STUPIDSID
1(a)
Find the Laplace Transform of cos t cos 2t cos 3t.
5 M
1(b)
If \( A=\begin{bmatrix}
1 & 4\\
1 & 1
\end{bmatrix} \), find A7-9A2+I.
5 M
1(c)
Show that \( \oint _c \log z dz=2\pi i \) , 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 \( \int ^{\infty}_0e^{-1}\int ^t_0\dfrac{\sin u}{u}du\ dt \)
6 M
2(c)
Find the orthogonal matrix that will diagonalise the matrix \( A=\begin{bmatrix}
7 & 0 & -2\\
0 & 5 & -2\\
-2 & -2 & 6
\end{bmatrix} \)
8 M
3(a)
Find the inverse Laplace Transform of \( \dfrac{s+4}{(s^2-1)(s+1)} \)
6 M
3(b)
Check whether the matrix \( A=\begin{bmatrix}
5 & -6 & -6\\
-1 & 4 & 2\\
3 & -6 & -4
\end{bmatrix} \) is derogatory or not.
6 M
3(c)
Using Kuhn-Tucker conditions solve the following NLPP
Maximize Z=2x1 +x2 -\( x_{1}^{2} \) Subject to 2x1+3x2≤6; 2x1+x2≤4; x1, x2≥0.
Maximize Z=2x1 +x2 -\( x_{1}^{2} \) 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-\( x_{1}^{2} \) -\( x_{2}^{2} \)
Subject to x1+x2=4; x1, x2 ≥0.
Optimise Z=4x1+8x2-\( x_{1}^{2} \) -\( x_{2}^{2} \)
Subject to x1+x2=4; x1, x2 ≥0.
8 M
5(a)
Find the Eigen values and Eigen vectors of \( A=\begin{bmatrix}
2 & -1 & 1\\
1 & 2 & -1\\
1 & -1 & 2
\end{bmatrix} \).
6 M
5(b)
Evaluate \( \oint _{c} \dfrac{x+2}{x^3-2x^2}dz \) where C is the circle |z-2-i|=2.
6 M
5(c)
Find the inverse Laplace Transform of (i) \( \log \left ( \dfrac{s^2+a^2}{s^2+b^2} \right ) \)
(ii) \( \dfrac{(s+1)e^{-s}}{(s^2+s+1)} \)
(ii) \( \dfrac{(s+1)e^{-s}}{(s^2+s+1)} \)
8 M
6(a)
Find the poles and calculate the residues at them for \( f(z)=\dfrac{z}{(z-1)(z+2)^2} \).
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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