CUET Mathematics 2023 PYQs23 May Shift 3. Free, no login required.

Q1:

Calculus > Differential Equations

Medium

common

In the context of differential equation Match List I with List II | LIST I | LIST II | |---|---| | A. $\frac{dy}{dx} = \frac{1+y^2}{1+x^2}$ | I. Not a differential equation | | B. $x^2 \frac{dy}{dx} = x^2 - 2y^2 + xy$ | II. Linear first order | | C. $\sin x + y = \cos(x+y)$ | III. Variable separable | | D. $(x+y)\frac{dy}{dx} = 1$ | IV. Homogenous | Choose the correct answer from the options given below:

A-I, B-II, C-III, D-IV

A-II, B-IV, C-III, D-I

A-III, B-IV, C-I, D-II

A-IV, B-I, C-III, D-II

Correct Answer

Option 3

Correct Answer

Explanation

Q2:

Calculus > Application of Derivatives

Easy

common

The critical points of $f(x) = x^3 + x^2 + x + 1$ are

2, 1

-2, -1

2, -1

do not exist

Correct Answer

Option 4

Correct Answer

Explanation

Q3:

Calculus > Application of Derivatives

Medium

common

The volume of a cube is increasing at the rate of 27 cm$^3$/s. How fast is the surface area increasing when the length of the cube is 12 cm.

9 cm$^2$/s

$\frac{9}{4}$ cm$^2$/s

$\frac{4}{9}$ cm$^2$/s

$\frac{9}{2}$ cm$^2$/s

Correct Answer

Option 1

Correct Answer

Explanation

Q4:

Algebra > Probability

Easy

common

If a fair coin is tossed 10 times, then the probability of obtaining at least one head is :

$\frac{1}{1024}$

$\frac{17}{1024}$

$\frac{1023}{1024}$

$\frac{23}{1024}$

Correct Answer

Option 3

Correct Answer

Explanation

Q5:

Algebra > Probability

Hard

common

Two cards are drawn simultaneously from a well shuffled pack of 52 cards. Then variance of the number of kings is

$\frac{680}{(221)^2}$

$\frac{6080}{(221)^2}$

$\frac{680}{221}$

$\frac{6800}{(221)^2}$

Correct Answer

Option 4

Correct Answer

Explanation

Q6:

Calculus > Integrals

Medium

common

The integral $\int e^x \left(\frac{x-1}{2x^2}\right) dx$ is equal to:

$\frac{e^x}{x} + C$, where C is constant of integration

$\frac{e^x}{2x} + C$, where C is constant of integration

$e^x x + C$, where C is constant of integration

$x^2 e^x + C$, where C is constant of integration

Correct Answer

Option 2

Correct Answer

Explanation

Q7:

Algebra > Matrices & Determinants

Easy

common

The order of a null matrix is :

any order

Correct Answer

Option 4

Correct Answer

Explanation

Q8:

Calculus > Continuity & Differentiability

Easy

common

If $y = x^x$, $\frac{dy}{dx}$ will be:

$x^x$

$x^x(1 + \log x)$

$x^{x-1}$

$x^{x+1}$

Correct Answer

Option 2

Correct Answer

Explanation

Q9:

Algebra > Linear Programming

Medium

common

If $5x + y \leq 100$, $x + y \leq 60$, $x \geq 0$, $y \geq 0$. Then one of the corner points of the feasible region is :

(60, 0)

(0, 100)

(10, 50)

(0, 20)

Correct Answer

Option 3

Correct Answer

Explanation

Q10:

Calculus > Integrals

Easy

common

The value of $\int \left(\sqrt{x} + \frac{1}{\sqrt{x}}\right)^2 dx$ is:

$\frac{x^2}{2} + \log|x| + C$, (where C is constant of integration)

$\frac{x^2}{2} + \log|x| + 2x + C$, (where C is constant of integration)

$\frac{3}{2}\left(\sqrt{x} + \frac{1}{\sqrt{x}}\right) + C$, (where C is constant of integration)

$\frac{2}{3} x^{\frac{3}{2}} + 2\sqrt{x} + C$, (where C is constant of integration)

Correct Answer

Option 2

Correct Answer

Explanation

Q11:

Calculus > Differential Equations

Hard

common

Which of the following differential equation represents the family of circles touching the x-axis at the origin ?

$(x^2 - y^2)dy - 2xy\, dx = 0$

$(x^2 + y^2)dy + 2xy\, dx = 0$

$(x^2 - y^2)dx + 2xy\, dy = 0$

$(x^2 + y^2)dy - 2xy\, dx = 0$

Correct Answer

Option 1

Correct Answer

Explanation

Q12:

Algebra > Matrices & Determinants

Easy

common

Let $A = \begin{bmatrix} 3 & -1 \\ 2 & 4 \end{bmatrix}$, then adjoint (A) is:

$\begin{bmatrix} 3 & 2 \\ -1 & 4 \end{bmatrix}$

$\begin{bmatrix} 4 & -1 \\ 2 & 3 \end{bmatrix}$

$\begin{bmatrix} -4 & 1 \\ -2 & -3 \end{bmatrix}$

$\begin{bmatrix} 4 & 1 \\ -2 & 3 \end{bmatrix}$

Correct Answer

Option 4

Correct Answer

Explanation

Q13:

Algebra > Linear Programming

Medium

common

In a LPP, let R be the feasible region. A. If R is unbounded then a max./min. value of objective function may not exist. B. If R is bounded then a max. and min. value of objective function will always exist. C. If a solution exists, it must occur at a corner point. D. If R is bounded then max. will exist but min. may or may not exist for an objective function. Choose the correct answer from the options given below:

A, B, C only

B only

A, C only

D, C only

Correct Answer

Option 1

Correct Answer

Explanation

Q14:

Algebra > Matrices & Determinants

Easy

common

Let A be a square matrix of order 3 then |3A| is equal to

3|A|

$3^2$|A|

$|A|^3$

$3^3$|A|

Correct Answer

Option 4

Correct Answer

Explanation

Q15:

Algebra > Matrices & Determinants

Easy

common

The inverse of the matrix $A = \begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}$ is:

$\begin{bmatrix} -2 & 3 \\ 1 & -2 \end{bmatrix}$

$\begin{bmatrix} 2 & -3 \\ -1 & 2 \end{bmatrix}$

$\begin{bmatrix} -2 & -3 \\ -1 & 2 \end{bmatrix}$

$\begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}$

Correct Answer

Option 2

Correct Answer

Explanation

Q16:

Geometry > 3D Geometry

Medium

core

The angle between the lines $\vec{r} = 3\hat{i} + 2\hat{j} - 4\hat{k} + \lambda(\hat{i} + 2\hat{j} + 2\hat{k})$ and $\vec{r} = 5\hat{j} - 2\hat{k} + \mu(3\hat{i} + 2\hat{j} + 6\hat{k})$ is :

$\sin^{-1}\left(\frac{19}{21}\right)$

$\cos^{-1}\left(\frac{19}{23}\right)$

$\cos^{-1}\left(\frac{19}{21}\right)$

$\sin^{-1}\left(\frac{19}{23}\right)$

Correct Answer

Option 3

Correct Answer

Explanation

Q17:

Calculus > Continuity & Differentiability

Medium

core

If $y = \sin^{-1}\left(\frac{1-x^2}{1+x^2}\right)$ then $\frac{dy}{dx} =$

$-\frac{2}{1+x^2}$

$\frac{2}{1+x^2}$

$-\frac{1}{1+x^2}$

$\frac{1}{1+x^2}$

Correct Answer

Option 1

Correct Answer

Explanation

Q18:

Algebra > Relations & Functions

Easy

core

The number of all onto functions from the set {1, 2, .......n} to itself is

$2^n$

$n^2$

(2n)!

Correct Answer

Option 3

Correct Answer

Explanation

Q19:

Geometry > 3D Geometry

Hard

core

The value of $\lambda$, so that the lines $\frac{1-x}{3} = \frac{7y-14}{2\lambda} = \frac{z-3}{2}$ and $\frac{7-7x}{3\lambda} = \frac{y-5}{1} = \frac{6-z}{5}$ are perpendicular is:

$-\frac{70}{11}$

$\frac{70}{11}$

$\frac{11}{70}$

$-\frac{11}{70}$

Correct Answer

Option 2

Correct Answer

Explanation

Q20:

Calculus > Application of Derivatives

Medium

core

The intervals for which $f(x) = x^4 - 2x^2$ is increasing are :

$(-\infty, 1)$

$(-1, \infty)$

$(-\infty, -1) \cup (0, 1)$

$(-1, 0) \cup (1, \infty)$

Correct Answer

Option 4

Correct Answer

Explanation

Q21:

Calculus > Application of Integrals

Easy

core

The area enclosed between the curves $y = x^2$ and $x = y^2$ is

$\frac{1}{2}$

$\frac{1}{3}$

$\frac{1}{4}$

Correct Answer

Option 3

Correct Answer

Explanation

Q22:

Geometry > 3D Geometry

Easy

core

Match List I with List II | LIST I | LIST II | |---|---| | A. $lx + my + nz = d$ is | I. Equation of plane passing through a given point and normal to given vector | | B. $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$ is | II. Equation of plane in normal form | | C. $(\vec{r} - \vec{a}) \cdot \vec{n} = 0$ | III. Plane passing through the intersection of two planes | | D. $(a_1x + b_1y + c_1z + d_1) + \lambda(a_2x + b_2y + c_2z + d_2) = 0$ | IV. Intercept from of plane | Choose the correct answer from the options given below:

A-I, B-III, C-IV, D-II

A-IV, B-III, C-I, D-II

A-II, B-IV, C-I, D-III

A-I, B-II, C-III, D-IV

Correct Answer

Option 3

Correct Answer

Explanation

Q23:

Algebra > Vector Algebra

Easy

core

If $\vec{a} = 5\hat{i} - \hat{j} - 3\hat{k}$ & $\vec{b} = \hat{i} - 3\hat{j} + 5\hat{k}$ the angle between $\vec{a} + \vec{b}$ and $\vec{a} - \vec{b}$ is :

60$^\circ$

90$^\circ$

120$^\circ$

30$^\circ$

Correct Answer

Option 2

Correct Answer

Explanation

Q24:

Calculus > Application of Derivatives

Easy

core

The slope of the normal to the curve $y = 2x^2 - 4$ at P (1, -2) is :

-4

$-\frac{1}{4}$

Correct Answer

Option 3

Correct Answer

Explanation

Q25:

Geometry > Trigonometry

Easy

core

The principal value of $\cot^{-1}\left(-\frac{1}{\sqrt{3}}\right)$

$\frac{\pi}{3}$

$-\frac{\pi}{6}$

$\frac{1}{\sqrt{3}}$

$\frac{2\pi}{3}$

Correct Answer

Option 4

Correct Answer

Explanation

Q26:

Algebra > Relations & Functions

Medium

core

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

Reflexive and symmetric

Neither reflexive nor symmetric

Transitive

Reflexive but not symmetric

Correct Answer

Option 2

Correct Answer

Explanation

Q27:

Algebra > Matrices & Determinants

Easy

core

If $\begin{vmatrix} 2x & 2 \\ 4 & x \end{vmatrix} = 10$, then $x$ is:

$\pm 2$

$\pm 3$

$\pm 4$

Correct Answer

Option 2

Correct Answer

Explanation

Q28:

Algebra > Probability

Hard

core

A doctor is to visit a patient. It is known that the probabilities that he will come by train, bus, scooter or by other means of transport are respectively $\frac{3}{10}, \frac{1}{5}, \frac{1}{10}$ and $\frac{2}{5}$. The probabilities that he will be late are $\frac{1}{4}, \frac{1}{3}$ and $\frac{1}{12}$, if he comes by train, bus and scooter respectively, but if he comes by other means of transport, then he will not be late. When he arrives, he arrives late. The probability that he comes by bus is:

$\frac{4}{9}$

$\frac{1}{18}$

$\frac{1}{3}$

$\frac{1}{2}$

Correct Answer

Option 1

Correct Answer

Explanation

Q29:

Algebra > Matrices & Determinants

Easy

core

The matrix $\begin{bmatrix} 2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$ is a

zero matrix

Identity matrix

Scalar matrix

Diagonal matrix

Correct Answer

Option 4

Correct Answer

Explanation

Q30:

Calculus > Differential Equations

Medium

core

The solution of $y' - y'' = 2x$ is: A. $y = x^2 + 2x + 2$ B. $y = x^2 + 2x + 1$ C. $y = x + 2$ D. $y = x^2 - 2x + 1$ Choose the correct answer from the options given below:

A and B only

B only

C only

A and D only

Correct Answer

Option 1

Correct Answer

Explanation

Q31:

Calculus > Application of Derivatives

Easy

core

The minimum value of $f(x) = |2x - 1|$ is

$-\infty$

$\frac{1}{2}$

Correct Answer

Option 2

Correct Answer

Explanation

Q32:

Algebra > Matrices & Determinants

Easy

core

Let A be the square matrix of order 3, then |kA|, where k is a scalar, is equal to:

3k|A|

$k^3$|A|

$k^2$|A|

k|A|

Correct Answer

Option 2

Correct Answer

Explanation

Q33:

Calculus > Continuity & Differentiability

Easy

core

Points of discontinuity of the greatest integer function $f(x) = [x]$, where $[x]$ denotes integer less than or equal to $x$, are

all natural numbers

all rational numbers

all integers

all real numbers

Correct Answer

Option 3

Correct Answer

Explanation

Q34:

Geometry > Trigonometry

Medium

core

The simplest form of $\tan^{-1}\left\{\frac{x}{\sqrt{a^2 - x^2}}\right\}$ is, where $-a < x < a$.

$\tan^{-1}\frac{x}{a}$

$\tan^{-1}(ax)$

$a \tan^{-1}\frac{x}{a}$

$\sin^{-1}\frac{x}{a}$

Correct Answer

Option 4

Correct Answer

Explanation

Q35:

Calculus > Application of Integrals

Medium

core

The area of the region $\{(x, y) : y \geq x^2 \text{ and } y \leq |x|\}$ is

$\frac{1}{2}$

$\frac{1}{3}$

Correct Answer

Option 4

Correct Answer

Explanation

Q36:

Calculus > Differential Equations

Easy

core

The general solution of $\frac{dy}{dx} = 1 + x^2 + y^2 + x^2y^2$ is: (given that $C$ is the constant of integration)

$\tan^{-1} x = y + \frac{y^3}{3} + C$

$\tan^{-1} y = x + \frac{x^3}{3} + C$

$\tan^{-1} x = \tan^{-1} y + C$

$\tan^{-1} x + \tan^{-1} y = C$

Correct Answer

Option 2

Correct Answer

Explanation

Q37:

Algebra > Probability

Medium

core

If a fair coin is tossed 10 times the probability of atleast 6 heads is:

$\frac{105}{512}$

$\frac{53}{128}$

$\frac{53}{64}$

$\frac{193}{512}$

Correct Answer

Option 4

Correct Answer

Explanation

Q38:

Algebra > Matrices & Determinants

Medium

core

Which one of the following options is incorrect? For a square matrix A in the matrix equation AX = B.

If $|A| \neq 0$, then there exists a unique solution

If $|A| = 0$ and (adj A)B $\neq$ 0 then there is no solution

If $|A| \neq 0$ and (adj A)B $\neq$ 0 then there is no solution

If $|A| = 0$ and (adj A)B = 0 then system has infinitely many solutions

Correct Answer

Option 3

Correct Answer

Explanation

Q39:

Algebra > Matrices & Determinants

Easy

core

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

$\begin{bmatrix} x^{-1} & 0 & 0 \\ 0 & y^{-1} & 0 \\ 0 & 0 & z^{-1} \end{bmatrix}$

$xyz \begin{bmatrix} x^{-1} & 0 & 0 \\ 0 & y^{-1} & 0 \\ 0 & 0 & z^{-1} \end{bmatrix}$

$\frac{1}{xyz} \begin{bmatrix} x^{-1} & 0 & 0 \\ 0 & y^{-1} & 0 \\ 0 & 0 & z^{-1} \end{bmatrix}$

$\frac{1}{xyz} \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$

Correct Answer

Option 1

Correct Answer

Explanation

Q40:

Algebra > Vector Algebra

Easy

core

The unit vector in the direction of $\vec{a} + \vec{b}$ if $\vec{a} = 2\hat{i} - \hat{j} + 2\hat{k}$ & $\vec{b} = -\hat{i} + \hat{j} + -\hat{k}$ is :

$\hat{i} + 0\hat{j} + \hat{k}$

$\hat{i} - \hat{j} + \hat{k}$

$\hat{i} + \hat{j} + \hat{k}$

$\frac{1}{\sqrt{2}}\hat{i} + \frac{1}{\sqrt{2}}\hat{k}$

Correct Answer

Option 4

Correct Answer

Explanation

Q41:

Geometry > Trigonometry

Easy

core

The maximum value of $\sin x + \cos x, x \in R$ is:

$\sqrt{2}$

$\frac{1}{\sqrt{2}}$

Not known

Correct Answer

Option 2

Correct Answer

Explanation

Q42:

Calculus > Continuity & Differentiability

Medium

core

If $y = x^{(x \sin x)}$ then $\frac{dy}{dx} = ?$

$x^{x \cos x}$

$x^{(x \sin x)}[\sin x + \sin \log x]$

$x^{(x \sin x)}[\sin (1 + \log x) + x \log x \cos x]$

$x^{(x \sin x)}[\sin \log x + x \log x \cos x]$

Correct Answer

Option 3

Correct Answer

Explanation

Q43:

Geometry > Trigonometry

Medium

core

The simplest form of $\tan^{-1} \frac{\sqrt{1+x^2} - 1}{x}$, $x \neq 0$ is:

$\tan^{-1} x$

$x$

$\frac{1}{2} \tan^{-1} x$

$\sqrt{1+x^2}$

Correct Answer

Option 3

Correct Answer

Explanation

Q44:

Algebra > Matrices & Determinants

Medium

core

The value of $\begin{vmatrix} 1 & a & b+c \\ 1 & b & c+a \\ 1 & c & a+b \end{vmatrix}$ is

(a + b + c)

(a - b)(b - c)(c - a)

$a^2 + b^2 + c^2$

Correct Answer

Option 1

Correct Answer

Explanation

Q45:

Calculus > Integrals

Hard

core

$\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{dx}{1 + \sqrt{\tan x}}$ is equal to :

$\frac{\pi}{6}$

$\frac{\pi}{12}$

$\frac{\pi}{4}$

$\frac{\pi}{3}$

Correct Answer

Option 2

Correct Answer

Explanation

Q46:

Calculus > Integrals

Medium

core

The value of integral $\int \sqrt{4x^2 + 9}\, dx$ is

$\frac{x}{2}\sqrt{4x^2 + 9} + \frac{9}{2}\log|2x + \sqrt{4x^2 + 9}| + C$

$\frac{x}{2}\sqrt{4x^2 + 9} + \frac{3}{2}\log|2x + \sqrt{4x^2 + 9}| + C$

$2x\sqrt{4x^2 + 9} + \frac{9}{2}\log|2x + \sqrt{4x^2 + 9}| + C$

$x\sqrt{4x^2 + 9} + \frac{9}{4}\log|2x + \sqrt{4x^2 + 9}| + C$

Correct Answer

Option 1

Correct Answer

Explanation

Q47:

Algebra > Relations & Functions

Easy

core

Let $f(x) = x^3$ be a function with domain {0, 1, 2, 3} then domain of $f^{-1}$ is :

{3, 2, 1, 0}

{0, -1, -2, -3}

{0, 1, 8, 27}

{0, -1, -8, -27}

Correct Answer

Option 3

Correct Answer

Explanation

Q48:

Algebra > Vector Algebra

Easy

core

The area of the parallelogram determined by the vectors $\hat{i} + 2\hat{j} + 3\hat{k}$ and $3\hat{i} - 2\hat{j} + \hat{k}$ is

$8\sqrt{3}$

$4\sqrt{3}$

$16\sqrt{3}$

$2\sqrt{3}$

Correct Answer

Option 1

Correct Answer

Explanation

Q49:

Algebra > Linear Programming

Easy

core

A manufacturing company makes two models M$_1$ and M$_2$ of a product. Each piece of M$_1$ requires 9 labour hours for fabricating and one labour hour for finishing. Each piece of M$_2$ require 12 labour hours for fabricating and 3 labour hours for finishing. For fabricating and finishing, the maximum labour hours available are 180 and 30 respectively. The company makes a profit of Rs.800 on each piece of M$_1$ and Rs.1200 on each piece of M$_2$ The above Linear Programming Problem [LPP] is given by

Maximize Z = 800x + 1200y Subject to constraints, $3x + 4y \leq 60$ $x + 3y \leq 30$ $x, y \geq 0$

Maximize Z = 800x + 1200y Subject to constraints, $3x + 4y \geq 60$ $x + 3y \geq 30$ $x, y \geq 0$

Minimize Z = 800x + 1200y Subject to constraints, $3x + 4y \leq 60$ $x + 3y \geq 30$ $x, y \geq 0$

Minimize Z = 800x + 1200y Subject to constraints, $3x + 4y \geq 60$ $x + 3y \leq 30$ $x, y \geq 0$

Correct Answer

Option 1

Correct Answer

Explanation

Q50:

Algebra > Linear Programming

Medium

core

(0, 10)

(20, 0)

(12, 6)

(0, 0)

Correct Answer

Option 3

Correct Answer

Explanation

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