SPPU Engineering Maths 3 - May 2017 Exam Question Paper

SPPU Computer Engineering (Semester 4)
Engineering Maths 3
May 2017

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

Solve any two question Q.1(a)(i, ii, iii) Solve any one question from Q.1(a,b) & Q.2(a,b,c)

1(a)(i)

\frac{d^{2} y}{d x^{2}} - y = \frac{2}{l + e^{x}} (use method  of variation of parameters)

$\frac{d^{2}y}{dx^{2}}-y=\frac{2}{l+e^{x}}\text{(use method of variation of parameters)}$ /

4 M

1(a)(ii)

(D^{2} - 4) y = e^{4 x} + 2 x^{3}

$\left ( D^{2} -4\right )y=e^{4x}+2x^{3}$

4 M

1(a)(iii)

{(2 x + 1)}^{2} \frac{d^{2} y}{d x^{2}} - 2 (2 x + 1) \frac{d y}{d x} - 12 y = 24 x .

$\left ( 2x+1 \right )^{2}\frac{d^{2}y}{dx^{2}}-2\left ( 2x+1 \right )\frac{dy}{dx}-12y=24x.$

4 M

1(b) Solve the following intergral equation using Fourier transform:

\begin{aligned} \int_{0}^{\infty} f (x) \sin λ x d λ & = 1 - λ, 0 \leq λ \leq 1 \\ = 0, λ \geq 1. \end{aligned}

$\begin{align*} \int_{0}^{\infty }f(x)\sin \lambda xd\lambda &=1-\lambda ,0\leq \lambda \leq 1\\ \\\ \ \ \ \ &=0 , \lambda \geq 1.\end{align*}$

4 M

2(a) A electrical circuit consists of an inductance 0.1 henry, a registance R of 20 ohms and a condenser of capacitance C of 25 microfarads. If the differential equation of electric circuit is:

L \frac{d^{2} q}{d t^{2}} + R \frac{d q}{d t} + \frac{q}{C} = 0,

$L\frac{d^{2}q}{dt^{2}}+R\frac{dq}{dt}+\frac{q}{C}=0,$ / then find the charge q and current i at any time t, given that when t = 0, q = 0.05 columbs and

i = \frac{d q}{d t} = 0

$i=\frac{dq}{dt}=0$

4 M

Solve any two question Q.2(b)(i, ii)

2(b)(i) Find:

z^{- 1} {\frac{1}{(z - 4) (z - 5)}}

$z^{-1}\left \{ \frac{1}{\left ( z-4 \right )\left ( z-5 \right )} \right \}$ / by inversion integral method.

4 M

2(b)(ii) Find z transform of:

f (k) = (k + 1) a^{k}, k \geq 0.

$f(k)=\left ( k+1 \right )a^{k},k\geq 0.$

4 M

2(c) Using z transform, solve the following difference equation:

\begin{aligned} f (k + 1) + \frac{1}{2} f (k) {(\frac{1}{2})}^{k}, k \geq 0 \\ f (0) = 0 \end{aligned}

$\begin{align*} f(k+1)+\frac{1}{2}f(k)\left ( \frac{1}{2} \right )^{k}, k\geq 0\\ \ f(0)=0\end{align*}$

4 M

Solve any one question from Q.3(a,b,c) &Q.4(a,b,c)

3(a) The first four moments of a distribution about the value 4 of the variable are -1.5, 17, -30 and 108. Find the central moments, β₁ and β₂

4 M

3(b) By the method of least squares, find the straight line that best fits the following data:

x	y
1	14
2	27
3	40
4	55
5	68

4 M

Solve any one question from Q.3(c) (i,ii)

3(c) There is small chance of 1/1000 for any computer produced to be defective. Determin in a sample of 2000 computers, the probability:
i) no defective and
i) 2 defectives.

4 M

Solve any one question from Q.4(a, b, c)

4(a) Team A has a probability of

\frac{2}{3}

$\frac{2}{3}$ / of wining whenever the team plays a particular game. If team A plays 4 games, find the probability that the team wins:
i) exactly two games and
ii) at least two games.

4 M

4(b) The lifetime of certain component has a normal distribution with mean 400 hours and standard deviation of 50 hours Assuming a normal sample of 1000 components, determine approximately the number of components whose lifetime lies between 340+465 hours. Given:
Z = 1.2 Area = 0.3849
Z = 1.3 Area = 0.4032

4 M

4(c) Calculate the coefficient of correlation for the following data:

x	y
10	18
14	12
18	24
22	6
22	30
30	36

4 M

Solve any two question from Q.5(a,b,c) & Solve any one question from Q.5(a, b,c) &Q.6(a, b, c)

5(a) Find the directional derivative of a function:

ϕ = 2 x^{2} + 3 y^{2} + z^{2} at (2, 1, 3)

$\phi =2x^{2}+3y^{2}+z^{2}\ \ \text{at}\left ( 2, 1, 3 \right )$ / in the direction of (i+j+k).

4 M

5(b) Show that the vector field:

\bar{F} = (x + 2 y + 4 z) i + (2 x - 3 y - z) j + (4 x - y + 2 z) k

$\bar{F}=\left ( x+2y+4z \right )i+\left ( 2x-3y-z \right )j+\left ( 4x-y+2z \right )k$ / is irrotational and hence find a scalar potential function φ such that

\bar{F} = \nabla ϕ .

$\bar{F}=\nabla\phi .$

4 M

5(c) Find the work done by a force field:

\bar{F} = x^{2} i + (x - y) j + (y + z) k

$\bar{F}=x^{2}i+\left ( x-y \right )j+\left ( y+z \right )k$ / along a straight line from (0, 0, 0) to (2, 1, 2).

5 M

6(a) Find the directional derivative of :

ϕ = 4 x z^{3} - 3 x^{2} y^{2} z at (1, 1, 1)

$\phi =4xz^{3}-3x^{2}y^{2}z\ \ \text{at}\ \ (1, 1, 1)$ / in the direction of a vector 3i-2j+k.

4 M

Solve any one question from Q.6 (b) (i, ii)

6(b)(i)

\[\nabla (\frac{\bar{a} . \bar{r}}{r^{3}}) = \frac{\bar{a}}{r^{3}} - \frac{3 (\bar{a} . \bar{r}) \bar{r}}{r^{5}} \]

$\nabla\left ( \frac{\bar{a}.\bar{r}}{r^{3}} \right )=\frac{\bar{a}}{r^{3}}-\frac{3\left ( \bar{a}.\bar{r} \right )\bar{r}}{r^{5}}$ / where

\bar{a}

$\bar{a}$ / is a constant vector.

4 M

6(b)(ii)

\nabla^{4} (r^{4}) = 120.

$\nabla^{4}\left ( r^{4} \right )=120.$

4 M

6(c) Evaluate the integral:

\int_{C} \bar{F} . d \bar{r}

$\int _{C}\bar{F}.d\bar{r}$ / along the curve x = y = z t from t = 0 to t = 2 where

\bar{F} = (x^{2} + y z) i + (y^{2} + z x) j + (z^{2} + x y) k

$\bar{F}=\left ( x^{2}+yz \right )i+\left ( y^{2}+zx \right )j+\left ( z^{2}+xy \right )k$ /

5 M

Solve any one question from Q.7(a,b,c) & Q.8(a,b,c)

7(a) If

u = 3 x^{2} y - y^{3},

$u=3x^{2}y-y^{3},$ / find v such that

f (z) = u + i v

$f(z)= u +iv$ / is analytic.

4 M

7(b) Evaluate :

\oint \frac{z + 4}{(z + 1) (z + 2)} d z,

$\oint \frac{z+4}{\left ( z+1 \right )\left ( z+2 \right )}dz,$ / where C is the circle |z| < 3.

5 M

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

4 M

8(a) If

u = 3 x^{2} - 3 y^{2} + 2 y,

$u = 3x^{2}-3y^{2}+2y,$ / find v such that f(z) = u +iv is analytic. Determine f(z) in terms of z.

4 M

8(b) Evaluate:

\oint_{C} \frac{A z^{2} + z}{z^{2} - 1} d z,

$\oint _C\frac{Az^{2}+z}{z^{2}-1}dz,$ / where C is the contour

| z - 1 | = \frac{1}{2} .

$|z-1|=\frac{1}{2}.$ /

5 M

8(c) Find the map of straight line y = x under the transformation

w = \frac{z - 1}{z + 1}

$w=\frac{z-1}{z+1}$ /

4 M

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