) not more than 2 workers suffer.

4 M

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

4 M

4 (a) A random sample of 200 screws is drawn from a population which represents size of screws. If a sample is normally distributed with a mean 3.15 cm and S.D. 0.025 cm, find expected number of screws whose size falls between 3.12 cm and 3.2 cm.
[Give: For z=12, area =0.3849; for z=2, area = 0.4772]

4 M

4 (b) show that (any one):

i) \nabla \cdot (\frac{\bar{a} \times \bar{r}}{r}) = 0 i i) \nabla^{4} (r^{2} \log r) = \frac{6}{r^{2}}

$i) \ \nabla \cdot \left ( \dfrac {\overline a \times \overline r}{r} \right ) = 0 \\ ii) \ \nabla^4 (r^2 \log r) = \dfrac {6}{r^2}$

4 M

4 (c) A fluid notaion is given by:

\bar{v} = (y \sin z - \sin x) \hat{i} + (x \sin z + 2 y z) \hat{j} + (x y \cos z + y^{2}) \hat{h}

$\overline v = (y \sin z - \sin x)\widehat i + (x \sin z + 2yz)\widehat j + (xy \cos z + y^2)\widehat h$ Is the motion irrotational? If so, find the scalar velocity potential.

4 M

Answer any one question from Q5 and Q6

5 (a) Find the work done by the force:

\bar{F} (x^{2} - y z) i + (y^{2} - z x) j + (z^{2} - x y) k

$\overline F (x^2 - yz)i + (y^2 -zx)j + (z^2 - xy)k$ in taking a particle from (1,1,1) to (3, -5, 7).

4 M

5 (b) Use divergence theorem to evaluate:

\iin_{s} (y^{2} z^{2} i + z^{2} x^{2} j + x^{2} y^{2} k) \cdot o v e r l i n e d s

$\iin_s (y^2z^2 i + z^2x^2j+x^2y^2k) \cdot overline {ds}$ where s is the upper half of the sphere x²+y²+z²=9 above the xoy plane.

5 M

5 (c) Apply Stoke's theorem to evaluate:

\int_{c} (4 y d x + 2 z d y + 6 y d z

$\int_c (4y \ dx + 2z \ dy \ + 6y \ dz$ Where C is the curve x²+y²+z²=6z, z=x+3.

4 M

6 (a) Find the work done in moving a particle from (0, 1, -1) to

(\frac{π}{2}, - 1, 2) i n a f o r c e f i e l d : \[\bar{F} (y^{2} \cos x + z^{3}) i + (2 y \sin x - 4) j + (3 x z^{2} + 2) k .

$\left ( \dfrac {\pi}{2}, -1, 2 \right ) in a force field: \[ \overline F (y^2 \cos x + z^3)i + (2y \sin x -4)j+(3xz^2 + 2)k.$

4 M

6 (b) Evaluate:

\iint_{s} [(x + y^{2}) i - 2 x j + 2 y z k] \cdot \bar{d s}

$\iint_s [(x+y^2)i-2x \ j + 2yz \ k] \cdot \overline {ds}$ Where s is the plane 2x+y+2z-6=0 considered as one of the bounding planes of the tetrahedron x=0, y=0, z=0, 2x+y+2z=6.

5 M

6 (c) Verify Stoke's theorem for: F= -y³i + x³ j and closed curve c is the boundary of the circle x²+y²=1.

4 M

Answer any one question from Q7 and Q8

7 (a) Find the condition under which:
u=ax³+ bx²y+cxy²+dy³ is harmonic.

4 M

7 (b) Evaluate:

\oint_{c} \frac{4 z^{2} + z}{z^{2} - 1} d z,

$\oint_c \dfrac {4z^2 + z}{z^2 -1}dz,$ where C: |z-1|=3.

5 M

7 (c) Show that:

w = \frac{z - i}{1 - i z}

$w= \dfrac {z-i} {1-iz}$ \] maps upper half of z-plane onto interior of unit circle in w-plane.

4 M

8 (a) Find the harmonic conjugate of:
u=r³ cos 3θ + r sin θ.

4 M

8 (c) Evaluate:

\oint_{c} \frac{\sin 2 z}{{(z + \frac{π}{3})}^{4}} d z,

$\oint_c \dfrac {\sin 2z}{\left ( z+\frac {\pi}{3} \right )^4}dz,$ where C: |z|=2.

5 M

8 (c) Find the bilinear transformation which maps the points 1, 0, i of the z-plane onto the points \[ \infty, -2, -\dfrac {1}{2} (1+i) of the w-plane.

4 M

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