STUPIDSID
1 (a)
Find the value of μ which satisfy the equation A100x=μ X, where \[ A= \begin{bmatrix}2 &1 &-1 \\0 &-2 &-2 \\ 1 &1 &0 \end{bmatrix} \]
5 M
1 (b)
\[ Evaluate \ \int^{1+i}_{0} (x^2+iy)dz \ along \ y=x \ and \ y=x^2 \]
5 M
1 (c)
Find the external of the function \[ \int^{x_2}_{x_1} [y^2-y^2 -2y\cosh x]dx \]
5 M
1 (d)
Verify Cauchy-Schwartz inequality for the vectors.
u=(-4, 2, 1) & V=(8, -4, -2)
u=(-4, 2, 1) & V=(8, -4, -2)
5 M
2 (a)
Determine the function that gives the shortest distance between two given points
6 M
2 (b)
Find Eigen values and Eigen vectors of :- \[ A= \begin{bmatrix} 2 & 1 &1 \\2 &3 &2 \\3 &3 &4 \end{bmatrix} \]
6 M
2 (c)
Obtain Taylor's and two distinct Laurent's series expansion of \[ f(z) = \dfrac {z-1}{z^2-2z-3} \] about z=0 indicating the region of convergence.
8 M
3 (a)
Verify Caley-Hamilton theorem for \[ A= \begin{bmatrix}1 &2 &0 \\2 &-1 &0 \\0 &0 &-1 \end{bmatrix} \ hence \ find \ A^{-2} \]
6 M
3 (b)
Evaluate by using Residue theorem. \[ \int^{2\pi}_0 \dfrac {d\theta}{(2+\cos \theta )^2} \]
6 M
3 (c)
Solve the boundary value problem. \[ I= \int^1_0 \left ( 2xy -y^2 -y^{1^{2 }} \right )dx \] given y(0)=y(1)=0 by Rayleigh-Ritz method.
8 M
4 (a)
Reduce the following Quadratic form \[ Q=3x^2_1 + 5x_2^2 + 3x^2_3 -2x_1x_2-2x_2x_3 +2x_3x_1 \] into canonical form. Hence find its rank index and signature.
6 M
4 (b)
Show that the matrix \[ A= \begin{bmatrix} 7 &4 &-1 \\4 &7 &-1 \\-4 &-4 &4 \end{bmatrix} \ is \ derogatory \]
6 M
4 (c) (i)
Show that the set W={(1,x)|x∈R} is a subspace of R2 under operations [1,x]+[1,y]=[1, x+y]; k[1,x]=[1,kx]; k is any scalar:
4 M
4 (c) (ii)
Is the set W={[a,1,1]|a∈R} a subspace of R3 under the usual addition and scalar multiplication?
4 M
5 (a)
Find the plane curve of fixed perimeter and maximum area.
6 M
5 (b)
Construct an orthonormal basis of R2 by applying Gram Schmidt orthogonalization to S={[3,1],[2,2]}
6 M
5 (c)
Show that the matrix \[ A= \begin{bmatrix} -9 &4 &4 \\-8 &3 &4 \\-16 &8 &7 \end{bmatrix} \] is diagonable. Also find diagonal form and diagonalising matrix.
8 M
6 (a)
Evaluate \[ \int^{\infty}_{-\infty} \dfrac {\cos 3x}{(x^2+1)(x^2+4)} dx \] using Cauchy Residue Theorem.
6 M
6 (b)
\[ If \ \phi(\alpha)= \oint_{c}\dfrac {ze^z}{z-\alpha}dz \ where \ c \ is \ |z-2i|=3 \\ find \phi (1), \ \phi '(2), \ \phi ' (3), \ \phi ' (4) \]
6 M
6 (c)
Show that the set V of position real numbers with operations Addition : x+y=xy, Scalar multiplication: kx=xk is a vector space where x, y are any two real numbers and k is any scalar.
8 M
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