MATHEMATICS—II (Linear Algebra, Transform Calculus and Numerical Method)

If $ A $ is a 3-rowed square matrix such that $ |A|=2 $, then $ |\text{adj}\{\text{adj}(\text{adj } A^2)\}| $ is equal to

If 3, -2 are the eigenvalues of a non-singular matrix $ A $ and $ |A|=4 $, then eigenvalues of $ \text{adj } A $ are

Let $ A $ be a skew-symmetric matrix of order $ n $, then

If $ A $ is non-zero column matrix of the type $ n \times 1 $ and $ B $ is non-zero row matrix of the type $ 1 \times n $, then $ \rho(AB) $ is

In regula-falsi method, the first approximation is given by

While evaluating the definite integral by trapezoidal rule, the accuracy can be increased by taking

Various types of Runge-Kutta methods are classified according to their

The value of $ L\left\{ \frac{\cos 10t}{t} \right\} $ is

Laplace transform of unit step function is

Which function has Laplace transform even it is not piecewise continuous in the range?

Investigate for what value of $ \lambda $ and $ \mu $ do the system of equations $ x+y+z=6 $, $ x+2y+3z=10 $ and $ x+2y+\lambda z=\mu $ have (i) no solution, (ii) unique solution and (iii) infinite number of solution.

Find the eigenvalues and eigenvectors of the matrix

$ A = \begin{bmatrix} -2 & 2 & -3 \\ 2 & 1 & -6 \\ -1 & -2 & 0 \end{bmatrix} $

Verify Cayley-Hamilton theorem for the matrix

$ A = \begin{bmatrix} 1 & 2 & 0 \\ -1 & 1 & 2 \\ 1 & 2 & 1 \end{bmatrix} $

Also obtain (i) $ A^{-1} $, (ii) eigenvalues of $ A $ and $ A^2 $, and (iii) spectral radius of $ A $.

Diagonalize the matrix

$ A = \begin{bmatrix} 2 & 0 & 4 \\ 0 & 6 & 0 \\ 4 & 0 & 2 \end{bmatrix} $

by means of an orthogonal transformation.

Find a real root of the equation $ x \log_{10} x = 1.2 $ by using regula-falsi method correct to four significant digits.

Show that the following two sequences both have convergence of the second order with the same limit $ \sqrt{a} $:

$ x_{n+1} = \frac{1}{2} x_n \left( 1 + \frac{a}{x_n^2} \right) $ and $ x_{n+1} = \frac{1}{2} x_n \left( 3 - \frac{x_n^2}{a} \right) $

Derive Newton's forward interpolation formula.

Find the value of $ \cos 51^\circ 43' $ by Gauss's backward interpolation formula. Given that

Solve the differential equation $ \frac{dy}{dx} = y - x^2 $ by Milne's method and compute $ y $ at $ x = 0.80 $. Given that

Using Adams-Moulton-Bashforth method, find $ y(1.4) $. Given $ \frac{dy}{dx} = x^2(1+y) $, $ y(1)=1 $, $ y(1.1)=1.233 $, $ y(1.2)=1.548 $, $ y(1.3)=1.979 $.

Solve $ u_{xx} = u_t $ in $ 0 < x < 2 $, $ t > 0 $, $ u(0, t) = u(2, t) = 0 $, $ t > 0 $ and $ u(x, 0) = \sin(\pi x / 2) $, $ 0 \le x \le 2 $ using $ \Delta x = 0.5 $, $ \Delta t = 0.25 $ for one time step by Crank-Nicolson implicit finite difference method.

Write an implicit method for solving the one-dimensional wave equation $ u_{tt} = c^2 u_{xx} $, $ 0 \le x \le l $, $ t > 0 $.

Evaluate $ \int_0^\infty \{ \cos t \cdot \delta(t - \pi/4) \} dt $ by using Laplace transform.

Find the Fourier transform of the function $ f(t) = e^{-a|t|} $, $ -\infty < t < \infty $, $ a > 0 $.

Find the inverse Laplace transform of $ \tan^{-1} \left( \frac{2}{s^2} \right) $.

Solve the given partial differential equation by Laplace transform : $ x \frac{\partial u}{\partial t} + \frac{\partial u}{\partial x} = xt $, if $ u(x, 0) = 0 $, $ u(0, t) = t $.