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QUESTION PAPER

2023 BOARD EXAM

CBSE CLASS 12-PCM MATHEMATICS Board Paper 2023 — Set 3

2023

MATHEMATICS

CLASS 12-PCM

CBSE EXAMINATION PAPER-2023 MATHEMATICS

CBSE EXAMINATION PAPER-2023

MATHEMATICS

(Solved)

Time allowed : 3 hours

Maximum Marks : 87

General Instructions :

Read the following instructions carefully and follow them :

This question paper contains 43 questions. All questions are compulsory.
This question paper is divided into 5 sections.
Section A – questions number 1 to 3 are case based questions
Section B – questions number 4 to 22 are multiple choice questions
Section C – questions number 23 to 29 are very short answer
Section D – questions number 30 to 37 are short answer
Section E – questions number 38 to 43 are long answer
There is no overall choice given in the question paper. However, an internal choice has been provided in few questions.
Use of calculator is NOT allowed.

Section A

Question 1.

Two schools ‘P’ and ‘Q’ decided to award prizes to their students for two games of Hockey ₹ x per student and Cricket ₹ y per student. School ‘P’ decided to award a total of ₹ 9,500 for the two games to 5 and 4 students respectively; while school ‘Q’ decided to award ₹ 7,370 for the two games to 4 and 3 students respectively.

Based on the above information, answer the following questions :

(1)

What is the prize amount for hockey?

[2 Marks]

(2) Represent the following information algebraically (in terms of x and y).

[1 Marks]

(3) What will be the total prize amount if there are 2 students each from two games?

[1 Marks]

(4)

Prize amount on which game is more and by how much?

[2 Marks]

Question 2.

Jagdish has a field which is in the shape of a right angled triangle AQC. He wants to leave a space in the form of a square PQRS inside the field for growing wheat and the remaining for growing vegetables (as shown in the figure). In the field, there is a pole marked as O.

Based on the above information, answer the following questions :

(1) Taking O as origin, coordinates of P are (–200, 0) and of Q are (200, 0). PQRS being a square, what are the coordinates of R and S?

[1 Marks]

(2)

What is the area of square PQRS?

[2 Marks]

(3) If S divides CA in the ratio K:1, what is the value of K, where point A is (200, 800)?

[1 Marks]

(4)

What is the length of diagonal PR in square PQRS?

[2 Marks]

Question 3.

Governing council of a local public development authority of Dehradun decided to build an adventurous playground on the top of a hill, which will have adequate space for parking. After survey, it was decided to build rectangular playground, with a semi-circular area allotted for parking at one end of the playground. The length and breadth of the rectangular playground are 14 units and 7 units, respectively. There are two quadrants of radius 2 units on one side for special seats.

Based on the above information, answer the following questions :

(1) What is the total perimeter of the parking area?

[1 Marks]

(2)

What is the total area of parking and the two quadrants?

[2 Marks]

(3) Find the cost of fencing the playground and parking area at the rate of ₹ 2 per unit.

[1 Marks]

(4)

What is the ratio of area of playground to area of parking area?

[2 Marks]

Section B

Question 4. The ratio of HCF to LCM of the least composite number and the least prime number is:

[1 Marks]

(A) 1:2
(B) 2:1
(C) 1:1
(D) 1:3

Question 5. The roots of the equation x² + 3x – 10 = 0 are:

[1 Marks]

(A) 2, –5
(B) –2, 5
(C) 2, 5
(D) –2, –5

Question 6.

The next term of the A.P. : √6, √24, √54 is:

[1 Marks]

(A) √60
(B) √96
(C) √72
(D) √216

Question 7.

The distance of the point (–1, 7) from x-axis is:

[1 Marks]

(A) 7
(B) –1
(C) 6
(D) √50

Question 8. What is the area of a semi-circle of diameter ‘d’?

[1 Marks]

(A) 1/4 π d²
(B) 1/16 π d²
(C) 1/8 π d²
(D) 1/2 π d²

Question 9. The empirical relation between the mode, median and mean of a distribution is:

[1 Marks]

(A) Mode = 3 Mean – 2 Median
(B) Mode = 3 Median – 2 Mean
(C) Mode = 2 Mean – 3 Median
(D) Mode = 2 Median – 3 Mean

Question 10.

The pair of linear equations 2x = 5y + 6 and 15y = 6x – 18 represents two lines which are:

[1 Marks]

(A) parallel
(B) intersecting
(C) coincident
(D) either intersecting or parallel

Question 11. If α, β are zeroes of the polynomial x² – 1, then value of (α + β) is:

[1 Marks]

(A) 1
(B) 2
(C) –1
(D) 0

Question 12.

If a pole 6 m high casts a shadow 2√3 m long on the ground, then sun’s elevation is:

[1 Marks]

(A) 60°
(B) 45°
(C) 90°
(D) 30°

Question 13.

sec θ when expressed in terms of cot θ, is equal to:

[1 Marks]

(A) √1-cot²θ/ cot θ
(B) √1 + cot² θ
(C) 1+ cot²θ / cot θ
(D) √1 + cot² θ / cot θ

Question 14.

In the given figure, △ABC ~ △QPR. If AC = 6 cm, BC = 5 cm, QR = 3 cm and PR = x; then the value of x is:

[1 Marks]

(A) 10 cm
(B) 2.5 cm
(C) 3.6 cm
(D) 3.2 cm

Question 15.

The distance of the point (–6, 8) from origin is:

[1 Marks]

(A) 6
(B) –6
(C) 8
(D) 10

Question 16.

In the given figure, PQ is a tangent to the circle with centre O. If ∠OPQ = x, ∠POQ = y, then x + y is:

[1 Marks]

(A) 60°
(B) 90°
(C) 45°
(D) 180°

Question 17.

In the given figure, TA is a tangent to the circle with centre O such that OT =4 cm, ∠OTA = 30°, then length of TA is :

[1 Marks]

(A) 2√2 cm
(B) 2 cm
(C) 2√3 cm
(D) √3 cm

Question 18.

In △ABC, PQ || BC. If PB = 6 cm, AP = 4 cm, AQ = 8 cm, find the length of AC.

[1 Marks]

(A) 6 cm
(B) 14 cm
(C) 20 cm
(D) 12 cm

Question 19.

If α, β are the zeroes of the polynomial p(x) = 4x² – 3x – 7, then (1/α + 1/β) is equal to:

[1 Marks]

(A) -7/3
(B) 7/3
(C) -3/7
(D) 3/7

Question 20. A card is drawn at random from a well-shuffled pack of 52 cards. The probability that the card drawn is not an ace is:

[1 Marks]

(A) 1/13
(B) 9/13
(C) 12/13
(D) 4/13

Question 21.

Assertion (A) : The probability that a leap year has 53 Sundays is 2/7.

Reason (R) : The probability that a non-leap year has 53 Sundays is 5/7.

[1 Marks]

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is not the correct explanation of Assertion (A).
(B) Assertion (A) is false but Reason (R) is true.
(C) Assertion (A) is true but Reason (R) is false.
(D) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

Question 22.

Assertion (A) : a, b, c are in AP. if and only if 2b=a +c.

Reason (R) : The sum of first n odd natural numbers is n².

[1 Marks]

(A) Assertion (A) is false but Reason (R) is true.
(B) Both Assertion (A) and Reason (R) are true and Reason (R) is not the correct explanation of Assertion (A).
(C) Assertion (A) is true but Reason (R) is false.
(D) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

Section C

Question 23. Two numbers are in the ratio 2 : 3 and their LCM is 180. What is the HCF of these numbers?

[2 Marks]

Question 24. If one zero of the polynomial p(x) = 6x² + 37x – (k – 2) is reciprocal of the other, then find the value of k.

[2 Marks]

Question 25. Find the sum and product of the roots of the quadratic equation 2x² – 9x + 4 = 0.

[2 Marks]

Question 26. Find the discriminant of the quadratic equation 4x² – 5 = 0 and hence comment on the nature of roots of the equation.

[2 Marks]

Question 27. If a fair coin is tossed twice, find the probability of getting ‘atmost one head’.

[2 Marks]

Question 28.

Evaluate 5 cos²60°+ 4 sec² 30°- tan²45° / sin² 30° + cos²30°.

[2 Marks]

Question 29. If A and B are acute angles such that sin(A – B) = 0 and 2 cos(A + B) – 1 = 0, then find angles A and B.

[2 Marks]

Section D

Question 30. How many terms are there in an A.P. whose first and fifth terms are –14 and 2, respectively, and the last term is 62?

[3 Marks]

Question 31. Which term of the A.P. : 65, 61, 57, 53, .......... is the first negative term?

[3 Marks]

Question 32.

Prove that √5 is an irrational number.

[3 Marks]

Question 33. Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the points of contact at the centre.

[3 Marks]

Question 34.

Prove that: sin A - 2 sin³ A / 2 cos³ A - cos A = tan A

[3 Marks]

Question 35. Prove that sec A (1 – sin A)(sec A + tan A) = 1.

[3 Marks]

Question 36. Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.

[3 Marks]

Question 37. Find the value of ‘p’ for which the quadratic equation px(x – 2) + 6 = 0 has two equal real roots.

[3 Marks]

Section E

Question 38. A straight highway leads to the foot of a tower. A man standing on the top of the 75 m high tower observes two cars at angles of depression of 30° and 60°, which are approaching the foot of the tower. If one car is exactly behind the other on the same side of the tower, find the distance between the two cars. (Use √3 = 1.73)

[5 Marks]

Question 39. From the top of a 7 m high building, the angle of elevation of the top of a cable tower is 60° and the angle of depression of its foot is 30°. Determine the height of the tower.

[5 Marks]

Question 40.

D is a point on the side BC of a triangle ABC such that ∠ADC = ∠BAC. Prove that CA² = CB.CD

[5 Marks]

Question 41.

If AD and PM are medians of triangles ABC and PQR respectively where △ABC ~ △PQR, prove that AB/PQ = AD/PM.

[5 Marks]

Question 42. A student was asked to make a model shaped like a cylinder with two cones attached to its ends by using a thin aluminium sheet. The diameter of the model is 3 cm and its total length is 12 cm. If each cone has a height of 2 cm, find the volume of air contained in the model.

[5 Marks]

Question 43.

The monthly expenditure on milk in 200 families of a Housing Society is given below:

Find the value of x and also, find the median and mean expenditure on milk.

[5 Marks]

Paper Details

CBSE Board Exam 2023

Class

CLASS 12-PCM

Subject

MATHEMATICS

Year

2023

Set

Set 3

Other Years — MATHEMATICS

2021 Set-4

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2022 Set-1

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2022 Set-2

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Solutions

2022 Set-3

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2023 Set-1

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Solutions

2023 Set-3

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2023 Set-4

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Solutions

2024 Set-1

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2024 Set-2

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Solutions

2024 Set-3

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Solutions

2024 Set-4

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Solutions

2025 Set-1

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Solutions

2025 Set-2

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Solutions

2025 Set-3

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Solutions

2025 Set-4

Paper

Solutions