SPPU Engineering Maths 3 - May 2014 Exam Question Paper

SPPU Electronics and Telecom Engineering (Semester 4)
Engineering Maths 3
May 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) (D²-1)y=x sin x + (1+x² e^x
ii) d²y/dx²+y = cosec x (by variation of parameters)
iii) x² d²y/dx² -4x dy/dx+6y=x⁵

8 M

1 (b) Find Fourier cosine transform of the function

$f(x)= \left\{\begin{matrix} \cos x &0 < x< a \\0 &x>a \end{matrix}\right.$

4 M

2 (a) A resistance of 50 ohms, an inductor of 2 Henry and farad capacitor are all in series with an e.m.f. of 40 volt. Find the instantaneous change and current after the switch is closed at t=0, assuming that at that time the change on the capacitor is 4 Coulomb.

4 M

2 (b) Solve (any one)
i) Find z transform for f(k)=(1/3)^|k|
Find inverse z transform of [Z² / (Z/14)(Z-1/5)] for |Z|<1/5

4 M

2 (c) Solve f(k+2)+3 f(k+1)+2f(k)=0 Given f(0)=0, f(1)=1

4 M

Answer any one question from Q3 and Q4

3 (a) Solve the following differential equation to get y(0.1) dy/dx=x-y², y(0)=1.

4 M

3 (b) Find Lagrange's long interpolating polynomial passing through set of points

x	0	1	2
y	2	3	6

Use it to find y at x=1.5 and find ∫₀² y dx.

4 M

3 (c) Find the directional derivative of ϕ=3 log (x+y+z) at (1,1,1) in the direction of tangent to the curve x=b sint, y=b cost, z=bt at t=0

4 M

4 (a) Show that (any one)

$i) \ \ \nabla^2 \left [ \nabla \cdot (\overline r / r^2) \right ] = 2/r^4 \\ ii) \ \ \nabla (\overline a \cdot \overline r / r^3)= \overline a /r^3 - 3 (\overline a \cdot \overline r)/r^5 \overline r$

4 M

4 (b) If φ, ψ satisfy Laplace equation then prove that the vector (φ ∇ ψ - ∇ ψ φ) is solenoidal.

4 M

4 (c) Use Simpson's 1/3^rd rule to find.

$\int^{0.6} _0 e^{-x^3} \ dx$ by taking seven ordinates.

4 M

Answer any one question from Q5 and Q6

5 (a) Find the work done by F=(2x + y²) i + (3y-4x)j in taking particle around the parabolic arc y=x² from (0,0) to (1,1).

4 M

5 (b) Apply Stoke's theorem to evaluate

$\oint_c (4y \ dx + 2 zdy + 6 \ ydz)$ where is curve of intersection of x²+y²+z² = 2z and z=x+1.

5 M

5 (c) Evaluate

$\iint_s (2y\widehat{i} + yzj + 2xz\widehat{k} )\cdot d\overline s$ over the surface of region bounded by y=0, y=3, x=0, z=0, x+2z=6.

4 M

6 (a) Using Green's Lemma, evaluate

$\int_c \overline F . d\overline r \ where \ \overline F = 3 y \ i + 2xj$ and c is boundary of region bounded by y=0, y=sinx fpr 0≤x≤π.

4 M

6 (b) Evaluate

$\iint_s (z^2-x) dydz - xydxdz + 3zdxdy$ Where us closed surface of region bounded by x=0, x=3, z=0, z=4-y².

5 M

6 (c) Show that

$\overline E = - \nabla \phi ? 1 /c \dfrac {\partial \overline A} {\partial t}; \ \ \overline H = \nabla \times \overline A$ are solutions of Maxwell's equations

$i) \ \nabla \cdot \overline H = 0 \\ ii) \ \nabla \times \overline H = 1/c \dfrac {\partial \overline E}{\partial t} \ if \ \nabla \cdor \cdot \overline A + 1/c \dfrac {\partial \phi}{\partial t} \ and \ \nabla^2 \overline A=1/c \dfrac {\partial ^2 A}{\partial t^2}$

4 M

Answer any one question from Q7 and Q8

7 (a) Show that the anaytic function with constant amplitude is constant.

4 M

7 (b) By using Cauchy's integral formula evaluate

$\oint_c 2Z^2+ z/z^2 -1 \ dx$ where C is the circle |z-1|=1.

5 M

7 (c) Find the bilinear transformation which maps the points z=1,0,1 on the points W=0,i,3i of w-plane.

4 M

8 (a) If u=cos hz cosy then find the harmonic conjugate v such that f(z)=u+iv is analytical function.

4 M

8 (b) Evaluate

$\int_c \dfrac {12 z-7} {(z-1)^2 (2z+3)} dz$ where c is the circle |z|=2 using Cauchy's residue theorem.

5 M

8 (c) Show that the transformation w=z+1/z-2i maps the circle |z|=2 into an ellipse.

4 M

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