Algebra PYQ — Page 3

Assume $P$, $Q$, $R$ and $W$ are matrices of order $3 \times 3$, $a \times 4$, $b \times c$ and $d \times a$ respectively. If $PQ + WR$ is well defined, then the value of $ab + cd$ is:

Consider the matrices $A = \begin{bmatrix} 9 & 0 & 0 \\ 0 & 16 & 0 \\ 0 & 0 & 25 \end{bmatrix}$ and $B = \begin{bmatrix} \frac{1}{5} & 0 & 0 \\ 0 & \frac{1}{4} & 0 \\ 0 & 0 & \frac{1}{3} \end{bmatrix}$. The value of $|(AB)^{-1}|$ is

Let $A = [a_{ij}]_{2 \times 3}$ and $B = [b_{ij}]_{3 \times 2}$, then $|5AB|$ is equal to

Consider the Linear Programming Problem Maximize $z = x + y$ Subject to the constraints $x - y \leq -1$, $x \geq y$, $x \geq 0, y \geq 0$ Then which one of the following is TRUE?

The probability of not getting 53 Sundays in a leap year is

Let E and F be two events such that $P(E) = \frac{1}{3}$, $P(F) = \frac{1}{4}$ and $P(E \cap F) = \frac{1}{5}$. Then the value of $P(F|E)$ is equal to

If $A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}$ and $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, then the value of $A^2 - 5A + 6I$ is

If $x, y \in \mathbb{R}$ then match List-I with List-II | List-I | List-II | | --- | --- | | (A) $\vert x\vert < \vert y\vert $ | (I) iff $x^2 > y^2$ | | (B) $\vert x\vert > \vert y\vert $ | (II) iff $x^2 \le y^2$ | | (C) $\vert x\vert \le \vert y\vert $ | (III) iff $x^2 < y^2$ | | (D) $\vert x\vert \ge \vert y\vert $ | (IV) iff $x^2 \ge y^2$ | Choose the correct answer from the options given below:

The value of $\lambda$, for which the two vectors $2\hat{i} - \hat{j} + 2\hat{k}$ and $3\vec{i} + \lambda\vec{j} + \hat{k}$ are perpendicular, is:

The system of equations $x - 3y - 8z = -10$ $2x + 5y + \lambda z = 13$ $3x + y - 4z = 0$ has infinite number of solutions if the value of $\lambda$ is equal to:

For the relation $R = \{(a, b): a \leq b\}$ in $\mathbb{R}$, which of the following is correct?

If R be a relation on the set of integers Z, given by R = {(a, b) : (a - b) is a multiple of 3}, then R is:

If the points $(2, -3)$, $(\lambda, -1)$ and $(0, 4)$ are collinear, then the value of $\lambda$ is

If $X$ is a random variable which can assume values $0, 1, 2, 3$ or $4$ such that $P(X = 1) = P(X = 2)$ and $3P(X = 3) = 4P(X = 4) = P(X = 0) = \frac{1}{8}$, then $P(X > 0)$ is:

The region represented by the constraints $x \geq 0, y \geq 0$ of an LPP is

If $A = \begin{bmatrix} 0 & 1 & 3 \\ 1 & 2 & x \\ 2 & 3 & 1 \end{bmatrix}$ and $A^{-1} = \begin{bmatrix} \frac{1}{2} & -4 & \frac{5}{2} \\ -\frac{1}{2} & 3 & -\frac{3}{2} \\ \frac{1}{2} & y & \frac{1}{2} \end{bmatrix}$, then the value of $8x + 5y$ is:

The probability of a shooter of hitting the target is $\frac{1}{4}$. The minimum number of fire needed so that the probability of hitting the target atleast once is greater than $\frac{7}{16}$ is:

If $P(A) = \frac{3}{10}$, $P(B) = \frac{2}{5}$ and $P(A \cup B) = \frac{3}{5}$ then the value of $P(B|A) + P(A|B)$ is:

Let X denote the number of hours a student studies on a selected day. The probability distribution of X is given by (where k is some unknown constant) $P(X = x_i) = \begin{cases} 0.5, & \text{if } x_i = 0, \\ kx_i, & \text{if } x_i = 1, \\ k(4 - x_i), & \text{if } x_i = 2 \text{ or } 3, \\ 0, & \text{otherwise}. \end{cases}$ Then the value of k is

If A and B are two events such that $P(A|B) = P(B|A)$, and $A \cap B \neq \phi$ then

The range of function $f(x) = 4x^2 + 12x + 7, x \in \mathbb{R}$ is

The corner points of the feasible region of a LPP with the constraints $x + 2y \leq 40$, $3x + y \geq 30$, $4x + 3y \geq 60$, $x, y \geq 0$ are

If $A = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}, B = \begin{bmatrix} 0 & 0 \\ 3 & 0 \end{bmatrix}$ then

A box contains 2 black and 4 red balls and another box contains 4 black and 3 red balls. If a ball drawn at random from one of the two boxes, then the probability of getting a black ball is