STUPIDSID
1(a)
Find the Laplace Transform of cos t cos 2t cos 3t.
5 M
1(b)
If A=[1411], find A7-9A2+I.
5 M
1(c)
Show that ∮clogzdz=2πi , where C is the unit circle |z|=1.
5 M
1(d)
The incidence of an occupational disease in an industry is such that workers have 20% chance of suffering from it. What is the probability that out of 6 workers 4 or more will catch the disease?
5 M
2(a)
Find an analytic function whose real part is e-x (y cos y-x sin y).
6 M
2(b)
Evaluate ∫∞0e−1∫t0sinuudu dt
6 M
2(c)
Find the orthogonal matrix that will diagonalise the matrix A=[70−205−2−2−26]
8 M
3(a)
Find the inverse Laplace Transform of s+4(s2−1)(s+1)
6 M
3(b)
Check whether the matrix A=[5−6−6−1423−6−4] is derogatory or not.
6 M
3(c)
Using Kuhn-Tucker conditions solve the following NLPP
Maximize Z=2x1 +x2 -x21 Subject to 2x1+3x2≤6; 2x1+x2≤4; x1, x2≥0.
Maximize Z=2x1 +x2 -x21 Subject to 2x1+3x2≤6; 2x1+x2≤4; x1, x2≥0.
8 M
4(a)
Find the bilinear transformation which maps the points z=2, i, -2 onto the points w=1, i, -1.
6 M
4(b)
Find the orthogonal trajectory of the family of curves given by ex cos y-xy=c.
6 M
4(c)
Solve the following NLPP using Lagrange's multipliers method
Optimise Z=4x1+8x2-x21 -x22
Subject to x1+x2=4; x1, x2 ≥0.
Optimise Z=4x1+8x2-x21 -x22
Subject to x1+x2=4; x1, x2 ≥0.
8 M
5(a)
Find the Eigen values and Eigen vectors of A=[2−1112−11−12].
6 M
5(b)
Evaluate ∮cx+2x3−2x2dz where C is the circle |z-2-i|=2.
6 M
5(c)
Find the inverse Laplace Transform of (i) log(s2+a2s2+b2)
(ii) (s+1)e−s(s2+s+1)
(ii) (s+1)e−s(s2+s+1)
8 M
6(a)
Find the poles and calculate the residues at them for f(z)=z(z−1)(z+2)2.
6 M
6(b)
From the following data calculate Spearman's rank correlation coefficient between X and Y
X: 36 56 20 42 33 44 50 15 60
Y:50 35 70 58 75 60 45 80 38 .
X: 36 56 20 42 33 44 50 15 60
Y:50 35 70 58 75 60 45 80 38 .
6 M
6(c)
Reduce the following quadratic form to canonical form. Also find it's rank and signature x2 +2y2 +2z2 -2xy -2yz+zx.
8 M
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