This article will cover the CBSE 12th Maths Answer Key 2023, including Maths sets 1, 2, 3, and 4, as well as the PDF version of the question paper and the steps to download it.

## CBSE 12th Maths Answer Key 2023

We have provided a sample question paper below in this article which can benefit most of the students. Students can practice for their examinations by solving these questions. The Central Board of Secondary Education has officially released these sample papers.

**Section A**

**Q1.****Β **If A is a square matrix of order 3, |π΄β²| = β3, then |π΄π΄β²| =

(a) 9 (b) -9 (c) 3 (d) -3

**Q2. **If A =[aij] is a skew-symmetric matrix of order n, then

**Q3. **The scalar projection of the vector 3π€Μβ π₯Μβ 2π on the vector π€Μ+ 2π₯Μβ 3π is

(a) 7/**β**14 (b) 7/14 (c) 6/13 (d) 7/2

**Q4. **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 does not contain the line 3x + 5y = 4.

(d) a closed half-plane containing the origin

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**Q5.**Β Given two independent events A and B such that P(A) =0.3, P(B) = 0.6, and P(π΄β² β© π΅β²) is

(a) 0.9 (b) 0.18 (c) 0.28 (d) 0.1

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

**Q6. **The general solution of the differential equation π¦ππ₯ β π₯ππ¦ = 0 ππ (a) π₯π¦ = πΆ (b) π₯ = πΆπ¦Β² (c) π¦ = πΆπ₯ (d) π¦ = πΆπ₯Β²

**Q7. **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) ππππ¦

(b) (3, 0) only

(c) (0.6, 1.6) and (3, 0) only

**Q8.**Β If A is a square matrix of order 3 and |A| = 5, then |ππππ΄| =

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

**Q9. **P is a point on the line joining the points π΄(0,5, β2) and π΅(3, β1,2). If the x-coordinate of P is 6, then its z-coordinate is

(a) 10 (b) 6 (c) -6 (d) -10

**Section B**

**Q10**. 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?

**Section CΒ **

**Q11. **Three friends go for coffee. They decide who will pay the bill by each tossing a coin and then letting the βodd personβ 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 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

**Section D**

**Q12**. Make a rough sketch of the region {(π₯, π¦): 0 β€ π¦ β€ π₯Β², 0 β€ π¦ β€ π₯, 0 β€ π₯ β€ 2} and find the area of the region using integration

**Q13.**Β Define the relation R in the set π Γ π as follows: For (a, b), (c, d) β π Γ π, (a, b) R (c, d) if ad = bc. Prove that R is an equivalence relation in π Γ π. Given a non-empty set X, define the relation R in P(X) as follows: For A, B β π(π), (π΄, π΅) β symmetric.

**SectionE**

**Q14.**Β 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 aeroplane is 0.3 and 0.2, respectively. Both of them fired one shell at an aeroplane at the same time.

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

(ii) If it is known that the shell fired from precisely one of them hit the plane, then what is the probability that it was fired from B?

**Q15. **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 π(π₯) = β0.1π₯Β² + ππ₯ + 98, the temperature in Β°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.

**Q16. **Case Study 2: Read the following passage and answer the questions 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 π₯Β²/aΒ²+ yΒ²/bΒ² = 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.

### Steps to Download CBSE Sample Paper

We have mentioned a few steps below, following which you can download CBSE Class 12th Maths sample paper.

- Visit the official site we have mentioned below in this article.
- Search for the notification βSample question papers of class XII exams 2022-23β.
- Search for the Maths sample question paper.
- Press on the download button to download the sample question paper PDF.
- After downloading it, save it in your device storage.