CBSE Class 12 2024 Mathematics Set 1 Question Paper (Q.P. Code: 65/3/1) is available for download. The exam was successfully conducted by CBSE on March 9 in the morning session from 10:30 AM to 1:30 PM. As per the student’s initial reactions, the CBSE Class 12 2024 Mathematics Set 1 Question Paper was reported as Moderate. The Calculus section was reported as Challenging, the Algebra section as Moderate, and the Probability & Statistics section as Easy to Moderate.

CBSE Class 12 2024 Mathematics Set 1 65/3/1 Question Paper with Answer Key PDF

Candidates can download the CBSE Class 12 Mathematics Question Paper with Solution and Answer Key PDFs for Set 1 Question Paper (Code: 65/3/1) using the link below.

CBSE Class 12 2024 Mathematics Questions with Solutions

SECTION A

Question 1:

If A = [a_ij] is an identity matrix, then which of the following is true?

(A) a_ij = { 0, if i = j,
                1, if i ≠ j
(B) a_ij = 1, ∀i, j
(C) a_ij = 0, ∀i, j
(D) a_ij = { 0, if i ≠ j,
                1, if i = j

Question 2:

Let R₊ denote the set of all non-negative real numbers. Then the function f : R₊ → R₊ defined as f(x) = x² + 1 is:

(A) one-one but not onto
(B) onto but not one-one
(C) both one-one and onto
(D) neither one-one nor onto

Question 3:

Let A = [[a, b], [c, d]] be a square matrix such that adj A = A. Then, (a + b + c + d) is equal to:

(A) 2a
(B) 2b
(C) 2c
(D) 0

Question 4:

A function f(x) = |1 - x + |x|| is:

(A) discontinuous at x = 1 only
(B) discontinuous at x = 0 only
(C) discontinuous at x = 0, 1
(D) continuous everywhere

Question 5:

If the sides of a square are decreasing at the rate of 1.5 cm/s, the rate of decrease of its perimeter is:

(A) 1.5 cm/s
(B) 6 cm/s
(C) 3 cm/s
(D) 2.25 cm/s

Question 6:

∫_-a^a f(x) dx = 0, if:

(A) f(-x) = f(x)
(B) f(-x) = -f(x)
(C) f(a - x) = f(x)
(D) f(a - x) = -f(x)

Question 7:

x log x dy/dx + y = 2 log x is an example of a:

(A) variable separable differential equation
(B) homogeneous differential equation
(C) first-order linear differential equation
(D) differential equation whose degree is not defined

Question 8:

If ⃗a = 2⃗i - ⃗j + ⃗k and ⃗b = ⃗i - 2⃗j + ⃗k, then ⃗a and ⃗b are:

(A) collinear vectors which are not parallel
(B) parallel vectors
(C) perpendicular vectors
(D) unit vectors

Question 9:

If α, β, and γ are the angles which a line makes with the positive directions of x, y, z axes respectively, then which of the following is not true?

(A) cos² α + cos² β + cos² γ = 1
(B) sin² α + sin² β + sin² γ = 1
(C) cos 2α + cos 2β + cos 2γ = -1
(D) cos α + cos β + cos γ = 1

Question 10:

The restrictions imposed on decision variables involved in an objective function of a linear programming problem are called:

(A) feasible solutions
(B) constraints
(C) optimal solutions
(D) infeasible solutions

Question 11:

Let E and F be two events such that P(E) = 0.1, P(F) = 0.3, P(E∪F) = 0.4. Then P(F | E) is:

(A) 0.6
(B) 0.4
(C) 0.5
(D) 0

Question 12:

If A and B are two skew-symmetric matrices, then AB + BA is:

(A) A skew-symmetric matrix
(B) A symmetric matrix
(C) A null matrix
(D) An identity matrix

Question 13:

If [[1, 3, 1], [k, 0, 1], [1, 0, 1]] has a determinant of ±6, then the value of k is:

(A) 1
(B) -2
(C) 2
(D) ±2

Question 14:

The derivative of 2^x w.r.t. 3^x is:

(A) (2^x log 2) / (3^x log 3)
(B) (3^x log 2) / (2^x log 3)
(C) (2^x log 2) / (3^x log 3)
(D) (x log 3) / (x log 2)

Question 15:

If |a| = 2 and -3 ≤ k ≤ 2, then |a||k| ∈:

(A) [-6, 4]
(B) [0, 6]
(C) [4, 6]
(D) [0, 6]

Question 16:

If a line makes an angle of π/4 with the positive directions of both x-axis and z-axis, then the angle which it makes with the positive direction of y-axis is:

(A) 0
(B) π/4
(C) π/2
(D) π

Question 17:

Of the following, which group of constraints represents the feasible region given below?

(A) x + 2y ≤ 76, 2x + y ≥ 104, x, y ≥ 0
(B) x + 2y ≤ 76, 2x + y ≤ 104, x, y ≥ 0
(C) x + 2y ≥ 76, 2x + y ≤ 104, x, y ≥ 0
(D) x + 2y ≥ 76, 2x + y ≥ 104, x, y ≥ 0

Question 18:

If A = [[2, 0, 0], [0, 3, 0], [0, 0, 5]], then A^-1 is:

(A) [[1/2, 0, 0], [0, 1/3, 0], [0, 0, 1/5]]
(B) [[30, 0, 0], [0, 10, 0], [0, 0, 6]]
(C) 1/30 [[2, 0, 0], [0, 3, 0], [0, 0, 5]]
(D) 1/30 [[1, 0, 0], [0, 1, 0], [0, 0, 1]]

Question 19:

Assertion (A): Every scalar matrix is a diagonal matrix.
Reason (R): In a diagonal matrix, all the diagonal elements are 0.

Question 20:

Assertion (A): Projection of ⃗a on ⃗b is the same as the projection of ⃗b on ⃗a.
Reason (R): The angle between ⃗a and ⃗b is the same as the angle between ⃗b and ⃗a numerically.

SECTION B

Question 21:

Evaluate: sec²(tan^-1 1/2) + csc²(cot^-1 1/3)

Question 22(a):

If x = e^y2, prove that dy/dx = (log x - 1) / (2y (log x)²).

Question 22(b):

Check the differentiability of f(x) at x = 1, where: f(x) = { x² + 1, 0 ≤ x < 1,
3 - x, 1 ≤ x ≤ 2.

Question 23(a):

Evaluate: ∫₀^π/2 sin 2x cos 3x dx

Question 23(b):

Given d/dx F(x) = 1/√(2x - x²) and F(1) = 0, find F(x).

Question 24:

Find the position vector of point C which divides the line segment joining points A and B having position vectors ⃗i + 2⃗j - ⃗k and -⃗i + ⃗j + ⃗k, respectively, in the ratio 4:1 externally. Further, find |⃗AB| : |⃗BC|.

Question 25:

Let ⃗a and ⃗b be two non-zero vectors. Prove that |⃗a x ⃗b| ≤ |⃗a||⃗b|. State the condition under which equality holds, i.e., |⃗a x ⃗b| = |⃗a||⃗b|.

SECTION C

Question 26(a):

If x cos(p + y) + cos p sin(p + y) = 0, prove that cos p dy/dx = -cos²(p + y), where p is a constant.

Question 26(b):

Find the value of a and b so that the function f defined as:

f(x) =
         (x - 2)/|x - 2| + a,   x < 2,
         a + b,   x = 2,
         (x - 2)/|x - 2| + b,   x > 2.

Question 27(a):

Find the intervals in which the function f(x) = (log x)/x is strictly increasing or strictly decreasing.

Question 27(b):

Find the absolute maximum and minimum values of the function f(x) = (1/2)x + 2/x on the interval [1,2].

Question 28:

Find:

∫ (x² + 1)/[(x² + 2)(x² + 4)] dx

Question 29(a):

Find:

∫ (2 + sin 2x)/(1 + cos 2x) e^x dx

Question 29(b):

Evaluate:

∫₀^π/4 (1/(sin x + cos x)) dx

Question 30:

Solve the following linear programming problem graphically:

Maximise z = 4x + 3y, subject to the constraints:
x + y ≤ 800, 2x + y ≤ 1000, x ≤ 400, x, y ≥ 0.

Question 31:

The chances of P, Q, and R getting selected as CEO of a company are in the ratio 4:1:2. Given the probabilities of profit increase under the new CEO, find the probability that the increase in profits is due to R's appointment.

SECTION D

Question 32:

A relation R on set A = {-4, -3, -2, -1, 0, 1, 2, 3, 4} is defined as R = {(x, y) : x + y is an integer divisible by 2}. Show that R is an equivalence relation. Also, write the equivalence class [2].

Question 33(a):

It is given that the function \( f(x) = x^4 - 62x^2 + ax + 9 \) attains a local maximum value at \( x = 1 \). Find the value of \( a \), hence obtain all other points where the given function \( f(x) \) attains local maximum or local minimum values.

Question 33(b):

The perimeter of a rectangular metallic sheet is 300 cm. It is rolled along one of its sides to form a cylinder. Find the dimensions of the rectangular sheet so that the volume of the cylinder formed is maximum.

Question 34:

Using integration, find the area of the region enclosed between the circle x² + y² = 16 and the lines x = -2 and x = 2.

Question 35(a):

Find the equation of the line passing through the point of intersection of the lines:

(x - 1)/1 = (y - 2)/2 = (z - 2)/3,    (x - 1)/0 = (y - 3)/-3 = (z - 7)/2

Question 35(b):

Find the distance between the skew lines AB and CD.

SECTION E

Question 36:

Self-study helps students to build confidence in learning. It boosts the self-esteem of the learners. Recent surveys suggested that close to 50 learners were self-taught using internet resources and upskilled themselves.

A student may spend 1 hour to 6 hours in a day in upskilling self. The probability distribution of the number of hours spent by a student is given below:

P(X = x) =
    kx², for x = 1, 2, 3,
    2kx, for x = 4, 5, 6,
    0, otherwise.

Where x denotes the number of hours. Based on the above information, answer the following questions:

1. Express the probability distribution given above in the form of a probability distribution table.
2. Find the value of k.
3. (a) Find the mean number of hours spent by the student.
4. (b) Find P(1 < X < 6).

Question 37:

A bacteria sample is observed to grow exponentially in a given amount of time. Using the exponential growth model, the rate of growth of this sample of bacteria is calculated.

The differential equation representing the growth of bacteria is given as:

dP/dt = kP,

where P is the population of bacteria at any time t.

Based on the above information, answer the following questions:

1. Obtain the general solution of the given differential equation and express it as an exponential function of t.
2. If the population of bacteria is 1000 at t = 0, and 2000 at t = 1, find the value of k.

Question 38:

A scholarship is a sum of money provided to a student to help them pay for education. Some students are granted scholarships based on their academic achievements, while others are rewarded based on their financial needs.

Every year, a school offers scholarships to girl children and meritorious achievers based on certain criteria. In the session 2022-23, the school offered a monthly scholarship of 3,000 each to some girl students and 4,000 each to meritorious achievers in academics as well as sports.

In all, 50 students were given scholarships, and the monthly expenditure incurred by the school on scholarships was 1,80,000.

Based on the above information, answer the following questions:

1. Express the given information algebraically using matrices.
2. Check whether the system of matrix equations obtained is consistent or not.
3. (a) Find the number of scholarships of each kind given by the school using matrices.
4. (b) Had the amount of scholarship given to each girl child and meritorious student been interchanged, what would be the monthly expenditure incurred by the school?