CUET 2023 Mathematics July 2 Shift 3 Question Paper (Available):Download Slot-wise Answer Key with Solutions PDF

Question 1:

If \[ A = \begin{bmatrix} 1 & 2 & 3
0 & 1 & 4
0 & 0 & 1 \end{bmatrix} \quad and \quad B = \begin{bmatrix} 1 & -2 & 1
0 & 1 & 0
0 & 0 & 1 \end{bmatrix}, \]
where \( I_3 \) is the unit matrix of order \(3 \times 3\), and \( AB = I_3 \), then \( x + y \) is equal to:

• (1) 0
• (2) -1
• (3) 2
• (4) -2

If \[ A = \begin{bmatrix} x & 1
0 & 1 \end{bmatrix} \quad and \quad A = A^{-1}, \]
then the value of \( x \) is:

• (1) 0
• (2) 1
• (3) 2
• (4) 3

If \( x, y, z \) are different and \[ A = \begin{bmatrix} x & x^2 & 1 + x^2
x^2 & y & 1 + y^2
z^2 & 1 + z^2 & z \end{bmatrix} \]
then the value of \( xyz \) is:

• (1) 1
• (2) 0
• (3) -1
• (4) 2

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

• (1) \( \frac{1}{3} \)
• (2) 3
• (3) \( \frac{7}{10} \)
• (4) \( \frac{10}{7} \)

If \( y = 10^{10x} \), then \( \frac{dy}{dx} \) is:

• (1) \( 10^{10x} \cdot \log(10) \)
• (2) \( 10^{10x} \cdot (\log(10))^2 \)
• (3) \( 10^{10x} \cdot \log(10) \)
• (4) \( 10^{10x} \cdot 10x \cdot \log(10) \)

The tangent to the parabola \( x^2 = 2y \) at the point \( \left( \frac{1}{2}, 1 \right) \) makes an angle with the x-axis of:

• (1) \( 0^\circ \)
• (2) \( 45^\circ \)
• (3) \( 30^\circ \)
• (4) \( 60^\circ \)

The function \( f(x) = x^3 \), where \( x \in \mathbb{R} \), has:

• (1) Maximum value at \( x = 0 \)
• (2) Minimum value at \( x = 0 \)
• (3) Neither maximum nor minimum value at \( x = 0 \)
• (4) Maximum value and minimum value at \( x = 0 \)

If \[ f(x) = \begin{cases} 2x + 8 & for 1 \leq x \leq 2,
6x & for 2 < x \leq 4, \end{cases} \]
then the value of \( \int_1^4 f(x) \, dx \) is:

• (1) 43
• (2) 45
• (3) 47
• (4) 46

The area of the region bounded by the line \( 2y = 5x + 7 \), the x-axis, and the lines \( x = 1 \) and \( x = 3 \) is:

• (1) 15
• (2) 17
• (3) 16
• (4) 19

The integrating factor of the differential equation \( (1 + y^2) \, dx - (\tan^{-1} y) \, dy = 0 \) is:

• (1) \( \tan^{-1} y \)
• (2) \( e^{\tan^{-1} y} \)
• (3) \( \frac{1}{1 + y^2} \)
• (4) \( \frac{1}{x |1 + y^2|} \)

The order and the degree of the differential equation \[ \frac{d^2y}{dx^2} = \left( 1 + \left( \frac{dy}{dx} \right)^2 \right) \]
respectively are:

• (1) Order = 2, Degree = 1
• (2) Order = 2, Degree = 2
• (3) Order = 1, Degree = 2
• (4) Order = 1, Degree = 1

Which of the following is correct?

• (1) Every LPP admits an optimal solution.
• (2) Every LPP admits a unique solution.
• (3) The optimal value does not occur at a corner point of the feasible region only.
• (4) If an LPP admits an optimal solution at two points, then it has an optimal solution at an infinite number of points.

The corner points of the feasible region determined by the system of linear constraints are \( (0, 3) \), \( (1, 1) \), and \( (3, 0) \). Let \( Z = px + qy \), where \( p, q > 0 \). The conditions on \( p \) and \( q \) so that the minimum of \( Z \) occurs at \( (3, 0) \) and \( (1, 1) \) are:

• (1) \( p = 2q \)
• (2) \( p = \frac{q}{2} \)
• (3) \( p = 3q \)
• (4) \( p = q \)

The mean of the numbers obtained on throwing a die having written 1 on three faces, 2 on two faces, and 5 on one face is:

• (1) 1
• (2) 2
• (3) 3.5
• (4) \( \frac{8}{3} \)

If \( P(A) = \frac{3}{10} \), \( P(B) = \frac{2}{5} \), and \( P(A \cup B) = \frac{3}{5} \), then \( P\left( \frac{B}{A} \right) + P\left( \frac{A}{B} \right) \) is equal to:

• (1) \( \frac{1}{4} \)
• (2) \( \frac{1}{3} \)
• (3) \( \frac{5}{12} \)
• (4) \( \frac{7}{12} \)

The relation \( R \) in the set \( A = \{1, 2, 3, 4\} \) is given by \[ R = \{(1, 2), (2, 1), (1, 1), (4, 4), (1, 3), (3, 2)\} \]
which is:

• (1) Reflexive and symmetric but not transitive
• (2) Reflexive and transitive but not symmetric
• (3) Symmetric and transitive but not reflexive
• (4) An equivalence relation

If \( f(x) = 2x^3 \) and \( g(x) = x^3 \), then \( g \circ f(x) \) is:

• (1) \( x \)
• (2) \( 2x \)
• (3) \( 3x \)
• (4) \( 0 \)

Principal value of \( \tan^{-1} \left( \sqrt{3} \right) + \tan^{-1}(1) \) is:

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

The principal value of \( \sin^{-1} \left( -\dfrac{1}{2} \right) \) is:

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

If \( A \) is an invertible matrix, such that \[ A^2 - A + I = 0, \]
then the inverse of \( A \) is:

• (1) \( A^{-2} \)
• (2) \( I - A \), where \( I \) is the identity matrix of order 2
• (3) \( 0 \)
• (4) \( A \)

The determinant \[ \left| \begin{array}{ccc} x & \sin \theta & \cos \theta
\sin \theta & -x & 1
\cos \theta & 1 & x \end{array} \right| \]
is:

• (1) independent of \( \theta \) only
• (2) independent of \( x \) only
• (3) independent of both \( \theta \) and \( x \)
• (4) independent of \( x \) but not of \( \theta \)

The value of \( k \) for which the matrix \[ \begin{bmatrix} 0 & 2 & k
2 & 0 & 5
-3 & 5 & 0 \end{bmatrix} \]
is a symmetric matrix is given by:

• (1) 3
• (2) \(-3\)
• (3) 0
• (4) 1

The value of \( z \) for which the matrix \[ \begin{bmatrix} 1 & 0 & z
0 & 1 & 0
1 & 0 & 1 \end{bmatrix} \]
is a singular matrix is:

• (1) 0
• (2) \(-1\)
• (3) 1
• (4) 2

If the order of a matrix \( A \) is \( 2 \times 3 \), the order of matrix \( B \) is \( 3 \times 4 \), and the order of matrix \( C \) is \( 3 \times 4 \), then the order of the matrix \( (AB)C^\top \) is:

• (1) \( 2 \times 3 \)
• (2) \( 3 \times 3 \)
• (3) \( 3 \times 4 \)
• (4) \( 4 \times 3 \)

The value of the determinant \[ \begin{vmatrix} x + y & y + z & z + x
x & x & y
1 & 1 & 1 \end{vmatrix} \]
is:

• (1) 1
• (2) 2
• (3) 0
• (4) -1

Match List I with List II:

List I (Functions)                                    List II (Derivatives)
A.   \(f(x) = \sin^{-1}\frac{1}{x}\)    1.   \(\displaystyle \frac{-1}{x\sqrt{x^2 - 1}}, \; x \in \mathbb{R}\)
B.   \(f(x) = \tan^{-1}\frac{1}{x}\)    2.   \(\displaystyle \frac{-1}{1 + x^2}, \; x \in (-\infty, -1) \cup (1, \infty)\)
C.   \(f(x) = \cos^{-1}\frac{1}{x}\)   3.   \(\displaystyle \frac{1}{x\sqrt{x^2 - 1}}, \; x \in (-\infty, -1) \cup (1, \infty)\)
D.   \(f(x) = \cot^{-1}\frac{1}{x}\)    4.   \(\displaystyle \frac{1}{1 + x^2}, \; x \in \mathbb{R}\)
 

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

If \( y = A \sin x + B \cos x \), where \( A \) and \( B \) are constants, then \( \frac{d^2y}{dx^2} \) is equal to:

• (1) \( y \)
• (2) \( -y \)
• (3) \( x \)
• (4) \( -x \)

If \[ f(x) = \begin{cases} \frac{1 - \cos 4x}{x^2}, & x \neq 0
k, & x = 0 \end{cases} \]
is continuous at \( x = 0 \), then the value of \( k \) is:

• (1) 8
• (2) 7
• (3) 6
• (4) 4

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

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

If the function \[ f(x) = \frac{k \sin x + 2 \cos x}{\sin x + \cos x} \]
is increasing for all values of \( x \), then:

• (1) \( k < 1 \)
• (2) \( 1 \leq k \)
• (3) \( k > 2 \)
• (4) \( k < 2 \)

For fencing of a flower bed with 100 cm long wire in the form of a circular sector, the maximum area of the flower bed is:

• (1) 1000 cm²
• (2) 225 cm²
• (3) 625 cm²
• (4) 500 cm²

Match List I with List II:

      List I (Integrals)                                                                                                                 List II (Values)

A.   \(\displaystyle \int_{0}^{\frac{\pi}{2}} \frac{\sin^2 x}{\sin^2 x + \cos^2 x} \, dx\)       1.  \(\displaystyle \frac{1}{2}\)
B.   \(\displaystyle \int_{0}^{\frac{\pi}{2}} \frac{\sin x}{1 + \cos^2 x} \, dx\)                       2.  \(0\)
C.   \(\displaystyle \int_{0}^{\frac{\pi}{2}} x \cos x \, dx\)                                                     3.  \(\displaystyle \frac{\pi}{4}\)
D.   \(\displaystyle \int_{0}^{\frac{\pi}{2}} \sin^2 x \, dx\)                                                     4.  \(\displaystyle \frac{\pi^2}{4}\)

Choose the correct answer from the options given below:

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

The area of the region bounded by \( |x| + |y| = 1 \), \( x \geq 0 \), \( y \geq 0 \) is:

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

The area of the region bounded by the curve \( y = \sqrt{3x + 10} \), the x-axis, and between the lines \( x = -3 \) and \( x = 2 \) is:

• (1) \( \frac{110}{9} \)
• (2) \( \frac{252}{9} \)
• (3) \( \frac{114}{9} \)
• (4) \( \frac{126}{9} \)

The value of the integral \[ \int \frac{1 - \sin x}{\cos^2 x} \, dx \]
is:

• (1) \( \sec x - \tan x + C \), where \( C \) is an arbitrary constant.
• (2) \( \tan x - \sec x + C \), where \( C \) is an arbitrary constant.
• (3) \( \sec x \tan x + C \), where \( C \) is an arbitrary constant.
• (4) \( \tan x \sec x + C \), where \( C \) is an arbitrary constant.

Match List I with List II:

List I (Differential Equation)                                                                                                    List II (Order and Degree)

A.  \(\left(\frac{d^2 y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^2 + 1 + x^2 = 0\)               1.  Order 2, Degree 1
B.  \(\displaystyle \frac{dy}{dx} = \frac{1}{y^2 \left(1 + x^2\right)^{1/2}}\)                           2.  Order 1, Degree not defined
C.  \(\displaystyle \frac{d^2 y}{dx^2} = \cos 3x \cdot \sin 3x\)                                              3.  Order 2, Degree 4
D.   \(\displaystyle \frac{dy}{dx} + 2\frac{dy}{dx} + y \cdot \log\left(\frac{dy}{dx}\right)\)  4. & Order 1, Degree 2
 

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

The general solution of the differential equation \( x \, dy - y \, dx = 0 \) represents:

• (1) A rectangular hyperbola
• (2) A parabola whose vertex is at the origin
• (3) A straight line passing through the origin
• (4) A circle whose center is at the origin

The vectors \( 3\hat{i} - \hat{j} + 2k\hat{k} \) and \( \hat{i} + 3\hat{j} + k\hat{k} \) are coplanar if \( k \) is:

• (1) -2
• (2) 0
• (3) 2
• (4) any real number

ABCD is a rhombus, whose diagonals intersect at E. Then \( \overrightarrow{EA} + \overrightarrow{EB} + \overrightarrow{EC} + \overrightarrow{ED} \) equals to:

• (1) \( \mathbf{0} \)
• (2) \( \overrightarrow{AD} \)
• (3) \( 2 \overrightarrow{BC} \)
• (4) \( 2 \overrightarrow{AD} \)

The angle at which the normal to the plane \( 4x + 8y + z = 7 \) is inclined to the \( y \)-axis is:

• (1) \( \cos^{-1} \left( \frac{4}{9} \right) \)
• (2) \( \cos^{-1} \left( \frac{-1}{9} \right) \)
• (3) \( \cos^{-1} \left( \frac{8}{9} \right) \)
• (4) \( \cos^{-1} \left( \frac{5}{9} \right) \)

If each side of a cube is \( x \), then the angle between the diagonals of the cube is:

• (1) \( \cos^{-1} \left( \frac{1}{\sqrt{3}} \right) \)
• (2) \( \cos^{-1} \left( \frac{1}{3} \right) \)
• (3) \( \cos^{-1} \left( \frac{1}{\sqrt{5}} \right) \)
• (4) \( \cos^{-1} \left( \frac{-1}{3} \right) \)

Which of the following statements is true?

- A. If the feasible region for a LPP is unbounded, maximum or minimum of the objective function \( Z = ax + by \) may or may not exist.
- B. Maximum value of the objective function \( Z = ax + by \) in a LPP always occurs at only one corner point of the feasible region.
- C. In a LPP, the minimum value of the objective function \( Z = ax + by \) (where \( a, b > 0 \)) is always 0 if origin is one of the corner points of the feasible region.
- D. In a LPP, the max value of the objective function \( Z = ax + by \) is always finite.

Choose the correct answer from the options given below:

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

The corner points of the feasible region determined by the system of linear inequalities are \( (0, 0) \), \( (4, 0) \), \( (2, 4) \), and \( (0, 5) \). If the maximum value of \( Z = ax + by \), where \( a, b > 0 \), occurs at both \( (2, 4) \) and \( (4, 0) \), then:

• (1) \( a = 2b \)
• (2) \( 2a = b \)
• (3) \( a = b \)
• (4) \( 3a = b \)

In a box consisting of 100 bulbs, 10 are defective. The probability that out of a sample of 5 bulbs, none is defective is:

• (1) \( 10^{-1} \)
• (2) \( \left( \frac{1}{2} \right)^5 \)
• (3) \( \left( \frac{9}{10} \right)^5 \)
• (4) \( \frac{9}{10} \)

Which of the following are not the probability distributions of a random variable?

A      X       0      1      2
     P(X)    0.4   0.4   0.2
 B.    X        0      1     2         3
      P(X)   0.4   0.4   0.2   -0.05
C.      Y      -1      0      1
      P(Y)    0.6   0.2   0.1
D.      Z        3      1       0      -1
      P(Z)    0.3    0.2   0.4   0.05
E.      X        0       1      2
      P(X)   25/36 10/36 1/36
 

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

Set \( A \) has 4 elements and set \( B \) has 6 elements, then the number of injective mappings that can be defined from \( A \) to \( B \) is:

• (1) 360
• (2) 1
• (3) 24
• (4) 1296

The maximum value of the function \( y = 2 - |x - 3| \) is:

• (1) 0
• (2) 3
• (3) 2
• (4) 5

If \( y = e^{(x - 1)} \), then the value of \( \frac{dy}{dx} \) at \( (1, 1) \) is:

• (1) \( \frac{1}{2} \)
• (2) 0
• (3) \( \frac{1}{8} \)
• (4) 1

If \( \mathbf{a} \) and \( \mathbf{b} \) are two non-zero vectors such that \( |\mathbf{a}| = 10 \), \( |\mathbf{b}| = 2 \), and \( \mathbf{a} \cdot \mathbf{b} = 12 \), then the value of \( |\mathbf{a} \times \mathbf{b}| \) is:

• (1) 5
• (2) 10
• (3) 14
• (4) 16

The direction cosines of a line which makes equal angles with the co-ordinate axes are:

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

The value of \( 2^{49} \mod 15 \) is:

• (1) 1
• (2) 0
• (3) -1
• (4) 7

A retailer has 900 kg of wheat, a part of which he sells at 10% loss and the remaining at a profit of 8%. Overall, he makes a profit of 6%. The quantity sold at profit is:

• (1) 100 kg
• (2) 450 kg
• (3) 600 kg
• (4) 800 kg

A tank can be filled by two pipes, A and B, in 18 minutes and 24 minutes respectively. Another tap, C, can empty the full tank in 36 minutes. If tap C is opened 6 minutes after pipes A and B are opened, the tank will become full in a total of:

• (1) 6 minutes
• (2) 12 minutes
• (3) 18 minutes
• (4) 36 minutes

A tank can be filled by two pipes A and B in 18 minutes and 24 minutes respectively. Another tap C can empty the full tank in 36 minutes. If tap C is opened 6 minutes after pipes A and B are opened, the tank will become full in a total of:

• (1) 4 hours 50 minutes
• (2) 5 hours
• (3) 5 hours 40 minutes
• (4) 6 hours 50 minutes

₹3,60,000, ₹4,20,000, and ₹4,80,000 were invested by three friends A, B, and C respectively in a business. If they earned a net profit of ₹2,10,000, the share of B's profit is:

• (1) ₹60,000
• (2) ₹80,000
• (3) ₹75,000
• (4) ₹70,000

The solution set of inequalities: \[ x + 3 \leq 0 \quad and \quad 2x + 5 \leq 0, if \, x \in \mathbb{R}, \]
is:

• (1) \( (-\infty, -3] \)
• (2) \( (-\infty, -4, -3] \)
• (3) \( (-\infty, -3] \)
• (4) \( \{-3, -2, -1, 0, \dots \} \)

If \( A \) is a symmetric matrix and \( n \in \mathbb{N} \), then \( A^n \) is:

• (1) Symmetric matrix
• (2) Skew-symmetric matrix
• (3) Diagonal matrix
• (4) Zero matrix

If the transpose of matrix \( A \) is matrix \( B \), where \[ A = \begin{pmatrix} 1 & 2 & a
5 & 6 & 0 \end{pmatrix} \quad and \quad B = \begin{pmatrix} 1 & 2 & 6
3 & 2 & 9
0 & 4 & 0 \end{pmatrix} \]
then the value of \( 3a + 2b + 4c \) is:

• (1) \( \frac{23}{4} \)
• (2) 9
• (3) 15
• (4) 17

A matrix \( P \) of order \( 2 \times 3 \) with each entry 0 or 1 and \( \alpha \) is a scalar which is 3 or 4. If \( R = \alpha P \), then the number of matrices \( R \) formed is:

• (1) 63
• (2) 64
• (3) 128
• (4) 127

If \( f(x) \) is a function that is derivable in an interval containing a point \( c \), then match List I with List II.
LIST I                                                                                                                                     LIST II
A. f''(x) has second order derivative at x = c such that f'(c) = 0 and f''(c) \neq 0 then   point of inflection of f(x)
B. Necessary condition for point x = c to be extreme point of f(x)                                 c is point of local minima of f(x)
C. f'(x) does not change its sign as x crosses the point x = c then it is called             c is a critical point of f(x)
D. f''(x) has second order derivative at x = c such that f'(c) = 0 and f''(c) > 0                c is point of local maxima of f(x)
 

If \( y = \sqrt{\log x + \sqrt{\log x + \sqrt{\log x + \dots}}} \), then \( \frac{dy}{dx} \) is:

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

The price per unit of a commodity produced by a company is given by \( P = 92 - 2x^2 \), where \( x \) is the quantity demanded. The marginal revenue of producing 3 units of such a commodity shall be:

• (1) 28
• (2) 38
• (3) 26
• (4) 44

For the function \( f(x) = 2^x + 10 \), which of the following is the most appropriate option?

• (1) The minimum value of \( f \) is 10
• (2) \( f \) has no maximum possible value
• (3) \( f \) has no minimum possible value
• (4) \( f \) has neither maximum nor minimum possible value

If \( x \sqrt{y} + y \sqrt{x} = 0 \), where \( x \neq y \), then the value of \( \frac{d^2y}{dx^2} + 2 \frac{dy}{dx} \) at \( x = 1 \) is:

• (1) 0
• (2) \( \frac{1}{4} \)
• (3) \( -\frac{1}{4} \)
• (4) \( \frac{1}{8} \)

For a discrete random variable \( X \), whose probability distribution is defined as:
\[ P(x) = \begin{cases} \frac{2x + 1}{4}, & x = 0,
\frac{3x}{36}, & x = 2,
\frac{5(5 - x)}{7}, & x = 3. \end{cases} \]

The value of the mean will be:

• (1) \( \frac{6}{7} \)
• (2) \( \frac{15}{7} \)
• (3) \( \frac{12}{7} \)
• (4) \( \frac{11}{7} \)

Consider the following statements with respect to probability distributions:


A. When mean (\( \mu \)) = 1 and standard deviation (\( \sigma \)) = 0 for a data set, normal distribution is called standard normal distribution.
B. In a normal distribution of data, \( z \) is given by \( z = \frac{x - \mu}{\sigma} \).
C. \( P(r success) \) is the \( (r - 1)^{th} \) term in the binomial expansion of \( (q + p)^n \).
D. In a random experiment, a collection of trials is called Bernoulli, if trials are dependent by nature.
E. When a random variable whose value is obtained by measuring and it takes many values between two values, it is called a continuous random variable.

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

A die is thrown \( n \) times. A random variable \( X \) denotes the number of times the number on the dice is greater than 4 and \( P(X = 1) = 2P(X = 2) \). The value of \( n \) is:

• (1) 2
• (2) 3
• (3) 4
• (4) 5

Two balls are chosen randomly from an urn containing 8 white and 4 black balls by a player. Suppose that he wins ₹30 for each black ball selected and loses ₹15 for each white ball selected. The expected value of winning amount is:

• (1) ₹12.72
• (2) ₹14.72
• (3) ₹15
• (4) ₹60

Consider the following statements:


A. Cost of living at two different cities can be compared with volume index.
B. When the prices of rice are to be compared we use price index.
C. In Laspeyres’s price index number weight is considered as price in current year.
D. Purchasing power of money can be assessed through consumer price index.
E. Fisher Index number is called ideal index number.

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

Match List I with List II
 LIST I                                                           LIST II
A. Marshall Edgeworth's Index Number  \sum P_0 q_0 \times 100 / \sum P_0 q_0
B.  Laspeyres's Index Number                  \sum P_0 q_0 / \sum P_1 q_0 \times 100
C.  Fisher's Ideal Index Number                \sum P_1 q_1 / P_0 q_0 \times 100
D.  Paasche's Index Number                      \sum P_1 q_1 / \sum P_0 (q_0 + q_1) \times 100
 

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

Consider the table below on the quantities of commodities alongside their prices in the year 2020 and 2022.
 

Commodity   Price in 2020  Price in 2022  Quantity in 2020   Quantity in 2022
       A                        1                      2                            5                            6
       B                        3                      4                            3                            4
       C                        4                      5                            3                            5
       D                        2                      5                            1                            3
       E                        3                      4                            4                            6
The value of \( \sum P_i Q_i \) is:

• (1) 40
• (2) 54
• (3) 73
• (4) 97

The following data is taken from a simple random sample: \[ 3, 7, 5, 9, 14, 11, 8, 4, 6, 2 \]

The point estimate of the population standard deviation is:

• (1) \( \frac{2}{3} \)
• (2) \( \frac{4}{9} \)
• (3) \( \frac{2}{3} \sqrt{35} \)
• (4) \( \frac{4}{9} \sqrt{35} \)

For a certain data test statistic \( t \) is calculated as: \[ \left| \frac{65 - 68}{\sqrt{15}} \right| = 2.00 \]
Then select the correct option:

• (1) \( \bar{x} = 68, \mu = 65, n = 16 \)
• (2) \( \bar{x} = 65, \mu = 68, n = 16 \)
• (3) \( \bar{x} = 15, n = 4, \mu = 3 \)
• (4) \( \bar{x} = 65, \mu = 68, n = 14 \)

Suppose that a 95% confidence interval states that the population mean is greater than 100 and less than 300. Then the value of sample mean (\(\bar{x}\)) and margin of error (E) respectively are:

• (1) \( \bar{x} = 150, E = \pm 100 \)
• (2) \( \bar{x} = 100, E = \pm 100 \)
• (3) \( \bar{x} = 250, E = \pm 50 \)
• (4) \( \bar{x} = 200, E = \pm 100 \)

Shyam takes a loan of ₹5,00,000 with 5% annual interest rate for 10 years. The value of EMI under the flat rate system is:

• (1) ₹4166.67
• (2) ₹7500
• (3) ₹8333.32
• (4) ₹50000

A machine costing ₹1 lakh depreciates at a constant rate of 10%. The estimated useful life of the machine is 8 years.
Match List I with List II:

       LIST I                                                               LIST II
A.  Total depreciation in 2nd and 3rd year is      81,000
B.  Value of machine after one year is                 17,200
C.  Value of machine after 2 years is                    43,050
D.  Scrap value of machine is                                90,000
Given  (1.3)^{3} = 2.144 and (0.9)^{3} = 0.4305 

Choose the correct answer from the options given below:

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

A bond of face value ₹1000 matures in 10 years and interest is paid annually at 4% per annum. If the present value of the bond is ₹838, find the yield to maturity, given that \( (1.04)^{10} \approx 0.676 \).

• (1) 1.6% p.a.
• (2) 2.0% p.a.
• (3) 2.6% p.a.
• (4) 3.2% p.a.

Consider the following feasible region. Which of the following constraints represents the feasible region?

A. \( 2x + 3y \leq 6 \)

B. \( x - 2y \leq 2 \)

C. \( x + 3y \leq 1 \)

D. \( x - 2y \geq -3 \)

E. \( x - 2y = -1 \)

1. A. C and E only
2. B. D and E only
3. B and C only
4. A. B and D only

The graph of the inequality \( 3x - 2y > 6 \) is:

1. Half plane that contains origin
2. Half plane that neither contains origin nor the points on the line 3x − 2y = 6
3. Whole XOY-plane excluding points on 3x − 2y = 6
4. Entire XOY-plane

An electric company has 300 Transistors, 400 Capacitors, and 500 Inductors. The company wishes to make electronic goods using two circuits A and B. The requirement by the circuit is as follows:

Component   Circuit A   Circuit B
Transistor         175            125
Capacitor          300            100
Inductor            200             300

The profit from circuit A and B is ₹2000 and ₹3000 respectively, then constraints of the Linear Programming Problem (LPP) based on this data are:

1. 7x + 5y ≤ 12; 3x + y ≤ 4; 2x + 3y ≤ 5; x, y ≥ 0
2. 7x + 5y ≤ 12; 3x + y ≤ 4; 2x + 3y ≤ 4; x, y ≥ 0
3. 7x + 5y ≤ 12; 3x + y ≤ 4; 2x + 3y ≤ 5; x, y ≥ 0
4. 7x + 5y ≤ 12; 3x + y = 4; 2x + 3y ≤ 5; x, y ≥ 0

In a 1000 m race, A beats B by 50 meters or 10 seconds. The time taken by A to complete the race is:

1. 150 seconds
2. 200 seconds
3. 190 seconds
4. 250 seconds

If \( y = 4t^2 \) and \( y = \frac{3}{t^3} \), then \( \frac{d^2y}{dt^2} \) at \( t = 1 \) is:

1. 15/8
2. 2/3
3. 15/16
4. 45/64

In a binomial distribution, the probability of getting a success is \( \frac{1}{3} \) and the standard deviation is 4. Then its mean is:

1. 8
2. 24
3. 16
4. 32

For the given five values 17, 26, 20, 35, 44, the three years moving averages are:

1. 19, 25, 31
2. 18, 20, 32
3. 15, 17, 22
4. 21, 27, 33

A vehicle whose cost is ₹7,00,000 will depreciate to a scrap value of ₹1,50,000 in 5 years. Using linear method of depreciation, the book value of the vehicle at the end of third year is:
1. 1,10,000
2. 3,70,000
3. 2,70,000
4. 2,50,000