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

The inverse of a symmetric matrix is

If A is a matrix such that there exists a square sub-matrix of order r and every square sub-matrix of order r + 1 or more is singular, then

Let a, b, c be positive real numbers, then the following system of equations in x, y and z

$ \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1, \frac{x^2}{a^2} - \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1, -\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 $

has

If $ f(x) = 0 $ is an algebraic equation, then Newton-Raphson method is given by $ x_{n+1} = x_n - \frac{f(x_n)}{?} $

Gauss forward interpolation formula involves

How many prior value(s) is/are required to predict the next value in Adams-Bashforth-Moulton method?

What is/are the order(s) of the predictor and corrector in Milne's predictor-corrector method?

A sufficient condition for convergence for the system of equations $ Au = d $ obtained, when we apply finite difference methods for Laplace's or Poisson equation is

$ L(\sqrt{t}) = $

$ L^{-1}(e^{-as} F(s)) = $

Reduce the matrix to normal form and find its rank.

$ A = \begin{bmatrix} 2 & 3 & 4 & 5 \\ 3 & 4 & 5 & 6 \\ 4 & 5 & 6 & 7 \\ 5 & 6 & 7 & 8 \end{bmatrix} $

For what values of k, the equation $ x + y + z = 1, 2x + y + 4z = k, 4x + y + 10z = k^2 $ has a solution?

Verify Cayley-Hamilton theorem for the matrix

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

Hence prove that

$ A^8 - 5A^7 + 7A^6 - 3A^5 + A^4 - 5A^3 + 8A^2 - 2A + I = \begin{bmatrix} 8 & 5 & 5 \\ 0 & 3 & 0 \\ 5 & 5 & 8 \end{bmatrix} $

Find a matrix P which transforms the matrix $ A = \begin{bmatrix} 1 & 0 & -1 \\ 1 & 2 & 1 \\ 2 & 2 & 3 \end{bmatrix} $ to diagonal form. Hence find $ A^4 $.

Find the smallest positive root of $ x^4 - x = 10 $ by using the Newton-Raphson method, correct to three decimal places.

Evaluate $ \Delta^2 \left( \frac{5x + 12}{x^2 + 5x + 6} \right) $; the interval of differencing being unity.

Derive the Newton's forward difference formula using the operator relations.

Using the trapezoidal rule, evaluate the integral $ I = \int_0^1 \frac{dx}{x^2 + 6x + 10} $, with 2 and 4 sub-intervals. Compare with the exact solution. Comment on the magnitudes of the errors obtained.

Using the Adams-Bashforth predictor-corrector equations, evaluate $ y(1.4) $, if y satisfies $ \frac{dy}{dx} + \frac{y}{x} = \frac{1}{x^2} $ and $ y(1) = 1, y(1.1) = 0.996, y(1.2) = 0.986, y(1.3) = 0.972 $.

Solve the equation $ u_{xx} = u_t $ subject to $ u(x, 0) = 0, u(0, t) = 0 $ and $ u(l, t) = t $, for two time steps by Crank-Nicolson method.

Find the inverse Laplace transform of $ \frac{s}{(s^2 + a^2)^3} $ with the help of convolution theorem.

Find the values of the integral $ \int_0^\infty e^{-3t} \frac{\sin^2 t}{t} dt $ with the help of Laplace transform.

$ t \frac{d^2y}{dt^2} + (1 - 2t) \frac{dy}{dt} - 2y = 0 $, when $ y(0) = 1, y'(0) = 2 $

$ \frac{\partial^2 y}{\partial t^2} = 9 \frac{\partial^2 y}{\partial x^2} $, where $ y(0, t) = 0, y(2, t) = 0 $ and $ y(x, 0) = 20 \sin 2\pi x - 10 \sin 5\pi x $

The eigenvalues of a matrix $ A $ are 2, 3, 1, then find the eigenvalues of $ A^{-1} + A^2 $.

If $ A $ and $ B $ are symmetric matrices, then prove that $ AB - BA $ is a skew-symmetric matrix.

Prove that the matrix

$ \frac{1}{3} \begin{bmatrix} -2 & 1 & 2 \\ 2 & 2 & 1 \\ 1 & -2 & 2 \end{bmatrix} $

is orthogonal.

Prove that $ \Delta = \frac{1}{2} \delta^2 + \delta \sqrt{1 + (\delta^2 / 4)} $.

Find the missing values in the table :

Use Euler's method to obtain an approximate value of $ y(0.4) $ for the equation $ y' = x + y, \ y(0) = 1 $ with $ h = 0.1 $.

Obtain the approximate value of $ y(1.2) $ for the initial value problem $ y' = 2xy^2, \ y(1) = 1 $ using Taylor series second-order method with step size $ h = 0.1 $.

Find the Laplace transform of $ \cos^2 3t $.

Find the inverse Laplace transform of $ \frac{s^3}{s^4 - a^4} $.

Evaluate : $ \int_0^\infty e^{-x^2} dx $.

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

Find the rank of the matrix

$ \begin{bmatrix} 3 & 2 & -1 \\ 4 & 2 & 6 \\ 7 & 4 & 5 \end{bmatrix} $

by reducing it to normal form.

Verify Cayley-Hamilton theorem for the matrix

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

Hence compute $ A^{-1} $.

Reduce the matrix

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

to diagonal form.

Find a real root of the equation $ x \log_{10} x = 1.2 $ by Regula-Falsi method, correct to four decimal places.

The following table gives the population of a town during the last six censuses. Estimate the population in 1913 by Newton's forward difference formula :

Find $ \int_0^6 \frac{e^x}{1+x} dx $ approximately by using Simpson's 3/8 rule on integration.

Evaluate $ \int_0^8 x \sec x dx $ using eight intervals by Trapezoidal rule.

State convolution theorem of Laplace transform and using it find $ L^{-1} \left\{ \frac{1}{(s^2+4)(s+2)} \right\} $.

Use Laplace Transform to solve :

$ \frac{dx}{dt} + y = \sin t, \ \frac{dy}{dt} + x = \cos t $

given that $ x=2, \ y=0 $ at $ t=0 $.

Find the Fourier transform of $ f(x) $, defined by

$ f(x) = \begin{cases} 1, & |x| < a \\ 0, & |x| > a \end{cases} $

and hence evaluate $ \int_{-\infty}^\infty \frac{\sin as \cos sx}{s} ds $.

Evaluate : $ \int_0^\infty e^{-st} t^3 \sin t dt $.

Solve the initial value problem $ yy' = x, \ y(0)=1 $, using the Euler method in $ 0 \le x \le 0.8 $, with $ h=0.2 $ and $ h=0.1 $. Compare the results with the exact solution at $ x=0.8 $. Extrapolate the result.

Given the initial value problem $ \frac{dy}{dx} = \frac{y-x}{y+x}, \ y(0)=1 $. Find $ y(1) $ by Runge-Kutta fourth-order method taking $ h=0.5 $.

Obtain the approximate value of $ y(0.2) $ for the initial value problem $ y' = x^2 + y^2, \ y(0)=1 $. Using the methods $ y_{n+1} = y_n + h f(x_n, y_n) $ as predictor and $ y_{n+1} = y_n + \frac{h}{2} [f(x_n, y_n) + f(x_{n+1}, y_{n+1})] $ as corrector, with $ h=0.1 $. Perform two corrector iterations per step.