List of practice Questions

Consider the function \[ f(x,y)=(x-2)^2(y+3). \] Then

The eigen values of a \(3\times3\) real matrix \(A\) are \(1,-2,3\). Then

The value of the definite integral \[ \int_{0}^{1}\frac{xe^{x}}{(x+1)^2}\,dx \] is equal to

The value of the constant \(a\) so that the vector \[ \vec{V}=(x+3y)\hat{i}+(y-2z)\hat{j}+(x+az)\hat{k} \] is solenoidal is

Let \[ I=\int_C z\,dx+y\,dy+xz\,dz, \] where \(x,y,z\) are real and \(C\) is the straight line segment from the point \((0,2,1)\) to the point \((4,1,-1)\). The value of the line integral \(I\) is

The value of \[ \int_{0}^{2+i}\left(\frac{\bar z}{2}\right)^2dz \] along \(x=2y\), is

Residue of \(\tan z\) at \(z=\dfrac{\pi}{2}\) is

Let \(M = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}\) be a \(2 \times 2\) matrix, where \(\theta = \dfrac{2\pi}{5}\), and \(I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\).
Which of the following options is equal to \(M^{2026}\)?

Consider a set \(S_1 = \{x = (x_1, x_2, x_3)^T \in \mathbb{R}^3 \mid x^Tx \leq 16\}\). Let \(S_2\) be another set which is a subspace of \(\mathbb{R}^3\) with dimension two.
Which of the following gives the area of \(S_1 \cap S_2\)?

Let \(\gamma_1, \gamma_2, \gamma_3\) be the eigenvalues of the matrix
\[ \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos t & \sin t \\ 0 & -\sin t & \cos t \end{bmatrix} \]
where \(t \in [-\pi, \pi]\) is in radians.

Which one of the following options lists all the possible values of \(t\) satisfying \(\gamma_1 + \gamma_2 + \gamma_3 = 1 + \sqrt{2}\)?

Let \(M = \left(I_n - \frac{1}{n}\mathbf{1}\mathbf{1}^T\right)\) be a matrix, where \(\mathbf{1} = (1,1,1,\ldots,1)^T \in \mathbb{R}^n\) and \(I_n\) is the identity matrix of order \(n\).

Which of the following options is/are correct?

Let \(A = \left(I_n - \frac{1}{n}\mathbf{1}\mathbf{1}^T\right)\) be a matrix, where \(\mathbf{1} = (1,1,1,\ldots,1)^T \in \mathbb{R}^n\) and \(I_n\) is the identity matrix of order \(n\).

The value of \(\max_{S} x^TAx\), where \(S = \{x \in \mathbb{R}^n \mid x^Tx = 1\}\), is __________. (Answer in integer)

Consider an \(n\times n\) orthogonal matrix \(A\) with real entries and each column having unit Euclidean norm.
Which of the following statements is/are correct?

Consider the system of linear equations \(Ax=b\), where \(A\) is an \(n\times n\) matrix, and \(x\) and \(b\) are \(n\)-dimensional column vectors.
Suppose this system of equations has a unique solution. Which of the following statements is/are correct?

Which one of the following statements is ALWAYS correct about a collection of \(p\) column vectors, each having \(n\) real-valued entries?

The determinant of a \(2 \times 2\) square matrix is 1.6. If the sum of the eigenvalues is \(-2.8\), the eigenvalues are

Two square matrices \( \mathbf{A} \) and \( \mathbf{B} \) of the same order are such that \( \det[\mathbf{A} \cdot \mathbf{B}] = -80 \) and \( \det[\mathbf{A}] = 5 \). Here, \( \mathbf{A} \cdot \mathbf{B} \) is the product of \( \mathbf{A} \) and \( \mathbf{B} \). The value of \( \det[\mathbf{B}^T] \) is

The system of equation \(x - y + 3z = 0, x + z = 0, x + y - z = 0\) has _______

Let \(A = \begin{pmatrix} 2026 & 2025 & 2024 \\ 2025 & 2024 & 2023 \\ 2024 & 2023 & 2023 \end{pmatrix}\) be a matrix. Then, \(\det(A)\) is equal to _______

If a \(2 \times 2\) matrix \(A\) has eigenvalues 1 and 4 with the corresponding eigenvectors \(\begin{pmatrix} 1 \\ -1 \end{pmatrix}\) and \(\begin{pmatrix} 2 \\ 1 \end{pmatrix}\), respectively, then \(A\) is ________

Let \(A = \begin{pmatrix} 2026 & 2025 & 2024 \\ 2025 & 2024 & 2023 \\ 2024 & 2023 & 2023 \end{pmatrix}\) be a matrix. Then, \(\det(A)\) is equal to _______

If a \(2 \times 2\) matrix \(A\) has eigenvalues 1 and 4 with the corresponding eigenvectors \(\begin{pmatrix} 1 \\ -1 \end{pmatrix}\) and \(\begin{pmatrix} 2 \\ 1 \end{pmatrix}\), respectively, then \(A\) is ________

The system of equation \(x - y + 3z = 0, x + z = 0, x + y - z = 0\) has _______