STUPIDSID
1 (a)
\[ \text {Evaluate }\int^\infty_0 e^{-t} \left ( \dfrac {\cos 3t - \cos 2t}{t} \right )dt \]
5 M
1 (b)
Obtain the Fourier Series expression for f(x)=2x-1 in (0,3).
5 M
1 (c)
Find the value of 'p' such that the function \[ f(x)=\dfrac {1}{2} \log (x^2 + y^2)+t\tan^{-1}\left ( \dfrac {py}{x} \right )\]
5 M
1 (d)
\[ \text {If} \overline {F}(y \sin z-\sin x)\widehat{i}+ (x\sin z+2yz)\widehat{j}+ (xy\cos z+y^2)\widehat{k} \] Show that \(\overline{F}\) is irrotational. Also find its scalar potential.
5 M
2 (a)
Solve the differential equations using Laplace Transform \[ \dfrac {d^2y}{dt^2} + 2 \dfrac {dy}{dt}+ y = 3te^{-t} \] given y(0)=4 and y'(0)=2.
6 M
2 (b)
Prove that \[ J_4 (x) \left ( \dfrac {48}{x^2} - \dfrac {8}{x} \right )J(x)- \left ( \dfrac {24}{x^2}-1 \right )J_0(x) \]
6 M
2 (c) (i)
In what direction is the directional derivative of ϕ=x2y2z4 at (3, -1, 2) maximum. Find its magnitude.
4 M
2 (c) (ii)
\[ \text{If } \overline{r}=x\widehat{i}+y\widehat{j}+z\widehat{k} \ \text{prove that, } \nabla r^n=nr^{n-2}\overline{r}\]
4 M
3 (a)
Obtain the Fourier Series expansion for the function \[\begin {align*}
f(x)&=1+\dfrac {2x}{\pi}, \ \pi\le x \le 0 \\ &=1-\dfrac {2x}{\pi}
, \ 0\le x\le \pi
\end{align*} \]
6 M
3 (b)
Find an analytic function f(z)=u+iv where. \[ u-v = \dfrac {x-y}{x^2 + 4xy - y^2} \]
6 M
3 (c)
Find Laplace transform of \[ i) \ \cosh t \int^1_0 e^u \sinh u \\ ii) \ t\sqrt{1+\sin t} \]
8 M
4 (a)
Obtain the complex from of Fourier series for f(x)=em in (-I, L)
6 M
4 (b)
Prove that \[ \int x^4 J_1 (x)dx = x^4 J_2(x)-2x^3J_3 (x) +c \]
6 M
4 (c)
Find \[ i) \ L^{-1} \left [ \dfrac {2s-1}{s^2 + 4s + 29} \right ] \\
ii) \ \L^{-1} \left [ \cot ^{-1} \left ( \dfrac {s+3}{2} \right ) \right ] \]
8 M
5 (a)
Find the Bi linear Transformation which maps the points 1, i, -1 of z plane onto 0, 1, ∞ of w-plane.
6 M
5 (b)
Using Convolution theorem find \[ L^{-1} \left [ \dfrac {s^2}{(s^2+4)^2} \right ] \]
6 M
5 (c)
Verify Green's Theorem for \(\int_c \overline{F} \cdot \overline {dr}\) where \( \overline {F}= (x^2 - y^2)\widehat {i} + (x+y)\widehat{j}\) and C is the triangle with vertices (0,0), (1,1) and (2,1).
8 M
6 (a)
Obtain half range sine series for \[ \begin {align*} f(x) & = x, 0\le x \le 2 \\ &=4-x, 2\le x \le 4 \end{align*} \]
6 M
6 (b)
Prove that the transformation \(w=\dfrac {1}{z+1}\) transforms the real axis of the z-plane into a circle in the w-plane.
6 M
6 (c) (i)
Use Stroke's theorem to evaluate \(\int_c \overline {F} \cdot \overline {dr} \) where \( \overline {F} = (x^2 - y^2)\widehat{i} + 2xy \widehat{j} \) and C is the rectangle in the plane z=0, bounded by x=0, y=0, x=a and y=b.
4 M
6 (c) (ii)
Use Gauss Divergence Theorem to evaluate \(\iint_s \overline {F} \cdot \widehat {n}ds\) where \(\overline {F} 4x\widehat{i} + 3y\widehat{j},\) 2zk and S is the surface bounded by x=0, y=0, z=0 and 2x+2y+z=4.
4 M
More question papers from Applied Mathematics - 3