List of top Questions asked in ICAR AIEEA (PG)- 2021

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If $u$ and $v$ are functions of two independent variables $x$ and $y$, then the determinant \[ \begin{vmatrix} \dfrac{\partial u}{\partial x} & \dfrac{\partial u}{\partial y} \\ \dfrac{\partial v}{\partial x} & \dfrac{\partial v}{\partial y} \end{vmatrix} \] is called:

Let $f(x) = \begin{cases} 2^{1/x}, & x \neq 0 \\ 0, & x = 0 \end{cases}$ Then,

\[ \begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix} \] is an eigenvector of the matrix \[ \begin{pmatrix} 3 & 6 & 7 \\ 3 & 3 & 7 \\ 5 & 6 & 5 \end{pmatrix}. \] With respect to which eigenvalue?

Let \[ A= \begin{pmatrix} 1 & -2 & -9 & 5 \\ 0 & 1 & 2 & -6 \end{pmatrix}. \] Then, the solution set of the system of equations \[ Ax=0 \] is:

Let \[ X= \begin{pmatrix} 1 & a & 3 \\ 1 & 3 & 3 \\ 4 & 2 & 2 \end{pmatrix} \] be the adjoint of a $3 \times 3$ matrix $A$, and let \[ \det(A)=6. \] Then, the value of $a$ is:

The number of distinct real roots of \[ \begin{vmatrix} \sin x & \cos x \\ \cos x & \sin x \end{vmatrix} = 0 \] in the interval \[ -\frac{\pi}{4} \leq x \leq \frac{\pi}{4} \] is:

Rank of the matrix $A = \begin{pmatrix} 2 & 3 & -1 \\ -8 & -7 & 6 \\ 6 & -1 & -7 \end{pmatrix}$ is

Let \[ v_1 = \begin{pmatrix} 1 \\ -2 \\ 7 \end{pmatrix}, \quad v_2 = \begin{pmatrix} 2 \\ -2 \\ 3 \end{pmatrix} \] be two vectors in $\mathbb{R}^3$. Then which of the following is true?

A number is chosen randomly from each of the two following sets
Set 1: {1,2,3,4,5,6,7 }
Set 2: {3,4,5,6,7,8}
Find the probability that the sum of the two numbers is 6.

Match List I with List II

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Match List I with List II