## 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 xy^{2}+yz^{3} 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 = r^{2}r 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=x^{2} 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

x^{2} + y^{2} + z^{2} - 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 x^{2} + y^{2}=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 x^{2} + y^{2}=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=x^{2}+y^{2} 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 x^{3}-3xy^{2}+3x^{2} - 3y^{2}.(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)