CUET Mathematics 2023 15 June Shift 2 PYQs

Let $A = [a_{ij}]$ be a $2 \times 2$ matrix such that $a_{ij} = \frac{|-3i+j|}{2}$ then $a_{21}$ is :

If $A = \begin{bmatrix} -2 & 6 \\ -5 & -1 \end{bmatrix}$ then $A^{-1}$ is :

The value of $\begin{vmatrix} \sqrt{3}/2 & 1/2 \\ \sqrt{3}/2 & 1/2 \end{vmatrix}$

If A is a square matrix of order 3 such that $|A|=2$, then the value of $|adj(adj A)|$ is :

If $y = \log\left[\frac{x^2}{e^2}\right]$ then value of $\frac{d^2y}{dx^2}$ is :

The condition on a and b, such that for $y = \frac{a}{x} - \frac{b}{x^2}$, $\frac{dy}{dx} = 0$ at $x=1$ is :

The interval in which the function $f(x) = 10 - 6x - 2x^2$ is decreasing is :

Area of the region bounded by the curve $|x| + |y| = 1$ and x-axis is :

The value of the integral $\int_{2}^{4} \frac{x}{x^2+1} dx$ is :

The sum of order and degree of the differential equation $\frac{\left\{1+\left(\frac{dy}{dx}\right)^2\right\}^{\frac{5}{2}}}{\frac{d^2y}{dx^2}} = p$ is :

The solution of the differential equation $\frac{dy}{dx} = \frac{6}{x^2}$; $y(1) = 3$ is :

The random variable X has a probability distribution P(X) of the following form, where k is some number. $P(X=x) = \begin{cases} k, & \text{if } x=0 \\ 2k, & \text{if } x=1 \\ 3k, & \text{if } x=2 \\ 0, & \text{otherwise} \end{cases}$ Then $P(x \leq 2)$ is :

The mean of the number of heads in a simultaneous toss of three coins is :

For the LPP Maximise $z = x + y$ subject to $x - y \leq -1$, $-x + y \leq 2$, $x, y \geq 0$, $z$ has :

Choose the wrong statement from the following :

Given relation $R = \{(x, y) : y = x + 5, x < 4, x, y \in N\}$. Where N is a set of natural numbers then :

Let $f : R \to R$ defined by $f(x) = 2x^3 - 7$ for $x \in R$. Then : (A) $f$ is one-one function (B) $f$ is many to one function (C) $f$ is bijective function (D) $f$ is into function Choose the correct answer from the options given below :

Match List - I with List - II. | List - I | List - II | |----------|-----------| | (A) Range of $y = \text{cosec}^{-1}x$ | (I) $R - (-1, 1)$ | | (B) Domain of $\sec^{-1}x$ | (II) $(0, \pi)$ | | (C) Domain of $\sin^{-1}x$ | (III) $[-1, 1]$ | | (D) Range of $y = \cot^{-1}x$ | (IV) $\left[\frac{-\pi}{2}, \frac{\pi}{2}\right] - \{0\}$ | Choose the correct answer from the options given below :

Let $\tan^{-1}y = \tan^{-1}x + \tan^{-1}\left(\frac{2x}{1-x^2}\right)$. Then $y$ is :

The value of $2y - 3x$, if $2\begin{bmatrix} x & 5 \\ 7 & y-3 \end{bmatrix} + \begin{bmatrix} 3 & -4 \\ 1 & 2 \end{bmatrix} = \begin{bmatrix} 7 & 6 \\ 15 & 14 \end{bmatrix}$ is :

Match List - I with List - II. If $A = \begin{vmatrix} 3 & -2 & 3 \\ 2 & 1 & -1 \\ 4 & -3 & 2 \end{vmatrix}$ | List - I | List - II | |----------|-----------| | (A) $M_{23}$ | (I) $-17$ | | (B) $A_{32} + a_{13}$ | (II) $-1$ | | (C) A | (III) 0 | | (D) $a_{13}A_{12} + a_{23}A_{22} + a_{33}A_{32}$ | (IV) 12 | Choose the correct answer from the options given below :

The number of square matrices of order 2 using numbers 1 and $-1$ exactly once and the number 0 twice is :

Let $\begin{vmatrix} 3x & -7 \\ 1 & 4 \end{vmatrix} = \begin{vmatrix} 3 & 2 \\ 4 & x \end{vmatrix}$, then value of $x$ is :

The value of the determinant $\begin{vmatrix} a\cos\theta & b\sin\theta & 0 \\ -b\sin\theta & a\cos\theta & 0 \\ 0 & 0 & c \end{vmatrix}$ is :

If the points (2, 1), $(-1, 4)$ and (a, 3) are collinear then the value/(s) of a is/(are) :

The points of discontinuity of the function f defined by $f(x) = \begin{cases} x+2 & x \leq 1 \\ x-2 & 1 < x < 2 \\ 0 & x \geq 2 \end{cases}$ are :

If $\cos y = x\cos(a+y)$, then $\frac{dy}{dx} = $

The value of C which satisfies Rolle's Theorem for $f(x) = \sin^4 x + \cos^4 x$ in $\left[0, \frac{\pi}{2}\right]$. Then C is :

Angle between tangents to the curve $y = x^2 - 5x + 6$ at the points (2, 0) and (3, 0) is :

The rate of change of the area of a circular disc with respect to its circumference when radius is 3 is :

The interval in which the $f(x) = \sin x - \cos x$, $0 \leq x \leq 2\pi$ is strictly decreasing is :

The value of $\int_{0}^{3} |2x - 6| dx$ is :

The integral $\int \frac{dx}{x^2(x^4+1)^{\frac{3}{4}}}$ equals __________.

The area of the shaded portion <img src="https://balti.afterboards.in/0Orw0ReJYISKyt0" width="300px"/> is :

Area of the region bounded by $y = -1$, $y = 2$, $x = y^3$ and $x = 0$ is :

The differential equation whose solution is $Ax^2 + By^2 = 1$ where A and B are arbitrary constant is of : (A) first order and first degree (B) second order and first degree (C) second order and second degree (D) second order Choose the correct answer from the options given below :

Integrating factor of the differential equation $(1 - y^2)\frac{dx}{dy} + xy = ay$ is :

Let $\vec{a} = 4\hat{i} - \hat{j} + 3\hat{k}$ and $\vec{b} = -2\hat{i} + \hat{j} - 2\hat{k}$. Then (A) $\vec{a}$ is a unit vector (B) $\vec{a} \times \vec{b} = -\hat{i} + 2\hat{j} + 2\hat{k}$ (C) $\vec{a}$ and $\vec{b}$ are parallel vectors (D) $\vec{a}$ and $\vec{b}$ are neither parallel nor perpendicular vectors Choose the correct answer from the options given below :

Let $\vec{a}$ and $\vec{b}$ be two unit vectors. If the vectors $\vec{c} = 5\vec{a} - 4\vec{b}$ and $\vec{d} = \vec{a} + 2\vec{b}$ are perpendicular to each other, then the angle between $\vec{a}$ and $\vec{b}$ is :

The equation of plane which cuts equal intercepts of unit length on the coordinate axes is :

If the straight lines $x = 1 + s$, $y = -3 - \lambda s$, $z = 1 + \lambda s$ and $x = \frac{t}{2}$, $y = 1 + t$, $z = 2 - t$ with parameters $s$ and $t$ respectively, are coplanar, then $\lambda$ is equal to :

Match List - I with List - II. | List - I | List - II | |----------|-----------| | (A) The common region determined by all the constraints of LPP is called | (I) objective function | | (B) Minimize $z = c_1x_1 + c_2x_2 + ..... + c_nx_n$ is | (II) convex set | | (C) A solution that also satisfies the non-negative restrictions of a LPP is called | (III) feasible region | | (D) The set of all feasible solutions of a LPP is a | (IV) feasible solution | Choose the correct answer from the options given below :

If corner points of a feasible region are (0, 0), (2, 0), $\left(\frac{20}{19}, \frac{45}{19}\right)$ and (0, 3), then (A) Maximum value of $z = 5x + 3y$ is 10 (B) Minimum value of $z = 5x + 3y$ is 0 (C) Maximum value of $z = 5x + 3y$ is $\frac{235}{19}$ and minimum value is 0 (D) Maximum value of $z = 5x + 3y$ is 10 and minimum value is 0 Choose the correct answer from the options given below :

If in a binomial distribution $n = 4$, $P(X=0) = \frac{16}{81}$, then $P(X=4)$ equals :

A and B throw a die alternatively till one of them gets a number more than 4 and wins the game. Then the probability of winning the game by B, if A starts first :

The inverse of the function $f : R \to R$ given by $f(x) = 2x + 7$ is :

If $f(x) = \begin{cases} \frac{x^2 - 9}{x - 3}, & x \neq 3 \\ 5, & x = 3 \end{cases}$ then $f(x)$ :

The integral $\int_{0}^{1} x(1-x)^n dx$ is equal to :

The set of value of $x$ for which the angle between the $\vec{a} = 2x^2\hat{i} + 4x\hat{j} + \hat{k}$ and $\vec{b} = 7\hat{i} - 2\hat{j} + x\hat{k}$ is obtuse is :

The shortest distance between the lines $\frac{x+3}{1} = \frac{y-2}{2} = \frac{z+4}{3}$ and $\frac{x+3}{-3} = \frac{y+7}{2} = \frac{z-6}{4}$ is :

If $x$ is the least positive integer satisfying $100 \equiv x \pmod{6}$, then $(2x+1)$ is equal to :

The quantity of water that must be added to 36 litres of milk at $2\frac{1}{2}$ litres for ₹ 120 so as to have mixture worth ₹ 36 for a litre is :

A motor boat covers 16 km in 2 hours downstream and 14 km in 2 hours upstream. The speed of the motor boat is :

Two pipes A and B can fill a cistern in 15 minutes and 30 minutes respectively. Both pipes are opened together, but after 5 minute pipe B is turned off. The cistern will be full in total :

In a partnership, A invests one-fourth of the capital for one-third of the time, B invests one-third of the capital for one-fourth of the time and C invests the rest of the capital for the whole time. Out of a profit of ₹ 3,500, A's share is :

Match List - I with List - II. | List - I | List - II | |----------|-----------| | (A) The solution set of the inequality $-5x > 3$, $x \in R$, is | (I) $\left[\frac{20}{7}, \infty\right)$ | | (B) The solution set of the inequality is, $\frac{-7x}{4} \leq -5$, $x \in R$ is, | (II) $\left[\frac{4}{7}, \infty\right)$ | | (C) The solution set of the inequality $7x - 4 \geq 0$, $x \in R$ is, | (III) $\left(-\infty, \frac{7}{5}\right)$ | | (D) The solution set of the inequality $9x - 4 < 4x + 3$, $x \in R$ is, | (IV) $\left(-\infty, -\frac{3}{5}\right)$ | Choose the correct answer from the options given below :

If $\begin{bmatrix} 3 & 2x+5y & -2 \\ x+4y & 7 & -5 \end{bmatrix} = \begin{bmatrix} 3 & 10 & -2 \\ 2 & 7 & -5 \end{bmatrix}$ Then the values of $x$ and $y$ are :

If A is a square matrix of order 3 and $|A| = 5$, then $|adj(adjA)|$ is :

If the matrix $A = \begin{bmatrix} x & -2 & -5y \\ 2 & 0 & -9 \\ 10 & 3z & 0 \end{bmatrix}$ is skew-symmetric, then the value of $(2x - 3y + 4z)$ is :

If $y = \log\left(\frac{x^5}{e^5}\right)$, then $\frac{d^2y}{dx^2}$ is,

The point on the curve $y^2 = 16x$ for which the y-coordinate is changing 2 times as fast as the x-coordinate is :

The total cost function for $x$ units of a commodity is given by $C(x) = \frac{25x^3}{3} - 75x^2 + 48x + 34$. The output $x$ at which the marginal cost is minimum is :

Match List - I with List - II. | List - I | List - II | |----------|-----------| | (A) The minimum value of $f(x) = 8x^2 - 4x + 7$ is | (I) 48 | | (B) The maximum value of $f(x) = x + \frac{1}{x}$, $x < 0$ is | (II) 13 | | (C) The maximum slope of the curve $y = -2x^3 + 6x^2 + 7x + 26$ is | (III) $-2$ | | (D) The minimum value of $f(x) = x^2 + \frac{128}{x}$ is | (IV) $\frac{13}{2}$ | Choose the correct answer from the options given below :

A product costs the manufacturer ₹ 20 per unit. The demand function is given by $p(x) = 1000 - 20x$, then the quantity for maximum profit is :

A discrete random variable X has the following probability distribution : | X: | 0 | 1 | 2 | 3 | 4 | 5 | |----|----|----|----|----|----|----| | P(X): | b | 3b | 5b | 3b | 4b | 6b | The value of b is :

A discrete random variable X takes the values 0, 1, 2, 3, 4 and its mean is 1.6. If $P(X=1) = 0.4$, $P(X=4) = P(X=2)$ and $P(X=3) = 2P(X=2)$, then $P(X=0)$ is :

A telephone exchange receives on an average 5 calls per minute. The probability of receiving 3 or less calls per minute is :

Match List - I with List - II. | List - I | List - II | |----------|-----------| | (A) In a binomial distribution, if $n = 10$, $q = 0.25$, then its mean is | (I) 12 | | (B) If the mean of a binomial distribution is 6 and its variance is 3, then p is | (II) 7.5 | | (C) In a binomial distribution, the probability of getting a success is $\frac{1}{4}$ and the standard distribution is 3, then its mean is | (III) 16 | | (D) If the mean and variance of a binomial distribution are 4 and 3 respectively, then the number of trials is | (IV) $\frac{1}{2}$ | Choose the correct answer from the options given below :

If Paasche's index number is 160 and Laspeyre's index number is 250, then Fisher's index number is :

The prices and the quantities of three commodities are given are : | Commodity | Price (₹) in Year 2006 | Price (₹) in Year 2009 | Quantities in Year 2006 | Quantities in Year 2009 | |-----------|------------------------|------------------------|--------------------------|--------------------------| | P | 100 | 90 | 12 | 10 | | Q | 80 | $x$ | 8 | 7 | | R | 60 | 50 | 4 | 6 | The Laspeyre's price index number for year 2009 with year 2006 as base is 200. The value of $x$ is :

Consider the following data : | Year | 2012 | 2013 | 2014 | 2015 | 2016 | |------|------|------|------|------|------| | Sales (in ₹ crores) | 8 | 10 | 7 | 9 | 12 | The equation of the straight line trend by the method of least squares is :

Consider the following hypothesis test : $H_0 : \mu \leq 20$ $H_1 : \mu > 20$ A sample of 81 produced a sample mean of 20.55. The population standard deviation is 3. The value of the test statistic is :

A simple random sample consists of four observations 7, 8, 10, 7. The point estimate of population standard deviation is :

Match List - I with List - II. | List - I | List - II | |----------|-----------| | (A) A special characteristic of a population is called | (I) Sample Size | | (B) The number of statistical individuals in a sample is called | (II) Statistic | | (C) A special characteristic of a sample is called | (III) Standard error | | (D) The standard deviation of the sampling distribution of a statistic is known as its | (IV) Parameter | Choose the correct answer from the options given below :

The present value of a perpetual income of ₹ $x$ payable at the end of each 6 months is ₹ 1,80,000. If the money is worth 5% compounded semi-annually, then the value of $x$ is ₹ :

A person buys a flat for which he makes down payment of ₹ 7,50,000 and the balance is to be paid in 10 years by monthly instalments of ₹ 22,000 each. If the bank charges interest at the rate of 12% per annum, then the actual price of the flat using flat rate system is :

A car costing ₹ 8,00,000 has scrap value of ₹ 3,00,000. If the book value of car at the end of fourth year is ₹ 6,00,000, then the useful life of the car is :

The minimum value of $z = 3x + 6y$ subject to the constraints $2x + 3y \leq 180$, $x + y \geq 60$, $x \geq 3y$, $x \geq 0$, $y \geq 0$ is :

A carpenter earns a profit of ₹ 50 and ₹ 80 on one chair and one table respectively. The requirement and availability of wood and labour are tabled as : | Required | Chair | Table | Available quantity | |----------|-------|-------|--------------------| | Wood | 3 | 5 | 150 | | Labour | 1 | 2 | 56 | The number of chairs and tables in appropriate units to be manufactured for maximum profit are, respectively :

Match List - I with List - II. | List - I | List - II | |----------|-----------| | (A) The common region determined by all the linear constraints of a L.P.P. is called | (I) corner point | | (B) A point in the feasible region which is the intersection of two boundary lines is called, | (II) non-negative | | (C) The feasible region for an LPP is always a | (III) feasible region | | (D) The constraints $x, y \geq 0$ describes that the variables involved in a LPP are | (IV) convex polygon | Choose the correct answer from the options given below :

The set of all positive integers less than 50 forming the equivalence class of 8 for modulo 11 is :

If $x = 3at^2$, $y = 3at^4$ then $\frac{dy}{dx}$ is :

In the equation of trend line $y_t = a + bx$, a and b represent :

If Mr. Ravi borrows a sum of ₹ 1,50,000 at an interest rate of 10% (flat) for a tenure of 3 years, then his EMI based on above data is (approximately) ₹ :

Which of the following statements are true ? (A) Central limit theorem states that the sampling distribution of the mean $(\bar{x})$ approaches a normal distribution as the sample size increases. (B) As per Central Limit Theorem, when the sample size increases, the mean $(\bar{x})$ for the data becomes closer to the mean of overall population. (C) The shape of t-distribution does not depend on degree of freedom. Choose the correct answer from the options given below :