CUET UG 2022 — Algebra

A. A relation $R$ on a set $A$ is called an equivalence relation, if it is reflexive, symmetric and transitive. B. The function $f : R \to R$ defined by $f(x) = e^x$ is not one-one. C. The one-one function is also known as injective function. D. The onto function is also known as subjective function. E. A function $f : X \to Y$ is said to be many-one, if two or more than two elements in set $X$ have the different image in set $Y$. Choose the correct answer from the option given below:

A card is picked at random from a pack of 52 playing cards. If the picked card is a queen, then probability of card to be of spade type also, is

A die is tossed four times. The probability of getting an odd number at least once, is

A feasible solution is :

A function f: R $\to$ R is given by $f(x) = x^3 + 3$. If $f(x) = -24$, then the value of x is:

A letter is expected to come either from city 'SURAT' or from city 'RAMPUR' through post office. If on the way, envelope containing the letter is damaged and only two consecutive alphabets RA are visible on it, then the probability that letter comes from the city 'SURAT' is :

A man is known to speak truth 4 out of 5 times. He throws a die and reports that five appears. Then the probability that actual five appears on the dice is

A manufacturer of electronic circuit has a stock of 200 resistors, 120 transistors and 150 capacitors and is required to produce two types of circuits A and B. Type A requires 20 resistors, 10 transistors and 10 capacitors. Type B requires 10 resistors, 20 transistors and 30 capacitors. If the profit on type A circuit is Rs. 50 and that on type B circuit is Rs. 60, identify the constraints for this LPP, if it was assumed that x circuit B of type A and y circuits of type B was produced by the manufacturer. A. $x + 2y \geq 15$ B. $2x + y \leq 20$ C. $x + 2y \leq 12$ D. $x, y \leq 0$ Choose the correct answer from the options given below

A random variable X has the following probability distribution: | x | 1 | 2 | 3 | 4 | |---|---|---|---|---| | p(x) | 2k | 4k | 3k | k | The value of E(X) is:

A relation R = {(1,1),(2,2),(3,3),(1,2),(2,1),(1,3),(2,3)} on A = {1,2,3} will be an equivalence relation, if we delete: Choose the correct answer from the options given below:

A shopkeeper sells three types of flower seeds $A_1, A_2, A_3$. They are sold as a mixture where the proportions are 4 : 4 : 2 respectively. The germination rates of the three types of seeds are 45%, 60% and 35% respectively. Calculate the probability in the following cases. The probability that seed will not germinate, given that the seed is of type $A_3$.

A shopkeeper sells three types of flower seeds $A_1, A_2, A_3$. They are sold as a mixture where the proportions are 4 : 4 : 2 respectively. The germination rates of the three types of seeds are 45%, 60% and 35% respectively. Calculate the probability in the following cases. The probability that seed is of type $A_1$ given that seed doesn't germinate.

A shopkeeper sells three types of flower seeds $A_1, A_2, A_3$. They are sold as a mixture where the proportions are 4 : 4 : 2 respectively. The germination rates of the three types of seeds are 45%, 60% and 35% respectively. Calculate the probability in the following cases. The probability of a randomly chosen seed to germinate is :

A shopkeeper sells three types of flower seeds $A_1, A_2, A_3$. They are sold as a mixture where the proportions are 4 : 4 : 2 respectively. The germination rates of the three types of seeds are 45%, 60% and 35% respectively. Calculate the probability in the following cases. The probability that seed is not of type $A_1$, given that seed germinates.

A shopkeeper sells three types of flower seeds $A_1, A_2, A_3$. They are sold as a mixture where the proportions are 4 : 4 : 2 respectively. The germination rates of the three types of seeds are 45%, 60% and 35% respectively. Calculate the probability in the following cases. The probability that seed is of type $A_2$ given that seed germinate.

A vector perpendicular to a plane containing a triangle ABC having vertices as $A(1,1,0)$, $B(2,1,1)$ and $C(0,3,2)$, is:

Area of the rectangular plot is:

Arrange the vectors in descending order of their magnitudes. (A) $\hat{i} + \hat{j} + \hat{k}$ (B) $2\hat{i} - 3\hat{j}$ (C) $\frac{1}{2}\hat{i} - \frac{1}{3}\hat{j}$ (D) $2\hat{i} - \hat{k}$ Choose the correct answer from the options given below :

Assume $P$, $Q$, $R$ and $S$ are matrices of order $2 \times m$, $k \times n$, $m \times 2$ and $2 \times 3$ respectively. The restrictions on $k$, $m$ and $n$, so that $PQ + RS$ is defined are

Bag A contains 2 red and 3 white balls, Bag B contains 3 red and 2 white balls. If a ball is drawn at random and is found to be red, then the probability that it was drawn from bag B, is :

Bag I contains 3 red and 4 black balls while another Bag II contains 5 red and 6 black balls. One ball is drawn at random from one of the bags and it is found to be red. The probability that it was drawn from Bag II is :

Bag I contains 4 red and 5 black balls, while another Bag II contains 5 red and 6 black balls. One ball is drawn at random from one of the bags and it is found to be black. Then the probability that it was drawn from Bag II, is

Choose the correct statement A. If any two rows or any two columns are identical or proportional, then value of determinant is Zero. B. Minor of an element $a_{ij}$ of the determinant of matrix A is the determinant obtained by deleting $i^{th}$ row and $j^{th}$ column C. If $A = \begin{bmatrix} 1 & 5 \\ 6 & 7 \end{bmatrix}$, then A is Skew-symmetric matrix D. If $A = \begin{bmatrix} 1 & 2 \\ 4 & 5 \end{bmatrix}$, then $(A + A')$ is Symmetric matrix E. If $|A| = 0$, then A is non-singular matrix

Consider an experiment of tossing 3 coins simultaneously. Define the following events: $E$ = [Three heads or three tails appear] $F$ = [At least two heads appear] and $G$ = [At most two heads appear] Choose the correct option:

Consider the linear programming problem : Minimize $z = 50x + 70y$ Subject to $2x+y \geq 8$, $x+2y \geq 10$, $x \geq 0$, $y \geq 0$ The minimum value of objective function is :

The 10th term of AP: 3, 7, 11, 15, ... is

Five numbers taken out from numbers 1-30 and arrange them in ascending order. The probability that the third number will be 20 is

Which of the following is true on the basis of above diagram?

Identify the correct statements. (A) If $A$ is a non-singular matrix, then $A^{-1} = \frac{|A|}{(\text{adj } A)}$ (B) If $A$ is an invertible matrix then $\frac{1}{|A^{-1}|} = |A|$ (C) If $A$ and $B$ are two invertible matrices of the same order then $AB$ is also invertible matrix and $(BA)^{-1} = A^{-1}B^{-1}$ (D) If $A$ is an invertible matrix, then $A^T$ is also invertible and $(A^T)^{-1} = \frac{1}{(A^{-1})^T}$ Choose the correct answer from the options given below :

If A and B are square matrices of same order n, then identify correct statements from the statements given below: A. $|adj\ A| = |A|^{n-1}$ B. $|A \cdot B| = |B| \cdot |A|$ C. $adj\ A' = (adj\ A)'$ D. $adj\ AB = (adj\ A) \cdot (adj\ B)$ E. $|A^n| = |A|^n$ Choose the correct answer from the options given below:

If A and B are square matrices of order 3 such that $|A| = 2$, $|B| = 3$, and $|2A \cdot \text{adj}(3(\text{adj}B))| = 2^\alpha \cdot 3^\beta$, then value of $\alpha + \beta$ is:

If A and B are square matrices of same order, then $A'B - B'A$ is a:

If A and B are two independent events such that $P(A) = 0.4$, and $P(B) = 0.5$, then P (neither A nor B) is

If A and B are two independent events with $P(A) = \frac{1}{5}$ and $P(B) = \frac{1}{3}$, then $P(A'/B)$ is:

If a fair coin is tossed 10 times, then the probability of getting all heads or all tails, is :

If A is a matrix of order $m \times n$ and B is another matrix such that $A'B$ and $BA'$ are both defined, then the order of matrix B is

If A is a square matrix of order 3 and $|A|$ is 2, then value of $|adj(A)|$ is :

If A is a square matrix of order 3 and $|adj A| = 49$, then $|7A^{-1}|^2$

If a relation R is defined on the set $X = \{1, 2, 3, 4\}$ as $R = \{(1,1), (2,2), (3,4), (4,3)\}$, then R is

If an unbiased coin is tossed 10 times, probability of obtaining more head than tail is :

If $\begin{vmatrix} -1 & a & a^2 \\ -1 & b & b^2 \\ -1 & c & c^2 \end{vmatrix}^2 = \lambda$ and $a - b = 1$, $b - c = 2$ and $c - a = 3$, then the value of $\lambda$ is

If $|\vec{a}| = 3|\vec{b}|$, $|\vec{b}| = 2$ and angle between $\vec{a}$ and $\vec{b}$ is $60^\circ$, then $|\vec{a} - \vec{b}|$ is equal to:

If $A$ and $B$ are independent events such that $0 < P(A) < 1$ and $0 < P(B) < 1$, then identify the correct statements. (A) $A$ and $B'$ are independent (B) $A'$ and $B$ are independent (C) $A$ and $B$ are mutually exclusive (D) $A'$ and $B'$ are independent Choose the correct answer from the options given below :

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

If $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$ and $\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$ are two non zero vectors inclined at an angle $\theta$, then identify the correct option out of the given options. (a) $\cos\theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| \cdot |\vec{b}|}$ (b) $\vec{a}$ and $\vec{b}$ are perpendicular, if $a_1 b_1 + a_2 b_2 + a_3 b_3 = 0$ (c) $\vec{a}$ and $\vec{b}$ are perpendicular, if $\frac{a_1}{b_1} = \frac{a_2}{b_2} \neq \frac{c_1}{c_2}$ (d) for $\theta = \pi$, $\vec{a} \times \vec{b} = 0$ (e) $\cos\theta = \frac{|\vec{a} \times \vec{b}|}{|\vec{a}| \cdot |\vec{b}|}$ Choose the most appropriate answer from the options given below

If $\vec{a}$ and $\vec{b}$ are two perpendicular vectors such that $|\vec{a}| = 3$, $|\vec{b}| = 4$ and $\theta$ is the angle between $\vec{a}$ and $(\vec{a} - \vec{b})$, then $\cos\theta$ is equal to:

If $A = \begin{bmatrix} 1 & -1 & 1 \\ 1 & -2 & -2 \\ 2 & 1 & 3 \end{bmatrix}$ and square matrix $B$ satisfy $AB = 8I$, then the value of $|B|$ is:

If $0 < x < \pi$ and the matrix $\begin{bmatrix} 4\sin x & -1 \\ -3 & \sin x \end{bmatrix}$ is singular, then the values of $x$ are :

If $A = \begin{bmatrix} 6 & -8 \\ -2 & 5 \end{bmatrix}$ and $A^2 - 10A = C$ then C is equal to

If $-3 \leq k \leq 1$ and $|\vec{a}| = 2$ then $|k\vec{a}|$ is

If $A = \begin{bmatrix} 6 & 4 \\ 5 & 3 \end{bmatrix}$ and $B = adj(A)$, then $|B|$ is equal to:

If $3A = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ x & 2 & y \end{bmatrix}$ and $AA^T = I$, then $x + y$ is equal to

If $\vec{p} = \hat{i} + \hat{j} - 2\hat{k}$ and $\vec{q} = 2\hat{i} + \hat{j} - \hat{k}$, then the area of parallelogram having diagonals $(\vec{p} + \vec{q})$ and $(\vec{p} - \vec{q})$ is

If $A = \begin{bmatrix} 2x & 0 \\ x & x \end{bmatrix}$ and $A^{-1} = \begin{bmatrix} 1 & 0 \\ -1 & 2 \end{bmatrix}$, then the value of $x$ is

If $|\vec{a}| = 8$, $|\vec{b}| = 3$ and $|\vec{a} \times \vec{b}| = 12$, then the value of $\vec{a} \cdot \vec{b}$ is

If $x-y=2$ and $y-z=3$ then value of $\begin{vmatrix}1 & x & x^2 \\ 1 & y & y^2 \\ 1 & z & z^2\end{vmatrix} =$

If $\vec{a} = \hat{i} - \hat{j} + \hat{k}$, $\vec{b} = 2\hat{i} + \hat{j} - 3\hat{k}$, $\vec{c} = 2\hat{i} - \hat{j} + 7\hat{k}$ and $\vec{a} \times (\vec{b} \times \vec{c}) = \lambda \vec{b} + \mu \vec{c}$ (When $\lambda$, $\mu$ are scalars), then the value of $\lambda + \mu$ is:

If answer is correct, the probability that he guesses, is :

If $R$ is a relation on $Z$ (set of all integers) defined by $xRy$, iff $|x - y| \leq 1$, then (a) $R$ is reflexive (b) $R$ is symmetric (c) $R$ is transitive (d) $R$ is not symmetric (e) $R$ is not transitive Choose the most appropriate answer from the options given below

If $A$ is a skew-symmetric matrix and $n$ is an odd positive integer, then $A^n$ is :

If $\vec{a}$ is a unit vector and $(\vec{x} - \vec{a}) \cdot (\vec{x} + \vec{a}) = 8$ then $|\vec{x}|$ is

If $f:\mathbb{R} \to [-5, \infty)$ is defined as $f(x) = x^2 - 5$, then the function f is A. one-one B. many-one C. onto D. into Which of the above statements are true? Choose the correct answer from the options given below:

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

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

If $A$ is square matrix of order 3 and $A \cdot (Adj.(A)) = 10I$, then the value of $\frac{1}{25}|Adj.(A)|$ is

If $\theta$ is the acute angle between two unit vectors $\vec{a}$ and $\vec{b}$, then $\cos\frac{\theta}{2} =$

If P and Q are symmetric matrices of same order, then $(PQ - QP)$ is

If P is matrix of order $m \times n$ and Q is a matrix such that PQ' and Q'P are both computable, then the order of matrix Q is

If R is a relation on $A = \{a, b, c\}$ such that $R = \{(a,a), (b,b)\}$, which element/elements should be included to make R an equivalence relation. A. (c, c) B. (c, c), (a, c), (c, a) C. (a, b), (b, c), (a, c) D. (b, c), (c, c), (c, a), (b, a) Choose the correct answer from the options given below:

If the area of a triangle with vertices A(1,3), B(0,0) and C(k,0) is 3 sq. units, then k is:

If the matrix $\begin{bmatrix} 0 & -1 & 3x \\ 1 & y & -5 \\ -6 & 5 & 0 \end{bmatrix}$ is skew-symmetric, then

If the mean and variance of a binomially distributed random variable X are 4 and 2 respectively, then $P(X = 2)$ is equal to

If the objective function for an LPP is max.$(z) = 300x + 700y$ and the corner points for the bounded feasible region are $(6,0)$ $(5,0)$ $(0,6)$ $(4,4)$ and $(0,4)$, then the maximum values of z occurs at :

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

If the system of linear equations $x + 2y - 3z = 1$ $(2p+1)y + z = 2$ $3x + 3z = 5$ has a unique solution, then p can not be equal to

If $|\vec{a}| = |\vec{b}| = |\vec{a} + \vec{b}| = 1$, then $|\vec{a} - \vec{b}|$ is equal to:

If $A=\begin{bmatrix}2 & 0 & 0 \\ -1 & 2 & 3 \\ 3 & 3 & 5\end{bmatrix}$, then $A(\text{adj } A)$ is equal to :

If $n(A) = 3$, $n(B) = 2$, then number of all possible surjective function from set A to set B are :

If $\begin{vmatrix} (a-x)^2 & (a-y)^2 & (a-z)^2 \\ (b-x)^2 & (b-y)^2 & (b-z)^2 \\ (c-x)^2 & (c-y)^2 & (c-z)^2 \end{vmatrix} = \lambda(a-b)(b-c)(c-a) \cdot (x-y)(y-z)(z-x)$ then the value of $\lambda$ is:

If $2\begin{bmatrix}3 & 4 \\ 5 & x\end{bmatrix} + \begin{bmatrix}1 & y \\ 0 & 1\end{bmatrix} = \begin{bmatrix}7 & 0 \\ 10 & 5\end{bmatrix}$, then the value of $x-y$ is :

If $2 \begin{bmatrix} a & d \\ b & c \end{bmatrix} + 3 \begin{bmatrix} 1 & -1 \\ 0 & 2 \end{bmatrix} = 3 \begin{bmatrix} 3 & 5 \\ 4 & 6 \end{bmatrix}$, then the value of $|a + b - c - d|$ is

If $x = \begin{vmatrix}1 & 2 & 3 \\ 2 & 3 & 1 \\ 3 & 1 & 2\end{vmatrix}$ then value of $9-2x$ is :

If $A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 1 & 4 \\ 3 & 2 & 1 \end{bmatrix}$, then which of the following is the value of $(\text{adj } A)^{-1}$

If two numbers are selected at random from the first 25 natural numbers, then the probability that their sum will be odd is:

If $A^2 - A + I = O$, where O is the zero matrix and I is the identity matrix, then $A^{-1}$ is

In a hospital, there are 300 patients, out of which 120 are female. It is known that out of 120 females, 10% of the patients are below 40 years of age. What is the probability that a patient chosen randomly is below 40 yrs of age given that the chosen patient is a female.

In a triangle, $\triangle ABC$, the sides AB and AC are represented by vectors $\hat{i} + \hat{j} + \hat{k}$ and $2\hat{i} - \hat{k}$ respectively. The length of median drawn from vertex A to BC is:

In Binomial distribution with parameters $n = 100$ and p, Variance of distribution is maximum when p is equal to :

In Binomial distribution with parameters $n = 12$ and $p = \frac{1}{3}$, value of $E(X^2) + E(X)$ is :

In linear programming, the optimal value of the objective function is attained at the points given by

In the family mother, father and son stand up at random for a family picture. Define following two events : $E$ = [Son stands at one of the two ends in the picture] $F$ = [Father stands in the middle of the picture] The value of $P(F/E)$ is :

$[\vec{a} + \vec{b},\ \vec{b} + \vec{c}, \vec{a} + \vec{b} + \vec{c}]$ is equal to

It is given that only 0.1% of a large population have COVID infection. In this population, the reliability of COVID RTPCR-test is specified as follows : For persons having COVID, 90% of the test detects the disease but 10% goes undetected. For persons not having COVID, 99% of the test is judged COVID negative but 1% are diagnosed as COVID positive. Based on the above informations, answer the question : The probability that the person is actually having COVID given that he is tested as COVID positive is :

It is given that only 0.1% of a large population have COVID infection. In this population, the reliability of COVID RTPCR-test is specified as follows : For persons having COVID, 90% of the test detects the disease but 10% goes undetected. For persons not having COVID, 99% of the test is judged COVID negative but 1% are diagnosed as COVID positive. Based on the above informations, answer the question : The probability of the person to be tested as COVID positive, given that he is actually not having COVID is :

It is given that only 0.1% of a large population have COVID infection. In this population, the reliability of COVID RTPCR-test is specified as follows : For persons having COVID, 90% of the test detects the disease but 10% goes undetected. For persons not having COVID, 99% of the test is judged COVID negative but 1% are diagnosed as COVID positive. Based on the above informations, answer the question : The probability that randomly selected person from a population, not having COVID is :

It is given that only 0.1% of a large population have COVID infection. In this population, the reliability of COVID RTPCR-test is specified as follows : For persons having COVID, 90% of the test detects the disease but 10% goes undetected. For persons not having COVID, 99% of the test is judged COVID negative but 1% are diagnosed as COVID positive. Based on the above informations, answer the question : The probability of the person tested as COVID positive, given that he is actually having COVID is :

It is given that only 0.1% of a large population have COVID infection. In this population, the reliability of COVID RTPCR-test is specified as follows : For persons having COVID, 90% of the test detects the disease but 10% goes undetected. For persons not having COVID, 99% of the test is judged COVID negative but 1% are diagnosed as COVID positive. Based on the above informations, answer the question : The probability that the selected person will be diagonosed as COVID positive is :

It is given that student marked the answer correctly, the probability that he guesses is

Let A and B be square matrices of order 3 and k is a constant. If $|A| \neq 0$ and $|B| \neq 0$, where $|A|$ represents the determinant of A, then which of the following statements are true ? (where A' denotes the transpose of matrix A) (A) $(AB)^{-1} = B^{-1} A^{-1}$ (B) $(A+B)' = A' \times B'$ (C) $(AB)' = A'B'$ (D) $|kA| = k^3 |A|$ (E) $|A'| = |A|$ Choose the correct answer from the options given below :

Let A be a non singular square matrix of $2 \times 2$. Then, $|adj\ A|$ is equal to

Let $A = (a_{ij})$ and $B = (b_{ij})$ are square matrices of same order. (A) The number of possible matrices of order $2 \times 2$ with entries $-1, 0, 1$ is 81. (B) $A + A'$ is skew symmetric matrix (C) $A \cdot A^{-1} = 0$, $|A| \neq 0$ (D) A is skew symmetric matrix if $a_{ij} = -a_{ji}$ for all $i, j$ (E) $(AB)' = A'B'$ Choose the correct answer from the options given below :

Let $A$ and $B$ be two non-singular, square matrices of same order, and A. $(AB)^{-1} = B^{-1} \cdot A^{-1}$ B. $(A+B)^{-1} = B^{-1} + A^{-1}$ C. $adj. A = |A| \cdot A^{-1}$ D. $det(A^{-1}) = [det A]^{-1}$ Choose the correct answer from the options given below

Let $\vec{a} = 2\hat{i} + 3\hat{j} - 4\hat{k}$ and $\vec{b} = 3\hat{i} - 5\hat{j} + 6\hat{k}$. Let $\vec{c}$ be a vector such that $\vec{c} \times \vec{a} = \vec{b} \times \vec{c}$ and $\vec{c}\cdot(2\vec{a}-3\vec{b}) = 238\sqrt{2}$ then $|\vec{c}|^2$ is equal to :

Let $\vec{OA} = 2\hat{i} - \hat{j} + \hat{k}$ and $\vec{OB} = \hat{i} + \hat{j} - \hat{k}$. Then A. The magnitude of vector $\vec{OA}$ is 6 B. The magnitude of vector $\vec{OB}$ is $\sqrt{3}$ C. The vector $\vec{AB}$ is $(-\hat{i} + 2\hat{j} - 2\hat{k})$ D. $\vec{OA} \cdot \vec{OB} = 0$ E. $\vec{OA} \parallel \vec{OB}$ Choose the correct answer from the options given below:

Let $f : [2, \infty) \to \mathbf{R}$ be a function defined by $f(x) = x^2 - 4x + 5$. The range of $f$ is :

Let L be the set of all lines in a plane and R be the relation in L defined as $R = \{(l_1, l_2) : l_1 \text{ is perpendicular to } l_2, \text{ where } l_1, l_2 \in L\}$. Choose the correct answer :

Let the matrix $A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}$ and $AB = \begin{bmatrix} 5 & 6 & 7 \\ 8 & 9 & 10 \end{bmatrix}$ then order of B is :

Let $R : \mathbb{R} \to \mathbb{R}$, where $\mathbb{R}$ is the set of real numbers and R be a relation. An element $(x, y) \in R$ if $x + y - \sqrt{2}$ belongs to the set of irrational numbers. Then the relation R is :

Let X be a discrete random variable and probability distribution X is | X | $-1$ | 0 | 1 | |---|---|---|---| | P(X) | $\frac{1}{2}$ | $\frac{1}{5}$ | $\frac{3}{10}$ | Then E(X) is equal to :

Let X be the random variable with probability distribution given by the following table. | X | 0 | 1 | 2 | 3 | |---|---|---|---|---| | P(X = x) | $\frac{1}{8}$ | k | $\frac{3}{8}$ | $\frac{1}{8}$ | The value of $P(X \leq 1)$ is:

Match List - I with List - II. | List - I | List - II | |---|---| | (A) The value of $\hat{i}\cdot(\hat{j}\times\hat{k}) + \hat{j}\cdot(\hat{i}\times\hat{k}) + \hat{k}\cdot(\hat{i}\times\hat{j})$ | (I) 16 | | (B) If $\lvert\vec{a}\rvert=10$, $\lvert\vec{b}\rvert=2$ and $\vec{a}\cdot\vec{b}=12$, then the value of $\lvert\vec{a}\times\vec{b}\rvert$ is | (II) $\frac{\pi}{4}$ | | (C) If $\theta$ is the angle between two vectors $\vec{a}$ and $\vec{b}$, then the value of $\theta$, for which $\vec{a}\cdot\vec{b} = \lvert\vec{a}\times\vec{b}\rvert$ is | (III) 14 | | (D) If $\vec{a}$ and $\vec{b}$ are perpendicular and $\vec{a} = 2\hat{i}+4\hat{j}+\lambda\hat{k}$ and $\vec{b} = 3\hat{i}-5\hat{j}+\hat{k}$, then the value of $\lambda$ is | (IV) 1 | Choose the correct answer from the options given below :

Match List - I with List - II. | | List - I (Two given vector) | | List - II (Projection of $\vec{a}$ on $\vec{b}$) | |---|---|---|---| | (A) | $\vec{a} = \hat{i} - \hat{j}$, $\vec{b} = \hat{i} + \hat{j}$ | (I) | $\frac{2}{\sqrt{5}}$ | | (B) | $\vec{a} = \hat{i} + \hat{j}$, $\vec{b} = 2\hat{i} - \hat{k}$ | (II) | 0 | | (C) | $\vec{a} = \hat{j} + \hat{k}$, $\vec{b} = \hat{i} + \hat{k}$ | (III) | $\sqrt{2}$ | | (D) | $\vec{a} = 2\hat{i} + 3\hat{j}$, $\vec{b} = \hat{i} - \hat{k}$ | (IV) | $\frac{1}{\sqrt{2}}$ | Choose the correct answer from the options given below :

Match List - I with List - II. | List - I | List - II | |---|---| | (A) $A$ is a square matrix of order $3$ and $\lvert 2A \rvert = k\lvert A \rvert$, then $k$ is | (I) $0$ | | (B) Value of $\begin{vmatrix}1 & a & b+c \\ 1 & b & c+a \\ 1 & c & a+b\end{vmatrix}$ is | (II) $3$ | | (C) Matrix $\begin{bmatrix}5-x & x+1 \\ 2 & 4\end{bmatrix}$ is singular, then $x =$ | (III) $8$ | | (D) If $A = (a_{ij}) = \begin{bmatrix}5 & 3 & 8 \\ 2 & 0 & 1 \\ 1 & 2 & 3\end{bmatrix}$, then the minor of the element $a_{23}$ is | (IV) $7$ | Choose the correct answer from the options given below :

Match List - I with List - II. | List - I | List - II | |---|---| | (A) $\begin{bmatrix}0 & -5 & 9 \\ 5 & 0 & -3 \\ -9 & 3 & 0\end{bmatrix}$ | (I) Scalar matrix | | (B) $\begin{bmatrix}5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5\end{bmatrix}$ | (II) Diagonal matrix | | (C) $\begin{bmatrix}1 & 0 & 0 \\ 0 & -5 & 0 \\ 0 & 0 & 7\end{bmatrix}$ | (III) Symmetric matrix | | (D) $\begin{bmatrix}3 & -2 & 1 \\ -2 & -5 & 6 \\ 1 & 6 & 0\end{bmatrix}$ | (IV) Skew-symmetric matrix | Choose the correct answer from the options given below :

Match List - I with List - II. | | List - I | | List - II | |---|---|---|---| | (A) | $A = \begin{bmatrix} 6 & 9 \\ 2 & 3 \end{bmatrix}$, $B = \begin{bmatrix} 2 & 6 & 0 \\ 7 & 9 & 8 \end{bmatrix}$, AB will be | (I) | $\begin{bmatrix} 75 & 117 & 72 \\ 35 & 39 & 24 \end{bmatrix}$ | | (B) | $P = \begin{bmatrix} 8 & 45 & 30 \\ 17 & 19 & 5 \end{bmatrix}$, $Q = \begin{bmatrix} 67 & 72 & 42 \\ 8 & 30 & 19 \end{bmatrix}$, P+Q will be | (II) | $\begin{bmatrix} 75 & 117 & 72 \\ 25 & 39 & 24 \end{bmatrix}$ | | (C) | $\begin{bmatrix} 85 & 42 & 69 \\ 73 & 42 & 50 \end{bmatrix} - \begin{bmatrix} 10 & -75 & -3 \\ 38 & 3 & 26 \end{bmatrix}$ | (III) | $\begin{bmatrix} 75 & 117 & 72 \\ 25 & 49 & 24 \end{bmatrix}$ | | (D) | $2 \begin{bmatrix} 34 & 36 & 30 \\ 12 & 18 & 20 \end{bmatrix} + \begin{bmatrix} 7 & 55 & 12 \\ 1 & 13 & -16 \end{bmatrix}$ | (IV) | $\begin{bmatrix} 75 & 127 & 72 \\ 25 & 49 & 24 \end{bmatrix}$ | Choose the correct answer from the options given below :

Match List - I with List - II. | List - I | List - II | |---|---| | (A) Two events E and F will be independent if $P(E'F')$ is equal to | (I) $1 - P(E/F)$ | | (B) If $P(F) \neq 0$, then $P(E'/F)$ is equal to | (II) $P(E) = P(F)$ | | (C) If E and F are independent events, then | (III) $P(E \cap F') = P(E) \cdot P(F')$ | | (D) If $P(E \cap F) \neq 0$ and $P(E/F) = P(F/E)$, then | (IV) $[1-P(E)][1-P(F)]$ | Choose the correct answer from the options given below :

Match List I with List II: Given that A and B are invertible matrices of size $3 \times 3$ | List I | List II | |---|---| | A. $\lvert AB \rvert$ | I. $\frac{1}{\lvert A \rvert}$ | | B. $\lvert \operatorname{Adj} A \rvert$ | II. $\lvert A \rvert \lvert B \rvert$ | | C. $\lvert A^{-1} \rvert$ | III. $B^{-1} \cdot A^{-1}$ | | D. $(AB)^{-1}$ | IV. $\lvert A \rvert^2$ | Choose the correct answer from the options given below:

Match List I with List II | List I | List II | |---|---| | A. Range of $\lvert x\rvert$ | I. $(-5, \infty)$ | | B. Range of $9x^2 + 6x - 5$ for all $x \geq 0$ | II. $[0, \infty)$ | | C. Domain of $\dfrac{1}{\sqrt{x+5}}$ | III. $\{(1,1), (2,2), (3,3)\}$ | | D. Smallest equivalence relation on Set $\{1,2,3\}$ | IV. $[-5, \infty)$ | Choose the correct answer from the options given below:

Objective function of LPP is:

$\begin{vmatrix} a+b & 1 & 0 \\ a^2-b^2 & a-b & 1 \\ a^3+b^3 & a^2+b^2+ab & a^2-b^2 \end{vmatrix} =$

Read the following statements carefully: A. Determinant is a square matrix B. If A be any given square matrix of order n, then $A(\text{adj}A) = (\text{adj}A)A = |A|I$. C. If A and B are nonsingular matrices of the same order, then AB and BA are also nonsingular matrices of the same order. D. If A is a nonsingular matrix, then its inverse does not exist Which of the above statements are true? Choose the correct answer from the options given below:

The angle between the vectors $\hat{i} - \hat{j}$ and $\hat{j} - \hat{k}$ is:

The area of rectangular field is:

The area of the parallelogram whose adjacent sides are determined by the vectors $\vec{a} = \hat{i} - \hat{j} + 3\hat{k}$ and $\vec{b} = 2\hat{i} - 7\hat{j} + \hat{k}$ is :

The breadth of plot is:

The conditional probability that his answer is correct when it is given that he knew it :

The constraints are - A. $2x + y \leq 8$ B. $2x + y \geq 8$ C. $x + 2y \leq 10$ D. $x + 2y \geq 10$ E. $x, y \geq 0$ Choose the correct answer from the options given below:

The constraints to the LPP are: A. $3x + 2y \leq 720$ B. $2x + 3y \leq 720$ C. $x + y \leq 300$ D. $x \geq 0$ and $y \geq 0$ E. $x + y \geq 300$ Choose the correct answer from the options given below:

The corner points of feasible region are: A. (0, 240) B. (0, 0) C. (300, 0) D. (120, 180) E. (180, 120) Choose the correct answer from the options given below:

The corner points of the feasible region determined by inequalities of LPP are $(4, 10)$, $(6, 8)$ and $(6, 5)$. Let $z = 3x + 4y$ be the objective function. Then the sum of maximum value of z and minimum value of z is :

The corner points of the feasible region for an L.P.P. are $(0, 10)$, $(5, 5)$, $(15, 15)$ and $(0, 20)$. If the objective function is $z = px + qy$; $p, q > 0$, then the condition on $p$ and $q$ so that the maximum of $z$ occurs at $(15, 15)$ and $(0, 20)$ is

The equation of the line passing through (-2, 3, 4) and parallel to the vector $2\hat{i} - \hat{j} + \hat{k}$ is:

The equations in terms of $x$ and $y$ are:

The feasible region and optimal solution of a LPP with objective function, max.$(z) = 600x + 400y$, subject to : $x + 2y \leq 12$, $2x + y \leq 12$, $x + 1.25y \geq 5$, $x \geq 0$ and $y \geq 0$, is: The feasible region and optimal solution is _____

The feasible solution to the LPP is-

The length ($x$) and breadth ($y$) of plot satisfy equations:

The length of the plot is:

The linear equation involving $x$ and $y$ are written in matrix form as:

The matrix $\begin{bmatrix} 4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4 \end{bmatrix}$ is: A. a square matrix B. a scalar matrix C. a diagonal matrix D. an identity matrix Which of the above statements are true? Choose the correct answer from the options given below:

The maximum profit per week is :

The maximum value of $z = 4x + 3y$, if the feasible region for an LPP is as shown below is:

The minimum cost of the mixture is-

The modulus function $f : R \to R$, given by $f(x) = |x|$, is :

The number of all different possible matrices of order $2 \times 2$ with each entry $-1$, $0$ or $1$ is :

The number of all possible matrices of order 3 x 3 with each entry belonging to the set {0, 1} is:

The number of all possible non-singular matrices of order $2 \times 2$ with each entry 0 or 1 is :

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

The objective function for a L.P.P. is $Z = 5x + 7y$ and the corner points of the bounded feasible region are (0, 0), (7, 0), (3, 4) and (0, 2), then the maximum value of Z occurs at

The objective function (z) to maximize the profit is:

The optimal solution of the Linear Programming problem Maximize $Z = 3x_1 + 5x_2$, s.t. $3x_1 + 2x_2 \leq 18$ $x_1 \leq 4$ $x_2 \leq 6$ $x_1 \geq 0, x_2 \geq 0$ is

The optimal value of linear programming problem maximum $Z = 3x + 4y$, subject to, $x + 3y \leq 12$ $x + y \geq 8$ $x, y \geq 0$ is

The probability distribution of a discrete random variable X is given as : | x | 0 | 1 | 2 | 3 | 4 | |---|---|---|---|---|---| | P(X=x) | 0.1 | k | 2k | 2k | k | The value of K is:

The probability distribution of a random variable $X$ is | x | 0 | 1 | 2 | 3 | | --- | --- | --- | --- | --- | | P(X = x) | $\frac{1}{4}$ | $\frac{1}{8}$ | $\frac{1}{8}$ | $\frac{1}{2}$ | The variance of $X$ is

The probability distribution of number of doublets in three throws of a pair of dice is

The probability of answering a question correctly, is :

The probability of 'the student knows the answer given that he answered it correctly' is

The probability that a question is guessed by a student and found to be correct is.

The probability that a student knows the answer, is:

The probability that B alone complete the task on time is:

The probability that exactly one of them complete the task on time is

The probability that exactly two of them complete the task on time is

The probability that he copied it given that his answer is correct :

The probability that task is completed on time by at least one of them is:

The probability that the study time of students is not more than one hour.

The probability that the study time of students is at least 1 hour

The probability that the study time of students is at least 3 hours

The probability that the study time of students is exactly 2 hours

The probability that the task is completed on time by none of them is

The random variable $X$ has a probability distribution $P(X=x) = \begin{cases} 5k, & x=0 \\ 2k, & x=1 \\ 3k, & x=2 \\ 0, & \text{otherwise.} \end{cases}$ Then, the value of $E(X)$ is :

The Relation $R = \{(x, y) : x \leq y^2\}$ defined on the set $\mathbf{R}$ of Real numbers is : (A) reflexive but not symmetric (B) neither reflexive nor symmetric (C) neither reflexive nor transitive (D) reflexive but not transitive (E) not reflexive but symmetric Choose the correct answer from the options given below :

The set of all values of $\alpha$ for which the system of linear equations $x + y + z = 1$ $x + 2y + 4z = \alpha$ $x + 4y + 10z = \alpha^2$ is consistent, is

The solution of LPP max.(z) = $5x + 3y$ subject to $2x + y \leq 6$ $x + y \leq 4$ $x \geq 0, y \geq 0$, is

The system of equations $3x + 4y = 5$, $6x + 7y = -8$ is written in matrix form as

The type of feasible region and its corner points are. A. bounded B. unbounded C. (0,5), (2,4), (10,0) D. (0,8), (2,4), (4,0) E. (0,8), (2,4), (10,0) Choose the correct answer from the options given below:

The unit vector in the direction of the sum of vectors $\vec{a} = 2\hat{i} + 2\hat{j} - 5\hat{k}$ and $\vec{b} = 2\hat{i} + \hat{j} + 3\hat{k}$ is :

The value $x$ is:

The value of $y$ is

The value of $P(E/E_1)$ is

The value of $k$ is

The value of $P(E)$ is

The value of $\hat{i} \cdot (\hat{k} \times \hat{j}) + \hat{j} \cdot (\hat{i} \times \hat{k}) + \hat{k} \cdot (\hat{j} \times \hat{i})$ is

The value of the expression $\frac{x^2 + y^2}{x - y}$ is:

The values of $x$ and $y$ in the equation $2\begin{bmatrix} x & 1 \\ 4 & -3 \end{bmatrix} + 3\begin{bmatrix} -2 & 1 \\ 2 & y-2 \end{bmatrix} = \begin{bmatrix} 2 & 5 \\ 14 & 3 \end{bmatrix}$ are respectively:

The vertices of a closed convex polygon representing the feasible region of the LPP with, objective function $z = 5x + 3y$ are $(0, 0)$, $(3, 1)$, $(1, 3)$ and $(0, 2)$. The maximum value of $z$ is

There are three identical boxes I, II and III, each containing two balls. In box I, both balls are red, In box II, both balls are blue and box III contains one blue ball and one red ball. A boy randomly chooses a box and takes out a ball at random from it. If the ball is red, then the probability that the other ball in the box is also red colour is:

Three friends A, B and C are playing with a pair of dice. They throw two dice alternately. Coming of a doublet on two dice leads to a success and the game stops. If A starts the game, then the probability of his winning, is :

Three urns contain 6 red, 4 black; 4 red, 6 black and 5 red, 5 black marbles respectively. One of the urns is selected at random and a marble is drawn from it. If the marble drawn is red, then the probability that it is drawn from the first urn is

Two cards are drawn successively with replacement from a well shuffled deck of 52 cards. The probability distribution of the number of kings will be:

Two numbers are selected at random (without replacement) from the first three positive integers. Let $X$ denotes the larger of the two integers, then the probability distribution of $X$ is

Value(s) of $x$ for which, $\begin{vmatrix} x & 1 \\ 5 & x \end{vmatrix} = \begin{vmatrix} 8 & 2 \\ 2 & 1 \end{vmatrix}$ is:

What is the probability that the doctor arrives late ?

When the doctor arrives late, what is the probability that he comes by metro?

When the doctor arrives late, what is the probability that he comes by cab?

When the doctor arrives late, what is the probability that he comes by other means of transport?

When the doctor arrives late, what is the probability that he comes by bike?

Which is the most suitable definition for random variable among the options given below:

Which of the following figures shows a bijective function from set $X_1$ to set $X_2$ :  Choose the correct answer from the options given below :

Which of the following relations on the set $A = \{1, 2, 3\}$ are equivalence ? (A) $R = \{(1,1), (2,2), (1,2), (2,1)\}$ (B) $R = \{(1,1), (2,2), (3,3)\}$ (C) $R = \{(1,1), (1,2), (2,1)\}$ (D) $R = \{(1,1), (2,2), (3,3), (1,2), (2,1), (2,3), (3,2), (1,3), (3,1)\}$ (E) $R = \{(1,1), (2,2), (3,3), (1,2)\}$ Choose the correct answer from the options given below :

Which of the following statements are true? (A) $f:\mathbb{R} \to \mathbb{R}$ given by $f(x) = 3x$ is one-one onto. (B) $f:\mathbb{R} \to \mathbb{R}$ given by $f(x) = x^4$ is one-one and onto. (C) $f:\mathbb{Z} \to \mathbb{Z}$ given by $f(x) = x^2$ is neither one-one nor onto. (D) $f:\mathbb{R} \to \mathbb{R}$ given by $f(x) = |x|$ is neither one-one nor onto. (Where $\mathbb{R}$ is the set of all real numbers and $\mathbb{Z}$ is the set of all integers) Choose the correct answer from the options given below: