STUPIDSID
Please log in to continue.
Please log in to continue.
Please log in to continue.
Please log in to continue.
Please log in to continue.
Please log in to continue.
Please log in to continue.
1 (a)
Find the Laplace transform of t-t cosh2t.
5 M
1 (b)
Find the fixed points of \[w=\dfrac{3z-4}{z-1}\]. Also express it in the normal form \[\dfrac{1}{w-\alpha}=\dfrac{1}{z-\alpha}+\lambda \ where\ \lambda\] is a constant and α is the fixed point is this transformation parabolic?
5 M
1 (c)
Evaluate \[\int_{0}^{1+i}\limits (x^{2}+iy)dz\] along the path i) y=x, ii) y=x2
5 M
1 (d)
Prove that \[f_{1}(x)=1,f_{2}(x)=x,f_{3}(x)=\dfrac{3x^{2}-1}{2}\] are orthogonal over (-1,1).
5 M
2 (a)
Find inverse Laplace transform of \[\dfrac{2s}{s^{4}+4}\]
6 M
2 (b)
Find the image of the triangular region whose vertices are i,1+i,1-1 under the transformation w=z+4-2i. Draw the sketch.
6 M
2 (c)
Obtain fourier expansion of \[f(x)=\left | \cos x \right |in \ (-\pi,\pi)\]
8 M
3 (a)
Obtain complex form of fourier series for f(x)=cosh 2x+sinh2x in (-2,2).
6 M
3 (b)
Using Carnk -Nicholson simplified formula solve \[\dfrac{\partial^2 u}{\partial x^2}-\dfrac{\partial u}{\partial t}=0\ given\ u(0,t)=0,u(4,t)=0,u(x,0)=\dfrac{x}{3}\] ujj for i=0,1,2,3,4 and j=0,1,2.
6 M
3 (c)
Solve the equation \[y+\int_{0}^{t}\limits ydt=1-e^{-t}\]
8 M
4 (a)
Evaluate \[\int_{0}^{2x}\limits \dfrac{d\theta}{5+3 \sin\theta}\].
6 M
4 (b)
Find half- range cosine series for f(x)=ex,0
6 M
4 (c)
Obtain two distinct Laurent's series for \[f(z)=\dfrac{2z-3}{z^{2}-4z-3}\] in powers of (z-4) indicating the regions of convergence.
8 M
5 (a)
Solve \[\dfrac{\partial^2 u}{\partial x^2}-2\dfrac{\partial u}{\partial t}=0\] by Bender-schmidt method, given u(0,t)=0,u(4,t)=0.u(x,0)=x(4-x). Assume h=1 and find the value of u upto t=5.
6 M
5 (b)
Find the Laplace transform of \[e^{-4t}\int_{0}^{t}\limits u\sin3udu\]
6 M
5 (c)
Evaluate \[\int_{c}\limits \dfrac{z+3}{z^{2}+2z+5}dz\] where C is the circle i) |z|=1, ii)|z+1-i|=2
8 M
6 (a)
Find inverse Laplace transform of \[\dfrac{s}{(s^{2}-a^{2})^{2}}\] by using convolution theorem.
6 M
6 (b)
Find an analytic function f(z)=u+iv where u+v=ex(cosy+siny).
6 M
6 (c)
Solve the equation \[\dfrac{\partial u}{\partial t}=k\dfrac{\partial^2 u}{\partial ^2x^2}\] for the conduction of heat along a rod of length l subjected to following conditions
(i) u is intinity for t→∞
(ii) \[\dfrac{\partial u}{\partial x}=0\] for x=0 and x=l for any time t
(iii) u=lx-x2 for t=0 between x=0 and x=l.
(i) u is intinity for t→∞
(ii) \[\dfrac{\partial u}{\partial x}=0\] for x=0 and x=l for any time t
(iii) u=lx-x2 for t=0 between x=0 and x=l.
8 M
More question papers from Applied Mathematics - 3