CUET 2023 Mathematics Question Paper (Available):Download Slot-wise Answer Key with Solutions PDF

Question 1:

• (1) only AB is defined.
• (2) only BA is defined.
• (3) AB and BA both are defined.
• (4) Neither AB nor BA is defined.

• (1) 8
• (2) 7
• (3) 9
• (4) 6

The area of a triangle with vertices \( (0, -3) \), \( (0, 3) \), and \( (k, 0) \) is 27 sq. units. The value of \( k \) is:

• (1) -27
• (2) -6
• (3) -18
• (4) -9

If \( A \) is a non-singular square matrix of order 3 such that \( A^3 = 4A^2 \), then the value of \( |A| \) is:

• (1) 16
• (2) 64
• (3) 4
• (4) 8

If \( A \) is a non-singular square matrix of order 3 such that \( A^3 = 4A^2 \), then the value of \( |A| \) is:

• (1) 16
• (2) 64
• (3) 4
• (4) 8

The slope of normal to the curve \( y = 3x^2 - 6x \) at \( x = 0 \) is:

• (1) -6
• (2) \( -\frac{1}{6} \)
• (3) \( \frac{1}{6} \)
• (4) \( -\frac{2}{3} \)

If \( f(x) = -3x^2 \), then \( f(x) \) is:

• (1) Increasing in \( (0, \infty) \), decreasing in \( (-\infty, 0) \)
• (2) Increasing in \( (-\infty, 0) \), decreasing in \( [0, \infty) \)
• (3) Increasing in \( \left[ -\frac{1}{3}, \infty \right) \), decreasing in \( (-\infty, -\frac{1}{3}) \)
• (4) Decreasing for all real values of \( x \)

The value of \( \int (5x - 2)^3 \, dx \) is:

• (1) \( 3(5x - 2) + C \) ; C is constant of integration.
• (2) \( 15(5x - 2)^2 + C \) ; C is constant of integration.
• (3) \( (5x - 2)^4 + C \) ; C is constant of integration.
• (4) \( \frac{(5x - 2)^4}{20} + C \) ; C is constant of integration.

The area bounded by \( y = |x - 5| \) and the x-axis between \( x = 2 \) and \( x = 4 \) is:

• (1) 5
• (2) 4
• (3) 3
• (4) 6

If \( \frac{d}{dx} \left( \frac{d^2 y}{dx^2} \right)^3 = 7 \), then the sum of the order and degree of the differential equation is:

• (1) 3
• (2) 2
• (3) 5
• (4) 4

The equation of the curve whose slope is given by \( \frac{dy}{dx} = \frac{4x}{y} \), \( x > 0 \), \( y > 0 \), and which passes through the point \( (2, 2) \) is:

• (1) \( y^2 = 4x^2 - 12 \)
• (2) \( x^2 = 4y^2 - 12 \)
• (3) \( y^2 = 4x^2 + 12 \)
• (4) \( x^2 = 4y^2 + 12 \)

A random variable has the following probability distribution:



The value of \( P(X < 3) \) is:

• (1) \( \frac{1}{12} \)
• (2) \( \frac{1}{3} \)
• (3) \( \frac{1}{4} \)
• (4) \( \frac{5}{12} \)

If a discrete random variable \( X \) has the following probability distribution:

Then \( c \) is:

• (1) \( \frac{1}{2} \)
• (2) \( \frac{1}{3} \)
• (3) \( \frac{1}{4} \)
• (4) \( \frac{1}{5} \)

The maximum value of \( Z = 3x + 4y \) subject to the constraint \( x + y \leq 6 \), \( x \geq 0 \), \( y \geq 0 \) is:

• (1) 18
• (2) 20
• (3) 22
• (4) 24

For the problem max \( Z = ax + by \), \( x \geq 0 \), \( y \geq 0 \), which of the following is NOT a valid constraint to make it a linear programming problem?

• (1) \( x \leq 5, \, y \leq 10 \)
• (2) \( 2x + 3y \leq 60 \)
• (3) \( x + 2y \leq 40 \)
• (4) \( x^2 + y \leq 50 \)

For real numbers \( a \) and \( b \), define \( aRb \) if \( b - a + \sqrt{5} \) is an irrational number. Then the relation \( R \) is:

• (1) Symmetric
• (2) Reflexive
• (3) Transitive
• (4) Equivalence

The domain of the function \( f(x) = \log(x^2 - 4) \) is:

• (1) \( [2, \infty) \)
• (2) \( (-\infty, -2) \cup (2, \infty) \)
• (3) \( (2, \infty) \)
• (4) \( (-\infty, -2] \cup (-2, -1) \)

Domain of function \( f(x) = \cos^{-1} \sqrt{2x - 1} \) is:

• (1) \( [-1, 1] \)
• (2) \( \left[ \frac{1}{2}, 1 \right] \)
• (3) \( \left[ \frac{1}{2}, 1 \right] \)
• (4) \( \left[ -1, \frac{1}{2} \right] \)

The value of \( \tan(\cos^{-1}(x)) \) is:

• (1) \( \frac{1}{x} \)
• (2) \( \frac{\sqrt{1 - x^2}}{x} \)
• (3) \( \frac{\sqrt{1 + x^2}}{x} \)
• (4) \( \frac{x}{\sqrt{1 - x^2}} \)

• (1) \( \left(\frac{1}{2}, 1\right) \)
• (2) \( (1, \frac{1}{2}) \)
• (3) \( (1, 1) \)
• (4) \( (-1, -1) \)

• (1) -5
• (2) 3
• (3) 5
• (4) -3

If \[ A = \begin{pmatrix} 1 & \sqrt{3} & 0
0 & 2 & 0 \end{pmatrix} \quad and \quad B = \begin{pmatrix} \sqrt{3} & 1 & 0
0 & 0 & 2 \end{pmatrix} \]
then \( AB \) is equal to:

• (1) \( 4I \)
• (2) \( -4I \)
• (3) \( \begin{pmatrix} 0 & 0 & 4
  0 & 0 & 4 \end{pmatrix} \)
• (4) \( \begin{pmatrix} \sqrt{3} & 1 & 2\sqrt{3}
  0 & 0 & 4 \end{pmatrix} \)

Which of the following is a correct statement?

• (1) Determinant is a rectangular arrangement of numbers.
• (2) Determinant is a square arrangement of numbers.
• (3) Determinant is a value assigned to a square arrangement of numbers in a specific manner.
• (4) Determinant is the sum of diagonal elements of a square arrangement of numbers.

The value of the determinant \[ det \begin{pmatrix} \cos^2(\theta) & \cos(\theta) \sin(\theta) & 0
-\sin(\theta) & \cos(\theta) & 0
0 & 0 & 1 \end{pmatrix} \]
is equal to:

• (1) 1
• (2) \( \cos(\theta) \)
• (3) \( 2 \cos(\theta) \)
• (4) \( \cos(\theta) - \sin(\theta) \)

If \[ A = (2, 3), B = (-1, 0), C = (4, 6) \]
then the area of the parallelogram ABCD is:

• (1) 3
• (2) \( \frac{3}{2} \)
• (3) 6
• (4) 2

Let \[ f(x) = \begin{cases} 2x - 1 & if x < 1
1 & if x = 1
x^2 & if x > 1 \end{cases} \]
then at \( x = 1 \):

• (1) \( f(x) is continuous from left only \)
• (2) \( f(x) is continuous from right only \)
• (3) \( f(x) is continuous \)
• (4) \( f(x) has removable discontinuity \)

If \[ x = e^y + e^y + \dots \quad then \quad \frac{d^2y}{dx^2} = ? \]

The options are:

• (1) \( -x^2 \)
• (2) \( \frac{-1}{x^2} \)
• (3) \( \frac{1}{x^2} \)
• (4) \( x^2 \)

The derivative of \( f(\cot(x)) \) with respect to \( g(\csc(x)) \) at \( x = \frac{\pi}{4} \) where \( f'(1) = 2g'(\sqrt{2}) = 4 \) is:

• (1) \( \sqrt{2} \)
• (2) 1
• (3) \( \frac{1}{\sqrt{2}} \)
• (4) \( \frac{1}{2\sqrt{2}} \)

The values of \( b \) for which the function \[ f(x) = \cos(x) + bx + a \]
decreases on \( \mathbb{R} \) are:

• (1) \( [-1, 1] \)
• (2) \( (-\infty, 1) \)
• (3) \( (-\infty, -1] \)
• (4) \( (-1, 1) \)

The equation of the normal to the curve \( y = 2\sin(x) \) at \( (0, 0) \) is:

• (1) \( x + \frac{1}{2} y = 0 \)
• (2) \( x - 2y = 0 \)
• (3) \( x - \frac{1}{2} y = 0 \)
• (4) \( x + 2y = 0 \)

The equation of the normal to the curve \( y = 2\sin(x) \) at \( (0, 0) \) is:

• (1) \( x + \frac{1}{2} y = 0 \)
• (2) \( x - 2y = 0 \)
• (3) \( x - \frac{1}{2} y = 0 \)
• (4) \( x + 2y = 0 \)

The function \[ f(x) = \frac{x^4}{4} - \frac{x^2}{2} \]
has:

• (1) 2 points of local maxima
• (2) 2 points of local minima and one point of local maxima
• (3) 1 point of local minima and 2 points of local maxima
• (4) 1 point of local maxima and 1 point of local minima

The value of \[ \int_{-1}^{1} x^2 \left\lfloor x \right\rfloor dx \]
is:

• (1) \( \frac{1}{3} \)
• (2) \( \frac{2}{3} \)
• (3) 1
• (4) \( -\frac{1}{3} \)

The value of \[ \int_{-3}^{2} x^2 |2x| dx \]
is:

• (1) \( \frac{65}{2} \)
• (2) \( \frac{2}{3} \)
• (3) 97
• (4) \( \frac{97}{2} \)

The value of the area lying between the curves \( y^2 = 9x \) and \( y = 3x \) is:

• (1) \( \frac{1}{4} \) sq. units
• (2) \( \frac{1}{2} \) sq. units
• (3) \( \frac{2}{3} \) sq. units
• (4) \( \frac{3}{4} \) sq. units

The area lying in the first quadrant and bounded by the circle \( x^2 + y^2 = 9 \) and the lines \( x = 1 \) and \( x = 3 \) is:

• (1) \( \frac{9\pi}{2} - \frac{9}{4} \) sq. units
• (2) \( \frac{9\pi}{2} - \frac{9}{2} \) sq. units
• (3) \( \frac{9\pi}{2} + \frac{9}{2} \) sq. units
• (4) \( \frac{9\pi}{2} - \frac{9}{2} \sin^{-1}(1/3) \) sq. units

The integrating factor of \[ \sin x \frac{dy}{dx} + 2y \cos x = 4 \]
is:

• (1) \( |\sin x| \)
• (2) \( |\sin x|^2 \)
• (3) \( |\sin x^2| \)
• (4) \( \cos x \)

The solution of the differential equation \[ \frac{dy}{dx} = -\frac{x}{y} \]
is:

• (1) \( x^2 + y^2 = 2C \), where \( C \) is constant of integration.
• (2) \( x - y^2 = 2C \), where \( C \) is constant of integration.
• (3) \( x^2 + y = 2C \), where \( C \) is constant of integration.
• (4) \( x^2 - y^2 = 2C \), where \( C \) is constant of integration.

If \( |a| = 5 \), \( |b| = 2 \) and \( |a \cdot b| = 8 \), then the value of \( |a \times b| \) is:

• (1) 5
• (2) 6
• (3) 36
• (4) \( \pm 6 \)

If \( |a + b| = 15 \), \( |a - b| = 10 \), \( |a| = \frac{11}{2} \), then the value of \( |b| \) is:

• (1) \( \frac{23}{2} \)
• (2) \( \pm \frac{23}{2} \)
• (3) \( \pm 23 \)
• (4) \( \frac{23}{\sqrt{2}} \)

The angle between the straight lines \[ \frac{x+4}{2} = \frac{y+5}{5} = \frac{z+6}{3} \quad and \quad \frac{x-4}{10} = \frac{y-5}{2} = \frac{z-6}{-10} \]
is:

• (1) 45°
• (2) 90°
• (3) 30°
• (4) 60°

The direction ratios of the line perpendicular to the lines \[ \frac{x - 7}{-6} = \frac{y + 17}{4} = \frac{z - 6}{2} \quad and \quad \frac{x + 5}{6} = \frac{y + 3}{3} = \frac{z - 4}{-6} \]
are proportional to:

• (1) 7, 4, 5
• (2) 7, 5, 4
• (3) 5, 7, 4
• (4) 5, 4, 7

The maximum value of Z = 2x + 3y subject to the constraints x  0, y  0, x + y 10, 3x + 4y  36  is:

• (1) 20
• (2) 27
• (3) 30
• (4) 0

For the LPP, Min Z = 5x + 7y subject to x 0, y  0, 2x + y  8,  x + 2y 10, the basic feasible solutions are:

• (1) (0, 0), (10, 0), (2, 4) and (0, 8)
• (2) (10, 0), (2, 4) and (0, 8)
• (3) (0, 0), (0, 10), (2, 4) and (8, 0)
• (4) (0, 10), (4, 2) and (8, 0)

A bag contains 12 white and 18 red balls. Two balls are drawn in succession without replacement. The probability that the first is red and second is white is:

• (1) \( \frac{63}{145} \)
• (2) \( \frac{36}{154} \)
• (3) \( \frac{36}{144} \)
• (4) \( \frac{36}{145} \)

If A and B are events such that
\[
P(A' \cup B') = \frac{1{3 \quad \text{and \quad P(A \cup B) = \frac{4{9 \quad \text{then the value of \quad P(A') + P(B') \text{ is:

• (1) 1
• (2) \( \frac{7}{9} \)
• (3) \( \frac{8}{9} \)
• (4) \( \frac{5}{9} \)

Let \( f : \mathbb{R} \to \mathbb{R} \) be defined by \[ f(x) = x^2, \quad for every \, x \in \mathbb{R}. \, Then \, f \, is: \]

• (1) one-one and onto
• (2) one-one and not onto
• (3) neither one-one nor onto
• (4) onto and not one-one

Match List I with List II \[ List I \quad \quad List II \]
\[ A. \frac{d}{dx}\left[ \tan^{-1}\left( \frac{3x - x^3}{1 - 3x^2} \right) \right] \quad \quad I. \frac{3}{1 + x^2} \] \[ B. \frac{d}{dx}\left[ \cos^{-1}\left( \frac{1 - x^2}{1 + x^2} \right) \right] \quad \quad II. \frac{-3}{1 + x^2} \] \[ C. \frac{d}{dx}\left[ \cos^{-1}\left( \frac{2x}{1 + x^2} \right) \right] \quad \quad III. \frac{-2}{1 + x^2} \] \[ D. \frac{d}{dx}\left[ \cot^{-1}\left( \frac{3x - x^3}{1 - 3x^2} \right) \right] \quad \quad IV. \frac{2}{1 + x^2} \]

Choose the correct answer from the options given below:

• (1) A-I, B-III, C-I, D-IV
• (2) A-IV, B-II, C-I, D-III
• (3) A-III, B-I, C-IV, D-II
• (4) A-I, B-IV, C-III, D-II

The points of non-differentiability of \( f(x) = |x - 2| + |x - 3| \) are:


A.1

B.2

C.3

D.4

E.5

• (1) A, B only
• (2) B, C only
• (3) A, C only
• (4) A, D only

The value of \[ \int \frac{dx}{x^2 - 6x + 13} \]
is:

• (1) \( \frac{1}{2} \tan^{-1} \left( \frac{x - 3}{2} \right) + C \), where \( C \) is the constant of integration.
• (2) \( \frac{1}{2} \cot^{-1} \left( \frac{x - 3}{2} \right) + C \), where \( C \) is the constant of integration.
• (3) \( \frac{1}{2} \tan^{-1} \left( \frac{x + 3}{2} \right) + C \), where \( C \) is the constant of integration.
• (4) \( \frac{1}{2} \cot^{-1} \left( \frac{x + 3}{2} \right) + C \), where \( C \) is the constant of integration.

If (x) and (y) \text{ are two collinear vectors, then which of the following are incorrect?

• (1) \( \mathbf{x} = \pm \mathbf{y} \)
• (2) \( \mathbf{y} = \lambda \mathbf{x}, for some scalar \lambda \)
• (3) Both the vectors \( \mathbf{x} \) and \( \mathbf{y} \) have the same direction, but different magnitudes.
• (4) The respective components of \( \mathbf{x} \) and \( \mathbf{y} \) are not proportional.

The present value (in \₹) of a perpetuity of \₹3600 payable at the end of each quarter, if the interest rate is 9% per annum compounded quarterly, is:

• (1) ₹2,40,000
• (2) ₹1,60,000
• (3) ₹2,00,000
• (4) ₹3,20,000

If the present value of a perpetuity of ₹600 payable at the end of every six months is ₹18000, then the rate of interest is:

• (1) \( \frac{20}{3} % \) per annum
• (2) \( \frac{22}{3} % \) per annum
• (3) \( \frac{17}{3} % \) per annum
• (4) \( \frac{10}{3} % \) per annum

In an 800 m race, A beats B by 74 m and in a 600 m race, B beats C by 50 m. By how many meters will A beat C in a race of 800 m?

• (1) 234.5 m
• (2) 84.06 m
• (3) 134.5 m
• (4) 665.5 m

For the data: 

The weighted price index number is:

• (1) 152.44
• (2) 150.44
• (3) 154.44
• (4) 156.44

A, B and C are partners in a business. A receives \( \frac{3}{5} \) of the total profit while B and C share the remainder equally. A's profit is increased by ₹1,500, when the rate of profit is increased from 10% to 12% in a year. Then, B's share in the total profit is:

• (1) ₹2,500
• (2) ₹3,000
• (3) ₹1,500
• (4) ₹1,000

A loan of ₹200,000 at the interest rate of 6% p.a. compounded monthly is to be amortized by equal payments at the end of each month for 5 years. The monthly payment is: \[ Given (1.005)^{60} = 0.74137220 \]

• (1) ₹1,866.57
• (2) ₹4,886.57
• (3) ₹3,866.57
• (4) ₹2,866.57

In reference to sampling, match List I with List II: \[ List I \quad List II \] \[ A. Measure of a characteristic of a sample \quad I. Parameter \] \[ B. An assumption made about a population \quad II. Standard Error \] \[ C. Standard deviation of the sample \quad III. Statistic \] \[ D. Measure of characteristic of a population \quad IV. Null Hypothesis \]

Choose the correct answer from the options given below:

• (1) A-I, B-III, C-I, D-IV
• (2) A-III, B-IV, C-II, D-I
• (3) A-II, B-IV, C-III, D-I
• (4) A-II, B-III, C-IV, D-I

Mr. A took a loan of ₹300000 at 10% annual interest rate and paid ₹5000 as monthly instalment under flat rate system. What is the term of the loan?

• (1) 12 years
• (2) 8 years
• (3) 10 years
• (4) 20 years

The percent income of a year on 6% debentures of face value of \₹100 available in the market for \₹200 is:

• (1) 3%
• (2) 4%
• (3) 5%
• (4) 6%

The effective rate equivalent to a nominal rate of 8% per annum compounded semi-annually is:

• (1) 8.20%
• (2) 8.24%
• (3) 8%
• (4) 8.16%

Ravi takes 40 seconds whereas Vishnu takes 60 seconds to complete a 240 metre race. By how many metres does Ravi defeat Vishnu?

• (1) 80 metres
• (2) 60 metres
• (3) 70 metres
• (4) 50 metres

The price of five different commodities for year 2018 and 2019 are as follows: 

The price index number for the year 2019 with 2018 as the base year, using simple average of price relatives is:

• (1) 230
• (2) 203
• (3) 233
• (4) 320

A boat can row upstream at 10 km/h and downstream at 18 km/h. If \( m \) is the speed of the boat in still water and \( n \) is the speed of the stream, then \( m+n \) is:

• (1) 16 km/h
• (2) 18 km/h
• (3) 20 km/h
• (4) 14 km/h

Pipes A and B can fill a tank in 40 hours and 50 hours respectively and pipe C can empty the full tank in 60 hours. If all the pipes are opened together, how much time (in hours) will be needed to fill the tank?

• (1) \(33 \dfrac{5}{17}\)
• (2) \(35 \dfrac{5}{17}\)
• (3) \(31 \dfrac{5}{17}\)
• (4) \(30 \dfrac{5}{17}\)

For the formula \( t = \frac{\mu_1 - \mu_2}{\sqrt{\frac{S_1^2}{n_1} + \frac{S_2^2}{n_2}}} \), consider the following statements:

A. \( \mu_1 \) and \( \mu_2 \) are sample mean and population mean respectively.

B. \( n_1 \) and \( n_2 \) are sample sizes of two samples from same population.

C. \( S_1 \) and \( S_2 \) are sample means of two samples from same population.

D. \( S_1 \) and \( S_2 \) are standard error of two samples from same population.

E. \( n_1 \) is the sample size and \( n_2 \) is the population size.


Choose the correct answer from the options given below:

• (1) B and D only
• (2) A, B and D only
• (3) B, C and D only
• (4) A and E only

10 litres of concentrated acid containing 75% of acid and rest water is mixed with \( X \) litres of diluted acid containing 40% of acid and rest water. If the final mixture contains 60% of acid then the value of \( X \) is:

• (1) 7.5 litres
• (2) 9 litres
• (3) 6 litres
• (4) 8 litres

Fisher’s price index number is:

• (1) A.M. of Laspeyres's and Paasche's index
• (2) G.M. of Laspeyres's and Paasche's index
• (3) H.M. of Laspeyres's and Paasche's index
• (4) Average of Laspeyres's and Paasche's index

A random variable \(X\) has the following probability distribution:


Then value of \( E(X) \) is:

• (1) 1.66
• (2) 2.66
• (3) 3.66
• (4) 4.66

With reference to time series, match List I with List II:


Choose the correct answer from the options given below:

• (1) A-III, B-I, C-IV, D-II
• (2) A-II, B-I, C-IV, D-III
• (3) A-III, B-II, C-IV, D-I
• (4) A-II, B-III, C-IV, D-I

The region represented by the system of inequalities \( x \geq 0, \ y \geq 0, \ y \leq 8, \ x + y \leq 4 \) is:

• (1) unbounded in first quadrant
• (2) unbounded in first and second quadrant
• (3) bounded in first quadrant
• (4) bounded in first and second quadrants

The solution of \( 7x \equiv 3 \pmod{5} \) is:

• (1) \( x = 5 \pmod{5} \)
• (2) \( x \equiv 4 \pmod{5} \)
• (3) \( x \equiv 1 \pmod{5} \)
• (4) \( x \equiv 2 \pmod{5} \)

For the LPP Max \( Z = 3x + 4y \), subject to the constraints: \[ x + y \leq 40, \quad x + 2y \leq 60, \quad x \geq 0, \quad y \geq 0 \]
The solution is:

• (1) \( x = 20, y = 20, Max Z = 140 \)
• (2) \( x = 40, y = 0, Max Z = 120 \)
• (3) \( x = 0, y = 60, Max Z = 240 \)
• (4) \( x = 10, y = 30, Max Z = 130 \)

If the mean of the probability distribution is 5, then the value of \( k \) is:

• (1) 1
• (2) 2
• (3) 4
• (4) 3

If \( A = \begin{bmatrix} 4 & 5 & 2
3 & -1 & 7 \end{bmatrix} \), then the sum of the elements of the matrix \( AA^\top \) is:

• (1) 156
• (2) 164
• (3) 146
• (4) 136

If \( x^{\frac{3}{4}} + y^{\frac{3}{4}} = a^{\frac{3}{4}} \) (where \( a \) is a constant), then \( \frac{dy}{dx} \) is equal to:

• (1) \( \left( \dfrac{y}{x} \right)^{\frac{1}{4}} \)
• (2) \( \left( \dfrac{x}{y} \right)^{\frac{1}{4}} \)
• (3) \( - \left( \dfrac{y}{x} \right)^{\frac{1}{4}} \)
• (4) \( - \left( \dfrac{x}{y} \right)^{\frac{1}{4}} \)

Let \( A \) and \( B \) be symmetric matrices of same order, then which of the following statements is true?

• (1) \( (A + B) \) is a symmetric matrix
• (2) \( (AB) \) is a symmetric matrix
• (3) \( (A + B) \) is a skew-symmetric matrix
• (4) \( (AB) \) is a skew-symmetric matrix

For matrix \( A = \begin{bmatrix} 3 & 1
7 & 5 \end{bmatrix} \), the values of \( x \) and \( y \) such that \( A^2 + xI = yA \) are:

• (1) \( x = 6, y = 8 \)
• (2) \( x = 8, y = 6 \)
• (3) \( x = 8, y = 8 \)
• (4) \( x = 6, y = 6 \)

If \( 8 + 3x < |8 + 3x| \), \( x \in \mathbb{R} \), then \( x \) lies in:

• (1) \( (-\infty, \infty) \)
• (2) \( \left( -\infty, -\dfrac{3}{8} \right) \)
• (3) \( \left( -\infty, -\dfrac{8}{3} \right) \)
• (4) \( \left( -\infty, \dfrac{8}{3} \right) \)

For \( x + y = 8 \), the maximum value of \( xy \) is:

• (1) 4
• (2) 8
• (3) 32
• (4) 16

The maximum profit that a company can make if the profit function is given by:
\[ P(x) = 32 + 24x - 18x^2 \]

is:

• (1) 48
• (2) 40
• (3) 36
• (4) 42

A motor boat goes 48 km downstream and comes back to the starting point in 16 hours. If the speed of the stream is 4 km/hr, then the speed of the motor boat in still water is:

• (1) 6 km/hr
• (2) 8 km/hr
• (3) 10 km/hr
• (4) 5 km/hr

If \( x = \frac{1}{t^2} \) and \( y = \frac{1}{t^3} \), then \( \frac{d^2y}{dx^2} \) at \( t = 1 \) is:

• (1) \( \frac{3}{2} \)
• (2) \( \frac{1}{2} \)
• (3) \( \frac{2}{3} \)
• (4) \( \frac{3}{4} \)

A coin is tossed 5 times. The probability of getting at least one head is:

• (1) \( \dfrac{31}{32} \)
• (2) \( \dfrac{15}{16} \)
• (3) \( \dfrac{1}{32} \)
• (4) \( \dfrac{63}{64} \)

The following data are from a simple random sample: 1, 4, 7

The point estimate of the population standard deviation is:

• (1) 2
• (2) 5
• (3) 1.5
• (4) 3

A car costing ₹8,50,000 has scrap value of ₹1,25,000. If annual depreciation charge is ₹1,45,000, then useful life of the car is:

• (1) 12 years
• (2) 5 years
• (3) 7 years
• (4) 10 years