STUPIDSID
1(a)
Evaluate \( \int ^{\infty}_0e^{-2t}\left ( \dfrac{\sinh t \sin t}{t} \right )dt \)
5 M
1(b)
Obtain the Fourier Series expression for
f(x) = 9-x2 in (-3,3)
f(x) = 9-x2 in (-3,3)
5 M
1(c)
Find the value of 'p' sucj that the function f(z) expressed in polar co-ordinates as
f(z) = r3 cos p&theta
f(z) = r3 cos p&theta
5 M
1(d)
If \( \bar{F}=(y^2-z^2+3yz-2x)\hat{i}+(3xz+2xy)\hat{j}+(3xy-2xz+2z)\hat{k}. \)
Show that \( \bar{F} \) is irrotational and solenoidal.
Show that \( \bar{F} \) is irrotational and solenoidal.
5 M
2(a)
Solve the differential equation using Laplace Transform \( \dfrac{d^2y}{dt^2}+4\frac{dy}{dt}+8y=1 \), given y(0)=0 and y'(0)=1
6 M
2(b)
Prove that \[J_4(x)=\left ( \dfrac{48}{x^3}-\dfrac{8}{x} \right )J_1(x)-\left ( \dfrac{24}{x^2}-1 \right )J_0(x)\]
6 M
2(c)(i)
Find the directional derivative of ϕ = 4xz3 - 3x2y2z at (2,-1,2) in the direction of \( 2\hat{i}+3\hat{j}+6\hat{k}. \)
8 M
2(c)(ii)
if \( \bar{r}=x\hat{i}+y\hat{j}+z\hat{k} \)
prove that \( \nabla \log r=\dfrac{\bar{r}}{r^2} \)
prove that \( \nabla \log r=\dfrac{\bar{r}}{r^2} \)
8 M
3(a)
Show that { cosx, cos2x, cos3x.....} is a set of orthogonal fundtions over (-π, π), Hence construct an orthogonal set.
6 M
3(b)
Find an analytic function f(z)=u+iy where. \[u=\dfrac{x}{2}\log(x^2+y^2)-y\tan^{-1}\left ( \dfrac{y}{x} \right )+\sin x \cosh y\]
6 M
Find the Laplace transform of
3(c)(i)
\( \int ^1_0 ue^{-3u}\cos^2u du \)
4 M
3(c)(ii)
\( t\sqrt{1+\sin t} \)
4 M
4(a)
Find the Fourier Series for \( f(x)=\dfrac{3x^2-6\pi x+2\pi^2}{12} \) in (0,2π)
Hence deduce that \( \dfrac{1}{1^2}+\dfrac{1}{2^2}+\dfrac{1}{3^2}\cdots=\dfrac{\pi^2}{6} \)
Hence deduce that \( \dfrac{1}{1^2}+\dfrac{1}{2^2}+\dfrac{1}{3^2}\cdots=\dfrac{\pi^2}{6} \)
6 M
4(b)
Prove that \[\int ^b_0 xJ_0(ax)dx=\dfrac{b}{a}J_1(ab)\]
6 M
Find
4(c)(i)
\( L^{-1}\left [ \log\left ( \dfrac{s^2+1}{s(s+1)} \right ) \right ] \)
4 M
4(c)(ii)
\( L^{-1}\left [ \log\left ( \dfrac{s+2}{s^2-2s+17} \right ) \right ] \)
4 M
5(a)
Obtain the half range cosine series for \[\begin {align*} f(x)&=x, 0
6 M
5(b)
Find the Bi-linear Transformation which maps the points 1,i,-1 of z plane onto i,0,-i of w-plane
6 M
5(c)
Verify Green's Theorem for \( \int _c \overline{F} .\overline{br} \) where \( \bar{F}=(x^2-xy)\hat{i}+(x^2-y^2)\hat{j} \) and C is the curve bounded by x2 = 2y and x=y
8 M
6(a)
Show that the transformation \( w=\dfrac{i-iz}{1+z} \) maps the unit circle |z|=1 into real axis of w plane.
6 M
6(b)
Using Convolution theorem find \[L^{-1}\left [ \dfrac{s}{(s^2+1)(s^2+4)} \right ]\]
6 M
6(c)(i)
Use Gauss Divergence Theorem to evaluate \( \iint _s \bar{F}.\hat{n}ds\ \text{where}\ \bar{F}=x\hat{i}+y\hat{j}+z\hat{k} \) and S is the sphere x2 + y2 + z2 = 9 and \( \hat{n} \) is the outward normal to S
8 M
6(c)(ii)
Use Stoke's Theorem to evaluate \( \int _c\overline{F}.\overline{dr}\ \text{where}\ \bar{F}=x^2\hat{i}-xy\hat{j} \) and C is the square in the plane z=0 and bounded by x=0, y=0, x=a and y=a.
8 M
More question papers from Applied Mathematics - 3