CUET UG 2025 — Algebra

If the roots of the equation x² - 5x + k = 0 are in the ratio 2:3, then the value of k is:

If $A = [a_{ij}]$ be square matrix of order 3, such that $a_{ij} = i + j$, $\forall i, j$ then which of the following are correct? (A) A is a skew-symmetric matrix. (B) A is a non-singular matrix. (C) The inverse of A does not exist. (D) A is a symmetric matrix. Choose the correct answer from the options given below:

A function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = \frac{x}{x^2+1}$ is (where $\mathbb{R}$ is a set of real number)

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:

If A is a square matrix such that $A^2 = A$ and I is the identity matrix of same order as A, then the value of $(A-2I)^2 - (2A + I)^2 + 11A$ is:

The graph given below represents which of the following function? 

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 value(s) of $K$, for which the system of linear equations $2x + y + z = 1, x + Ky - z = \frac{3}{2}$ and $3y - 5z = 9$ does not possess a unique solution is

If A (3, 2), B (1, -1) and C (2, 1) are three vertices of a parallelograms ABCD, then its area (in sq.units) is equal to

Which of the following are the properties of Normal Distribution function f(x) and Normal probability curve: (A) The probability of success remains the same in each trial and the number of trials is small in number. (B) The curve is bell-shaped and is symmetrical about the mean. (C) If set of n trials are repeated N times, then frequency f(r) of r successes is given by f(r) = N.p(r) = N$e^{-m\frac{m^r}{r!}}$, r=0,1,2,... (D) As x increases numerically, f(x) decreases rapidly and the maximum value of f(x) occurs at x=μ(mean) Choose the correct answer from the options given below:

In a linear programming problem, the constraints on decision variables $x$ and $y$ are $y-2x \leq 0$, $y \geq 0$, $0 \leq x \leq 5$. The feasible region of the above problem:

If A and B are symmetric matrices of order 3 x 3 then the matrix $2AB - BA$ is:

If $A$ is a $3 \times 3$ matrix such that $|adj A| = 9$ and $|kA^{-1}| = 9$, then the value of $k$ are:

If A is a skew-symmetric matrix, then which of the following statements is **NOT** true? (A) A is singular if order of A is odd (B) A is non-singular (C) $A^{2025}$ is a skew-symmetric matrix (D) $A^{2025}$ is a symmetric matrix (E) all diagonal elements of A are zeros Choose the correct answer from the options given below:

If $\vec{a}$, $\vec{b}$, $\vec{c}$ are unit vectors such that $\vec{a} \cdot \vec{b} = \vec{a} \cdot \vec{c} = 0$, and the angle between $\vec{b}$ and $\vec{c}$ is $\frac{\pi}{6}$, then

Which of the following functions from $\mathbb{Z}$ to $\mathbb{Z}$ is a bijective function? (where $\mathbb{Z}$ is set of integers)

The feasible region corresponding to the linear constraints of a Linear Programming Problem (LPP) is represented by the shaded region in the given figure. Which of the following is not a constraint to the given LPP? 

Let x denotes the number of heads in a simultaneous toss of three coins, then $P(0 < x \leq 3)$

If A is an invertible symmetric matrix, then A⁻¹ is

If $\vec{a} = 3\hat{i} - 6\vec{j} + \hat{k}$ and $\vec{b} = 2\hat{i} - 4\vec{j} + \lambda\hat{k}$ are such that $\vec{a} \parallel \vec{b}$, then $3\lambda + 2 =$

If $A = [a_{ij}]_{3×3}$ where $a_{ij} = \begin{cases} (-1)^{i+j} - 1, & i = j \\ (-1)^{i+j}, & i \neq j \end{cases}$ then the value of $A + A^T$ is:

Match List-I with List-II | List-I | List-II | | --- | --- | | Matrix Product | Order of resultant matrix | | --- | --- | | (A) $[a_{ij}]_{2 \times 3} \times [b_{ij}]_{3 \times 4}$ | (I) $2 \times 4$ | | (B) $[a_{ij}]_{2 \times 1} \times [b_{ij}]_{1 \times 3}$ | (II) Not possible | | (C) $[a_{ij}]_{3 \times 2} \times [b_{ij}]_{3 \times 2}$ | (III) $3 \times 3$ | | (D) $[a_{ij}]_{3 \times 3} \times [b_{ij}]_{3 \times 3}$ | (IV) $2 \times 3$ | Choose the correct answer from the options given below:

A couple has 3 children each child is equally likely to be a boy or a girl. The probability that the eldest child is a girl given that they have atleast one boy is:

If $\begin{bmatrix} 1 & 0 \\ b & 5 \end{bmatrix} + 2\begin{bmatrix} a & 0 \\ 1 & -2 \end{bmatrix} = I$, where $I$ is a unit matrix of order 2, then the value of $(a - b)$ is:

For the system of linear equations $x + y + z = 5000$ $6x + 7y + 8z = 35800$ $6x + 7y - 8z = 7000$ the values of x, y and z are:

Which of the following statements are correct? (A) If A is a square matrix, then $|A^2| = |A|^2$. (B) If A and B are square matrices of the same order, then det (AB) = det (A) + det (B). (C) If A is a square matrix of order 3 and $|A|=2$, then the value of $|-3A|$ is 54. (D) If the matrix $\begin{bmatrix} 5 -x & x -1 \\ 3 &5 \end{bmatrix}$ is singular, then the value of x is 7/2. Choose the correct answer from the options given below:

If the matrix $A = \begin{bmatrix} 1 & 3 \\ 2 & 1 \end{bmatrix}$, then the value of det $(A^2 - 2A)$ is equal to

The value of $\begin{vmatrix} 7! & 8! & 9! \\ 8! & 9! & 10! \\ 9! & 10! & 11! \end{vmatrix}$ is:

If $\begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix} \begin{bmatrix} 3 & 5 \\ -1 & 3 \end{bmatrix} = \begin{bmatrix} m & 14 \\ 2 & n \end{bmatrix}$, then $m + n$ is equal to

If $A = \begin{bmatrix} -1 & 2 & 3x \\ 2y & 4 & -1 \\ 6 & -1 & 0 \end{bmatrix}$ is a symmetric matrix, then the value of $2x - y$ is:

If $A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$ then $A^{-1}$ is equal to

Let $A = [a_{ij}]_{2 \times 4}$ and $B = [b_{ij}]_{4 \times 2}$, then $|3AB|$ is equal to

If a matrix P is both symmetric and skew-symmetric, then

If $A = \begin{bmatrix} a & 1 & -1 \\ 0 & b & 4 \\ 4 & 4 & c \end{bmatrix}$ and $abc = 12$, $b = 4a$, then the value of $|A(adjA)|$ is:

Match List-I with List-II | List-I | List-II | | --- | --- | | (A) A square matrix $P$ is said to be non-singular if | (I) $\vert P\vert = 0$ | | (B) A square matrix $P$ is said to be singular if | (II) $P P^T$ is symmetric | | (C) If a matrix $P$ is both symmetric and skew-symmetric, then | (III) $\vert P\vert \neq 0$ | | (D) If $P$ is a square matrix, then | (IV) $P$ is a null matrix | Choose the correct answer from the options given below:

$A = \begin{bmatrix} 1/3 & 2 \\ 0 & 2x - 3 \end{bmatrix}$ & $B = \begin{bmatrix} 3 & 6 \\ 0 & -1 \end{bmatrix}$ If $AB = I$ (Where I is an identity matrix of order 2) , then value of x is

For two matrices $A = \begin{bmatrix} 3 & 4 \\ -1 & 2 \\ 0 & 1 \end{bmatrix}$ and $B^T = \begin{bmatrix} -1 & 2 & 1 \\ 1 & 2 & 3 \end{bmatrix}$, A - B equals

If $P\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} = \begin{bmatrix} -7 & -8 & -9 \\ 2 & 4 & 6 \end{bmatrix}$, then matrix P is equal to:

If $A = \begin{bmatrix} 2 & 3 \\ 5 & -2 \end{bmatrix}$ be such that $A^{-1} = KA$, then the value of K is:

If $\vec{a}, \vec{b}, \vec{c}$ are three mutually perpendicular unit vectors, then $|\vec{a} + \vec{b} + \vec{c}|$ is equal to

If $A = \begin{bmatrix} a & 4 & -5 \\ d & b & -6 \\ 5 & e & c \end{bmatrix}$ is a skew symmetric matrix, then value of $a + b + c + d + e$ is equal to

Let $A = [a_{ij}]$ is given by $A = \begin{bmatrix} 1 & -1 & 2 \\ 3 & 4 & -5 \\ 2 & -1 & 3 \end{bmatrix}$. Then the matrix $B = [b_{ij}]$, where $b_{ij}$ = Minor of $a_{ij}$ is:

For the objective function Z=-4x + 6y subject to the constraints 3x + 2y ≥ 5, 7x + 2y ≤ 9, x ≥ 0, y ≥ 0, the maximum value of Z occurs at $(a, b)$ and the minimum value of Z occurs at $(p, q)$ then the value of $\frac{a}{p} + \frac{b}{q}$ is:

If the matrix $\begin{bmatrix}2 & -1 & 3\\ \lambda & 0 & 7\\-1 & 1 & 4\end{bmatrix}$ is not invertible, then value of $\lambda$ is

It is known that 3% of plastic bags manufactured in a factory are defective. Using the Poisson distribution on a sample of 100 bags, the probability of at most one defective bag is:

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

If $\hat{a}, \hat{b}$ and $\hat{c}$ are three unit vectors and $\hat{a} + \hat{b} + \hat{c} = \vec{0}$, then the angle between $\hat{a}$ and $(-\hat{b})$ is

If the matrix $\begin{bmatrix} -1 & x-y & 4 \\ 2 & 0 & 5 \\ x+y & z & 6 \end{bmatrix}$ is symmetric, then $x + 3y + 2z$ is equal to

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

Let $A = \begin{bmatrix} 2 & -1 & 3 \\ 1 & 2 & -1 \\ 4 & 1 & 2 \end{bmatrix}$. $M_{ij}$ and $A_{ij}$ respectively denote the minor and cofactor of an element $a_{ij}$ of matrix $A = [a_{ij}]$ (A) $M_{23} = 6$ (B) $A_{22} = -8$ (C) $A_{13} = 7$ (D) $M_{32} = -5$ Choose the correct answer from the options given below:

If $\vec{p}$ and $\vec{q}$ are two unit vectors such that $|\vec{p} + \vec{q}| = \sqrt{2}$, then which of the following are correct? (A) $|\vec{p}| = |\vec{q}| = 1$ (B) $\vec{p}$ and $\vec{q}$ are orthogonal vectors (C) $\vec{p}$ and $\vec{q}$ are collinear vectors (D) $(4\vec{p} - \vec{q}).(2\vec{p} + \vec{q}) = 7$ Choose the correct answer from the options given below:

Let $A = [a_{ij}]$ be a square matrix, where $a_{ij} = \begin{cases} 0, & \text{when } i = j \\ 1, & \text{otherwise} \end{cases}$. If |adj A| = |A|², then which of the following statements are correct? (A) A is a skew symmetric matrix. (B) A is a non-singular matrix. (C) A is a square matrix of order 4. (D) A is a symmetric matrix. Choose the correct answer from the options given below:

If $A = \begin{bmatrix}1 & 2\\ 0 & 3\end{bmatrix}$ then $|A. adj A|$ is

If $A = \begin{bmatrix} 2 & -2 & 1 \\ 0 & 4 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & -2 & 7 \\ 2 & 0 & 6 \end{bmatrix}$ are two matrices such that $3A - 2B + 4C = 0$, then matrix $C$ is equal to:

Mean and variance of a binomial distribution are 6 and 2 respectively. The probability of 2 successes will be

If $A = \begin{bmatrix} 4 & 3 \\ 2 & -1 \\ 1 & 0 \end{bmatrix}$ and $B^T = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 1 & -1 \end{bmatrix}$, then $A - B$ is equal to

If $A = \begin{bmatrix} 0 & l & -3 \\ -2 & 0 & 1 \\ m & -1 & 0 \end{bmatrix}$ is a skew symmetric matrix, then

If $A = \begin{bmatrix} 1 & 2 \\ 4 & -3 \end{bmatrix}$ and $f(x) = 2x^2 - 4x + 5$, the $f(A)$ is equal to

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

Consider the function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = \frac{x}{x^2+1}$ then

The matrix $A = \begin{bmatrix}0 & 0 & 5\\0 & 5 & 0\\5 & 0 & 0\end{bmatrix}$ is a (A) Diagonal matrix (B) Scalar matrix (C) Square matrix (D) Symmetric matrix Choose the correct answer from the options given below:

If $A = \begin{bmatrix} 2 & -1 & 0 \\ 1 & 1 & 2 \\ -1 & 0 & 1 \end{bmatrix}$, then which of the following statement(s) is/are correct? (A) A is singular matrix (B) |3A| = 135 (C) |adj A| = 125 (D) $|A^{-1}| = \frac{1}{5}$ Choose the correct answer from the options given below:

If $A = \begin{bmatrix}1 & -1 \\ 2 & -1\end{bmatrix}$, $B = \begin{bmatrix}a & 1 \\ b & -1\end{bmatrix}$ and $(A + B)^2 = A^2 + B^2$ then

If $\vec{a} = 2\hat{j} - \hat{k}$, $\vec{b} = 2\hat{i} - 3\hat{j} + \hat{k}$ and $\vec{c} = -\hat{i} + \hat{k}$ are three vectors, then the area (in sq. units) of the parallelogram whose diagonals are $(\vec{b} + \vec{c})$ and $(\vec{a} + \vec{c})$ is

Let $A = \begin{bmatrix} 152 & 105 & 3 \\ 149 & 25 & 35 \\ 2 & 1 & 0 \end{bmatrix}$. If $A_{ij}$ denotes the co-factor of an element $a_{ij}$ of the matrix A, then the value of $a_{11}A_{21} + a_{12}A_{22} + a_{13}A_{23}$ is equal to

The mean of the number of heads in the two tosses of a coin is

The probability that a leap year selected at random will have 53 Mondays is

If $P$, $Q$ and $R$ are three singular matrices given by $P = \begin{bmatrix} 2 & 3a \\ 4 & 3 \end{bmatrix}$, $Q = \begin{bmatrix} b & 5 \\ 2a & 6 \end{bmatrix}$ and $R = \begin{bmatrix} a^2 + b^2 - c & 1 - c \\ c + 1 & c \end{bmatrix}$, then the value of $(2a + 6b + 17c)$ is

If $A = \begin{bmatrix} 1 & 2 & 3 \\ -4 & -5 & -2 \end{bmatrix}$, $B = \begin{bmatrix} 2 & -3 \\ 4 & -5 \\ 2 & -1 \end{bmatrix}$ and $BA = [b_{ij}]$, then $(b_{23} - b_{31})$ is equal to

If the corner points of the bounded feasible region for a linear programming problem (LPP) are (0, 2), (3, 0), (6, 0), (6, 8) and (0,5), then which of the following are correct for the objective function $Z = 4x + 6y$? (A) The minimum value of the objective function occurs at (0, 2) and (3, 0) only. (B) The minimum value of the objective function occurs at the mid-point of the line segment joining the points (0, 2) and (3, 0) only. (C) The minimum value of the objective function occurs at every point of the line segment joining the points (0, 2) and (3, 0). (D) The difference between the maximum value and minimum value of the objective function is 60. Choose the correct answer from the options given below:

If $A = \begin{bmatrix} 1 & 5 \\ 7 & 12 \end{bmatrix}, B = \begin{bmatrix} 9 & 1 \\ 7 & 8 \end{bmatrix}$ and C are three matrices such that $3A + 5B + 2C = 0$, then the matrix C is equal to

If $|\vec{a} - \vec{r}| = |\vec{a}| = |\vec{r}| = 1$, then angle between $\vec{a}$ and $\vec{r}$ is

If $A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 0 & 2 \\ x & 1 & 1 \end{bmatrix}$ and $A^{-1} = \frac{1}{4}\begin{bmatrix} -2 & 0 & y \\ 5 & -2 & -1 \\ 1 & 2 & -1 \end{bmatrix}$, then values of x and y, are:

If $x, y$ and $z$ are real number such that $x + y + z = 0$, then value of $\begin{vmatrix}3x & -x+y & -x+z\\x-y & 3y & z-y\\x-z & y-z & 3z\end{vmatrix}$ is

If matrix $A = \begin{bmatrix} x & 2 & 3 \\ a & y & -5 \\ b & c & 0 \end{bmatrix}$ is a skew-symmetric matrix, then (A) $x + y + c = 5$ (B) $c = 5$ (C) $a + b + c = 0$ (D) $a + b - c = 10$ Choose the correct answer from the options given below:

If $A = \begin{bmatrix} 3 & -2 \\ 4 & -2 \end{bmatrix}$, $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ and $A² = KA - 2I$, then the value of K is

Let a pair of dice be thrown and the random variable X be the sum of the numbers that appear on the two dice. Match List-I with List-II | List-I | List-II | |---|---| | **X** | **Probability, P(X)** | | (A) 4 | (I) $\frac{1}{6}$ | | (B) 5 | (II) $\frac{5}{36}$ | | (C) 6 | (III) $\frac{1}{12}$ | | (D) 7 | (IV) $\frac{1}{9}$ | Choose the correct answer from the options given below:

If the system of equations $x - 3y + 5z = 3$ $x - 2y + 4z = 4$ $2x - 7y + \lambda z = 5$ has infinite number of solutions, then the value of $\lambda$ is:

For the matrix $A = \begin{bmatrix} 2 & -1 & -1 \\ 0 & 2 & 3 \\ 1 & -2 & 1 \end{bmatrix}$, which of the following statements are correct? (A) The order of the matrix is 3 × 3 (B) |A| = 21 (C) $|adj\ A| = 225$ (D) A is skew symmetric matrix Choose the correct answer from the options given below:

If the matrix $A = \begin{bmatrix} x & 2 & y \\ -2 & 0 & 3 \\ -1 & z & 0 \end{bmatrix}$ is skew-symmetric, then the value of $2x - 3y + 5z$ is equal to

If $A^T = \begin{bmatrix} -2 & 3 \\ 1 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} -1 & 0 \\ 1 & 2 \end{bmatrix}$, then the matrix $(A + 2B)^T$ is

Let $\begin{vmatrix} 3 & y \\ x & 1 \end{vmatrix} = \begin{vmatrix} 3 & 2 \\ 4 & 1 \end{vmatrix}$ and $x, y$ are natural numbers, then the number of solutions for the system is:

If $\left|\begin{matrix} x & 8 \\ 4 & x \end{matrix}\right| = \left|\begin{matrix} 6 & 2 \\ 18 & 6 \end{matrix}\right|$, then $x$ is/are equal to

Let $A = \begin{bmatrix} 1 & \sin \theta & 1 \\ -\sin \theta & 1 & \sin \theta \\ -1 & -\sin \theta & 1 \end{bmatrix}$, where $0 \leq \theta \leq 2\pi$. Then which of the following are true? (A) $|A| = 2 + 2 \sin^{2} \theta$ (B) $|A| = 2 + \sin^{2} \theta$ (C) minimum value of $|A|$ is $1$ (D) maximum value of $|A|$ is $4$ Choose the correct answer from the options given below:

A die is rolled in such a way that an even number is twice likely to occur as an odd number. If the die is rolled twice, then the mean of the number of perfect squares in two tosses is:

If $A = [a_{ij}]$ is a square matrix of order 2 such that $a_{ij} = \begin{cases} 2, & \text{when } i \neq j \\ 0, & \text{when } i = j \end{cases}$, then det $(A^2)$ is:

If $A = \begin{bmatrix} 2 & 1 & -1 \\ 0 & 1 & 2 \\ 2 & -1 & \lambda \end{bmatrix}$ is a singular matrix, then the value of $\lambda$ is

Let $\vec{a} = \hat{i} + 4\hat{j} + 2\hat{k}$, $\vec{b} = 3\hat{i} - 2\hat{j} + 7\hat{k}$ and $\vec{c} = 2\hat{i} + \hat{j} + 4\hat{k}$. A vector $\vec{d}$ which is perpendicular to both $\vec{a}$ and $\vec{b}$, and $\vec{c} \cdot \vec{d} = 14$, is:

If a person rides his motorbike $x$ km at 30 km per hour, he has to spend ₹ 3 per kilometer on petrol. If he rides $y$ km at a faster speed of 40 km per hour, the petrol cost increases to ₹ 4 per kilometer. If he has ₹ 100 to spend on petrol and wishes to find the maximum distance he can travel within one hour, then linear programming problem (LPP) formulation is:

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 $A = \begin{bmatrix} 0 & 0 & \sqrt{7} \\ 0 & \sqrt{7} & 0 \\ \sqrt{7} & 0 & 0 \end{bmatrix}$, then $|\text{adj } A|$ is equal to

For any events A and B of a sample space S, which of the following statements are TRUE? (A) $P(S | B) = 1$ (B) $P(A \cap B) = P(A) + P(B) + P(A \cup B)$ (C) $P(\bar{A} | B) = 1 - P(A | B)$ (D) $P(A | B) = \frac{P(A \cap B)}{P(B)}, P(B) \neq 0$ Choose the correct answer from the options given below:

The inverse of the matrix $\begin{bmatrix} 4 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 6 \end{bmatrix}$ is

If $\begin{bmatrix} ab & cd \\ a+c & b+d \end{bmatrix} = \begin{bmatrix} 2 & -3 \\ 4 & 1 \end{bmatrix}$ where $a$, $b$, $c$, $d$ are integers, then which of the following are true? (A) $a + d = 0$ (B) $b + d = 3$ (C) $b + d = 1$ (D) $c + d = 2$ Choose the **correct** answer from the options given below:

If $\begin{vmatrix} 1 & \cos \theta & 0 \\ \sin \theta & 1 & \cos \theta \\ |\cos \theta & 1 & -\sin \theta| \end{vmatrix} = A\sin \theta + B\cos \theta + C\sin \theta\cos \theta$ then:

If $\begin{bmatrix} x-y & 0 \\ x+y & 1 \end{bmatrix}$ is an identity matrix and $\begin{bmatrix} x & y \\ z & x \end{bmatrix}$ is a singular matrix then:

If $\vec{a} + \vec{b} + \vec{c} = \vec{0}$ and $|\vec{a}| = 5, |\vec{b}| = 3, |\vec{c}| = 7$, then the acute angle between $\vec{a}$ and $\vec{b}$ is

The value of $\begin{vmatrix} x & x+y & x+y+z \\ 2x & 3x+2y & 4x+3y+2z \\ 3x & 6x+3y & 10x+6y+3z \end{vmatrix}$ is

If $A^{-1}$ exists for the matrix $A = \begin{bmatrix} 1 & \lambda & -1 \\ -1 & 1 & 0 \\ \lambda & 1 & 1 \end{bmatrix}$ then

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

If $\begin{bmatrix} 1 & 3 & 9 \\ 1 & x & x^2 \\ 4 & 6 & 9 \end{bmatrix}$ is singular matrix, where $x \in \mathbb{N}$ (where N set of natural number), then x is equal to

Let $A = \begin{bmatrix} 0 & 2\alpha+1 \\ \ 1& \beta \end{bmatrix}$ and $B = \begin{bmatrix} b_{ij}\end{bmatrix}$ be a skew symmetric matrix of order 2 such that $b_{12} = 1$. If $AB = I_2$ where $I_2$ is identity matrix of order 2, then

For the system $\begin{bmatrix} 2 & -3 \\ -4 & 6 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 5 \\ -10 \end{bmatrix}$ which of the following statements are correct? (A) The system has no solution. (B) The system is consistent. (C) It has infinitely many solutions. (D) It has a unique solution. Choose the correct answer from the options given below:

The value of $\left|\begin{array}{cc}\log_5 10 & 2 \\[4pt] 2 & \log_{10} 5\end{array}\right|$ is

If $\vec{a}$ and $\vec{b}$ are two unit vectors and $\vec{a} + \vec{b}$ is also unit vector, the magnitude of $\vec{a} - \vec{b}$ is

If $A = \begin{bmatrix} 2 & 0 & 3 \\ -1 & 1 & 3 \\ 0 & -4 & 0 \end{bmatrix}$, then the value of det (2A) is

Let the matrix $A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}$. Then which of the following are true? (A) adj $A = \begin{bmatrix} 4 & -3 \\ -1 & 2 \end{bmatrix}$ (B) det $(A) = 5$ (C) det (adjA) = 25 (D) If $A^3 = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, then $a + b = c + d$ Choose the correct answer from the options given below:

If $A = \begin{bmatrix} 0 & 1 & -3 \\ -1 & 0 & 5 \\ 3 & -5 & 0 \end{bmatrix}$, then the value of $|A^{2025}|$ is

If the matrix $\begin{bmatrix}3 & 2a & -5\\4 & 0 & b\\-5 & 3 & 10\end{bmatrix}$ is symmetric, then the value of $5a + 2b$ is

If $\vec{a} \cdot \vec{b} = \vec{a} \cdot \vec{c}$, $\vec{a} \times \vec{b} = \vec{a} \times \vec{c}$ and $\vec{a} \neq {0}$, then the vector $\vec{b}$ in equal to.

If the corner points of bounded feasible region for an LPP are (0,2) (3,0) (6,0) (6,8) and (0, 5) then the minimum value of the objective function f=4x+6y occur at

If two dice are rolled 12 times and getting a total greater than 4 is considered as a success, then which of the following statements are correct? (A) The probability of getting a total greater than 4 in a single throw of the pair of dice is 5/6. (B) Mean = 10 (C) Variance = 3/5 (D) The probability of getting a total less than or equal to 4 in a single throw of the pair of dice is 1/6. Choose the correct answer from the options given below:

If $x, y, z$ are non-zero numbers, then the inverse of matrix $A = \begin{bmatrix}x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z\end{bmatrix}$ is

The vector equation of the line passing through points $A(3,4,-7)$ and $B(1,-1,6)$ is

If $A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$, then the value of $A^{20}$ is:

Let $A = \begin{bmatrix} 2 & -3 & 4 \\ 0 & 1 & 5 \\ -4 & 2 & 3 \end{bmatrix}$ and $a_{ij}$ be any element of matrix A, i, j ∈ {1,2,3}, then which of the following are TRUE? (A) Minor of $a_{23} = 16$ (B) Minor of $a_{23} = -8$ (C) Cofactor of $a_{23} = -16$ (D) Cofactor of $a_{23} = 8$ (E) Cofactor of $a_{13} = 4$ Choose the correct answer from the options given below:

If $A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}$ then $A^2 - 5A$ is equal to (where I is identity matrix of order 2)

If $A = \begin{bmatrix} 7 & 3 \\ 5 & -7 \end{bmatrix}$ be such that $A^{-1} = kA$, then $k$ equals

Probability that a man speaks truth is $\frac{3}{4}$. He throws a die and reports that it is a six. The probability that it is actually a six is

If $A = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$ and $B = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$ then the matrix AB is equal to

If $\vec{a}$ is a non-zero vector, then always

If matrix $A_p = \begin{bmatrix} p(p+1) \\ p(p-1) \end{bmatrix}_{p \in N} \ $(where N is the set of natural numbers), then the value of $|A_1| + |A_2| + |A_3| + ... + |A_{2025}|$ is:

If $\begin{bmatrix} 1 & 2 & 1\end{bmatrix}$ $\begin{bmatrix}1 & 2 & 0 \\ 2 & 0 & 1 \\ 1 & 0 & 2 \end{bmatrix} \begin{bmatrix} 0 \\ 2 \\ x\end{bmatrix} = 0, $then value of x is

If A and B are invertible matrices of same order, then which one of the following is NOT true?

If $\begin{bmatrix} x-y & t \\ 2x-y & w \end{bmatrix} = \begin{bmatrix} -1 & 2 \\ 0 & 1 \end{bmatrix}$, then $2x + y + 3t + w$ is equal to

If $\vec{a}, \vec{b}$ and $\vec{c}$ are three unit vectors such that $\vec{a} + 2\vec{b} - 3\vec{c} = \vec{0}$, then the value of $2\vec{a}.\vec{b} - 6\vec{b}.\vec{c} - 3\vec{c}.\vec{a}$ is

Let A be a square matrix of order 2 such that $\begin{bmatrix} 2 & 1 \\ 3 & 2 \end{bmatrix} A \begin{bmatrix} -3 & 2 \\ 5 & -3 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, then A is:

If $\begin{vmatrix} 1 & -2 & 5 \\ 2 & a & -1 \\ 0 & 4 & 2a \end{vmatrix} = 86$, then product of all values of $a$ 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 $A = \begin{bmatrix} 2 & -3 \\ -4 & 7 \end{bmatrix}$ 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 $\vec{a}, \vec{b}$ and $\vec{c}$ are three vectors such that $\vec{a} + \vec{b} + \vec{c} = \vec{0}$, $|\vec{a}| = 7$, $|\vec{b}| = 3$ and $|\vec{c}| = 5$, then angle between $\vec{b}$ and $\vec{c}$ is

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

The matrix $X$ in the equation $AX = B$, such that $A = \begin{bmatrix} 1 & 3 \\ 0 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & -1 \\ 0 & 2 \end{bmatrix}$ is given by

If $C_{ij}$ represents the cofactor of element $a_{ij}$ of the matrix $A = \begin{bmatrix} 2 & -1 & 3 \\ 1 & 2 & 0 \\ 4 & 1 & 5 \end{bmatrix}$ then the value of $C_{23} + C_{31} - C_{22}$ is

The value of $\begin{vmatrix} 2^x & 1 & 6^x \\ 4^x & 1 & 3^x \\ 2^x & 1 & 6^x \end{vmatrix}$, where $x \neq 0$ is:

Match List-I with List-II | List-I | List-II | |---|---| | (A) Corner point of a feasible region | (I) The line segment joining any two arbitrary points of the region always lies entirely within the region | | (B) Bounded feasible region | (II) can not be enclosed within a circle | | (C) Unbounded feasible region | (III) can be enclosed within a circle | | (D) Convex region | (IV) Is a point of intersection of two boundary lines in the feasible region | Choose the correct answer from the options given below:

If A and B are two non-singular matrices of order n, then which of the following statement/statements is/are not correct? (A) AB is non-singular. (B) AB is singular. (C) $(AB)^{-1} = A^{-1}B^{-1}$ (D) $(AB)^{-1}$ does not exist. Choose the correct answer from the options given below:

If $A = \begin{bmatrix} 5 & 6 \\ 3 & 2 \end{bmatrix}$ then which of the following is correct? (A) $|A|$ is positive (B) $|adj\ A| = -8$ (C) Cofactor of 3 is 6 (D) $|2A| = -32$ Choose the **correct** answer from the options given below:

The system of linear equations $kx + 5y = 5$, $2x + 3y = 5$ will be consistent if

If the matrix $\begin{bmatrix} 0 & 1 & 4x\ \\ -1 & 0 & -5 \\ 2 & 5 & y \end{bmatrix}$ is skew-symmetric, then

The minimum value of $z = 3x + 2y$ subjected to the constraints $2x + y \geq 7, x + 2y \geq 8, x, y \geq 0$ is

Consider the LPP: Maximize $z = 5x + 3y$ subject to $3x + 5y \leq 15$, $5x + 2y \leq 10$, $x,y \geq 0$. The optimal feasible solution occurs at

Let $AX = B$ be a system of three linear equations in three variables. Then the system has (A) a unique solutions if $|A| = 0$ (B) a unique solutions if $|A| \neq 0$ (C) no solutions if $|A| = 0$ and (adj A) $B \neq 0$ (D) infinitely many solutions if $|A| = 0$ and (adj A)$B = 0$ Choose the correct answer from the options given below:

If $\vec{a} = \hat{i} + \hat{k}$, $\vec{b} = \hat{j} - \hat{k}$ and $\vec{c} = \hat{i} + \hat{j} + \hat{k}$ such that $\vec{r} \times \vec{b} = \vec{c} \times \vec{b}$ and $\vec{r} \cdot \vec{a} = 0$, then $\vec{r}$ is:

Bag I contains 3 black and 2 white balls. Bag II contains 2 black and 4 white balls. A bag is selected at random and then a ball is drawn from it. The probability that the ball drawn is black is:

If $A = \begin{bmatrix}0 & x^2-6 & -3\\-x & 0 & -8\\x^2-2x & 8 & 0\end{bmatrix}$ is a skew symmetric matrix, then the value(s) of x is/ are - (A) 3 (B) -3 (C) -2 (D) -1 Choose the correct answer from the options given below:

A die is tossed once. If the random variable X is defined as $X = \begin{cases} 1, & \text{if the die result in an odd number} \\ -1, & \text{if the die result in an even number} \end{cases}$, then the variance of X is

If $A = \begin{bmatrix} 5 & 3 \\ 2 & 4 \end{bmatrix}$, then the matrix $A^2 - 6A + 14$ I is (where I is an identity matrix of order 2)

If $M = \begin{bmatrix} 2 \\ -1 \\ 3 \end{bmatrix}$ and $N = [7 \quad 1 \quad -4]$, then $(MN)^T$ will be equal to:

From the below-mentioned graph of shaded feasible region of a linear programming problem (LPP) with objective function $z = 1.50x + 1.00y$; the maximum value of $z$ will be: 

A random variable X has the following probability distribution | X | 0 | 1 | 2 | 3 | 4 | 5 | |---|---|---|---|---|---|---| | P(X) | 0.1 | k | 0.2 | 2k | 0.3 | k | Then P (X < 3) is

The value of $\begin{vmatrix} 1 & x & y \\ 1 & x+y & y \\ 1 & x & x+y \end{vmatrix}$ is

A function $f: \mathbb{R} \rightarrow \{x \in \mathbb{R}: -1 < x < 1\}$ is defined as $f(x) = \frac{x}{1+|x|}$, then $f$ is:

If $3\begin{bmatrix} x & y \\ z & w \end{bmatrix} = \begin{bmatrix} x & 6 \\ -1 & 2w \end{bmatrix} + \begin{bmatrix} 4 & x + y \\ z + w & 3 \end{bmatrix}$, then the values of $x, y, z$ and $w$ are

In the given figure, feasible region represented by the constraints $4x + y \geq 80$, $x + 5y \geq 115$, $3x + 2y \leq 150$, $x,y \geq 0$ is 

If $A = \begin{bmatrix} 4 & 5 \\ 2 & 1 \end{bmatrix}$ and $I$ is an identity matrix of order 2, then $A - 3I$ equals

For a Binomial distribution, B(n,p), where p+q=1, the sum and product of mean and variance are 8 and 12 respectively, when the value of n is:

The value of $\begin{vmatrix} x^2 - x + 1 & x - 1 \\ x + 1 & x + 1 \end{vmatrix}$ is equal to:

Consider the LPP: Minimize $Z = x + 2y$ subject to $2x + y \geq 3$, $x + 2y \geq 6$, $x, y \geq 0$. The optimal feasible solution occurs at

If $f(x) = \begin{vmatrix} 0 & x-1 & x-2 \\ x+1 & 0 & x-3 \\ x+2 & x+3 & 0 \end{vmatrix}$, then the value of $f(0)$ is equal to:

A bag contains 4 red and 6 green balls. A ball is drawn at random. Its colour is noted and is returned to the bag. One additional ball of the colour drawn is put in the bag. Again a ball is then drawn from the bag. The probability of this ball to be of green colour is

If $|\vec{a}| = 10$, $|\vec{b}| = 2$ and $\vec{a} \cdot \vec{b} = 12$, then value of $|\vec{a} \times \vec{b}|$ is :

If A and B are symmetric matrices of the same order, then

Match List-I with List-II | List-I | List-II | |---|---| | (A) Angle between î and -ĵ is | (I) $\frac{\pi}{6}$ | | (B) Angle between 2î + k̂ and 10î + 5k̂ is | (II) $\frac{\pi}{4}$ | | (C) Angle between î and î + ĵ is | (III) 2π | | (D) Angle between √3ĵ - k̂ and ĵ is | (IV) $\frac{\pi}{2}$ | Choose the correct answer from the options given below:

If the area of a triangle with vertices $(-3,0)$, $(3, 0)$ and $(0, k)$ is 9 sq. units, then k equals

If A is a square matrix of order 3 such that $|A| = 3$, then $|adj(adj A)|$ is equal to:

The corner points of the bounded feasible region for an LPP are (0,4), (4,4), (6,6), (0,12). If the objective function is $Z = px + qy, p > 0, q > 0$, then the condition on p and q so that maximum of Z occurs at (6,6) and (0,12) is

Which of the following terms are associated with a linear programming problem? (A) Constraints (B) Independent events (C) Feasible region (D) Objective function Choose the correct answer from the options given below:

Match List-I with List-II | List-I | List-II | |---|---| | (A) The vectors $\lambda\hat{i}$ $ + \hat{j} + 2\hat{k}$ and $\vec{i} + \lambda\hat{j} + \hat{k}$ are perpendicular if λ is equal to | (I) 1 | | (B) The vectors $3\hat{i} + 6\hat{j} - \hat{k}$ and $2\hat{i} + 4\hat{j} - \lambda\hat{k}$ are collinear if λ is equal to | (II) -1 | | (C) The number of vectors of unit-length which are perpendicular to both the vectors $\vec{a}$ = $\hat{i} + \hat{j} + 2\hat{k}$ and $\vec{b}$ = $3\hat{i} - \hat{j} + 5\hat{k}$ is | (III) 2/3 | | (D) If $\lvert \vec{a} \rvert = 1$ and $\vec{a} + \vec{b} = \vec{0}$, then $\lvert \vec{b} \rvert$ is equal to | (IV) 2 | Choose the correct answer from the options given below:

For independent events $A_1, A_2, A_3, ..., A_n$ if $P(A_i) = \frac{1}{i+1}$, $i = 1, 2, 3, ..., n$, then the probability that none of the events occur is:

Five dice are thrown simultaneously. If the occurrence of an even number in a single dice is considered a success, then the probability of at most 3 successes is

If, in a pair of consecutive positive integers, both numbers are greater than 5 and their sum is less than 23, then the number of such pairs are:

If $A$ is a square matrix such that $A^2 = A$ and $I$ is the identity matrix of the same order as $A$, then $(I + 2A)^2 - 5A$ is equal to

For the objective function $Z = 3x + 5y$ subject to constraints $x + 3y \geq 3$, $x + y \geq 2$, $x \geq 0$, $y \geq 0$:

The corner points of the bounded feasible region of the LPP: Maximize $z = x + y$ subject to constraints $2x + 5y \leq 100$, $8x + 5y \leq 200$, $x \geq 0$, $y \geq 0$ are

The corner points of a bounded feasible region determined by the following system of linear inequalities $x + 3y \leq 60, x + y \geq 10$, $x \leq y$, $x \geq 0$, $y \geq 0$ are (0,10), (5,5), (15, 15) and (0, 20). Let $z = 2px + qy$, $p, q > 0$. If maximum of z occurs at both (15, 15) and (0, 20), then the relation between p and q is

if $A = \begin{bmatrix}1 & 0\\3 & 1\end{bmatrix}$ and $A^4 = \begin{bmatrix}1 & 0\\k & 1\end{bmatrix}$ then value of $k$ is

The corner points of the bounded feasible region determined by a set of constraints in an LPP are $P(0, 5)$, $Q(3, 5)$, $R(5, 0)$ and $S(4, 1)$. If the objective function is $z = ax + 2by$, where, $a, b > 0$, then the condition on $a$ and $b$ such that the maximum value of $z$ occurs at $Q$ and $S$ is

The probability distribution of a random variable X is given by | X | 0 | 1 | 2 | |---|---|---|---| | P(X) | $1 - 7a^2$ | $\frac{1}{2}a + \frac{1}{4}$ | $a^2$ | If $a > 0$, then $P(0 < x \leq 2)$ is equal to

If the difference between mean and variance of a Binomial distribution is 1 and the difference of their squares is 5, then the probability of success is

The corner points of the bounded feasible region determined by the system of linear constraints are (15,0), (40,0), (4,18) and (6, 12). If objective function is Z = 30x + 20y, then the sum of the maximum and the minimum values of Z is

The value of $\begin{vmatrix} 1 & bc & bc(b+c) \\ 1 & ca & ca(c+a) \\ 1 & ab & ab(a+b) \end{vmatrix}$ is

If $\vec{a}$ is any vector, then $|\vec{a} \times \hat{i}|^2 + |\vec{a} \times \hat{j}|^2 + |\vec{a} \times \hat{k}|^2$ is equal to

The curve $y = f(x)$ is normal probability curve, then which of the following statements are correct? (A) mean, median and mode of the distribution coincide. (B) the area bounded by the curve $y = f(x)$ and $x$-axis is one unit. (C) The curve is symmetrical about the line $x = \mu$, where $\mu$ is the mean. (D) $y$-axis is an asymptote to the curve. Choose the correct answer from the options given below:

Match List-I with List-II Let A & B are two events such that P(A)=0.8, P(B)=0.5, P(B|A)=0.4 | List-I | List-II | | :--- | :--- | | (A) $P(A \cap B)$ | (I) 0.2 | | (B) $P(A \mid B)$ | (II) 0.32 | | (C) $P(A \cup B)$ | (III) 0.64 | | (D) $P(A')$ | (IV) 0.98 | Choose the correct answer from the options given below:

If the system of equations $kx + y + z = 0$, $x + ky - z = 0$, $x - y + z = 0$ has a non-zero solution, then the possible values of $k$ are:

Let f: $\mathbb{R} \rightarrow \mathbb{R}$ be defined as $f(x) = 10x$. Then (Where $\mathbb{R}$ is the set of real numbers)

A die is tossed 6 times and getting "1 or 5" is considered a success. The probability of getting at least one success in six tosses is:

The function $f: \mathbb{R} \rightarrow [-1, 1]$ defined by $f(x) = \cos x$ is:

The region represented by the system of inequalities $x, y \geq 0, y \leq 6, x + y \leq 3$

A random variable X has the following probability distribution | X | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |---|---|---|---|---|---|---|---|---|---| | P(X) | a | 3a | 5a | 7a | 9a | 11a | 13a | 15a | 17a | Then the values of 'a' and P(0 < X < 5) respectively are

Match List-I with List-II | List-I | List-II | |---|---| | (Matrix A) | (Determinant of adj A) | | (A) $\begin{bmatrix} 2 & 1 \\ -1 & 2 \end{bmatrix}$ | (I) 3 | | (B) $\begin{bmatrix} 3 & 4 \\ 3 & 6 \end{bmatrix}$ | (II) 6 | | (C) $\begin{bmatrix} 3 & 7 \\ -2 & -4 \end{bmatrix}$ | (III) 5 | | (D) $\begin{bmatrix} 4 & 3 \\ 3 & 3 \end{bmatrix}$ | (IV) 2 | Choose the correct answer from the options given below:

The corner points of the bounded feasible region associated with the LPP: Maximize $Z=px+qy$, $p,q>0$ are $(0, 0)$, $(3.5, 0)$, $\left(\frac{112}{59}, \frac{135}{59}\right)$ and $(0, 3)$. If the optimum value of Z occurs at both $\left(\frac{112}{59}, \frac{135}{59}\right)$ and $(0, 3)$, then

Let $A = [a_{ij}]_{3×2}$ and $B = [b_{ij}]_{3×4}$ be two matrices. Then the order of the matrix $(A^T . B)^T$ is:

If $A = \begin{bmatrix} 2 & -1 & -2 \\ 0 & 2 & -1 \\ 3 & -5 & 0 \end{bmatrix}$, then the value of det (adj (2A)) is:

The probabilities of occurrance of two events A and B are 0.45 and 0.20 respectively. The probability of their simultaneous occurrence is 0.06. The probability that neither A nor B occurs is

The area of triangle with vertices P, Q, R is given by (where $\vec{AB}$ = position vector of point B – position vector of point A)

A random variable X has the following probability distribution: | X | 0 | 1 | 2 | 3 | |---|---|---|---|---| | P(X) | 0.1 | 0.2 | 0.3 | 0.4 | The variance of the X will be:

If $A = \begin{bmatrix} a & a & a \\ o & a & a \\ o & o & a \end{bmatrix}$, then $|adj A|$ is equal to

If the system of equations $x + 2y + 3z = 10$ $-x + y + \lambda z = 20$ $2x + 3y + \lambda z = 0$ does not possess a unique solution, then $\lambda$ is equal to

If A and B are matrices of same order, then $(AB^T - BA^T)$ is always

If E and F are independent events associated with an experiment, then which one of the following statements is correct?

If $\vec{a}, \vec{b}$ and $\vec{c}$ be vectors such that $\vec{a} + \vec{b} + \vec{c} = \vec{0}$, $|\vec{a}| = 3$, $|\vec{b}| = 5$ and $|\vec{c}| = 7$, then the angle between $\vec{a}$ and $\vec{b}$ is

The corner points of the bounded feasible region determined by the system of linear constraints are (0, 0), (5, 0), (3, 4) and (0, 5). Let $Z = px + qy$ where $p, q > 0$. Condition on p and q so that the maximum of Z occurs at both (5, 0) and (3, 4) is

The probability distribution of a ramdom variable $X$ is: $P(X=x)=\begin{cases}kx^2,& x=1,2,3\\2kx,&x=4,5,6\\0,&\text{otherwise}\end{cases}$ Where $K$ is a constant Match List-I with List-II | List-I | List-II | |---|---| | (A) $k$ | (I) 7/22 | | (B) $P(X \geq 4)$ | (II) 1/44 | | (C) $P(X < 4)$ | (III) 95/22 | | (D) $E[X]$ | (IV) 15/22 | Choose the correct answer from the options given below:

The minimum value of $Z = 2x + y$ subjected to $x + y \geq 10, 2x + 3y \leq 26, x, y \geq 0$ is

A player participates in 3 matches against three teams T₁, T₂ and T₃.The probability of winning a match against teams T₁, T₂ and T₃ are 0.2, 0.3 and 0.9 respectively. If 'wins' can be regarded as independent events, then the probability that he (A) wins all the 3 matches is 0.054 (B) wins no match is 0.054 (C) wins exactly two matches is 0.348 (D) wins exactly one match is 0.542 Choose the correct answer from the options given below:

Which of the given values of $x$ and $y$ make the following pair of matrices equal ? $\begin{bmatrix}2x-1 & 4\\y-1 & 3+2x\end{bmatrix}$ and $\begin{bmatrix}0 & y-2\\5 & 4\end{bmatrix}$

The point which provides the optimal solution of the linear programming problem maximize $z = 21x + 35y$ $3x + 2y \leq 30$ $4x + 5y \leq 60$ $x \geq 0, y \geq 0$ has the coordinates

The corner points of the bounded feasible region determined by the system of linear inequalities are $(0,0)$, $(4,0)$, $(2,4)$ and $(0,5)$. If maximum value of $z = ax + by$, where $a,b > 0$, occurs at both $(2,4)$ and $(4,0)$ then

The probability distribution of a random variable x is given below. | x | 0 | 1 | 2 | 3 | |---|---|---|---|---| | P(x) | k/3 | k/2 | k/4 | k/7 | Then the value of k is

If X is a random variable with probability distribution as given below: | X | 0 | 1 | 2 | 3 | |---|---|---|---|---| | P(X) | k | 2k | k | 3k | Then, the variance of the distribution is

The minimum value of $\begin{vmatrix}2 & 2 & 2 \\ 2 & 2+x & 2 \\ 2 & 2 & 2+x\end{vmatrix}$, $x \in R$ is

Let A and B be two events. Then which of the following statements are TRUE? (A) $P(B|A) = \frac{P(A \cap B)}{P(A)}$, provided $P(A) \neq 0$ (B) $P(B') = 1 + P(B)$ (C) $P(A \cup B) = P(A) + P(B) + P(A \cap B)$ (D) $P(A \cap B) = P(A).P(B)$ If A and B are independent events Choose the correct answer from the options given below:

Let $f: \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = 100x + 1$, where $\mathbb{R}$ is a set of real numbers, then

If P and Q are non-singular square matrices of the same order, then $(PQ^{-1})^{-1}$ equals

Three bad eggs are mixed with 7 good ones. If two eggs are drawn one by one without replacement, then the probability distribution of the number (X) of bad eggs drawn is: | X | 0 | 1 | 2 | |---|---|---|---| | P(X) | 1/4 | 1/2 | 1/4 | | X | 0 | 1 | 2 | |---|---|---|---| | P(X) | 15/61 | 20/61 | 26/61 | | X | 0 | 1 | 2 | |---|---|---|---| | P(X) | 7/15 | 7/15 | 1/15 | | X | 0 | 1 | 2 | |---|---|---|---| | P(X) | 1/8 | 1/4 | 5/8 |

If A is an invertible matrix of order 2, then $\det(( {adj } A)^{-1})$ is equal to

The value of $\begin{vmatrix}265 & 240 & 219 \\ 240 & 225 & 198 \\ 219 & 198 & 181\end{vmatrix}$ is

Which of the following statements are true? (A) The vector joining the points P(2, 3, 0) and Q(-1,-2,-4) directed from P to Q is $\vec{PQ} = -3\hat{i} - 5\hat{j} - 4\hat{k}$ (B) Projection of a vector $\vec{a}$ on other vector $\vec{b}$ is $\frac{\vec{a}.\vec{b}}{|\vec{a}|}$ (C) If $\vec{a} = \hat{i} - 2\hat{j} + \hat{k}$ and $\vec{b} = -2\hat{i} + 4\hat{j} + 5\hat{k}$ then $\vec{a} + \vec{b} = -\hat{i} + 2\hat{j} + 6\hat{k}$ (D) If $\theta$ is the angle between $\vec{a}$ and $\vec{b}$ then $\cos \theta = \frac{\vec{a}.\vec{b}}{|\vec{a}||\vec{b}|}$ Choose the correct answer from the options given below:

The probability of a man hitting a target is 1/2. How many times must he fire so that the probability of hitting the target at least once is more than 90%?

If a random variable X has the following probability distribution: | X | 0 | 1 | 2 | 3 | |---|---|---|---|---| | P(X) | K | K/2 | K/4 | K/8 | then, Match List-I with List-II | List-I | List-II | |---|---| | (A) The value of K is | (I) 2/15 | | (B) P(0 < X < 2) is | (II) 1/15 | | (C) P(1 < X < 3) is | (III) 8/15 | | (D) P(X > 2) is | (IV) 4/15 | Choose the correct answer from the options given below:

If $a$, $b$ and $c$ are distinct prime numbers then the value of $\begin{vmatrix} a-b & b-c & c-a \\ b-c & c-a & a-b \\ c-a & a-b & b-c \end{vmatrix}$ is equal to

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:

If $A$ is a square matrix and $I$ is an identity matrix of same order such that $A^2 = A$, then $(2I + A)^2 - 5A$ is

If $\mathbb{Z}$ and $\mathbb{R}$ denote set of integers and set of real numbers respectively, then match List I with List II. | List-I | List-II | |---|---| | (A) $5x - 3 \leq 3x + 1$, $x \in \mathbb{Z}$ | (I) $x \in (-\infty, -3]$ | | (B) $3x + 17 \leq 2(1 - x)$, $x \in \mathbb{R}$ | (II) $x \in (-\infty, -1)$ | | (C) $13x + 17 \leq 2(1 - x)$, $x \in \mathbb{R}$ | (III) $\{......, -4, -3, .......,0,1\}$ | | (D) $\frac{2x + 3}{5} - 2 > \frac{3(x - 2)}{5}$, $x \in \mathbb{Z}$ | (IV) $\{......, -4, -3, -2\}$ | Choose the correct answer from the options given below:

Which of the following are correct? (A) If A and B are symmetric matrices such that AB = BA, then AB is symmetric. (B) If A and B are symmetric matrices of the same order, then (A+B) is a symmetric matrix. (C) If A and B are symmetric matrices of the same order, then (AB-BA) is a symmetric matix. (D) If A and B are symmetric matrices of the same order, then (AB+BA) is a skew symmetric matrix Choose the correct answer from the options given below:

The maximum value of the objective function $z = 10x + 15y$ of an L.P.P. subjected to the constraints $2x + 4y \leq 8$, $3x + y \leq 6$, $-x - y \geq -4$, $x \geq 0$, $y \geq 0$ is:

Match List-I with List-II Where ℝ is set of real numbers | List-I | List-II | |---|---| | (A) f: ℝ → ℝ s.t f(x) = x⁴ is | (I) one-one, Into | | (B) f: ℝ → [0, ∞) s.t f(x) = x⁴ is | (II) many-one, into | | (C) f: [0, ∞) → ℝ s.t f(x) = x⁴ is | (III) one-one, onto | | (D) f: [0, ∞) → [0, ∞) s.t f(x) = x⁴ is | (IV) many-one, onto | Choose the correct answer from the options given below:

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:

Let A, B, C be three events. If the probability of occurring exactly one out of A and B is $\frac{3}{5}$, exactly one of B and C is $\frac{1}{5}$, exactly one of C and A is $\frac{3}{5}$ and that of occurring of three events is $\frac{4}{25}$, then the probability of occurring at least one of them is

The relation R on the set of real numbers defined by $R = \{(a, b): a \leq b^2\}$ is

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

A and B are two independent events. The probability that both events A and B occur is $\frac{1}{6}$ and the probability that neither of them occur is $\frac{1}{3}$. If P(A) = x, P(B) = y then the value of x+y is.

The corner points of the bounded feasible region determined by the system of linear constraints are $(0,8)$, $(4,4)$, $(12,12)$, $(0,20)$. Let $z = px + qy$, where $p, q > 0$. Condition on $p$ and $q$ so that the maximum of $z$ occurs at both the points $(12,12)$, $(0,20)$ is

A pair of dice is thrown until the sum of numbers appeared is a perfect square or a non-perfect square sum appeared five times in succession. If random variable $X$ denotes the number of non perfect square sums appeared, then $P(X > 0)$ is

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

The relation R in $\mathbb{R}$ (set of real numbers) is defined by $R = \{(a,b): a \leq b^3\}$, then R is

As per the below-mentioned graph of shaded bounded feasible region of the LPP, the maximum value of the objective function $z = 2x + y$ is 

If R and S are two equivalence relations on a set A, then

Consider the following L.L.P. Minimize z = 30x - 30y + 1800; subject to x + y ≤ 30, x ≤ 15, y ≤ 20, x + y ≥ 15 and x, y ≥ 0. Then it attains its optimal value at the point

Given that $\vec{a} = -3\hat{i} - 6\hat{j} + 4\hat{k}$, $\vec{b} = 9\hat{i} - λ\hat{j} - 12\hat{k}$. If $\vec{a} \times \vec{b} = \vec{0}$, then the value of λ is

The probability distribution of a random discrete variable is given | X | -1 | 0 | 1 | 2 | 3 | |---|---|---|---|---|---| | P(X) | 0.1 | $p$ | 0.3 | $q$ | $r$ | If it is known that P(X=1) is the mean of P(X=0) and P(X=2). Then the value of r 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 $\vec{a}$ and $\vec{b}$ are two vectors such that $|\vec{a}| = 10, |\vec{b}| = 2$ and $\vec{a} \cdot \vec{b} = 12$, then $|\vec{a} \times \vec{b}|$ is equal to

Let $A = [a_{ij}]$ be a square matrix of order 2 with elements either 0 or 1. Then the difference between the possible number of singular and non-singular matrices 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

The solution set of the inequality $\frac{2x+3}{x-1} < 0$ is:

Which one of the following inequalities is redundant for the shaded feasible region (ABCDA) shown below? 

A coin is tossed and a die is thrown. The probability that the outcome will be a tail on the coin or a number greater than 3 on the die is

If the objective function $z = px + qy$ has its maximum value at the points (2, 1) and (0, 6), then the relationship between p and q is:

The number of all possible matrices of order $2 \times 3$ with each entry 0 or 1 is

If $\vec{a}$ is a unit vector perpendicular to both the vectors $\vec{b} = \hat{j} + \hat{2k}$ and $\vec{c} = \hat{i} + 2\hat{j}$, then $\hat{a}$ is equal to

A bag contains 4 red and 6 black balls. Two balls are drawn in succession without replacement. The probability that the first is red and the second is black is

The values of $\lambda$ for which the system of equation $x + 2y + z = 14, - x + y + z = 10, x + \lambda y + z = 2$ has unique solution is

The corner points of a bounded feasible region are (0, 5), (6, 1), (17, 2) and (4, 29). If the maximum value of objective function $z = px + qy$ where $p$ and $q > 0$ occurs at two points (17, 2) and (4, 29), then the relation between $p$ and $q$ is:

Let A be a 3 × 7 matrix, then each column of A contains:

If $\begin{bmatrix}3 & 1\\2 & 1\end{bmatrix}A\begin{bmatrix}2 & 1\\1 & 1\end{bmatrix} = \begin{bmatrix}1 & 1\\0 & 1\end{bmatrix}$, then matrix 'A' is

If A is a square matrix of order 3 and |A| = -5, then |3A| is equal to

If A and B are any two events such that P(B) = P(A and B), then which of the following is correct

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:

If $x, y$ and $z$ are non-zero distinct numbers, then $\begin{vmatrix} x+y & y+z & z+x \\ z & x & y \\ 1 & 1 & 1 \end{vmatrix}$ is equal to

A letter is known to have come from either TATAPUR or from CHAKRATA. On the envelope, only two letters 'TA' are visible consecutively. The probability that the letter has come from CHAKRATA is:

The corner points of the bounded feasible region determined by the system of linear constraints are $(0, 0)$, $(5, 0)$, $(6, 5)$, $(6, 8)$, $(4, 10)$, $(0, 8)$. Let $Z = 3x - 4y$ be the objective function. The minimum value of Z occurs at

Let $R = \{(L_1, L_2): L_1 \perp L_2\ $ where $L_1, L_2 \in L$ (set of straight line in a plane)}, then

Let A be a square matrix of order n, then which of the following are TRUE? (A) $|adj A| = |A|^{n-1}$ (B) $|A. adj A| = |A|^n$ (C) $A. (adj A) = |A|$ (D) $|KA| = K|A|$ (E) $|A^{-1}| = \frac{1}{|A|}, |A| \neq 0$ Choose the correct answer from the options given below:

A dice is thrown twice, the probability of occurence of 5 at least once is

Let X denotes the number of doublets obtained in 3 throws of a pair of dice. Match **List-I** with **List-II** | List-I | List-II | |---|---| | (A) $P(X = 0)$ | (I) $\frac{1}{216}$ | | (B) $P(X = 1)$ | (II) $\frac{15}{216}$ | | (C) $P(X = 2)$ | (III) $\frac{75}{216}$ | | (D) $P(X = 3)$ | (IV) $\frac{125}{216}$ | Choose the correct answer from the options given below:

A vector $\vec{a}$ of magnitude $3\sqrt{2}$ making an angle of $\frac{\pi}{3}$ with $\hat{i}$, $\frac{\pi}{4}$ with $\hat{j}$ and an actue angle $\theta$ with $\hat{k}$, is

A random variable X has the following probability distribution: | X | -2 | -1 | 0 | 1 | 2 | 3 | |---|---|---|---|---|---|---| | P(X) | 0.1 | 0.2 | k | 0.3 | 2k | 0.1 | then which of the following are TRUE? (A) $k=0.1$ (B) $P(X < 1) = 0.4$ (C) $P(X < 2) = 0.7$ (D) $P(0 < X < 3) = 0.5$ Choose the correct answer from the options given below:

The corner points of the feasible region determined by a system of linear constraints are $(0, 0)$, $(0, 40)$, $(20, 40)$, $(60, 20)$, $(60, 0)$. If the objective function is $z = 4x + 3y$, then which one of the following is true?

A and B are two sets such that $n(A) = 5$ and $n(B) = 7$. The number of one-one functions from A to B is

The number of equivalence relation on the set $\{1, 2, 3\}$ containing $(1, 2)$ and $(2, 1)$ is

Which of the following statements are correct? (A) If $\vec{a}$ and $\vec{b}$ represent the adjacent sides of a triangle, then its area is $\frac{1}{2}|\vec{a} \times \vec{b}|$ (B) If $\vec{a}$ and $\vec{b}$ represent the adjacent sides of a parallelogram, then its area is $|\vec{a} \times \vec{b}|$ (C) $|\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \cos\theta$ (D) If $\vec{a}$ and $\vec{b}$ represent the 'diagonals' of a parallelogram, then its area is $\frac{1}{2}|\vec{a} \times \vec{b}|$ Choose the correct answer from the options given below:

Consider the LPP: Max $Z = 5x + 3y$ subject to $3x + 5y \leq 15, 5x + 2y \leq 10, x \geq 0, y \geq 0$ Match **List-I** with **List-II** | List-I | List-II | |---|---| | (A) Objective function | (I) $3x + 5y \geq 15$ | | (B) One constraint | (II) $x, y \geq 0$ | | (C) Non-negative restrictions | (III) $Z = 5x + 3y$ | | (D) Point $(1, 2)$ does not lie in the region | (IV) $3x + 5y \leq 15$ | Choose the correct answer from the options given below:

If $\hat{i},\hat{j}$ and $\hat{k}$ are unit vectors along co-ordinates axes OX, OY and OZ respectively, then which of the following is/are true? (A) $\hat{i} \times \hat{i} = \vec{0}$ (B) $\hat{i} \times \hat{k} = \hat{j}$ (C) $\hat{i} \cdot \hat{i} = 1$ (D) $\hat{i} \cdot \hat{j} = 0$ Choose the correct answer from the options given below:

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:

The area (in sq. units) of the triangle whose vertices are $(0, 0)$, $(a, 0)$, $(0, b)$, is equal to

Let $A = \begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} 4 & -6 \\ -2 & 4 \end{bmatrix}$ (A) $\det(A^T) = 1$ (B) $AB = I$, where $I$ is the identity matrix of order 2. (C) $A^{-1} = \begin{bmatrix} 2 & -3 \\ -1 & 2 \end{bmatrix}$ (D) adj $(B) = \begin{bmatrix} 4 & 2 \\ 6 & 4 \end{bmatrix}$ Choose the correct answer from the options given below:

Two cards are drawn successively with replacement from a well-shuffled deck of 52 cards. The probability distribution of number of aces is given by:

The solution of the system of equations $2x + \frac{1}{2}y - z = 1$, $2y = 3$, $x + 2z = 4$ is:

Suppose X has Poisson distribution such that $3 P(X=1) = 2 P(X=2)$ then $P(X>0)$ is:

If A and B are independent events and $P(A) = \frac{1}{2}$ $P(B) = \frac{1}{3}$, then Match List-I with List-II | List-I | List-II | |---|---| | (A) $P(A \cap B)$ | (I) $\frac{1}{2}$ | | (B) $P(\bar{A})P(B) + P(A)P(\bar{B})$ | (II) $\frac{1}{3}$ | | (C) $P(A \mid B) + P(B \mid A)$ | (III) $\frac{1}{6}$ | | (D) $P(A \cap \bar{B})$ | (IV) $\frac{5}{6}$ | Choose the correct answer from the options given below:

Let A and B be 3×3 matrices such that $A \neq B$. If $A^3 = B^3$ and $A^2B = B^2A$, then the determinant of $A^2 + B^2$ is:

The maximum value of the determinant of the matrix $\begin{bmatrix} 1 & 1 & 1 \\ 1 & 1+\sin x & 1 \\ 1+\cos x & 1 & 1 \end{bmatrix}$ is: (where $x$ is real)

let $\vec{a}$ be a non-zero vector of magnitude '$a$' and $\lambda$ is a non-zero scalar, then $\lambda\vec{a}$ is a unit vector if

Let $|\vec{a}| = 5, |\vec{b}| = 2$ and $\vec{a}\cdot\vec{b} = 6$, then the value of $|\vec{a} \times \vec{b}|$ is equal to

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:

For any vector $\vec{a}$, the value of $|\vec{a} \times \hat{i}|^2 + |\vec{a} \times \hat{j}|^2 + |\vec{a} \times \hat{k}|^2$ is equal to:

If the objective function z = 4x + 3y has maximum value on a line joining points (3, a) and (b, 2) where a > 0, b > 0 such that a - b = 2, then the maximum value of z is:

If the random variable X has the following probability distribution: | X | 0 | 1 | 2 | otherwise | |---|---|---|---|---| | P(X) | k | 3k | 5k | 0 | Match List-I with List-II | List-I | List-II | |---|---| | (A) k | (I) $\frac{13}{9}$ | | (B) E (X) | (II) $\frac{4}{9}$ | | (C) P (X ≤ 1) | (III) $\frac{8}{9}$ | | (D) P (1 ≤ X ≤ 2) | (IV) $\frac{1}{9}$ | Choose the correct answer from the options given below: 1. (A) - (II), (B) - (I), (C) - (IV), (D) - (III) 2. (A) - (IV), (B) - (I), (C) - (II), (D) - (III) 3. (A) - (IV), (B) - (II), (C) - (I), (D) - (III) 4. (A) - (III), (B) - (II), (C) - (I), (D) - (IV)

If $\vec{a} = 2\hat{i} - 3\hat{j} + \hat{k}$ and $\vec{b} = 2\hat{i} + \hat{j} - \hat{k}$, then which of the following statements is/are correct? (A) $\vec{a}$ and $\vec{b}$ are collinear (B) $\vec{a}$ and $\vec{b}$ are perpendicular (C) Angle between $\vec{a}$ and $\vec{b}$ is $\frac{\pi}{4}$ (D) $|\vec{a} + \vec{b}| = 2\sqrt{5}$ Choose the **correct** answer from the options given below:

In a Binomial distribution, the probability of getting a success is $\frac{3}{4}$ and the variance is $\frac{3}{8}$ then the probability of no success 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:

The solution of $\frac{7x+12}{x-9} < 4$; $ \neq 9$ is:

The random variable X has the following probability distribution | X | 0 | 1 | 2 | 3 | |---|---|---|---|---| | P(X) | a | a | b | b | such that E(x²) = 2E(x), then the value of b is:

If $|\vec{a}| = a$, then the value of $|\vec{a} \times \hat{i}|^2 + |\vec{a} \times \hat{j}|^2 + |\vec{a} \times \hat{k}|^2$ is

If $\vec{a}$, $\vec{b}$ and $\sqrt{3}\vec{a} + \vec{b}$ are unit vectors, then the angle between $\vec{a}$ and $\vec{b}$ is:

If A and B are two distinct events such that P(A|B) = P(B|A), then which of the following is /are possible? (A) A= B (B) P (A) = P(B) (C) A ⊂ B but A ≠ B (D) A∩ B = ɸ Choose the correct answer from the options given below:

Consider the linear programming problem(LPP): *Minimize* $Z = x + y$ $x + 2y \leq 4,$ $3x + y \geq 3,$ $4x + 3y \geq 6,$ $x, y \geq 0.$ Which of the following is correct for the above linear programming problem (LPP): (A) The LPP has a bounded feasible region. (B) The LPP has a unique optimal solution. (C) The optimal value of the LPP exists at the point (3/2, 0) (D) The corner points of the feasible region are (3/2, 0), (3/5, 6/5), (2/5, 6/5) and (4, 0) Choose the **correct** answer from the options given below:

Two cards are drawn simultaneously at random from a well shuffled pack of 52 Cards. Let X be the random variable which denotes number of kings in the draw. Then the probability distribution of X is

The corner points of the bounded feasible region determined by a set of constraints (linear inequalities) are A(0, 5), B(3, 5), C(5, 0) and D(4, 1) and the objective function is z = $px$ + 2$qy$ where p,q > 0. The condition on $p$ and $q$ such that the maximum z occurs at B and D, is:

If A and B are symmetric matrices of the same order, then which of the following are true? (A) AB - BA is a skew symmetric matrix (B) AB is a symmetric matrix (C) AB is a scalar matrix (D) AB + BA is a symmetric matrix Choose the correct answer from the options given below:

If A and B are invertible matrices of the same order, then $(AB)^{-1}$ is equal to

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:

A fair coin is tossed a fixed number of times. If the probability of getting 11 heads is equal to the probability of getting 13 heads, then the probability of getting 2 heads is:

For an LPP: Maximize $z = 3x + 9y$, $x \geq 0, y \geq 0$, the feasible region OAB is shown in the figure, then the other constraints are 

If A and B are independent events, then which of the following is/are true? (A) $\bar{A}$ and B are independent events (B) $P(A \cap B) = 0$ (C) $\bar{A}$ and $\bar{B}$ are independent events (D) $P(A \cap B) = P(A) + P(B)$ Choose the correct answer from the options given below:

Match List-I with List-II | List-I | List-II | | --- | --- | | (A) If vector $\vec{a}$ and $\vec{b}$ are such that $\vec{a} = \lambda \vec{b}$ and $\vert \vec{a}\vert = \vert \vec{b}\vert $, then | (I) $\vec{a}$ and $\vec{b}$ are orthogonal | | (B) Projection vector of $\vec{a}$ on $\vec{b}$ | (II) $[0, 12]$ | | (C) $\vec{a}$ and $\vec{b}$ are non-zero vectors such that $\vert \vec{a} + \vec{b}\vert = \vert \vec{a} - \vec{b}\vert $, then | (III) $\vec{a} = \pm \vec{b}$ | | (D) If $\vert \vec{a}\vert = 4, -3 \le \lambda \le 2$, then the range of $\vert \lambda \vec{a}\vert $ | (IV) $(\dfrac{\vec{a} \cdot \vec{b}}{\vert \vec{b}\vert ^2}) \vec{b}$ | Choose the correct answer from the options given below:

The following system of equations: $x + y - z = 7$ $4x + \lambda y - \lambda z = 3$ $3x + 2y - 4z = 5$ does not possess a solution if the value of $\lambda$ is:

Let $\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}, \vec{b} = -\hat{i} + 2\hat{j} + \hat{k}, \vec{c} = 3\hat{i} + \hat{j}$ be three vectors. If $(\vec{a} + \lambda\vec{b})$ is perpendicular to $\vec{c}$, then the value of $\lambda$ is

The probability that in a year of the 22nd century choosen at random, there will be 53 Sundays is:

If $\begin{vmatrix} p-a & 0 & c-r \\ 0 & q-b & c-r \\ a & b & r \end{vmatrix} = 0$, then the value of $\dfrac{p}{p-a} + \dfrac{q}{q-b} + \dfrac{r}{r-c}$ is

If $A = \begin{bmatrix} -2 \\ -1 \\ -4 \end{bmatrix}$, $B = [-1 \quad 2 \quad 3]$, then the value of $A'B'$ is

The system of equation $2x + \lambda y = 8$, $\lambda x + 8y = 3$ has a unique solution if the value of $\lambda$ is (are):

If $z = 5x + 8y$ is the objective function of a LPP and (0, 0), (3, 1), (2, 4), (0, 3), (5, 0) are corner points of the bounded feasible region, then the maximum value of the objective function is

The probability distribution of a random variable X is given by | X | 0 | 1 | 2 | |---|---|---|---| | P(X) | k | 2k | 3k | If k > 0, then $P(0 < X \leq 2)$ is equal to

If $\frac{1}{|x| - 3} \leq \frac{1}{2}$, then value of $x$:

If A is a square matrix such that $A^2 = A$ then which of the following statements are TRUE ? (Where I is an identity matrix of same order as A) (A) $(I+A)^4 = I + 15A$ (B) $(I+A)^2 = I + 3A$ (C) $(I+A)^6 = I + 30A$ (D) $(I+A)^3 = I + 7A$ Choose the correct answer from the options given below:

A random variable X follow Poisson distribution such that P(X=1) = 2P(X=2) , then P(X=0) is

Match List-I with List-II | List-I | List-II | |---|---| | Matrix/equations | Values | | (A) $\begin{bmatrix} 2x+1 & 3y \\ 0 & y^2-5y \end{bmatrix} = \begin{bmatrix} x+3 & y^2+2 \\ 0 & -6 \end{bmatrix}$ | (I) $x = 2, y = -1$ | | (B) $\begin{bmatrix} 1 & 2 & -1 \\ x & 0 & 3 \\ y & 3 & 4 \end{bmatrix}$ is symmetric | (II) $x = 2, y = 2$ | | (C) $[x \ \ 1]\begin{bmatrix} 1 & 0 \\ -2 & -3 \end{bmatrix}\begin{bmatrix} 5 & 2 \\ 0 & y \end{bmatrix} = O$ | (III) $x = -2, y = 2$ | | (D) $\begin{bmatrix} x & 0 \\ 1 & 1 \end{bmatrix}\begin{bmatrix} x & 0 \\ 1 & 1 \end{bmatrix} = \begin{bmatrix} 4 & 0 \\ -1 & y/2 \end{bmatrix}$ | (IV) $x = 2, y = 0$ | Choose the correct answer from the options given below:

If $\vec{a}$ and $\vec{b}$ are two non-zero orthogonal vectors, then $|\vec{a} + \vec{b}|$ is equal to

Consider the following L.P.P. Minimize z = 400x + 300y subject to 100x + 200y ≥ 12000, 300x + 400y ≥ 20000, 200x + 100y ≥ 15000 and x, y ≥ 0. Then

If the points (a, b), (c, d) and (a + c, b + d) are collinear, then

The corner points of the feasible region with the constraints $x + y \leq 30$, $x + y \geq 15$, $y \leq 20$, $x \leq 15$ and $x$, $y \geq 0$ are

If $x \neq y \neq z$ then $\begin{vmatrix} 1 & x & x^2 \\ 1 & y & y^2 \\ 1 & z & z^2 \end{vmatrix}$ is equal to

The feasible region of a LPP is bounded. The corresponding objective function is Z= 6x - 7y. Then objective function attains:

Which of the following statement are correct? (A) $A = [a_{ij}]_{n \times n}$ is a diagonal matrix if $a_{ij} = 0$ when $i = j$ (B) A square matrix $A = [a_{ij}]$ is called a symmetric matrix if $a_{ij} = a_{ji}$ for all $i, j$ (C) A square matrix $A = [a_{ij}]$ is called a skew-symmetric matrix if $a_{ij} = -a_{ji}$ for all $i, j$ (D) For every square matrix $A$, there exist an identity matrix of the same order such that $IA = AI = I$ Choose the correct answer from the options given below:

If $A = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$ and $B = \begin{bmatrix} 0 & 2 \\ 2 & 0 \end{bmatrix}$, then the matrix AB is equal to

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:

If the corner points of the bounded feasible region of an LPP are (0,2), (3,0), (6,0), (6,8) and (0,5), then the minimum value of objective function F = 4x + 6y occurs at

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

The probability that A hits a target is $\frac{1}{5}$ and the probability that B hits it is $\frac{2}{3}$. The probability that the target will be hit if both A and B shoot at it independently is:

Let $A = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$, then $(A^{-1})^T$ equals

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' speaks the truth in 80% of the cases while 'B' in 90% of the cases. The probability that they contradict each other in stating the same statement is

A person can sell a maximum of 20 units of shirts and pants on which a profit of ₹40 is made on each shirt and a profit of ₹30 on each pant. A minimum of 2 shirts are being sold, while pants are sold at least 4 times as many as shirts. Then the maximum profit is:

Which one of the following set of constraints represents the shaded region given below? 

Match List-I with List-II Let $A$ and $B$ be two events such that $P(A) = 0.2$, $P(B) = 0.4$, $P(B|A) = 0.5$ | List-I | List-II | | --- | --- | | (A) $P(A \cap B)$ | (I) $0.5$ | | (B) $P(A\vert B)$ | (II) $0.8$ | | (C) $P(A \cup B)$ | (III) $0.25$ | | (D) $P(A')$ | (IV) $0.1$ | Choose the correct answer from the options given below:

If $\hat{i}$, $\hat{j}$ and $\hat{k}$ are unit vectors along the co-ordinate axes OX, OY and OZ respectively, then (A) $\hat{i} \times \hat{j} = \hat{k}$ (B) $\hat{k} \times \hat{i} = -\hat{j}$ (C) $\hat{j} \cdot \hat{j} = 1$ (D) $\hat{j} \cdot \hat{k} = 0$ Choose the correct answer from the options given below:

For a random variable x, probability distribution P(x) is given by $P(x) = \frac{k}{6}(3-x), x = 0, 1, 2$, then Match List-I with List-II | List-I | List-II | |---|---| | (A) k is equal to | (i) $\frac{1}{2}$ | | (B) P(x = 0) | (ii) 1 | | (C) P(x < 2) | (iii) $\frac{1}{6}$ | | (D) P(1 < x ≤ 2) | (iv) $\frac{5}{6}$ | Choose the correct answer from the options given below:

Given a matrix A of order 3x3. If |A|=3 then the value of |A(adj A)| is:

If $\begin{bmatrix}2x+1 & 5x \\ 0 & y^2+1\end{bmatrix} = \begin{bmatrix}x+3 & 10 \\ 0 & 26\end{bmatrix}$ then the possible values of x + y are:

For a linear programming problem, the feasible region is shown in the figure by shaded portion, then linear constraints are 

For the L.P.P. Maximize z = 10x + 6y subjected to 3x + y ≤ 12, 2x + 5y ≤ 34, x, y ≥ 0. Then the feasible region represented by system of inequalities 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:

Probability distribution of random variable X is | X | -2 | -1 | 0 | 1 | 2 | |---|---|---|---|---|---| | P(X) | 2/11 | 1/11 | 4/11 | 3/11 | 1/11 | Then the value of E(X) is

Which of the following statement('s) is/are TRUE? (A) Skew symmetric matrix of even order is always symmetric (B) Skew symmetric matrix of odd order is non-singular (C) Skew symmetric matrix of odd order is singular (D) Skew symmetric matrix is always square matrix Choose the correct answer from the options given below:

A random variable X has the following probability distribution: | X | 0 | 1 | 2 | 3 | |---|---|---|---|---| | P(X) | 0.2 | 0.1 | 0.3 | 0.4 | The variance of X will be

If $ \theta$ is the angle between two unit vectors $\hat{a}$ and $\hat{b}$ then $|\hat{a}-\hat{b}| =$

If $A = [a_{ij}]_{3 \times 2}$ where $a_{ij} = i + j$, then (A) A is a square matrix (B) $a_{21} + a_{32} = 8$ (C) Number of elements in A is 6 (D) Transpose of $A = \begin{bmatrix}2 & 3 \\ 3 & 4 \\ 4 & 5\end{bmatrix}$ Choose the correct answer from the options given below:

The linear inequalities satisfying the shaded feasible region given in the figure are  (A) $x \geq 0$, $y \geq 0$, $2x + y \geq 2$ (B) $x \geq 0$, $y \geq 0$, $2x + y \leq 2$ (C) $x \geq 0$, $y \geq 0$, $2x + y \geq 2$, $x + 2y \leq 8$, $x - y \leq 1$ (D) $x + 2y \geq 8$, $x - y \geq 1$ Choose the correct answer from the options given below:

If A is a singular matrix, then A{adj A} is equal to

If A and B are two events such that $P(A) = \frac{1}{2}$, $P(B) = \frac{1}{3}$ and $P(A \cap B) = \frac{1}{4}$, then which of the following statements are true? (A) A and B are independent events (B) $P(A | B) = \frac{3}{4}$ (C) $P(A' | B') = \frac{5}{8}$ (D) $P(A' | B) = \frac{1}{4}$ Choose the correct answer from the options given below:

If $\vec{a} + \vec{b} + \vec{c} = \vec{0}$ and $|\vec{a}| = 3, |\vec{b}| = 5, |\vec{c}| = 7$, then the angle between $\vec{a}$ and $\vec{b}$ is

The probability distribution of a random variable $x$ is, $P(x) = \frac{k}{2^x}, x = 0, 1, 2, 3$. Then Match List-I with List-II | List-I | List-II | |---|---| | (A) $k$ | (I) $\frac{2}{15}$ | | (B) $P(x = 1)$ | (II) $\frac{1}{5}$ | | (C) $P(1 < x < 3)$ | (III) $\frac{8}{15}$ | | (D) $P(x \geq 2)$ | (IV) $\frac{4}{15}$ | Choose the correct answer from the options given below:

Consider an LPP: Maximise $Z = 50x + 15y$ subjected to constraints $x + y \leq 60$, $5x + y \leq 100$, $x, y \geq 0$. If the maximum value of $Z$ occurs at $x = \alpha$ and $y = \beta$, then the value of $\alpha + \beta$ is

Linear inequalities corresponding to the shaded feasible region OABCO in the given figure are 

For the linear programing problem, $Minimize(Z) = 60x + 30y$ subject to: $2x - y \geq -5; 3x + y \geq 3; 2x - 3y \leq 12; x, y \geq 0$ the optimal value of $z$ is

The maximum value of $z$ for the linear programing problem maximize $z = x + y$ subject to the constraints $x + 4y \leq 8, 2x + 3y \leq 12, 3x + y \leq 9, x \geq 0, y \geq 0$ is:

Solution of the inequality $\frac{2x+3}{4x-5} \geq 0$ is

If $a_{ij}=i+3j$, then the matrix of order 2 with elements as $a_{ij}$ is

The domain of $y = \cos^{-1}(x^2 - 4)$ is

Relation R on the set $A = \{1, 2, ..., 15\}$ defined as $R = \{(x, y): y - 4x = 0\}$ is

In a college, 30% students fail in physics, 25% fail in Mathematics and 10% fail in both. One student is chosen at random. The probability that she fails in physics if she has failed in mathematics is

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:

If $A$ is an invertible matrix of order 3, then Match List-I with List-II | List-I | List-II | | --- | --- | | (A) $\vert \text{adj} A\vert $ | (I) $8\vert A\vert $ | | (B) $\vert A(\text{adj} A)\vert $ | (II) $\vert A\vert ^2$ | | (C) $\vert 2A\vert $ | (III) $\frac{1}{\vert A\vert }$ | | (D) $\vert A^{-1}\vert $ | (IV) $\vert A\vert ^3$ | Choose the correct answer from the options given below:

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:

Which of the following statements are true? (A) If $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$, then $x, y, z$ are called direction ratios of $\vec{r}$. (B) For any two vectors $\vec{a}$ and $\vec{b}$, $\vec{a} + \vec{b} = \vec{b} + \vec{a}$ (C) $\vec{a} \perp \vec{b}$ if and only if $\vec{a} \times \vec{b} = \vec{0}$ (D) Projection of $\vec{b}$ on $\vec{a}$ is $\frac{\vec{a} \cdot \vec{b}}{|\vec{a}|^2}$ Choose the correct answer from the options given below:

Let $\vec{a}$ and $\vec{b}$ are unit vectors. If $\sqrt{3}\vec{a} - \vec{b}$ is a unit vector, then the angle between $\vec{a}$ and $\vec{b}$ is

If the binomial distribution $X\sim B(n, p)$ of mean 3 and variance $\frac{3}{2}$, $(p + q) = 1$, then which of the following is/are TRUE? (A) $q = \frac{1}{2}$, $n = 6$ (B) $P(X \leq 5) = \frac{63}{64}$, $p = \frac{1}{2}$ (C) $q = \frac{1}{3}$, $p = \frac{2}{3}$ (D) $P(X = 4) = \frac{15}{64}$, $n = 6$ Choose the correct answer from the options given below:

A random variable X has the following probability distribution: | X | 2 | 3 | 4 | 5 | |---|---|---|---|---| | P(X) | 5/k | 7/k | 9/k | 11/k | Then the value of $\frac{k}{4}$ is:

If $A = [a_{ij}]$ is skew symmetric matrix of order 'n', then

Let A be a matrix such that $A = \begin{bmatrix} 1 & 2 \\ -2 & 3 \end{bmatrix}$. Then which of the following are TRUE? (A) A is non-singular matrix (B) $A^T = A$ (C) A is not invertible matrix (D) A is not skew-symmetric matrix Choose the *correct* answer from the options given below:

If the system of equations $2x + 5y = 7, 6x + \lambda y = 28$ is inconsistent, then

Which of the following statements are correct? (A) Inverse of a matrix, if it exists, is unique (B) $(kA)' = -kA'$ (where k is any real number) (C) For an invertible matrix $A$, $(A^{-1})^{-1} = A$ (D) For an invertible matrix $A$, $(A')^{-1} = (A^{-1})'$ Choose the **correct** answer from the options given below:

Two percent of the bolts manufactured in a factory are found to be defective. Using the Poisson distribution, the probability that in a sample of 100 bolts chosen at random, exactly two will be defective, is: [Given $e^{-2}=0.135$]

Which of the following statements are correct? (A) The mean and variance of the Poisson distribution are equal. (B) The mean and variance of a Binomial distribution are equal. (C) An unbiased die is thrown again and again until two sixes are obtained, then the probability of obtaining the second six in the 3rd throw is $\frac{5}{108}$. (D) If the variance of a Poisson distribution is 2, then P(X = 2) = $2e^{-2}$ Choose the **correct** answer from the options given below:

A Linear Programming Problem (LPP) consists of which of the following components? (A) Decision variables (B) The graphical compliment (C) The objective function (D) The linear constraints Choose the **correct** answer from the options given below:

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

For a square matrix $A$ of order 3, if $|A| = 2$, then $|adj\ 2A| =$

The value of k for which the system of equations $x + y + z = 1$ $x - ky + z = 1$ $x - y + z = 1$ has more than one solutions is

60% members of a committee favour a certain proposal and 40% members oppose the proposal. A member is selected and let the random variable X = 0 if he opposes and X = 1 if he is in favour. Then the variance of the random variable X is

The objective function of an LPP is $z = ax + \beta y, (a, \beta > 0)$ in that has to be maximized/minimized subject to constraints $x + y \leq 2$, $x \geq 0$, $y \geq 0$. Then max (z) $-$ min (z) is equal to

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

Two persons A and B throw a die alternately till one of them gets a 'three' and wins the game. The probability of A's winning if A starts first is

A random variable X has the following probability distribution: | X | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | |---|---|---|---|---|---|---|---|---| | P(X) | 0 | k | 2k | 2k | 3k | k² | 2k² | 7k² + k | The value of $P(4 < x < 7)$ is equal to

If $A = \begin{bmatrix} 2 & 1 & 3 \\ 4 & -3 & 5 \end{bmatrix}$ and $B = \begin{bmatrix} -2 & 3 \\ 4 & -5 \\ 1 & 2 \end{bmatrix}$, then which of the following statements are TRUE? (A) AB is defined (B) AB and BA both are defined and AB = I, where I is an identity matrix of order 2 (C) BA is defined (D) AB and BA both are defined and AB = BA Choose the correct answer from the options given below:

If A and B are square matrices of the same order, then (A+B) (A-B) is equal to

For the linear programming problem(LPP), Maximize $Z = 4x + y$ $x + y \leq 5$ $3x + y \leq 9$ $x,y \geq 0$. Which of the following are NOT true? (A) The given LPP has unbounded feasible region. (B) The corner points of the feasible region are (0,0), (0, 5), (3, 2) and (3, 0). (C) The optimal value of the objective function is 12. (D) The given LPP has a unique optimal solution. Choose the correct answer from the options given below:

If $A = [a_{ij}]_{2×2}$ where $a_{ij} = \begin{cases} 1, & i \neq j \\ 0, & i = j \end{cases}$ and $I$ is the identity matrix of order 2, then $(A^2 - 3A + 4I)$ is (A) Symmetric Matrix (B) Skew-symmetric Matrix (C) Non-singular Matrix (D) Square Matrix Choose the correct answer from the options given below:

If $a, b$ and $c$ are positive real numbers, then Match List-I with List-II | List-I | List-II | |---|---| | (Expression) | (The Least value of the expression) | | (A) $(a + b)(b + c)(c + a)$ | (I) $8abc$ | | (B) $(a + b + c)(ab + bc + ca)$ | (II) $9a^2b^2c^2$ | | (C) $(a^2b + b^2c + c^2a)(ab^2 + bc^2 + ca^2)$ | (III) $9abc$ | | (D) $(a + b)^2(b + c)^2(c + a)^2$ | (IV) $64a^2b^2c^2$ | Choose the correct answer from the options given below:

The matrix $\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$ is a (A) Null matrix (B) Unit matrix (C) Symmetric matrix (D) Skew-symmetric matrix Choose the correct answer from the options given below:

If three balls are drawn one by one without replacement from a bag containing 5 white and 4 red balls, then the probability distribution of the number of white balls drawn is

If $y$ is normal distribution random variable with mean $\mu = 10$ and standard deviation $\sigma = 2$. $z$ is standard normal variable and $F(Z)$ is cumulative distribution function, then which of the following are true? [Given that $F(1.5) = 0.9332$, $F(3) = 0.9986$, $F(2.25) = 0.9878$ and $F(1) = 0.8413$] (A) $P(X < 13) = 0.9332$ (B) $P(X > 16) = 0.9986$ (C) $P(12 < X < 14.5) = 0.1465$ (D) $P(X > 8) = 0.8413$ Choose the correct answer from the options given below:

Match List-I with List-II | List-I | List-II | |---|---| | (A) Angle between $\vec{i} - \vec{j}$ and $\vec{j} + \vec{k}$ | (I) 0 | | (B) Angle between $2\vec{j} - \vec{k}$ and $\vec{j} + 2\vec{k}$ | (II) $\frac{2\pi}{3}$ | | (C) Angle between $\vec{i} + 2\vec{j}$ and $5\vec{i} + 10\vec{j}$ | (III) $\frac{\pi}{6}$ | | (D) Angle between $\sqrt{3}\vec{i} + \vec{j}$ and $\vec{i}$ | (IV) $\frac{\pi}{2}$ | Choose the correct answer from the options given below:

If the probability of two successes is 9 times the probability of 3 successes in 3 trials of a binomial distribution, then the probability of success in each trial is:

The feasible region associated with the inequality $2x + 3y > 4$ is

If $\theta$ is an acute angle and the vector $\vec{a} = (\sin \theta)\vec{i} + (\cos\theta)\vec{j}$ is perpendicular to the vector $\vec{b} = i - \sqrt{3}j$ then $\theta$ is equal to

If matrices $A = [1 \quad 2 \quad 3]$ and $B = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$, then $BA$ is equal to:

Which of the following statements are correct? (A) If E and F are independent events then $P(E \cap F) = P(E) \cdot P(F)$ (B) If E and F are mutually exclusive events, then $P(E \cup F) = P(E) + P(F) - P(E) \cdot P(F)$ (C) The conditional probability of an event E, given the occurrence of the event F is given by $\frac{P(E \cap F)}{P(F)}, P(F) \neq 0$ (D) If E and F be the events associated with the sample space S of an experiment, then $P(\overline{E}|F) = 2 - P(E|F)$ Choose the correct answer from the options given below:

The projection of the vector $\vec{a} = \hat{i} + 2\hat{j} - 3\hat{k}$ on the vector $2\hat{i} + 6\hat{j} + 3\hat{k}$ is

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

If A be a square matrix of order 3 such that $|A| = 2$, then $|adj(2A)|$ is equal to

If A and B are symmetric matrices, then AB - BA is

If A is an invertible matrix, then which of the following statement(s) is/are TRUE? (A) $|A^{-1}| = |A|$ (B) $(A^{-1})^{-1} = A$ (C) $A^{-1} = \frac{adj A}{|A|}$ (D) $(A^T)^{-1} = (A^{-1})^T$ Choose the correct answer from the options given below:

Assume A, B and C are matrices of order $m \times n$, $n \times 3$ and $3 \times q$ respectively. The restrictions on $_{m,n}$ and $_q$ so that $AB + BC$ is defined are

If X is a random variable and $a$, $b$ are real numbers, then which of the following statements are true? (A) $Var(aX+b) = a^2 Var(X)$ (B) $E(aX+b)= a E(X) + b$ (C) $E(aX+b)= a E(X) - E(b)$ (D) $Var(aX+b)= a Var(X) + b$ Choose the correct answer from the options given below:

A random variable X has the following probability distribution: | X | 0 | 1 | 2 | |---|---|---|---| | P(X) | 1/4 | 1/2 | 1/4 | then, which of the following is correct?

The binomial distribution for which the mean is 5 and variance 4, is

If $A = \begin{bmatrix} 3 & 2a \\ 1 & 5 \end{bmatrix}$ and $B = \begin{bmatrix} 2 & 3 \\ b & 5 \end{bmatrix}$ both are singular matrices, then $a + b$ is equal to

A unit vector perpendicular to the vectors $\hat{i} - \hat{j}$ and $\hat{i} + \hat{j}$ is

If the area of a triangle whose vertices are $(-2, 4)$, $(2, -6)$ and $(k, 4)$, $(k > 0)$ is 35 squnits, then the value of k is

If $\begin{vmatrix} 2x & 5 \\ 8 & x \end{vmatrix} = \begin{vmatrix} 3 & 0 \\ 4 & -8 \end{vmatrix}$, then value(s) of x is/are

The diagonal elements of a skew symmetric matrix are all

The probability that it will rain on any particular day is 50%. The probability that it rains only on the first 4 days of the week is:

The constraints of the given shaded feasible region below of an L.P.P., for non-negative variable constraints $x$ and $y$ are 

If a random variable $X$ has the following probability distribution: | X | 0 | 1 | 2 | 3 | 4 | |---|---|---|---|---|---| | P(X) | k | 2k | 3k | k² | 6k² | , then Match List-I with List-II | List-I | List-II | |---|---| | (A) k | (I) 3/7 | | (B) $P(X < 2)$ | (II) 6/49 | | (C) $P(X > 3)$ | (III) 1/7 | | (D) $P(2 \leq X \leq 3)$ | (IV) 22/49 | Choose the correct answer from the options given below:

How many minimum number of times must a man toss a fair coin so that the probability of having at least one head is more than 90%?

The function $f: [-1, 1] \rightarrow R$ is given by $f(x) = \frac{x}{x + 2}$

If $A = \begin{bmatrix} x & 3 \\ 2 & 4 \end{bmatrix}$, $B = \begin{bmatrix} 2 & 3 \\ y & 3 \end{bmatrix}$ and $C = \begin{bmatrix} z & 1 \\ 8 & 2 \end{bmatrix}$ are singular matrices then: (A) $x > y$ (B) $y > z$ (C) $z > x$ (D) $x \neq y \neq z$ Choose the correct answer from the options given below:

The maximum value of the objective function $Z = 8x + 2y$ of an LPP subject to constraints $2x + y \leq 3, 2x + 3y \leq 6, x \geq 0, y \geq 0$ is:

Let $A = \{1, 2, 3\}$. Then, the number of relations containing $(1, 2)$ and $(1, 3)$ which are reflexive and symmetric but not transitive, is

If the sum and difference of squares of mean and variance of a Binomial distribution is $\frac{225}{256}$ and $\frac{63}{256}$ respectively, the $P(X \geq 2)$ is:

The corner points of the bounded feasible region for an LLP are: (5, 5), (15, 15), (0, 20) and (0, 10). Let $z = 3x + 9y$ be the objective function. Then the value of $maximum(z) - minimum(z)$ is

The probability distribution of a random variable X is | X | 0 | 1 | 2 | 3 | 4 | |---|---|---|---|---|---| | P(X) | 0.2 | k | k | 2k | k | Match List-I with List-II | List-I | List-II | |---|---| | (A) value of k | (I) $\frac{16}{25}$ | | (B) $P(x \geq 2)$ | (II) $\frac{9}{25}$ | | (C) $P(x = 3)$ | (III) $\frac{4}{25}$ | | (D) $P(x < 2)$ | (IV) $\frac{8}{25}$ | Choose the correct answer from the options given below:

The system of equations $x + y + z = 4$ $x + 2y + 3z = 12$ $x + 3y + \lambda z = \mu$ has a unique solution if

If the feasible region of an LPP is bounded and the corresponding objective function is $z = 5x - 9y$, then the objective function attains:

If $A = \begin{bmatrix} 1 & 2 & 1 \\ 2 & 3 & 1 \\ 0 & 0 & 1 \end{bmatrix}$ then $|adj(3A^T)|^2$ is equal to

A random variable 'X' denotes the number of sixes obtained in three throws of a die. Then, the mean of the distribution is:-

The objective function of an LPP is $z = ax + by$. If the maximum value of the objective function is 180, which occurs at two points (15,15) and (0,20), then which one of the following is true?

If $\vec{a} = 2\hat{i} - \hat{j} + 3\hat{k}$ and $\vec{b} = 2\hat{i} + 2\hat{j} + \hat{k}$, then Match List-I with List-II | List-I | List-II | |---|---| | (A) Projection of $\vec{a}$ on $\vec{b}$ is | (I) $-7\hat{i} + 4\hat{j} + 6\hat{k}$ | | (B) $\vec{a} \times \vec{b}$ is | (II) $\frac{1}{\sqrt{101}}(-7\hat{i} + 4\hat{j} + 6\hat{k})$ | | (C) unit vector along $\vec{a} + \vec{b}$ is | (III) $\frac{5}{3}$ | | (D) Unit vector perpendicular to both $\vec{a}$ & $\vec{b}$ is | (IV) $\frac{1}{\sqrt{33}}(4\hat{i} + \hat{j} + 4\hat{k})$ | Choose the correct answer from the options given below:

Let L be the set of all lines in a plane and R be the relation on set L defined by $R = \{(L_1, L_2): L_1 \perp L_2\}$ Then R is (A) an equivalence Relation (B) a symmetric Relation (C) not a transitive Relation (D) a reflexive Relation Choose the correct answer from the options given below:

If P, Q and R are matrices of order 2x3, 3x5 and 5x3 respectively. Then which of the following are valid? (A) P Q R (B) P R Q (C) Q R (D) R Q (E) P R 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 is a square matrix, then $(A^T - A)$ is-

If $\begin{bmatrix} a-b & 0 & 0 \\ 0 & b-c & 0 \\ 0 & 0 & c-2 \end{bmatrix}$ is a scalar matrix such that $a + b + c = 0$, then, which of the following are TRUE? (A) $a = 0$ (B) $b = 0$ (C) $a = 1$ (D) $c = 1$ Choose the correct answer from the options given below:

If the corner points of the bounded feasible region of an LPP with objective function Maximize $z = 2x + 3y$ are (0,0), (1,2) and (1,1), then its optimal value is

Match List-I with List-II An urn contains 4 white and 3 red balls. In a random draw of three balls, the probability of | List-I | List-II | |---|---| | (A) No red ball is | (I) $\frac{12}{35}$ | | (B) Only 1 red ball is | (II) $\frac{1}{35}$ | | (C) Exactly 2 red balls is | (III) $\frac{4}{35}$ | | (D) no white ball is | (IV) $\frac{18}{35}$ | Choose the correct answer from the options given below:

If A and B are two square symmetric matrices of same order, then AB-BA is

Two persons A and B throw a die alternately till one of them gets a six and wins the game. If A begins, then the probabilities of winning of A and B respectively are

If matrix $A = \begin{bmatrix} p & -3 \\ -4 & p \end{bmatrix}$ and $|A^3| = 64$, then the value of p is:

The system of equations $x + y + z = 7$ $x + 2y + 3z = 5$ $x + 3y + \lambda z = \mu$ has a unique solution, if

The vector in the direction of the vector $2\hat{i} - \hat{j} - 2\hat{k}$ that has magnitude 9 units is:

Let $\vec{a} = 2\hat{i} - \hat{j}, \vec{b} =- 4\hat{j} + k\,\text{and}\,\vec{c} = \hat{i} + 2\hat{k}$. If $\vec{d}$ is a vector perpendicular to both $\vec{a}$ and $\vec{b}$ such that $\vec{c} \cdot \vec{d} = 34$, then $|\vec{d}|$ is equal to

The solution set of inequality $3x + 5y < 4$ is

If $\begin{bmatrix} x - 2 & 3 & -2 \\ y & 0 & -4 \\ 2 & z & 0 \end{bmatrix}$ is a skew symmetric matrix, then the value of $x + y + z$ is

The solution set of the inequation $4x + 3y > 5$ is

If X is a random variable and a, b are real numbers, then which of the following statements are correct? (A) $E[aX+b] = a E(X) + b$ (B) $Var (aX + b) = a^2 Var (X) + b$ (C) $Var (aX + b) = a Var (X)$ (D) $Var (X) = E(X^2) - [E(X)]^2$ Choose the correct answer from the options given below:

Which of the following is NOT a basic requirement of the linear programming problem (LPP)?

Two numbers are selected without replacement at random, one at a time from the first six positive integers. Let x denotes the larger of the two numbers. Match List-I with List-II | List-I | List-II | |---|---| | (A) P(x = 2) | (i) $\frac{4}{15}$ | | (B) P(x = 3) | (ii) $\frac{1}{15}$ | | (C) P(x = 4) | (iii) $\frac{2}{15}$ | | (D) P(x = 5) | (iv) $\frac{1}{5}$ | Choose the correct answer from the options given below:

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 the probability that an individual suffers a bad reaction from an injection of a given serum is 0.001. The probability that out of 2000 individuals, more than two individuals suffer from bad reaction is: [Given that $e^{-2} \approx 0.13534$]

If $\frac{1}{x^2} - \frac{1}{x} > 0$, then $x$ lies in the interval

If $A = \begin{bmatrix} 5 & 2 \\ 4 & 3 \end{bmatrix}$ is a given matrix, then which of the following statements are correct? (A) $|A| = 7$ (B) minor of $3 = -5$ (C) co-factor of $2 = -4$ (D) $adj(A) = \begin{bmatrix} 3 & -2 \\ -4 & 5 \end{bmatrix}$ Choose the correct answer from the options given below:

If $A = \begin{bmatrix}2 & 3 & 1 \\ 2 & -1 & 0\end{bmatrix}$ and $B^T = \begin{bmatrix}4 & 4 \\ 6 & -2 \\ 2 & 0\end{bmatrix}$, then $4A + B$ is

A furniture trader deals in only two items - chairs and tables. He has Rs. 50,000 to invest and a space to store almost 35 items. A chair costs him Rs. 1000 and a table costs him Rs. 2000. The trader earns a profit of Rs. 150 and Rs. 250 on a chair and a table, respectively. Choose the correct option among following that describes the given linear programming problem (LPP) to maximize the profit ( where x and y are the number of chairs and tables that trader buys and sells)?

The sum of the x-coordinates of the corner points of the feasible region for the LPP: Minimize $z = 3x + 2y$ subject to constraints $x + y \leq 14$, $x \geq 4$, $x \leq 8, y \geq 0$ is

If A and B are independent events, then which of the following is **not** true?

If $A = \begin{bmatrix} 1 & 2 \\ 4 & 5 \end{bmatrix}$, then Match **List-I** with **List-II** | List-I | List-II | |---|---| | (A) det (A) | (I) $-\frac{1}{3}$ | | (B) det $(A^{-1})$ | (II) $-12$ | | (C) det (2A) | (III) $-3$ | | (D) det $(3A^T)$ | (IV) $-27$ | Choose the **correct** answer from the options given below:

If the objective function for a linear programming problem (LPP) is $Z = 4x + 5y$ and the corner points of the bounded feasible region are (9, 0), (4, 3), (2, 5), and (0,8), then the minimum value of Z is:

Let $A = [a_{ij}]_{3 \times 3}$ be a matrix, defined by $a_{ij} = \begin{cases} 2i+3j & , i < j \\6 &, i=j\\ 3i-2j & , i > j \end{cases}$. The number of elements in A which are greater than 6, is

Let A = {1, 2, 3}. The number of equivalence relations containing (1, 3) is

The solution set of the linear constraints $x - 2y \geq 0, 2x - y \leq -4, x \geq 0$ and $y \geq 0$ 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:

One person speaks truth in 60% of the cases and another person in 80% of the cases. They are likely to agree in stating the same fact in

If A is a square matrix and I is the identity matrix of same order such that $A^2 = I$, then $3(A - I)^3 + 3(A + I)^3 - 15A$ is equal to

If the matrix $\begin{bmatrix} 0 & 7 & -12 \\ -7 & 0 & -5 \\ 2a & 5 & 3b \end{bmatrix}$ is skew-symmetric, then the value of $(4a + 3b)$ is:

Which of the following statements are NOT correct about Standard Normal Distribution? (A) The probability curve of the Standard Normal Distribution is a bell-shaped curve. (B) The Standard Normal variate (Z) score describes the position of each data point in terms of its distance from the mean, when measured in standard deviation units. (C) The Z-score is negative if the data point lies above the mean, and positive if it lies below the mean. (D) There is a 95.45 % probability of randomly selecting a score between $\mu - \sigma$ and $\mu + \sigma$, when $\sigma$ is standard deviation and $\mu$ is mean. Choose the correct answer from the options given below:

A coin is tossed twice and outcomes are recorded. If the random variable X represents the number of heads in the experiment, then the expectation of X will be:

The feasible region of the linear programming problem is represented below:  The constraints of this LPP are

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

Let $\begin{vmatrix} x & 2 \\ 18 & x \end{vmatrix} = \begin{vmatrix} -4 & -2 \\ -8 & -4 \end{vmatrix}$. Then (A) $x = -4$ (B) $x = -6$ (C) $x = 4$ (D) $x = 6$ Choose the correct answer from the options given below:

Match List-I with List-II | List-I | List-II | |---|---| | (A) Angle between $\hat{i} - \hat{j}$ and $\hat{i} + \hat{j}$ | (I) $\pi$ | | (B) Angle between $\hat{i} - \hat{j} + \hat{k}$ and $-\hat{i} + \hat{j} - \hat{k}$ | (II) $\frac{3\pi}{4}$ | | (C) Angle between $\hat{i} + \hat{j}$ and $-\hat{i}$ | (III) $\frac{\pi}{4}$ | | (D) Angle between $\hat{i} + \hat{k}$ and $\hat{k}$ | (IV) $\frac{\pi}{2}$ | Choose the correct answer from the options given below:

The length of line segment joining the points with position vectors $2\hat{i} - 2\hat{j} + 3\hat{k}$ and $5\hat{i} + 2\hat{j} + 3\hat{k}$ is

If A and B are square matrices of order 3 such that $|A| = 3$ and $|B| = -1$, then $|3AB|$ is equal to

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