SPPU Engineering Maths 3 - May 2014 Exam Question Paper

SPPU Computer 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 of the following:
i) (D²+4)y=cos 3x. cos x
ii) (D²+6D+9) y=e^3x/x² ..... ( by variation of parameters method)
iii) x²(d³y/dx³)+2x² (d²y/dx²)+2y=20 (x+1/x)

8 M

1 (b) Obtain f(k) given that,
f(k+2)+5f(k+1)+6f (k)=0, k≥0 f(0)=0, f(1)=2 by using Z transform.

4 M

2 (a) An emf E sin (pt) is applied at t=0 to a circuit containing a condenser 'C' and Inductance 'L' in series. The current 'x' satisfies the equation.

L (d x / d t) + \frac{1}{2} \int x d t = E sin (p t) W h e r e = \frac{- d q}{d t} . I f p^{2} = \frac{1}{L C}

$L (dx/dt) + \dfrac {1} {2} \int x \ dt =E \sin (pt) \ Where \ = \dfrac {-dq}{dt}. \ If \ p^2 = \dfrac {1} {LC}$ and initially the current x and change q is zero then show that current in the circuit at any time is E/2L t sin (pt).

4 M

2 (b) Solve the integral equation

\int_{0}^{\infty} f (x) cos λ x d x = {\begin{matrix} 1 - λ & 0 \leq λ \leq 1 0 & λ > 1 \end{matrix}

$\int^\infty_0 f(x) \cos \lambda x \ dx = \left\{\begin{matrix} 1-\lambda & 0 \le \lambda \le 1\\ 0 & \lambda > 1 \end{matrix}\right.$ And hence show that

\int_{0}^{\infty} \frac{{sin}^{2} z}{z^{2}} d z = π / 2

$\int^\infty_0 \dfrac {\sin^2 z}{z^2} dz = \pi/2$

4 M

2 (c) Attempt any one
i) Find the z-transform of f(k)=e^-2k cos (5k+3)
ii) Find the inverse z-transform of

\frac{z (z + 1)}{z^{2} - 2 z + 1}, | z | > 1

$\dfrac {z(z+1)}{z^2 - 2z +1}, \ |z| >1$

4 M

Answer any one question from Q3 and Q4

3 (a) The first four moments of a distribution about 2 are 1, 2.5, 5.5 and 16. Calculate the first four moments about the mean, A, M, S, D, β₁ and β₂.

5 M

3 (b) In a certain examination 200 students appeared. Average marks obtained were 50% with standard 5%. How many students do you expect to obtain more than 60% of marks, supposing that the marks are distributed normally ?( Given z=2 ; A = 0.4772).

4 M

3 (c) Find the directional derivatives of:
ϕ xy²+yz²+zx² at (1,1,1) along the line 2(x-2)=y+1=z-1

4 M

4 (a) Calculate the coefficient of correlation for the following data.

x	1	2	3	4	5	6	7	8	9
y	9	8	10	12	11	13	14	16	15

5 M

4 (b) Prove the following (any one):

\nabla^{4} r^{4} = 120 \nabla \cdot [r^{\nabla} \frac{1}{r^{5}}] = \frac{15}{r^{6}}

$\nabla^4 r^4 =120 \\ \nabla \cdot \left [ r^\nabla \dfrac {1}{r^5} \right ] = \dfrac {15}{r^6}$

4 M

4 (c) Such that

¯ ¯ ¯ F = (x^{2} - y z) ¯ i (y^{2} - z x) ¯ J + (z^{2} - x y) ¯ k

$\overline {F}= (x^2-yz)\overline{i} (y^2 - zx)\overline J + (z^2 - xy)\overline k$ is irrotational. Also find ϕ such that F=∇ϕ.

4 M

Answer any one question from Q5 and Q6

5 (a) Find the work done in moving a particle along

x = a cos Θ, y = a sin Θ, z = b Θ, f r o m Θ = \frac{π}{4} t o Θ = \frac{π}{2}

$x=a \cos \Theta, \ y=a \sin \Theta, \ z=b\Theta, \ from \ \Theta = \dfrac {\pi}{4} \ to \ \Theta = \dfrac {\pi}{2}$ under a field of force given by

¯ ¯ ¯ F = - 3 a {sin}^{2} Θ cos Θ ˆ i + a (2 sin Θ - 3 sin Θ) ˆ J + b sin 2 Θ ˆ k

$\overline F = -3a \sin^2 \Theta \cos \Theta \widehat{i} +a ( 2 \sin \Theta - 3 \sin \Theta)\widehat J +b \sin 2 \Theta \widehat {k}$

4 M

5 (b) Evaluate

\iint_{s} (y z ˆ i + z x ˆ j + x y ˆ k) . d ¯ s,

$\iint_s ( yz\widehat{i}+ zx\widehat{j}+xy\widehat {k}).d\overline s,$ where s is the curved surface of the cone x²+y²=z², z=4

4 M

5 (c) Using Strokes Theorem to evaluate

\int_{c} (4 y ˆ i + 2 z ˆ j + 6 y ˆ k) . d ¯ k

$\int_c (4y \widehat i + 2z\widehat j + 6y\widehat k). d \overline k$ where C is the curve of intersection of x² + y² + z² = 2z and x=z-1

4 M

6 (a) A vector field is given by

¯¯¯F=(2x−cosy)ˆi+x(4+siny)ˆj, evaluate ∫c¯¯¯F.d\overlienr,

$\overline F = (2x - \cos y)\widehat i + x (4+\sin y)\widehat j, \ evaluate \ \int_c \overline F.d\overlien r,$ where C is the ellipse

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1, z = 0

$\dfrac {x^2}{a^2}+ \dfrac {y^2}{b^2}=1, \ z=0$

4 M

6 (b) Prove that

∭_{v} \frac{1}{r^{2}} d v = \iint_{s} \frac{1}{r^{2}} ¯ r . d ¯ s,

$\iiint_v \dfrac {1}{r^2}dv = \iint_s \dfrac {1}{r^2}\overline r . d\overline s,$ where s is closed surface enclosing the volume v. Hence evaluate

∬s\dfacxi+yj+zˆkr2⋅d¯s,

$\iint_s \dfac { xi + yj + z \widehat k}{r^2}\cdot d\overline s,$ k where s is the surface of the sphere x²+y²+z²=a².

4 M

6 (c)

I f ¯ ¯ ¯ E = \nabla ϕ a n d \nabla^{2} ϕ = - 4 π p .

$If \ \overline E = \nabla \phi \ and \ \nabla^2 \phi = - 4 \pi p.$ Prove that

$\iint_s \overline E\cdot d\overline s = - 4\pi \iiint_v \rho \ dv$

4 M

Answer any one question from Q7 and Q8

7 (a) Find the value of p such that the function. f(x) = r² cos 2 Θ + i r² sin p Θ becomes analytical function.

4 M

7 (b) Evaluate

$\oint_c \dfrac {z^2 + \cos ^2 z}{\left ( z - \frac {\pi}{4} \right )^3}dz$ where C is a circle x²+y²=1

5 M

7 (c) Find the bilinear transformation which maps 1, i, -1 from z plane into i, 0, -i from the w plane.

4 M

8 (a) Determine the analytic function f(z) whose real part is

$U=x^3 ? 3xy^2+3x^2-3y^2+1$

4 M

8 (b) Prove

$\oint _c \left [ \dfrac {\sin \pi z^2 + 2z}{(z-1)^2 (z-2)} \right ]dz$ where C is a circle x2+y²=16

5 M

8 (c) Show that the transformation ω=sin z transforms the straight lines x=c of z plane into hyperbola in the ω plane.

4 M

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