Question Paper: Engineering Maths 3 : Question Paper Jun 2015 - Computer Engineering (Semester 4) | Pune University (PU)
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## Engineering Maths 3 - Jun 2015

### Computer Engg (Semester 4)

TOTAL MARKS: 100
TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.
(2) Attempt any four from the remaining questions.
(3) Assume data wherever required.
(4) Figures to the right indicate full marks.

### 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 marks) 1 (b) Find Fourier sine transform of f(x)=e<ssup>-x cos x, x>0.</ssup>(4 marks) 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 marks) 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 marks)
2 (c) Solve difference equation:
f(k+2) -3f(k+1) + 2f(k)=0, f(0)=0, f(1)=1.
(4 marks)

### 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 marks) 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 marks) 3 (c) Find the directional derivative of xy2+yz3 at (2, -1, 1) along the line 2(x-2)=(y+1)=(z-1).(4 marks) 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 marks)
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 marks) 4 (c) Show that F = r2r is conservative. Obtain the scalar potential associated with it.(4 marks)

### 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 marks) 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 marks)
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 marks) 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 marks) 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 marks) 6 (c) Show that: $$\iiint_v \dfrac {2}{r} dv = \iint_s \dfrac {\overline s \cdot \widehat n } {r} ds$$(4 marks)

### 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 marks) 7 (b) Evaluate: $$\oint_c \dfrac {z^2 +1}{z^2 -1} dz$$ where C is the circle: |z-1|=1.(4 marks) 7 (c) Find the bilinear transformation which maps the points
z=-1, 0, 1 on to the points w=0, i, 3i respectively.
(4 marks)
8 (a) Show that analytic function f(z) with constant amplitude is constant.(4 marks) 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 marks) 8 (c) Find the image of the straight line y=3x under the transformation $$w=\dfrac {z-1} {z+1}$$(5 marks)