Engineering Mathematics 3 - Dec 2011

Computer Science Engg. (Semester 3)

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. 1 (a) Obtain the Fourier series for the function$$ f(x)=\left\{\begin{matrix} -\pi x&; 0\le x \le 1 \\\pi (2-x) &;1\le x\le 2 \end{matrix}\right. ]\ and deduce that $$ \dfrac {\pi^2}{8}=\sum^\infty_{n=1}\dfrac {1}{(2n-1)^2} $$\lt/span\gt\ltspan class='paper-ques-marks'\gt(7 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt1 (b)\lt/b\gt Obtain the half range Fourier sine for the function. $$ f(x)=\begin{Bmatrix}1/4-x &;0<x<1 2="" \\x-3="" 4&;1="" 2<x<1="" \end{matrix}="" $$\lt="" a=""\gt\ltbr\gt\ltbr\gt \lt/x\lt1\gt\lt/span\gt\ltspan class='paper-ques-marks'\gt(7 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt1 (c)\lt/b\gt Compute the constant term and the first two harmonics in the Fourier series of f(x) given by the following table. \ltbr\gt \lttable border="1" cellpadding="1" cellspacing="1" style="width: 500px;"\gt \lttbody\gt \lttr\gt \lttd\gt x\lt/td\gt \lttd\gt 0\lt/td\gt \lttd\gt 1\lt/td\gt \lttd\gt 2\lt/td\gt \lttd\gt 3\lt/td\gt \lttd\gt 4\lt/td\gt \lttd\gt 5\lt/td\gt \lt/tr\gt \lttr\gt \lttd\gt f(x)\lt/td\gt \lttd\gt 4\lt/td\gt \lttd\gt 8\lt/td\gt \lttd\gt 15\lt/td\gt \lttd\gt 7\lt/td\gt \lttd\gt 6\lt/td\gt \lttd\gt 2\lt/td\gt \lt/tr\gt \lt/tbody\gt \lt/table\gt\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt2 (a)\lt/b\gt Find the fourier transform of $$ f(x)= \left\{\begin{matrix} 1-x^2&for &|x|\le 1 \\0 &for &|x| >1 \end{matrix}\right. $$ and hence evaluate $$ \int^\infty_0 \left ( \dfrac {x \cos x - \sin x}{x^3} \right )\cos \dfrac {x}{2}dx $$\lt/span\gt\ltspan class='paper-ques-marks'\gt(7 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt2 (b)\lt/b\gt Find the Fourier cosine transform of $$ f(x)=\dfrac {1}{1+x^2} $$\lt/span\gt\ltspan class='paper-ques-marks'\gt(7 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt2 (c) \lt/b\gt Solve the integral equation $$ \int^{\infty}_0d(\theta)\cos \alpha \theta d\theta=\left\{\begin{matrix} 1-\alpha&;&0 \le \alpha \le 1 \\0 &; & a>1 \end{matrix}\right. \ hence \ evaluate \ \int^{\infty}_0 \dfrac {\sin^2 t}{t^2}dt. $$\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt3 (a)\lt/b\gt Solve two dimesional Laplace equation u\ltsub\gtxx\lt/sub\gt+u\ltsub\gtyy\lt/sub\gt=0, by the method of separation of variables.\lt/span\gt\ltspan class='paper-ques-marks'\gt(7 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt3 (b)\lt/b\gt Solve the one dimensional heat equations $$ \dfrac {\partial u}{\partial t}=\dfrac {c^2\partial^2 u}{\partial x^2}, 0<x<\pi $$="" under="" the="" conditions:="" \ltbr=""\gt (i) u(0,+)=0,u(?,t)=0 \ltbr\gt (ii) u(x,0)=u\ltsub\gt0\lt/sub\gt sinx where u\ltsub\gt0\lt/sub\gt = constant ± 0.\lt/x\lt\pi\gt\lt/span\gt\ltspan class='paper-ques-marks'\gt(7 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt3 (c) \lt/b\gt Obtain the D' Almbert's solution of one dimensional wave equation. \lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt4 (a) \lt/b\gt Fit a curve of the form y=ae\ltsup\gtbx\lt/sup\gt to the following data: \ltbr\gt \lttable border="1" cellpadding="1" cellspacing="1" style="width: 500px;"\gt \lttbody\gt \lttr\gt \lttd\gt x:\lt/td\gt \lttd\gt 77\lt/td\gt \lttd\gt 100\lt/td\gt \lttd\gt 185\lt/td\gt \lttd\gt 239\lt/td\gt \lttd\gt 285\lt/td\gt \lt/tr\gt \lttr\gt \lttd\gt y:\lt/td\gt \lttd\gt 2.4\lt/td\gt \lttd\gt 3.4\lt/td\gt \lttd\gt 7.0\lt/td\gt \lttd\gt 11.1\lt/td\gt \lttd\gt 19.6\lt/td\gt \lt/tr\gt \lt/tbody\gt \lt/table\gt\lt/span\gt\ltspan class='paper-ques-marks'\gt(7 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt4 (b)\lt/b\gt Using graphical method solve the L.P.P minimize z=20x\ltsub\gt1\lt/sub\gt=10x\ltsub\gt2\lt/sub\gt subject to the constraints \ltbr\gt x\ltsub\gt1\lt/sub\gt+2x\ltsub\gt2\lt/sub\gt?40; 3x\ltsub\gt1\lt/sub\gt+x\ltsub\gt2\lt/sub\gt?0; 4x\ltsub\gt1\lt/sub\gt+3x\ltsub\gt2\lt/sub\gt? 60; x\ltsub\gt1\lt/sub\gt?0; x\ltsub\gt2\lt/sub\gt?0\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt4 (c) \lt/b\gt Solve the following L.P.P miximize z=2x\ltsub\gt1\lt/sub\gt + 3x\ltsub\gt2\lt/sub\gt + x\ltsub\gt3\lt/sub\gt, subject to the constraints \ltbr\gt x\ltsub\gt1\lt/sub\gt+2x\ltsub\gt2\lt/sub\gt+5x\ltsub\gt3\lt/sub\gt?19, 3x\ltsub\gt1\lt/sub\gt+x\ltsub\gt2\lt/sub\gt+4x\ltsub\gt3\lt/sub\gt?25, x\ltsub\gt1\lt/sub\gt?0, x\ltsub\gt2\lt/sub\gt?0, x\ltsub\gt3\lt/sub\gt?0 using simplex method.\lt/span\gt\ltspan class='paper-ques-marks'\gt(7 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5 (a)\lt/b\gt Using the Regula - falsi method, find the root of the equation xe\ltsup\gtx\lt/sup\gt =cosx that lies between 0.4 and 0.6 Carry out four interations.\lt/span\gt\ltspan class='paper-ques-marks'\gt(7 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5 (b)\lt/b\gt Using relaxation method solve the equations. \ltbr\gt 10x-2y-3z=205; -2x+10y-2z=154; -2x-y+10z=120\lt/span\gt\ltspan class='paper-ques-marks'\gt(7 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5 (c) \lt/b\gt Using the Rayleigh's power method, find the dominant eigen value and the corresponding eigen ector of the matrix $$ A=\begin{bmatrix} 6&-2 &2 \\ -2&3 &-1 \\2 &-1 &3 \end{bmatrix} $$ starting with the initial vector [1, 1, 1]\ltsup\gtT\lt/sup\gt\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6 (a)\lt/b\gt From the following table, estimate the number of students who have obtained the marks between 40 and 45: \ltbr\gt \lttable border="1" cellpadding="1" cellspacing="1" style="width: 500px;"\gt \lttbody\gt \lttr\gt \lttd\gt Marks\lt/td\gt \lttd\gt 30-40\lt/td\gt \lttd\gt 40-50\lt/td\gt \lttd\gt 50-60\lt/td\gt \lttd\gt 60-70\lt/td\gt \lttd\gt 70-80\lt/td\gt \lt/tr\gt \lttr\gt \lttd\gt Number of student\lt/td\gt \lttd\gt 31\lt/td\gt \lttd\gt 42\lt/td\gt \lttd\gt 51\lt/td\gt \lttd\gt 35\lt/td\gt \lttd\gt 31\lt/td\gt \lt/tr\gt \lt/tbody\gt \lt/table\gt\lt/span\gt\ltspan class='paper-ques-marks'\gt(7 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6 (b)\lt/b\gt Using Lagrange's formula, find the interpolating polynomial that approximate the function described by following table: \ltbr\gt \lttable border="1" cellpadding="1" cellspacing="1" style="width: 500px;"\gt \lttbody\gt \lttr\gt \lttd\gt x\lt/td\gt \lttd\gt 0\lt/td\gt \lttd\gt 1\lt/td\gt \lttd\gt 2\lt/td\gt \lttd\gt 5\lt/td\gt \lt/tr\gt \lttr\gt \lttd\gt f(x)\lt/td\gt \lttd\gt 2\lt/td\gt \lttd\gt 3\lt/td\gt \lttd\gt 12\lt/td\gt \lttd\gt 147\lt/td\gt \lt/tr\gt \lt/tbody\gt \lt/table\gt\ltbr\gt Hence find f(3).\lt/span\gt\ltspan class='paper-ques-marks'\gt(7 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6 (c) \lt/b\gt A curve is drawn to pass through the points given by the following table: \ltbr\gt \lttable border="1" cellpadding="1" cellspacing="1" style="width: 500px;"\gt \lttbody\gt \lttr\gt \lttd\gt x\lt/td\gt \lttd\gt 1\lt/td\gt \lttd\gt 1.5\lt/td\gt \lttd\gt 2\lt/td\gt \lttd\gt 2.5\lt/td\gt \lttd\gt 3\lt/td\gt \lttd\gt 3.5\lt/td\gt \lttd\gt 4\lt/td\gt \lt/tr\gt \lttr\gt \lttd\gt y\lt/td\gt \lttd\gt 2\lt/td\gt \lttd\gt 2.4\lt/td\gt \lttd\gt 2.7\lt/td\gt \lttd\gt 2.8\lt/td\gt \lttd\gt 3\lt/td\gt \lttd\gt 2.6\lt/td\gt \lttd\gt 2.1\lt/td\gt \lt/tr\gt \lt/tbody\gt \lt/table\gt\ltbr\gt Using Weddle's rule, estimate the area bonded bt the curve, the x-axis and the lines x=1, x=4.\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt7 (a)\lt/b\gt Solve the Laplace's equation u\ltsub\gtxx\lt/sub\gt+u\ltsub\gtyy\lt/sub\gt=0, given that; \ltimg alt="" src="https://i.imgur.com/wWQDX30.png"\gt \lt/span\gt\ltspan class='paper-ques-marks'\gt(7 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt7 (b)\lt/b\gt $$ Solve \ \dfrac {\partial^2u}{\partial t^2}=4 \dfrac {\partial^2 u}{\partial x^2} $$ subject to u(0,t)=0; u(4,t)=0; u(x,0)=x (4-x). Take h=1, k=0.5\lt/span\gt\ltspan class='paper-ques-marks'\gt(7 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt7 (c) \lt/b\gt Solve the equation $$ \dfrac {\partial u}{\partial t}=\dfrac {\partial^2 u}{\partial x^2} $$ subject to the conditions u(x,0)=sinx, 0?x?1; u(0, t)=u(1, t)=0 using Schmidt's method. Carry out computations for two levels, taking h-1/3, k=1/36.\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt8 (a)\lt/b\gt Find the Z-transform of : $$ i) \ (2n-1)^2 \\ ii) \ \cos \left (\dfrac {n\pi}{2}+\pi/4 \right ) $$\lt/span\gt\ltspan class='paper-ques-marks'\gt(7 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt8 (b)\lt/b\gt Obtain the inverse Z-transform of $$ \dfrac {4x^2-2z}{z^3-5z^2+8z-4} $$</span>(7 marks) 8 (c) Solve the difference equation y_n+2 +6y_n+1+9y_n=2n with y₀=y₁=0 using Z transforms.(6 marks)