STUPIDSID
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: \( \begin{align*} &i)\left ( D^2-1 \right )y=\cos x\cosh x+3^x\\
&ii)\left ( D^2+3D+2 \right )y=e{^{e^x}}\\
&iii)\left ( 2x+3 \right )^2\frac{d^2y}{dx^2}+\left ( 2x+3 \right )\frac{dy}{dx}-2y=24x^2.\end{align*}\)/
8 M
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 \(\begin{align*}& L\frac{di}{dt}+\frac{1}{C}\int i dt = E\sin pt,\text{where}\\
&i=\frac{dq}{dt}, \text{if}\ p^2=\frac{1}{LC}\text{and}\end{align*} \)/ initiallly the current and the charge are zero, find current at any time t.
4 M
2(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.
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.
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(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=e2x 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=e2x 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 \(\int _C\bar{F}.d\bar{r}, \text{where}\bar{F}\left [ e^xy+\sin y \right ]i+\left [ e^x+x\left ( 1+\cos y \right ) \right ]j \text{where}\\ C \text{is the ellipse}\frac{x^2}{a^2}+\frac{y^2}{b^2}=1, z=0 \)/
4 M
6(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=ex 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=ex 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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