Algebra PYQ — Page 4

If $\begin{vmatrix} 1 & -2 & 5 \\ 2 & a & -1 \\ 0 & 4 & 2a \end{vmatrix} = 86$, then product of all values of $a$ is:

If $\vec{a}$ is a unit vector and $(\vec{x} - \vec{a}) \cdot (\vec{x} + \vec{a}) = 15$, then the value of $|\vec{x}|$ is:

A random variable y has the following probability distribution | y | 1 | 2 | 3 | 4 | 5 | |---|---|---|---|---|---| | P(y) | 2k | 3k | k | 4k | 5k | Match List-I with List-II | List-I | List-II | |---|---| | (A) $P(y > 2)$ | (I) $2/5$ | | (B) k | (II) $2/3$ | | (C) $P(y \leq 3)$ | (III) $8/15$ | | (D) $P(2 \leq y \leq 4)$ | (IV) $1/15$ | Choose the correct answer from the options given below:

The system of equations $x + ky = 0$ $3x + 5y = 0$ has infinitely many solutions, then k is equal to

Consider two independent events A and B such that $P(A) = 0.3$, $P(B) = 0.6$. Match **List-I** with **List-II** | List-I | List-II | |---|---| | (A) $P(A$ and $B)$ | (I) 0.28 | | (B) $P(A$ and not $B)$ | (II) 0.18 | | (C) $P(A$ or $B)$ | (III) 0.12 | | (D) $P$(neither A nor B) | (IV) 0.72 | Choose the correct answer from the options given below:

For the LPP: minimize $z = 6x + 3y$ subject to the constraints $4x + y \geq 80$ $x + 5y \geq 115$ $3x + 2y \leq 150$ $x \geq 0, y \geq 0$ then the minimum value of z is

If A and B are two matrices of order 2 × 2 such that A is a symmetric matrix and B is a skew-symmetric matrix, then:

A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn and are found to be heart then the probability of the missing card to be a heart is:

If A and B are two square matrices of same order such that $AB = A$ and $BA = B$, then the value of $A^{2024} + B^{2024}$ is equal to

A black and a red die are rolled simultaneously. The probability of obtaining a sum greater than 9, given that the black resulted in a 5 is

A relation R on the set $A = \{1, 2, 3, \ldots, 13, 14\}$ defined as $R = \{(x,y): 3x - y = 0\}$ is

Which of the following is incorrect about the Linear Programming Problem (LPP)?

The feasible region of a linear programming problem is bounded. The corresponding objective function is Z= 3x-4y. The objective function attains

If a person A speaks the truth in 80% cases and the person B speaks the truth in 75% cases, then the probability that they contradict each other in a statement is

For the given linear programming problem $z = ax + by; a, b > 0$ subject to the constraints $2x + y \leq 10, x + 3y \leq 15, x, y \geq 0$. If the corner points are (0,0), (5,0), (3,4) and (0,5) and z is maximum at both (3,4) and (0,5), then the relationship between a and b is

Consider the following L.P.P minimize $z = x - 7y + 190$ subject to $x + y \le 8, x + y \ge 4, x \le 5, y \le 5$ and $x, y \ge 0$. Then which of the following is/are true? (A) It's feasible region is unbounded (B) It's feasible region is bounded (C) It's feasible region has 5 corner points (D) It's feasible region has 6 corner points Choose the **correct** answer from the options given below:

If X is a normal variate with mean 16 and standard deviation 4, then the value of standard normal variate Z corresponding to X = 17 is:

If $A = \begin{bmatrix} 2 & -3 \\ -4 & 7 \end{bmatrix}$ and $2A^{-1} = KI - A$, where K is a real number and I is the identity matrix of order 2, then the value of K is:

If $\begin{bmatrix} -1 & 1 & 0 \\ a & b & 1 \\ 1 & 2 & 1 \end{bmatrix}$ is a singular matrix, then the relation between $a$ and $b$ is:

If $A = \begin{bmatrix} x & -3 & 4 \\ 3 & y & -5\\-4&z&0 \end{bmatrix}$ is a Skew-Symmetric matrix and $adj \ A = [a_{ij}]_{3 \times3}$, then $a_{11} + a_{22} + a_{33}$ is equal to

If $x = -4$ is a root of $\begin{vmatrix}x & 2 & 3 \\ 1 & x & 1 \\ 3 & 2 & x\end{vmatrix} = 0$, then the sum of the other 2 roots is

If $a > b$ and $c < 0$, then which of the following is NOT correct? (A) $ac < bc$ (B) $a + c < b + c$ (C) $a - c < b - c$ (D) $ac > bc$ Choose the correct answer from the options given below:

If $\begin{bmatrix} 1 & 0 & 0 \\ 0 & y+1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 2x & \\ -2 & \\ z-3 & \end{bmatrix} = \begin{bmatrix} 6 \\ 4 \\ 1 \end{bmatrix}$ then $x + y + z$ is

If $\vec{a}$ and $\vec{b}$ are two non-zero vectors such that $|\vec{a} \cdot \vec{b}| = |\vec{a} \times \vec{b}|$, then the angle $\theta$ between $\vec{a}$ and $\vec{b}$ is