CUET Mathematics 2022 17 Aug Shift 2 PYQs

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

The area (in square units) bounded by the curve $y^2 = 4x$ and the line $x=1$ is :

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

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

If the function $f(x) = x^2 - ax - 2$ is strictly decreasing on $(2, 3)$ then $a$ lies in the interval.

The slope of the tangent to the curve $y = 3x^2 + 2kx - 5$ at $x=1$ is $9$. The value of $k$ is :

If $\int(\sqrt{x+1} + \sqrt{x-1})^2 dx = \alpha x^2 + \beta x\sqrt{x^2-1} + \gamma \log|x+\sqrt{x^2-1}| + C$, then value of $\alpha + \beta - 2\gamma$ is :

If $g(x) = \int \frac{dx}{x^{1/2} + x^{1/6}}$, then $g(1) - g(0)$ is :

Consider the differential equation $\frac{dy}{dx} = \frac{y+1}{x+1}$, and $y=0$ when $x=2$. The value of $y$ at $x=3$ is :

If $x^y = e^{x-y}$, then $\frac{dy}{dx} =$

Let $y = \log(x + \sqrt{x^2+1})$, and $a\frac{d^2y}{dx^2} + b\frac{dy}{dx} = c$. Then identify the correct statements about the values of $a$, $b$ and $c$ : (A) $a = 1 + x^2$ (B) $b = 0$ (C) $c = 0$ (D) $b = x$ (E) $c = 2$ Choose the correct answer from the options given below :

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 :

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 :

The area of the smaller region bounded by the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$ and the line $\frac{x}{3} + \frac{y}{2} = 1$ 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 $A$ is a skew-symmetric matrix and $n$ is an odd positive integer, then $A^n$ is :

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 :

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 :

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 :

The value of $\sin\left[\frac{\pi}{2} - \sin^{-1}\left(-\frac{\sqrt{3}}{2}\right)\right]$ is :

Match List - I with List - II. | List - I | List - II | |---|---| | (A) $\tan^{-1}\sqrt{3} - \sec^{-1}(-2)$ | (I) $\frac{3\pi}{4}$ | | (B) $\cot^{-1}\left(\frac{-1}{\sqrt{3}}\right)$ | (II) $-\frac{\pi}{3}$ | | (C) $\cos^{-1}\left(-\frac{1}{\sqrt{2}}\right)$ | (III) $\frac{\pi}{2}$ | | (D) $\cos^{-1}\left(\frac{1}{2}\right) + \sin^{-1}\left(\frac{1}{2}\right)$ | (IV) $\frac{2\pi}{3}$ | Choose the correct answer from the options given below :

The direction ratio's of line of intersection of two planes : $2x + y + z + 47 = 0$ and $3x - 2y - z + 41 = 0$ are :

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 :

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

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 figures shows a bijective function from set $X_1$ to set $X_2$ : <img src="https://balti.afterboards.in/P356b70GmYqx15Z" width="500px"/> Choose the correct answer from the options given below :

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 :

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 :

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 :

A line makes angle $\theta$ with $x$-axes as well as $z$-axis. If the angle $\beta$, which it makes with $y$-axis is such that $\sin^2\beta = 3\sin^2\theta$, then $\cos^2\theta$ is :

The equation of the plane, parallel to the plane $3x + 4y - 12z = 3$ and passes through $(1, 1, -1)$, is :

The distance of plane $\vec{r}\cdot(6\hat{i} - 3\hat{j} - 2\hat{k}) + 1 = 0$ from origin 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.

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 :

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 :

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 :

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 :

The foot of perpendicular from point $(2, 4, -1)$ on the line $\frac{x+5}{1} = \frac{y+3}{4} = \frac{z-6}{-9}$ is :