Algebra PYQ

If a random variable y follows Poisson's distribution such that $P(X = 2) = 9 P(X = 4) + 90 P(X = 6)$, then sum of the mean and variance of X is

If $\vec{a}, \vec{b}, \vec{c}$ are vectors such that $\vec{a} + \vec{b} + \vec{c} = \vec{0}$ and $|\vec{a}| = 1, |\vec{b}| = 2, |\vec{c}| = 5$, then the expression $\vec{a}\cdot\vec{b} + \vec{b}\cdot\vec{c} + \vec{c}\cdot\vec{a}$ equals

The relation R in the set $\{1, 2, 3\}$ given by $R = \{(1, 1), (2, 2), (3, 3), (1, 2), (1, 3), (2, 3)\}$ is:

Let $A$ and $B$ be square matrices of order 3, then det $[(A - A^T) + (B - B^T)]$ is equal to

If A is an invertible matrix of order 3 and the determinant of A is 9, then the determinant of $A^{-1}$ is:

Suppose that A, B and C are matrices of order $m \times n$, $n \times 5$ and $5 \times q$ respectively. The restriction on $m$, $n$ and $q$ so that AB-BC is defined are

A die is thrown 4 times and getting 3 is considered a success. The probability of 2 successes is:

Match List-I with List-II If the random variable x has the following distribution: | x | 0 | 1 | 2 | otherwise | |---|---|---|---|-----------| | P(x) | k | k | 2k | 0 | | List-I | List-II | |---|---| | (A) k | (I) $\frac{3}{4}$ | | (B) P(x ≥ 2) | (II) $\frac{1}{4}$ | | (C) P(x ≤ 2) | (III) $\frac{1}{2}$ | | (D) P(0 < x ≤ 2) | (IV) 1 | Choose the correct answer from the options given below:

If A and B are symmetric matrices of the same order, then which one of the following is true?

The corner points of the bounded feasible region for an LPP are (0, 20), (3,12), (6,8), and (0,15). The objective function is $Z = \alpha x + \beta y$, where $\alpha, \beta > 0$. If the maximum of Z occurs at the corner points (3,12) and (6,8), then the relationship between $\alpha$ and $\beta$ is:

The solution set of $6 \leq -3(2x - 4) < 12$, $x \in R$ is:

Let $A$ be a non-singular square matrix of order $n$, then Match List-I with List-II | List-I | List-II | | --- | --- | | (A) $A(\text{adj} A)$ | (I) $\frac{1}{\vert A\vert }$ | | (B) $\vert \text{adj} A\vert $ | (II) $\vert A\vert ^n$ | | (C) $\vert A^{-1}\vert $ | (III) $\vert A\vert I$ | | (D) $\vert A(\text{adj} A)\vert $ | (IV) $\vert A\vert ^{n-1}$ | Choose the correct answer from the options given below:

If $\begin{bmatrix} x+y & 4 \\ 1+z & y \end{bmatrix} = \begin{bmatrix} 2 & 4 \\ 5 & 6 \end{bmatrix}$, then

If $A$ is a skew-symmetric matrix of order 5, then $|adjA|$ is equal to

Let $A$ and $B$ are square matrices of order 3 such that $A + B = \begin{bmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \end{bmatrix}$. If $A$ is a symmetric matrix, then the value of $|B|$ is

Let A be a non-singular square matrix of order 3 and $|adj A| = 5$ then $|A|$ is equal to

The number of all possible matrices of order $2 \times2$ with each entry $0, 1$ or $2$ are.

A fair coin is tossed 100 times. The probability of getting head an odd number of times is

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be defined as $f(x) = [x]$, where [x] denotes the greatest integer less than or equal to x. Then which of the following statements are correct? (A) f is one-one but not onto (B) f is not onto (C) f is not one-one (D) f is one-one and onto Choose the correct answer from the options given below:

For the system of equations AX = B, which of the following is correct?

The function $f: [-1, 1] \rightarrow R$ (set of real numbers) given by $f(x) = \frac{x}{x+3}$ is

If the matrix $M = \begin{bmatrix} 0 & -1 & 3\alpha \\ 1 & \beta & -5 \\ -6 & 5 & 0 \end{bmatrix}$ is skew-symmetric, then