SPPU Engineering Maths 3 - December 2016 Exam Question Paper

SPPU Computer Engineering (Semester 4)
Engineering Maths 3
December 2016

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 one question.Q1(a,b) Q2(a,b,c)

1(a) Solve any two:

$\begin{align*} &i)\left ( D^2-1 \right )y=\cos x\cosh x+3^x\\ &ii)\left ( D^2+3D+2 \right )y=e{^{e^x}}\\ &iii)\left ( 2x+3 \right )^2\frac{d^2y}{dx^2}+\left ( 2x+3 \right )\frac{dy}{dx}-2y=24x^2.\end{align*}$ /

8 M

Solve any one question.Q1(a,b) & Q2(a,b,c)

1(a) Solve any two:

8 M

1(b) Find the Fourier transform of

$f(x)=\left\{\begin{matrix} 1-x^2 & |x|\leq 1\\ 0, &|x|>1 \end{matrix}\right.$ / Hence evaulate

$\int_{0}^{\infty }\left ( \frac{x\cos x-\sin x}{x^3} \right )\cos \frac{x}{2}dx.$ /

4 M

1(b) Find the Fourier transform of

$f(x)=\left\{\begin{matrix} 1-x^2 & |x|\leq 1\\ 0, &|x|>1 \end{matrix}\right.$ / Hence evaulate

$\int_{0}^{\infty }\left ( \frac{x\cos x-\sin x}{x^3} \right )\cos \frac{x}{2}dx.$ /

4 M

2(a) An e.m.f E sin pt is applied at t =0 to a circuit containing a condenser C and inductance L in series, the current I satisfies the equation

$\begin{align*}& L\frac{di}{dt}+\frac{1}{C}\int i dt = E\sin pt,\text{where}\\ &i=\frac{dq}{dt}, \text{if}\ p^2=\frac{1}{LC}\text{and}\end{align*}$ / initiallly the current and the charge are zero, find current at any time t.

4 M

2(a) An e.m.f E sin pt is applied at t =0 to a circuit containing a condenser C and inductance L in series, the current I satisfies the equation

4 M

2(b) Find the inverse z-transform(any one):i)

$F(z)\frac{1}{\left ( z-3 \right )\left ( z-4 \right )}, |z|<3$
ii) Find inverse z-transform of

$F(z)=\frac{z^2}{z^2+1}$ / using inversion integral method.

4 M

2(b) Find the inverse z-transform(any one):i)

$F(z)\frac{1}{\left ( z-3 \right )\left ( z-4 \right )}, |z|<3$
ii) Find inverse z-transform of

$F(z)=\frac{z^2}{z^2+1}$ / using inversion integral method.

4 M

2(c) Solve the following difference equation to find f(k).

$12f\left ( k+2 \right )-7\left ( k+1 \right )+f(k)=0\\ k\geq 0, f(0)=0, f(1)=3.$ /

4 M

2(c) Solve the following difference equation to find f(k).

$12f\left ( k+2 \right )-7\left ( k+1 \right )+f(k)=0\\ k\geq 0, f(0)=0, f(1)=3.$ /

4 M

Solve any one question.Q3(a,b,c) Q4(a,b,c)

3(a) The first four moments of a distribution about 30.2 are 0.255 6.222, 30.211 and 400.25. Calculate the first four moments about the mean. Also calculate coefficient of skewness.

4 M

Solve any one question.Q3(a,b,c) & Q4(a,b,c)

3(a) The first four moments of a distribution about 30.2 are 0.255 6.222, 30.211 and 400.25. Calculate the first four moments about the mean. Also calculate coefficient of skewness.

4 M

3(b) Suppose heights of students follows normal distribution with mean 190 cm and variance 80 cm². In a school of 1000 students, how many would you expect to be above 200 cm tall? ( Given that : A₁(z>1.1180)= 0.13136).

4 M

3(c) Find the directional derivative of ϕ=e^2x cos(yz) at (0, 0, 0) in the direction of tangent to the curve

$x=a\sin t, y=a\cos t, z= at, \text{at} t =\frac{\pi }{4}$

4 M

3(c) Find the directional derivative of ϕ=e^2x cos(yz) at (0, 0, 0) in the direction of tangent to the curve

$x=a\sin t, y=a\cos t, z= at, \text{at} t =\frac{\pi }{4}$

4 M

4(a) Prove the following (any one)

$\begin{align*}&i)\nabla.\left ( \frac{\bar{a}\times \bar{r}}{r} \right )=0\\ &ii)\nabla^4\left ( r^2\log r \right )=\frac{6}{r^2}. \end{align*}$ /

4 M

4(a) Prove the following (any one)

$\begin{align*}&i)\nabla.\left ( \frac{\bar{a}\times \bar{r}}{r} \right )=0\\ &ii)\nabla^4\left ( r^2\log r \right )=\frac{6}{r^2}. \end{align*}$ /

4 M

4(b) Show that vector field given by

$\bar{F}=\left ( y^2 \cos x+z^2 \right )\bar{i}+\left ( 2y\sin x \right )\bar{j}+\left ( 2xz \right )\bar{k}$ / is conservative and find scalar field ϕ such that

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

4 M

4(b) Show that vector field given by

$\bar{F}=\left ( y^2 \cos x+z^2 \right )\bar{i}+\left ( 2y\sin x \right )\bar{j}+\left ( 2xz \right )\bar{k}$ / is conservative and find scalar field ϕ such that

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

4 M

4(c) If

$\sum x_i=30, \sum y_i=40, \sum x^2_i=220, n =5, \sum y^2_i=340 \\\text{and}\sum x_iy_i=214,$ / them obtain regression lines for this data.

4 M

4(c) If

$\sum x_i=30, \sum y_i=40, \sum x^2_i=220, n =5, \sum y^2_i=340 \\\text{and}\sum x_iy_i=214,$ / them obtain regression lines for this data.

4 M

Solve any one question.Q5(a,b,c) Q6(a,b,c)

5(a) Evaluate the integral

$\int \bar{F}.d\bar{r}, \text{where}\\ \bar{F}=\left ( y\sin z-\sin x \right )i+\left ( x\sin z+2yz \right )j+\left ( xy\cos z+y^2 \right )k$ / from the point (0, 0, 0) to

$\left ( \frac{\pi }{2} ,1,\frac{\pi }{2}\right )$ . Is &bar{F} conservative?

5 M

Solve any one question.Q5(a,b,c) & Q6(a,b,c)

5(a) Evaluate the integral

$\int \bar{F}.d\bar{r}, \text{where}\\ \bar{F}=\left ( y\sin z-\sin x \right )i+\left ( x\sin z+2yz \right )j+\left ( xy\cos z+y^2 \right )k$ / from the point (0, 0, 0) to

$\left ( \frac{\pi }{2} ,1,\frac{\pi }{2}\right )$ . Is &bar{F} conservative?

5 M

5(b) Using divergence theorem, evaluate

$\iint_s\bar{F}.\hat{n}ds, \text{where}\bar{F}=x\hat{i}-y\hat{j}+\left ( z^2-1 \right )\hat{k}$ / and s is the total surfaces of the cylinder bounded by z= 0, z=1, and

$x^2+y^2=4$

4 M

5(b) Using divergence theorem, evaluate

$\iint_s\bar{F}.\hat{n}ds, \text{where}\bar{F}=x\hat{i}-y\hat{j}+\left ( z^2-1 \right )\hat{k}$ / and s is the total surfaces of the cylinder bounded by z= 0, z=1, and

$x^2+y^2=4$

4 M

5(c) Use strokes' theorem of evaluate

$\iint _s\left ( \nabla\times \bar{F} \right ).\hat{n} ds, \text {where} \bar{F} =yi+\left ( x-2xz \right )j-xy\hat{k}$ / and S is the surface of the sphere

$x^2+y^2+z^2=a^2$ , above the xy plane.

4 M

5(c) Use strokes' theorem of evaluate

$\iint _s\left ( \nabla\times \bar{F} \right ).\hat{n} ds, \text {where} \bar{F} =yi+\left ( x-2xz \right )j-xy\hat{k}$ / and S is the surface of the sphere

$x^2+y^2+z^2=a^2$ , above the xy plane.

4 M

6(a) Evaluate the integral

$\int _C\bar{F}.d\bar{r}, \text{where}\bar{F}\left [ e^xy+\sin y \right ]i+\left [ e^x+x\left ( 1+\cos y \right ) \right ]j \text{where}\\ C \text{is the ellipse}\frac{x^2}{a^2}+\frac{y^2}{b^2}=1, z=0$ /

4 M

6(a) Evaluate the integral

4 M

6(b) Evaluate

$\iint _s\left ( yz\hat{i} +zx\hat{j}+xy\hat{z}\right ).\bar{n}dS,$ / where S is the curved surface of the cone

$x^2+y^2=z^2, z=4$ .

5 M

6(b) Evaluate

$\iint _s\left ( yz\hat{i} +zx\hat{j}+xy\hat{z}\right ).\bar{n}dS,$ / where S is the curved surface of the cone

$x^2+y^2=z^2, z=4$ .

5 M

6(c) If

$\bar{E}=\nabla\phi \\\text{and}\nabla^2\phi =-4\pi \rho ,$ / prove that:

$\iint_s\bar{E} .d\bar{s}=-4\pi \iint_v\int \rho dV.$ /

4 M

6(c) If

$\bar{E}=\nabla\phi \\\text{and}\nabla^2\phi =-4\pi \rho ,$ / prove that:

$\iint_s\bar{E} .d\bar{s}=-4\pi \iint_v\int \rho dV.$ /

4 M

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

7(a) Find the harmonic conjugate of ν=e^x sin y such that f(z)= u + iν is analytic. Find f(z) in terms of z.

4 M

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

7(a) Find the harmonic conjugate of ν=e^x sin y such that f(z)= u + iν is analytic. Find f(z) in terms of z.

4 M

7(b) Using Cauchy's Integral formula evaluate

$\oint _c\frac{3z^3+5z+z}{\left ( z-2 \right )^2} dz \\ \text{where} \\ C is \frac{X^2}{9}\frac{Y^2}{25}=1$ /

5 M

7(b) Using Cauchy's Integral formula evaluate

$\oint _c\frac{3z^3+5z+z}{\left ( z-2 \right )^2} dz \\ \text{where} \\ C is \frac{X^2}{9}\frac{Y^2}{25}=1$ /

5 M

7(c) Find the map of the strip x >0, 0

4 M

7(c) Find the map of the strip x >0, 0

4 M

8(a) Show that analytic function with constant amplitude is constant.

4 M

8(a) Show that analytic function with constant amplitude is constant.

4 M

8(b) Evaluate

$\oint _c\frac{z-3}{z^2+2z+5}dz, \text{where} 'C' \text{is}|z|=1$ /

5 M

8(b) Evaluate

$\oint _c\frac{z-3}{z^2+2z+5}dz, \text{where} 'C' \text{is}|z|=1$ /

5 M

8(c) Find the bilinear transformation which maps the points z= 0, 1,2 onto the points w=1,

$\frac{1}{2},\frac{1}{3}$ / respectively.

4 M

8(c) Find the bilinear transformation which maps the points z= 0, 1,2 onto the points w=1,

$\frac{1}{2},\frac{1}{3}$ / respectively.

4 M

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