MORE IN Engineering Maths 3
SPPU Computer Engineering (Semester 4)
Engineering Maths 3
June 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

Answer any one question from Q1 and Q2
1 (a) Solve (any two): $i) \ \ (D^2 +9) y=x^3 - \cos 3x \\ ii) \ (D^2 + 2D + 1 ) y =e^{-x} \log x \\ iii) \ (2x-1)^2 \dfrac {d^2y}{dx^2}- 6 (2x-1) \dfrac {dy}{dx}+ 16 =8 (2x +1)^2.$
8 M
1 (b) Find Fourier sine transform of f(x)=e-x cos x, x>0.
4 M

2 (a) A resistance of 50 ohms, an inductor of 2H and a 0.005 Farad capacitor are connected in series with e.m.f. of 40 volts and an open switch. Find the instantaneous charged and current after the switch is closed at t=0, assuming that at the time charge on capacitor is 4 Coulomb.
4 M
2 (b) Solve (any one):
i) Find z-transform of $f(k) = \dfrac {\sin ak}{k}, \ k >0$ ii) Find inverse z-transform of $\dfrac {3z^2+2z} {z^2+3z+2}, 1< |z| <2$
4 M
2 (c) Solve difference equation:
f(k+2) -3f(k+1) + 2f(k)=0, f(0)=0, f(1)=1.
4 M

Answer any one question from Q3 and Q4
3 (a) The first four moments of a distribution about the value 5 ae 2, 20, 40 and 50. Obtain the first four central moments, mean, standard deviation and coefficient of skewness and kurtosis.
4 M
3 (b) A manufacture of electronic goods has 4% of his product defective. He sells the articles in packets of 300 and guarantees 90% good quality. Determine the probability that a particular packet will violet the guarantee.
4 M
3 (c) Find the directional derivative of xy2+yz3 at (2, -1, 1) along the line 2(x-2)=(y+1)=(z-1).
4 M

4 (a) in an intelligence test administered to 1000 students the average score was 42 and standard deviation 24. Find the number of students with score lying between 30 and 54.
(Given: For z=0.5, area = 0.1915).
4 M
4 (b) Prove (any one): $i) \ \nabla ^2 \left ( \dfrac {\overline a \cdot \overline b}{r} \right ) =0 \\ ii) \ \nabla \times \left ( \dfrac {\overline a \times \overline r}{r} \right ) = \dfrac {\overline a}{r} + \dfrac {(\overline a \cdot \overline r)\overline r}{r^3}$
4 M
4 (c) Show that F = r2r is conservative. Obtain the scalar potential associated with it.
4 M

Answer any one question from Q5 and Q6
5 (a) Evaluate: $\int_c \overline F \cdot d\overlin r \\ where \ \overline F = (2x+y^2) \overlien i + (3y ? 4x) \overline j \ and$ C is the parabolic arc y=x2 joining (0, 0) and (1, 1).
4 M
5 (b) Using Strokes theorem, evaluate: $\int_c (x+y) dx + (2x ? z) dy + (y+z) dz)$ Where C is the curve given by
x2 + y2 + z2 - 2ax =0, x+y=2a.
5 M
5 (c) Use divergence theorem to evaluate: $\iint_s (x\overline i - 2y^2 \overline j + z^2 \overline k) \cdot d\overline s$ where s is the surface bounded by the region x2 + y2=1 and z=0 and z=1.
4 M

6 (a) Apply Green's theorem to evaluate: $\int_c (2x^2 - y^2) dx + (x^2 + y^2) dy$ where C is the boundary of the area enclosed by the x-axis and the upper-half of the circle x2 + y2=16.
4 M
6 (b) Using Strokes theorem, evaluate: $\iint_s (\nabla \times \overline F) \cdot d\overline \\ where \ \overline F = 3 y \overline i - xz\overline j + yz^2 \overline k$ and 's' is the surface of the paraboloid 2z=x2+y2 bounded by z=2.
5 M
6 (c) Show that: $\iiint_v \dfrac {2}{r} dv = \iint_s \dfrac {\overline s \cdot \widehat n } {r} ds$
4 M

Answer any one question from Q7 and Q8
7 (a) Find the imaginary part of the analytic function whose real part is x3-3xy2+3x2 - 3y2.
4 M
7 (b) Evaluate: $\oint_c \dfrac {z^2 +1}{z^2 -1} dz$ where C is the circle: |z-1|=1.
4 M
7 (c) Find the bilinear transformation which maps the points
z=-1, 0, 1 on to the points w=0, i, 3i respectively.
4 M

8 (a) Show that analytic function f(z) with constant amplitude is constant.
4 M
8 (b) Evaluate the following integral using residue theorem: $\oint_c \dfrac {4-3z}{z(z-1) (z-2) } dz$ where C is the circle: $|z|=\dfrac {3}{2}$
4 M
8 (c) Find the image of the straight line y=3x under the transformation $w=\dfrac {z-1} {z+1}$
5 M

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