List of practice Questions

Consider \(\mathbb{R}^2\) with the usual Euclidean metric. Let
\[ X = \left\{\left(x, x \sin\frac{1}{x}\right) \in \mathbb{R}^2 : x \in (0,1]\right\} \cup \{(0, y) \in \mathbb{R}^2 : -\infty < y < \infty\}, \]
\[ Y = \left\{\left(x, \sin\frac{1}{x}\right) \in \mathbb{R}^2 : x \in (0,1]\right\} \cup \{(0, y) \in \mathbb{R}^2 : -\infty < y < \infty\}. \]

Consider the following statements:
P: \(X\) is a connected subset of \(\mathbb{R}^2\).
Q: \(Y\) is a connected subset of \(\mathbb{R}^2\).
Then
Let \(T: \mathbb{R}^3 \to \mathbb{R}^3\) be a linear transformation satisfying
\(T(1, 0, 0) = (0, 1, 1)\), \(T(1, 1, 0) = (1, 0, 1)\) and \(T(1, 1, 1) = (1, 1, 2)\).
Then
Let \(P(x) = 1 + e^{2\pi i x} + 2 e^{3\pi i x}\), \(x \in \mathbb{R}\), \(i = \sqrt{-1}\). Then
\[ \lim_{N \to \infty} \frac{1}{N} \sum_{k=0}^{N-1} P(k\sqrt{2}) \] is equal to
Let \(\mathbb{R}^3\) be a topological space with the usual topology and \(\mathbb{Q}\) denote the set of rational numbers. Define the subspaces X, Y, Z and W of \(\mathbb{R}^3\) as follows:
\(X = \{(x, y, z) \in \mathbb{R}^3 : |x| + |y| + |z| \in \mathbb{Q}\}\)
\(Y = \{(x, y, z) \in \mathbb{R}^3 : xyz = 1\}\)
\(Z = \{(x, y, z) \in \mathbb{R}^3 : x^2 + y^2 + z^2 = 1\}\)
\(W = \{(x, y, z) \in \mathbb{R}^3 : xyz = 0\}\)
Which of the following statements is correct?
Let \( f, g: \mathbb{R}^2 \to \mathbb{R} \) be defined by
\[ f(x, y) = x^2 - \frac{3}{2}xy^2 \quad \text{and} \quad g(x, y) = 4x^4 - 5x^2y + y^2 \] for all \( (x, y) \in \mathbb{R}^2 \).
Consider the following statements:
P: \( f \) has a saddle point at (0, 0).
Q: \( g \) has a saddle point at (0, 0).
Then

An opaque cylinder (shown below) is suspended in the path of a parallel beam of light, such that its shadow is cast on a screen oriented perpendicular to the direction of the light beam. The cylinder can be reoriented in any direction within the light beam. Under these conditions, which one of the shadows P, Q, R, and S is NOT possible? 

Which one of the given figures P, Q, R and S represents the graph of the following function? 
\( f(x) = | |x + 2| - |x - 1| | \) 

 

In the given diagram, ovals are marked at different heights (h) of a hill. Which one of the following options P, Q, R, and S depicts the top view of the hill? 

 

Consider the ordinary differential equation (ODE) \[ 4 (\ln x) y'' + 3 y' + y = 0, \quad x>1. \] If \(r_1\) and \(r_2\) are the roots of the indicial equation of the above ODE at the regular singular point \(x=1\), then \(|r_1 - r_2|\) is equal to ................. (rounded off to 2 decimal places).
Let \(u(x, t)\) be the solution of the non-homogeneous wave equation \[ \frac{\partial^2 u}{\partial x^2} - \frac{\partial^2 u}{\partial t^2} = \sin x \sin(2t), \quad 0<x<\pi, \ t>0 \] \[ u(x, 0) = 0, \text{ and } \frac{\partial u}{\partial t}(x, 0) = 0, \quad \text{for } 0 \le x \le \pi, \] \[ u(0, t) = 0, \quad u(\pi, t) = 0, \quad \text{for } t \ge 0. \] Then the value of \( u \left( \frac{\pi}{2}, \frac{3\pi}{2} \right) \) is ................. (rounded off to 2 decimal places).
Let \(\sigma \in S_8\), where \(S_8\) is the permutation group on 8 elements. Suppose \(\sigma\) is the product of \(\sigma_1\) and \(\sigma_2\), where \(\sigma_1\) is a 4-cycle and \(\sigma_2\) is a 3-cycle in \(S_8\). If \(\sigma_1\) and \(\sigma_2\) are disjoint cycles, then the number of elements in \(S_8\) which are conjugate to \(\sigma\) is ..................
Let A be a \(3 \times 3\) real matrix with det(\(A + i I\)) = 0, where \(i = \sqrt{-1}\) and I is the \(3 \times 3\) identity matrix. If det(A) = 3, then the trace of \(A^2\) is ..................
Let \(\Omega\) be the disk \(x^2 + y^2<4\) in \(\mathbb{R}^2\) with boundary \(\partial\Omega\). If \(u(x, y)\) is the solution of the Dirichlet problem \[ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0, \quad (x, y) \in \Omega, \] \[ u(x, y) = 1 + 2x^2, \quad (x, y) \in \partial\Omega, \] then the value of \(u(0,1)\) is ..................
For every \(k \in \mathbb{N} \cup \{0\}\), let \(y_k(x)\) be a polynomial of degree k with \(y_k(1) = 5\). Further, let \(y_k(x)\) satisfy the Legendre equation \[ (1-x^2)y'' - 2xy' + k(k+1)y = 0. \] If \[ \frac{1}{2} \sum_{k=1}^{n} \int_{-1}^{1} (y_k(x) - y_{k-1}(x))^2 dx - \sum_{k=1}^{n} \int_{-1}^{1} (y_k(x))^2 dx = 24, \] for some positive integer n, then the value of n is ..................
Let C be the curve of intersection of the cylinder \( x^2 + y^2 = 4 \) and the plane \( z - 2 = 0 \). Suppose C is oriented in the counterclockwise direction around the z-axis, when viewed from above. If \[ \int_C ( \sin x + e^x ) dx + 4x dy + e^z \cos^2 z dz = \alpha\pi, \] then the value of \( \alpha \) equals ..................
Consider the transportation problem between five sources and four destinations as given in the cost table below. The supply and demand at each of the source and destination are also provided:
\begin{tabular}{|c|c|c|c|c|c|c|} \cline{3-7} \multicolumn{2}{c|}{} & \multicolumn{4}{c|}{DESTINATIONS} & Supply
\cline{3-7} \multicolumn{2}{c|}{} & P & Q & R & S &
\hline \multirow{5}{*}{\rotatebox[origin=c]{90}{SOURCES}} & 1 & 13 & 8 & 12 & 9 & 20
\cline{2-7} & 2 & 10 & 7 & 5 & 20 & 10
\cline{2-7} & 3 & 3 & 19 & 5 & 12 & 50
\cline{2-7} & 4 & 4 & 9 & 7 & 15 & 30
\cline{2-7} & 5 & 14 & 0 & 1 & 7 & 40
\hline \multicolumn{2}{|c|}{Demand} & 60 & 10 & 20 & 60 &
\hline \end{tabular}
Let \(C_N\) and \(C_L\) be the total cost of the initial basic feasible solution obtained from the North-West corner method and the Least-Cost method, respectively. Then \(C_N - C_L\) equals ..................
For \(h>0\), and \(\alpha, \beta, \gamma \in \mathbb{R}\), let \[ D_h f(a) = \frac{\alpha f(a-h) + \beta f(a) + \gamma f(a+2h)}{6h} \] be a three-point formula to approximate \(f'(a)\) for any differentiable function \(f: \mathbb{R} \to \mathbb{R}\) and \(a \in \mathbb{R}\).
If \(D_h f(a) = f'(a)\) for every polynomial f of degree less than or equal to 2 and for all \(a \in \mathbb{R}\), then
Let \( f \) be a twice continuously differentiable function on \( [a, b] \) such that \( f'(x)<0 \) and \( f''(x)<0 \) for all \( x \in (a, b) \). Let \( f(\zeta) = 0 \) for some \( \zeta \in (a, b) \). The Newton-Raphson method to compute \( \zeta \) is given by \[ x_{k+1} = x_k - \frac{f(x_k)}{f'(x_k)}, \quad k = 0, 1, 2, ...... \] for an initial guess \( x_0 \). If \( x_k \in (\zeta, b) \) for some \( k \geq 0 \), then which of the following statements is/are correct?
Let \(\phi\) and \(\psi\) be two linearly independent solutions of the ordinary differential equation \[ y'' + (2 - \cos x) y = 0, \quad x \in \mathbb{R}. \] Let \(\alpha, \beta \in \mathbb{R}\) be such that \(\alpha<\beta\), \(\phi(\alpha) = \phi(\beta) = 0\) and \(\phi(x) \neq 0\) for all \(x \in (\alpha, \beta)\).
Consider the following statements:
P: \(\phi'(\alpha)\phi'(\beta)>0\).
Q: \(\psi(x) \neq 0\) for all \(x \in (\alpha, \beta)\).
Then
Let \((\mathbb{R}, \tau)\) be a topological space, where the topology \(\tau\) is defined as \[ \tau = \{U \subseteq \mathbb{R} : U = \emptyset \text{ or } 1 \in U\}. \] Which of the following statements is/are correct?
Consider the Cauchy problem \[ x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = u; \] \[ u = f(t) \text{ on the initial curve } \Gamma = (t, t); t>0. \] Consider the following statements:
P: If \(f(t) = 2t + 1\), then there exists a unique solution to the Cauchy problem in a neighbourhood of \(\Gamma\).
Q: If \(f(t) = 2t - 1\), then there exist infinitely many solutions to the Cauchy problem in a neighbourhood of \(\Gamma\).
Then
For a fixed \(c \in \mathbb{R}\), let \(\alpha = \int_0^c (9x^2 - 5cx^4)dx\).
If the value of \(\int_0^c (9x^2 - 5cx^4)dx\) obtained by using the Trapezoidal rule is equal to \(\alpha\), then the value of c is .................. (rounded off to 2 decimal places).
If for some \(a \in \mathbb{R}\), \[ \int_1^4 \int_{-x}^x \frac{1}{x^2 + y^2} dydx = \int_{-\pi/4}^{\pi/4} \int_{\sec\theta}^{a \sec\theta} \frac{1}{r} drd\theta, \] then the value of \(a\) equals ................
Let S be the portion of the plane \(z = 2x + 2y - 100\) which lies inside the cylinder \(x^2 + y^2 = 1\). If the surface area of S is \(a\pi\), then the value of a is equal to ................
Consider the following Linear Programming Problem P:
Minimize \(3x_1 + 4x_2\)
subject to \[ x_1 - x_2 \le 1, \] \[ x_1 + x_2 \ge 3, \] \[ x_1 \ge 0, x_2 \ge 0. \] The optimal value of the problem P is ..................