List of practice Questions

Let \( T \) be the Möbius transformation that maps the points 0, \( \frac{1}{2} \), and 1 conformally onto the points -3, \( \infty \), and 2, respectively, in the extended complex plane. If \( T \) maps the circle centered at 1 with radius \( k \) onto a straight line given by the equation \( \alpha x + \beta y + \gamma = 0 \), then the value of \[ \frac{2k(\alpha + \beta) + \gamma}{\alpha + \beta - 2k\gamma} \] is equal to:

Let \( u(x,t) \) be the solution of the initial-value problem \[ \frac{\partial^2 u}{\partial t^2} - 9 \frac{\partial^2 u}{\partial x^2} = 0, \quad x \in \mathbb{R}, \quad t>0, \quad u(x, 0) = e^x, \quad \frac{\partial u}{\partial t}(x, 0) = \sin x. \] Then, the value of \( u\left(\frac{\pi}{2}, \frac{\pi}{6}\right) \) is:

Let \( C \) be the curve of intersection of the surfaces \( z^2 = x^2 + y^2 \) and \( 4x + z = 7 \). If \( P \) is a point on \( C \) at a minimum distance from the \( xy \)-plane, then the distance of \( P \) from the origin is:

Let \( y_1(x) \) and \( y_2(x) \) be the two linearly independent solutions of the differential equation

\[ (1 + x^2) y'' - x y' + (\cos^2 x) y = 0, \]
satisfying the initial conditions

\[ y_1(0) = 3, \quad y_1'(0) = -1, \quad y_2(0) = -5, \quad y_2'(0) = 2. \]
Define

\[ W(x) = \left| \begin{array}{cc} y_1(x) & y_2(x) \\ y_1'(x) & y_2'(x) \\ \end{array} \right|. \]

Then, the value of \( W\left( \frac{1}{2} \right) \) is:

Let \( y(x) \) be the solution of the differential equation \[ x^2 y'' + 7xy' + 9y = x^{-3} \log_e x, \quad x>0, \] satisfying \( y(1) = 0 \) and \( y'(1) = 0 \). Then, the value of \( y(e) \) is equal to:

Consider the linear system \( A x = b \), where \( A = [a_{ij}] \), \( i, j = 1, 2, 3 \), and \( a_{ii} \neq 0 \) for \( i = 1, 2, 3 \), is a matrix with entries in \( \mathbb{R} \). For

\( D = \begin{bmatrix} a_{11} & 0 & 0 \\ 0 & a_{22} & 0 \\ 0 & 0 & a_{33} \end{bmatrix} \), let

\[ D^{-1} A = \begin{bmatrix} 1 & 1 & -2 \\ 3 & 1 & 2 \\ 1 & 1 & 1 \end{bmatrix}, \quad D^{-1}b = \begin{bmatrix} 4 \\ 4 \\ 1 \end{bmatrix}. \]

Consider the following two statements:

S1: The approximation of \( x \) after one iteration of the Jacobi scheme with initial vector \( x_0 = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} \) is \( x_1 = \begin{bmatrix} 5 \\ -1 \\ -1 \end{bmatrix} \).

S2: There exists an initial vector \( x_0 \) for which the Jacobi iterative scheme diverges.

Then, which one of the following is correct?

Let \( y(x) \) be the solution of the initial value problem \[ \frac{dy}{dx} = \sin(\pi(x + y)), \quad y(0) = 0. \] Using Euler's method, with the step-size \( h = 0.5 \), the approximate value of \( y(1.5) + 2y(1) \) is equal to (in integer):

Let \( \alpha, \beta, \gamma, \delta \in \mathbb{R} \) be such that the quadrature formula \[ \int_{-1}^{1} f(x) \, dx = \alpha f(-1) + \beta f(1) + \gamma f'(-1) + \delta f'(1) \] is exact for all polynomials of degree less than or equal to 3. Then, \( 9(\alpha^2 + \beta^2 + \gamma^2 + \delta^2) \) is equal to (in integer):

Consider \[ I = \frac{1}{2\pi i} \int_C \frac{\sin z}{1 - \cos(z^3)} \, dz, \] where \( C = \{ z \in \mathbb{C} : z = x + iy, |x| + |y| = 1, x, y \in \mathbb{R} \} \) is oriented positively as a simple closed curve. Then, the value of \( 120I \) is equal to _________ (in integer).

Let \( \vec{F} = (y - z)\hat{i} + (z - x)\hat{j} + (x - y)\hat{k} \) be a vector field, and let \( S \) be the surface \( x^2 + y^2 + (z - 1)^2 = 9, 1 \leq z \leq 4 \).
If \( \hat{n} \) denotes the unit outward normal vector to \( S \), then the value of

\[ \frac{1}{\pi} \left| \iint_S (\vec{v} \times \vec{F}) \cdot \hat{n} \, dS \right| \]

is equal to _________ (in integer).

Let \( W \) be the vector space (over \( \mathbb{R} \)) consisting of all bounded real-valued solutions of the differential equation \[ \frac{d^4y}{dx^4} + 2 \frac{d^2y}{dx^2} + y = 0. \] Then, the dimension of \( W \) is _________ (in integer).

The volume of the region bounded by the cylinders \( x^2 + y^2 = 4 \) and \( x^2 + z^2 = 4 \) is _________ (rounded to TWO decimal places).

Let \( \hat{a} \) be a unit vector parallel to the tangent at the point \( P(1, 1, \sqrt{2}) \) to the curve of intersection of the surfaces \( 2x^2 + 3y^2 - z^2 = 3 \) and \( x^2 + y^2 = z^2 \). Then, the absolute value of the directional derivative of \[ f(x, y, z) = x^2 + 2y^2 - 2\sqrt{11} z \] at P in the direction of \( \hat{a} \) is _________ (in integer).

Let \( M \) be a \( 7 \times 7 \) matrix with entries in \( \mathbb{R} \) and having the characteristic polynomial \[ c_M(x) = (x - 1)^\alpha (x - 2)^\beta (x - 3)^2, \] where \( \alpha>\beta \). Let \( {rank}(M - I_7) = {rank}(M - 2I_7) = {rank}(M - 3I_7) = 5 \), where \( I_7 \) is the \( 7 \times 7 \) identity matrix.
If \( m_M(x) \) is the minimal polynomial of \( M \), then \( m_M(5) \) is equal to __________ (in integer).

Let \( f: \mathbb{R}^2 \setminus \{(0,0)\} \to \mathbb{R} \) be a function defined by

\[ f(x, y) = \frac{x^2 - y^2}{x^2 + y^2} + x \sin \left( \frac{1}{x^2 + y^2} \right). \]

Consider the following three statements:

S1: \( \lim_{x \to 0,\, y \to 0} f(x, y) \) exists.

S2: \( \lim_{y \to 0} \lim_{x \to 0} f(x, y) \) exists.

S3: \( \lim_{(x, y) \to (0, 0)} f(x, y) \) exists.

Then, which of the following is/are correct?

Consider the following regions: \[ S_1 = \{(x_1, x_2) \in \mathbb{R}^2 : 2x_1 + x_2 \leq 4, \quad x_1 + 2x_2 \leq 5, \quad x_1, x_2 \geq 0\} \] \[ S_2 = \{(x_1, x_2) \in \mathbb{R}^2 : 2x_1 - x_2 \leq 5, \quad x_1 + 2x_2 \leq 5, \quad x_1, x_2 \geq 0\} \] Then, which of the following is/are TRUE?

Consider the balanced transportation problem with three sources \( S_1, S_2, S_3 \), and four destinations \( D_1, D_2, D_3, D_4 \), for minimizing the total transportation cost whose cost matrix is as follows:

where \( \alpha, \lambda>0 \). If the associated cost to the starting basic feasible solution obtained by using the North-West corner rule is 290, then which of the following is/are correct?

Let \( p_A(x) \) denote the characteristic polynomial of a square matrix \( A \). Then, for which of the following invertible matrices \( M \), the polynomial \( p_M(x) - p_{M^{-1}}(x) \) is constant?

Let the functions \( f: \mathbb{R} \to \mathbb{R} \) and \( g: \mathbb{R}^2 \to \mathbb{R} \) be given by \[ f(x_1, x_2) = x_1^2 + x_2^2 - 2x_1x_2, \quad g(x_1, x_2) = 2x_1^2 + 2x_2^2 - x_1x_2. \] Consider the following statements:
S1: For every compact subset \( K \) of \( \mathbb{R} \), \( f^{-1}(K) \) is compact.
S2: For every compact subset \( K \) of \( \mathbb{R} \), \( g^{-1}(K) \) is compact. Then, which one of the following is correct?

Consider the function \( F: \mathbb{R}^2 \to \mathbb{R}^2 \) given by \[ F(x, y) = (x^3 - 3xy^2 - 3x, 3x^2y - y^3 - 3y). \] Then, for the function \( F \), the inverse function theorem is:

Let \( u(x, t) \) be the solution of the following initial-boundary value problem: \[ \frac{\partial u}{\partial t} - \frac{\partial^2 u}{\partial x^2} = 0, \quad x \in (0, \pi), \quad t>0, \] with the boundary conditions: \[ u(0, t) = u(\pi, t) = 0, \quad u(x, 0) = \sin 4x \cos 3x. \] Then, for each \( t>0 \), the value of \( u\left( \frac{\pi}{4}, t \right) \) is

The partial differential equation \[ (1 + x^2) \frac{\partial^2 u}{\partial x^2} + 2x(1 - y^2) \frac{\partial^2 u}{\partial x \partial y} + (1 - y^2) \frac{\partial^2 u}{\partial y^2} + x \frac{\partial u}{\partial x} + (1 - y^2) \frac{\partial u}{\partial y} = 0 \] is:

Let \( g(x, y) = f(x, y)e^{2x + 3y} \) be defined in \( \mathbb{R}^2 \), where \( f(x, y) \) is a continuously differentiable non-zero homogeneous function of degree 4. Then,

\[ x \frac{\partial g}{\partial x} + y \frac{\partial g}{\partial y} = 0 \text{ holds for} \]

Let \( X = \{ f \in C[0,1] : f(0) = 0 = f(1) \} \) with the norm \( \|f\|_\infty = \sup_{0 \leq t \leq 1} |f(t)| \), where \( C[0,1] \) is the space of all real-valued continuous functions on \( [0,1] \).

Let \( Y = C[0,1] \) with the norm \( \|f\|_2 = \left( \int_0^1 |f(t)|^2 \, dt \right)^{\frac{1}{2}} \). Let \( U_X \) and \( U_Y \) be the closed unit balls in \( X \) and \( Y \) centered at the origin, respectively. Consider \( T: X \to \mathbb{R} \) and \( S: Y \to \mathbb{R} \) given by

\[ T(f) = \int_0^1 f(t) \, dt \quad \text{and} \quad S(f) = \int_0^1 f(t) \, dt. \]

Consider the following statements:
S1: \( \sup |T(f)| \) is attained at a point of \( U_X \).
S2: \( \sup |S(f)| \) is attained at a point of \( U_Y \).

Then, which one of the following is correct?

Consider the system of ordinary differential equations \[ \frac{dX}{dt} = MX, \] where \( M \) is a \( 6 \times 6 \) skew-symmetric matrix with entries in \( \mathbb{R} \). Then, for this system, the origin is a stable critical point for