Algebra PYQ — Page 17

Match List-I with List-II | List-I | List-II | | --- | --- | | (Inequality) | (Solution Set) | | --- | --- | | (A) $2x - 3 < x + 2 \le 3x + 5, x \in \mathbb{R}$ | (I) $x \in (-1, \infty)$ | | (B) $\vert 2x + 3\vert < 7, x \in \mathbb{R}$ | (II) $x \in (-\infty, 120]$ | | (C) $\frac{1}{2}\left(\frac{3}{5}x + 4\right) \ge \frac{1}{3}(x - 6), x \in \mathbb{R}$ | (III) $x \in (-5, 2)$ | | (D) $\frac{\vert x + 1\vert }{x + 1} > 0, x \in \mathbb{R} - \{-1\}$ | (IV) $x \in \left[-\frac{3}{2}, 5\right)$ | Choose the correct answer from the options given below:

If A is a square matrix of order 3 such that $A( {adj } A) = \begin{bmatrix} -2 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & -2 \end{bmatrix}$, then $|A|$ is equal to

If $A = \begin{bmatrix} 2 & -3 & 4 \\ -3 & 5 & x \\ 4 & 3 & 0 \end{bmatrix}$ is a symmetric matrix and $B = \begin{bmatrix} 0 & 2 & -10 \\ -2 & z & 6 \\ y & -6 & 0 \end{bmatrix}$ is a skew-symmetric matrix, then the value of $(xy + yz + zx)$ is

Let $A$ be a non singular matrix of order $n \times n$, Then $|\text{adj }(3A)|$ is equal to:

Let $\vec{a} = \hat{i} + \hat{j}$, $\vec{b} = \hat{i} - \hat{j}$ and $\vec{c} = \hat{i} + \hat{j} + \hat{k}$. If $\hat{m}$ is a unit vector perpendicular to both $\vec{a}$ and $\vec{b}$, then $|\vec{c}.\hat{m}|$ is equal to

If the area of a triangle whose vertices are (-1, 3), (1, -5) and (k, 2) where $k > 0$ is 30 sq. units, then the value of k is

Let A be any square matrix of order 3 and $B = \begin{bmatrix} 0 & -4 & 2 \\ 4 & 0 & 3 \\ -2 & -3 & 0 \end{bmatrix}$. Then the matrix $ABA^T$ is a

Let the matrix $A = [a_{ij}]_{3\times3}$ be defined by $a_{ij} = \begin{cases} 2i + 3j, & i < j \\ 5, & i = j \\ 3i - 2j, & i > j \end{cases}$ The number of elements in the matrix A which are greater than 7, is:

If we take 8 identical slips of paper and write the number 0 on one of them, the number 1 on three of the slips, the number 2 on three of the slips and the number 3 on one of the slips. These slips are folded, put in a box and roughly mixed. One slip is drawn at random from the box. If X is the random variable denoting the number written on the drawn slip, the variance of X is:

The solution set of the linear inequation $|4x - 3| \leq \frac{3}{4}$ is:

Let $A = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}$ and $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$. If $A^T + A = I$, then

Which of the following is NOT correct?

The maximum value of $z = 5x + 7y$ subjected to constraints $x + y \leq 5$, $x \geq 0$, $y \geq 0$ is:

If A is square matrix of order 3 × 3 and |adj A| = 64, then the value of |5A| is

The following system of equations $2x - y + 3z = 5, 3x + 2y - z = 7, 4x + 5y - \lambda z = \mu$ is consistent. Then

Match List-I with List-II Let A and B be any two events | List-I | List-II | | --- | --- | | (A) $P(A')$ | (I) $\frac{P(A \cap B)}{P(A)}; P(A) \neq 0$ | | (B) $P(\phi)$ | (II) $\frac{P(A \cap B)}{P(B)}; P(B) \neq 0$ | | (C) $P(A\vert B)$ | (III) $1 - P(A)$ | | (D) $P(B\vert A)$ | (IV) 0 | Choose the correct answer from the options given below:

For the linear programming problem (LPP): Maximize $Z = x + 1.5y$, subject to constraints, $x + 2y \leq 40$, $2x + y \leq 40$, $x + y \leq 25$, $x \geq 0$, $y \geq 0$. Which of the following is NOT correct?

The probability distribution function of a normal variate with mean $\mu$ and variance $\sigma^2$ is given by: $f(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2}$, $-\infty < x < \infty$, $-\infty < \mu < \infty$, $\sigma > 0$ If $y = f(x)$ be the normal probability curve, then which of the following is correct? (A) The normal curve is symmetrical about the line $x = \mu$. (B) Mean, median and mode of the distribution coincide. (C) Y- axis is an asymptote to the normal curve. (D) If x increases numerically, $f(x)$ decreases rapidly. Choose the correct answer from the options given below:

The number of all possible matrices of order 3 with each entry either 0 or 1 is:

If $A$ and $B$ are square matrices of the same order, then which of the following statements are correct? (A) $|A^{-1}| = |A|^{-1}$ (B) $adj(A) = |A|A^{-1}$ (C) $(A + B)^{-1} = B^{-1} + A^{-1}$ (D) $(AB)^{-1} = B^{-1}A^{-1}$ Choose the correct answer from the options given below:

Match List-I with List-II | List-I (Matrix A) | List-II (Determinant of Adjoint of A) | |---|---| | (A) $\begin{bmatrix} 3 & 1 \\ 4 & 2 \end{bmatrix}$ | (I) 9 | | (B) $\begin{bmatrix} 5 & -1 \\ 4 & 2 \end{bmatrix}$ | (II) 8 | | (C) $\begin{bmatrix} 6 & -1 \\ 2 & 1 \end{bmatrix}$ | (III) 14 | | (D) $\begin{bmatrix} 4 & 1 \\ 3 & 3 \end{bmatrix}$ | (IV) 2 | Choose the correct answer from the options given below:

About linear programming problem (LPP), which of the following statements are correct? (A) In a LPP, the linear inequalities or restrictions on the variables are called linear constraints. (B) If the feasible region for an LPP is unbounded, then the maximum or minimum value of the objective function $Z = ax + by$ never exists. (C) The feasible region for an LPP is always convex. (D) The common region determined by all the linear constraints of an LPP is called the feasible region. Choose the correct answer from the options given below:

If the events A and B are independent, then which of the following statements are true? (A) P(A'B) = [1-P(A)] P(B) (B) A and B are mutually exclusive (C) P(A) = P(B) (D) P(A'B') = [1-P(A)] [1-P(B)] Choose the correct answer from the options given below:

If the matrix $A = \begin{bmatrix} \alpha & \beta & \gamma \\ 0 & 0 & 2 \\ 3 & -2 & 0 \end{bmatrix}$ is a skew symmetric matrix, then the value of $(\alpha + \beta + \gamma)^2$ is: