STUPIDSID
1 (a)
Find Laplace transform of t3 cost.
5 M
1 (b)
Find the image of |z-ai|=a under the transformation w=1/z.
5 M
1 (c)
Construct an analytic function, whose real part is 22x (x cos 2y-y sin 2y).
5 M
1 (d)
Show that the set of functions cos nx n=1,2,3,... is orthogonal on (0,2?).
5 M
2 (a)
By using Convolution Theorem. Find inverse Laplace transform pf 1s2(s+1)2.
6 M
2 (b)
Find bilinear transformation that maps the points 2,i,-2 onto the point 1,i,-1.
6 M
2 (c)
Find Fourier Series for f(x)=cos mx in (?, &-pi;) where m is not an integer. Deduce that cosmπ=2mπ(12m2+12m2−12+1m2−22⋯ ⋯1m2−n2) hence show that ∞∑119n2−1=12−π√318
8 M
3 (a)
Find complex form Fourier series f(x)-e3x in 0
6 M
3 (b)
Using Crank Nicholson method solve ∂2u∂x2=∂u∂t subject to 0 ≤ x ≤ 1 u(0,t)=0 u (1,t)=0, u(x,0)=100x(1-x) taking h=0.25 in one step.
6 M
3 (c)
Using Laplace solve (D2+2D+5)y=e-tsint when y(0)=0 and y'(0)=1.
8 M
4 (a)
Evaluate ? f(z)dz along the Parabola y=2x2 from z=0 to z=3+18i where f(z)=x2-2iy.
6 M
4 (b)
Find half range cosine series for f(x)=x0<xπ/2 =π−x π/2<x<π
6 M
4 (c)
Obtain two distinct Laurent's series of f(z)=1(1+z2)(z+2) for 1<|z|<2 and |z|>2.
8 M
5 (a)
By using Bender Schmitt method solve ∂2f∂x2=∂f∂tf(0,t)=f(5,t)=0. f(x,0)=x2(25−x2) find f in range taking h=1 and up to 5 seconds.
6 M
5 (b)
Evaluate ∫∞0e−tsin2ttdt.
6 M
5 (c)
Evaluate ∫2π0cos3θ5−4cosθdθ
8 M
6 (a)
A string is stretched and fastened to two points distance l apart, motion is started by displacing the string in the form y=asin(πxl) from which it is released at time t=0. Show that the displacement of a point at a distance x from on end at a distance x from one end at time t is given by y(x,t)−asin(πxl)cos(πctl)
6 M
6 (b)
If f(z)=u+iv is analytic and u-v=ex(cos y-sin y) find f(z) in terms of z.
6 M
6 (c)
Evaluate: L−1[s(s−2)6]
8 M
More question papers from Applied Mathematics - 3