STUPIDSID
1 (a)
Evaluate \( \int^\infty_0 e^{-l} \left ( \dfrac {\cos 3t - \cos 2t}{t} \right )dt \)
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 \( f(z) = \dfrac {1}{2} \log (x^2 + y^2)+ i \tan^{-1} \left ( \dfrac {py}{x} \right ) \) is analytic.
5 M
1 (d)
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 equation using Laplace Transform \( \dfrac {d^2y}{dt^2}+ 2 \dfrac {dy}{dt}+ y = 3te^{-l}, \) given y(0)=4 and y'(0)=2.
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)
In what direction is the directional derivative of φ x2, y2, z4 at (3, -1, 2) maximum. Find its magnitude.
4 M
2 (c) (ii)
if \( \overline r = x\widehat i + y \widehat j + z\widehat k \) Prove that ∇rn=mn-2r
8 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 form of Fourier series for f(x)=em in (-L, L)
6 M
4 (b)
Prove that \( \int x^4 J_1 (x)dx=x^4J_1(x)-2x^3J_3(x)+c \)
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 \( \int_c \overline F.\overline {dr} \) 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 \( w = \dfrac {1}{z+l} \) transforms the real axis of the z-plane into a circle in the w-plane.
6 M
6 (c) (i)
Use Stoke's Theorem to evaluate \( \int_c \overline F. \overline {dr} \) where \( \overline {F}=(x^3 - y^2) \widehat{i}+2xy\widehat{j} \) 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. \( \iint_s \overline F. \widehat{n}ds \text{ where }\overline F= 4x\widehat{i} + 3y\widehat{j} - 2z\widehat{k} \) and S is the surface bounded by x=0, y=0 and 2x+2y+z=4.
4 M
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