CUET Mathematics 2022 30 Aug Shift 1 PYQs

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

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

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

Consider the function $f(x) = x^{\frac{1}{x}}$. Its

The given function $f(x) = x^5 - 5x^4 + 5x^3 - 1$; has/have (a) local maxima at $x = 1$ (b) local maximum value is 0 (c) local minimum at $x = 3$ (d) local minimum value is $-28$ (e) The point of inflexion is $x = 1$ Choose the correct answer from the options given below

Match List-I with List-II | List-I | List-II | |---|---| | (a) If $x = t^2$ and $y = t^3$, then $\frac{d^2y}{dx^2}$ at $t = 1$ | (i) $-2$ | | (b) If $f(x) = \sqrt{x} + 1$, then $f''(1)$ | (ii) $-1$ | | (c) The minimum value of $f(x) = 9x^2 + 12x + 2$ is | (iii) $\frac{3}{4}$ | | (d) The point of inflexion of the function $f(x) = (x-2)^4 (x+1)^3$ is | (iv) $-\frac{1}{4}$ | Choose the correct answer from the options given below

The area enclosed by the curve $y^2 = 4ax$ and its latus-rectum is

$\int \frac{xe^x}{(x+1)^2} dx =$

The solution of the differential equation $(x+1)\frac{dy}{dx} = 1 + y$ is

Order and degree of the differential equation $y\frac{dy}{dx} + \frac{4}{\frac{dy}{dx}} = 5$ are

Derivative of $x^3 + 1$ with respect to $x^2 + 1$ is

Solution of the differential equation $(x + xy)dy - y(1 - x^2)dx = 0$ is

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

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

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

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 the vertices of a triangle ABC are $A(1, 2, 1)$, $B(4, 2, 3)$ and $C(2, 3, 1)$, then the equation of the median passing through the vertex $A$, is

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

If $x = 2\sin\theta$ and $y = 2\cos\theta$, then the value of $\frac{d^2y}{dx^2}$ at $\theta = 0$ is

If $x = e^{y + e^{y + e^{y + \ldots \infty}}}$, $x > 0$, then $\frac{dy}{dx}$ is equal to

$\sin^{-1}(1 - x) - 2\sin^{-1}x = \frac{\pi}{2}$, than $x$ is equal to (a) $0$ (b) $1$ (c) $\frac{1}{2}$ (d) $2$ Choose the most appropriate answer from the options given below:

The smaller of the areas enclosed by the circle $x^2 + y^2 = 4$ and the line $x + y = 2$ 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 :

$\int_{\frac{1}{3}}^{1} \frac{(x - x^3)^{\frac{1}{3}}}{x^4} dx =$

The function $f(x) = e^{|x|}$ is (a) continuous everywhere on $R$ (b) not continuous at $x = 0$ (c) Differentiable everywhere on $R$ (d) not differentiable at $x = 0$ (e) continuous and differentiable on $R$ Choose the most appropriate answer from the options given below :

<img src="https://balti.afterboards.in/1VZUCz5GIcv95ws" width="400px"/>Which of the following is true on the basis of above diagram?

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

The value of $\sin\left[2\cot^{-1}\left(\frac{-5}{12}\right)\right]$ is :

Let $y = m\sin rx + n\cos rx$. What is the value of $\frac{d^2y}{dx^2}$?

The integrating factor of the differential equation $\cos x \frac{dy}{dx} + y\sin x = 1$ is

The order and degree of the differential equation $\left[\left(\frac{d^2y}{dx^2}\right)^2 - 3\right]^{\frac{1}{3}} = 2\left(\frac{dy}{dx}\right)^{\frac{1}{4}}$ are

$\int \sqrt{1 - 49x^2} \, dx$ is equal to

The shortest distances of the point $(1, 2, 3)$ from $x$, $y$, $z$ axes respectively are

Distance between two planes $x + 2y - z = 5$ and $2x + 4y - 2z + 2 = 0$ 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 $\vec{p} = \hat{i} + \hat{j} - 2\hat{k}$ 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 $\vec{a}, \vec{b}$ 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

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

$\int x\sqrt{x + 2} \, dx$ is equal to :

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