CBSE Class 12 Maths Sample Paper 2023 with Solutions

Class Class 12 Maths Sample Paper 2023

Maths Sample Paper Class 12: Central Board of Secondary Education has released the official Maths Sample Paper Class 12 exam along with the other subject sample papers on www.cbseacademic.nic.in. All students are advised to buckle up their preparation and be punctual for the board exam. The Class 12 Maths Sample Paper would be of great help in practicing as per the revised exam pattern. In this article, we have provided direct links for downloading Maths Sample Paper Class 12 along with its solution PDF.

CBSE Class 12 Maths Sample Paper PDF

CBSE has released the Class 12 Maths Sample Paper as per the Class 12 Maths Syllabus along with the solution/marking scheme in PDF format. The students are advised to go through the Maths Sample Paper Class 12 at least once before the examination so as to get familiar with the new pattern to be followed. Download the CBSE Class 12 Maths Sample Paper PDF from the below link and practice. 

CBSE Class 12 Maths Sample Paper Download PDF

CBSE Class 12 Maths Sample Paper Solution

Class 12  Maths Sample Paper and Solution 2023

Maths is one of the important subjects in boards examination. It comprises a total of 100 Marks. CBSE has released the official sample paper, you can go through this sample question to understand the level of questions the examiner can ask in the board exam. Go through these questions so as to score well in exams.

Section A

Q1. If A =[aij] is a skew-symmetric matrix of order n, then 

Solution: (c) In a skew-symmetric matrix, the (i, j)th element is negative of the (j, i)th element. Hence, the (i, i)th element = 0 

Q2. If A is a square matrix of order 3, |𝐴′| = −3, then |𝐴𝐴′| = 

(a) 9 (b) -9 (c) 3 (d) -3 

Solution: (a) |𝐴𝐴′| = |𝐴||𝐴′| = (−3)(−3) = 9

Q7. The solution set of the inequality 3x + 5y < 4 is

(a) an open half-plane not containing the origin.

(b) an open half-plane containing the origin.

(c) the whole XY-plane not containing the line 3x + 5y = 4.

(d) a closed half-plane containing the origin

Solution: (b) The strict inequality represents an open half plane and it contains the origin as (0, 0) satisfies it.

Q8. The scalar projection of the vector 3𝚤̂− 𝚥̂− 2𝑘 on the vector 𝚤̂+ 2𝚥̂− 3𝑘 is  

(a) 7/√14 (b) 7/14 (c) 6/13 (d) 7/2

Solution: (a) 7/√14

Q11. The corner points of the shaded unbounded feasible region of an LPP are (0, 4), (0.6, 1.6) and (3, 0) as shown in the figure. The minimum value of the objective function Z = 4x + 6y occurs at 

(a)(0.6, 1.6) 𝑜𝑛𝑙𝑦 

(b) (3, 0) only 

(c) (0.6, 1.6) and (3, 0) only 

(d) at every point of the line segment joining the points (0.6, 1.6) and (3, 0)

Solution:  (d) The minimum value of the objective function occurs at two adjacent corner points (0.6, 1.6) and (3, 0) and there is no point in the half plane 4𝑥 + 6𝑦 < 12 in common with the feasible region. So, the minimum value occurs at every point of the line segment joining the two points. 

Q13. If A is a square matrix of order 3 and |A| = 5, then |𝑎𝑑𝑗𝐴| =

(a) 5 (b) 25 (c) 125 (d)1/5

Solution: (b) = 25

Q14. Given two independent events A and B such that P(A) =0.3, P(B) = 0.6, and P(𝐴′ ∩ 𝐵′) is 

(a) 0.9 (b) 0.18 (c) 0.28 (d) 0.1 

Solution: (c) P(𝐴′ ∩ 𝐵 ′) = 𝑃(𝐴′) × 𝑃(𝐵′) (As A and B are independent, 𝐴 ′𝑎𝑛𝑑 𝐵′ are also independent.) = 0.7 × 0.4 = 0.28 

Q15. The general solution of the differential equation 𝑦𝑑𝑥 − 𝑥𝑑𝑦 = 0 𝑖𝑠 (a) 𝑥𝑦 = 𝐶 (b) 𝑥 = 𝐶𝑦² (c) 𝑦 = 𝐶𝑥 (d) 𝑦 = 𝐶𝑥² 

Solution: 𝑦𝑑𝑥 − 𝑥𝑑𝑦 = 0 

⟹ 𝑦𝑑𝑥 − 𝑥𝑑𝑦 = 0

⟹ 𝑑𝑦/𝑦 = 𝑑𝑥/𝑥

Q18. P is a point on the line joining the points 𝐴(0,5, −2) and 𝐵(3, −1,2). If the x-coordinate of P is 6, then its z-coordinate is

 (a) 10 (b) 6 (c) -6 (d) -10 

Solution: (b) The line through the points (0, 5, -2) and (3, -1, 2) is 𝑥/0-3 = 𝑦-5/-1-5 = 𝑧+2/2-2   𝑜𝑟, 𝑥 /3 = 𝑦 − 5/ −6 = 𝑧 + 2/4.

Any point on the line is (3𝑘, −6𝑘 + 5,4𝑘 − 2), where k is an arbitrary scalar. 3𝑘 = 6 ⟹ 𝑘 = 2 

The z-coordinate of the point P will be 4 × 2 − 2 = 6

Section B

Q22. A man 1.6 m tall walks at the rate of 0.3 m/sec away from a street light that is 4 m above the ground. At what rate is the tip of his shadow moving? At what rate is his shadow lengthening? 

Solution: Let AB represent the height of the street light from the ground. At any time t seconds, let the man represented as ED of height 1.6 m be at a distance of x m from AB and the length of his shadow EC be y m.

Using the similarity of triangles, we have 4/1.6= x+y/y= 3y=2x

Section C 

Q27. Three friends go for coffee. They decide who will pay the bill, by each tossing a coin and then letting the “odd person” pay. There is no odd person if all three tosses produce the same result. If there is no odd person in the first round, they make the second round of tosses and they continue to do so until there is an odd person. What is the probability that exactly three rounds of tosses are made?

OR

Find the mean number of defective items in a sample of two items drawn one-by-one without replacement from an urn containing 6 items, which include 2 defective items. Assume that the items are identical in shape and size

Solution: P(not obtaining an odd person in a single round) = P(All three of them throw tails or All three of them throw heads)

= ½ × ½ × ½ ×2 = ¼ 

 P(obtaining an odd person in a single round)

= 1 − P(not obtaining an odd person in a single round)= ¾ 

The required probability

= P(‘In the first round, there is no odd person’ and ‘In the second round

there is no odd person’ and ‘In the third round, there is an odd person) 

= ¼ × ¼ × ¾ = 3/64 

OR

Let X denote the Random Variable defined by the number of defective items.

P(X=0) = 4/6 × 3/5 = 2/5

P(X=1) = 2 × (2/6 × 4/5 ) = 8/15

P(X=2) = 2/6×1/5 = 1/15

Section D

Q32. Make a rough sketch of the region {(𝑥, 𝑦): 0 ≤ 𝑦 ≤ 𝑥², 0 ≤ 𝑦 ≤ 𝑥, 0 ≤ 𝑥 ≤ 2} and find the area of the region using integration

Q33. Define the relation R in the set 𝑁 × 𝑁 as follows: For (a, b), (c, d) ∈ 𝑁 × 𝑁, (a, b) R (c, d) if ad = bc. Prove that R is an equivalence relation in 𝑁 × 𝑁. Given a non-empty set X, define the relation R in P(X) as follows: For A, B ∈ 𝑃(𝑋), (𝐴, 𝐵) ∈ symmetric.

SectionE

Q36. Case Study 1: Read the following passage and answer the questions given below.

The temperature of a person during an intestinal illness is given by 𝑓(𝑥) = −0.1𝑥² + 𝑚𝑥 + 98 the temperature in °F at x days.

(i) Is the function differentiable in the interval (0, 12)? Justify your answer.

(ii) If 6 is the critical point of the function, then find the value of the constant m.

(iii) Find the intervals in which the function is strictly increasing/strictly decreasing.

Q37. Case Study 2: Read the following passage and answer the questions given below. 

For an elliptical sports field, the authority wants to design a rectangular soccer field with the maximum possible area. The sports field is given by the graph of 𝑥²/a²+ y²/b² = 1

(i) If the length and the breadth of the rectangular field be 2x and 2y respectively, then find the area function in terms of x.

(ii) Find the critical point of the function.

(iii) Use the First derivative Test to find the length 2x and width 2y of the soccer field (in terms of a and b) that maximize its area.

OR

(iii) Use Second Derivative Test to find the length 2x and width 2y of the soccer field (in terms of a and b) that maximize its area.

Q38. Case Study 3: Read the following passage and answer the questions given below. There are two antiaircraft guns, named A and B. The probability that the shell fired from them hits an airplane is 0.3 and 0.2 respectively. Both of them fired one shell at an airplane at the same time.

(i) What is the probability that the shell fired from exactly one of them hit the plane?

(ii) If it is known that the shell fired from exactly one of them hit the plane, then what is

the probability that it was fired from B? 

Steps to download CBSE Class 12 Maths Sample Paper 2022-23

Download the CBSE Sample Paper Class 12 by following the below-mentioned steps or directly download CBSE Class 12 Maths Sample papers from the above link.

Step I- Visit the official website of CBSE Academic @ www.cbseacademic.nic.in. 

Step II- Click on the notification appearing in the academic section- “Sample Question Papers of Classes XII for Exams 2022-23”.

Step III- Click on the link under “Sample Papers Class XII”.

Step IV- Search for Maths in the list of “Class XII Sample Question Paper & Marking Scheme for Exam 2022-23” that appears on the screen. 

Step V- Click on “SQP” for all the subjects and download the sample paper pdf. 

Step VI- Check the marking scheme after attempting Class 12 Maths sample paper 2022-23.