CBSE Class 12 Sample Paper 2025 for Applied Maths

CENTRAL BOARD OF SECONDARY EDUCATION

SAMPLE PAPER
CLASS 12
2025

SAMPLE QUESTION PAPER (2024 - 25)
CLASS- XII
SUBJECT: Applied Mathematics (241)

Time: 3 Hours. Maximum Marks: 80

General Instructions:

Read the following instructions very carefully and strictly follow them:
(i) This Question paper contains 38 questions. All questions are compulsory.
(ii) This Question paper is divided into five Sections - A, B, C, D and E.
(iii) In Section A, Questions no. 1 to 18 are multiple choice questions (MCQs) and Questions
no. 19 and 20 are Assertion-Reason based questions of 1 mark each.
(iv) In Section B, Questions no. 21 to 25 are Very Short Answer (VSA)-type questions, carrying
2 marks each.
(v) In Section C, Questions no. 26 to 31 are Short Answer (SA)-type questions, carrying 3 marks
each.
(vi) In Section D, Questions no. 32 to 35 are Long Answer (LA)-type questions, carrying 5 marks
each.
(vii) In Section E, Questions no. 36 to 38 are case study-based questions carrying 4 marks
each.
(viii) There is no overall choice. However, an internal choice has been provided in 2 questions in
Section B, 2 questions in Section C, 2 questions in Section D and one sub-part each in 2
questions of Section E.
(ix) Use of calculators is not allowed.

SECTION-A 1  20 = 20
(This section comprises of multiple-choice questions (MCQs) of 1 mark each)

Select the correct option (Question 1 - Question 18):

Q.1. The area (in sq units) bounded by the curve 𝑦 = √𝑥, the x − axis, 𝑥 = 1 and 𝑥 = 4 is

𝟏𝟏 𝟏 𝟏𝟒 𝟏𝟑
(A) 𝟑 (B) 𝟒 (C) 𝟑 (D) 𝟑

Q.2. Sampling which provides for a known non-zero equal chance of selection is

(A) Systematic sampling (B) Convenience sampling

(C) Quota sampling (D) Purposive sampling

x3
Q.3. Let the cost function for a manufacturer is given by C ( x ) = − x 2 + 2 x (In rupees)
3

Which of the following statement is correct based on the above information?
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(A) The marginal cost decreases from 0 to 1 and then increases onwards.

(B) The marginal cost increases from 0 to 1 and then decreases onwards.

(C) Marginal cost decreases as production level increases from zero.

(D) Marginal cost increases as production level increases from zero.

1  9
Q.4. The absolute minimum value of the function f ( x ) = 4 x − x 2 in the interval  −2,  is:
2  2

(A) −8 (B) −9 (C) −10 (D) −16
Q.5. For the purpose of t − test of significance, a random sample of size ( n ) 2025 is drawn from a

normal population, then the degree of freedom (𝜐) is

(A) 2025 2025 (B) 2024 2025 (C) 2025 (D) 2024
Q.6. The constraints of a linear programming problem along with their graphs is shown below:
x + 2 y  3, x  10, y  0

Which of the following inequality may be removed so that the feasible region remains the
same in above graph?
(A) 𝑥 + 2𝑦 ≥ 3
(B) 𝑥 ≥ 10
(C) 𝑦 ≥ 0
(D) 𝑥 ≥ 0
Q.7. A player rolls one fair die. If the die shows an odd number, the player wins the value that
appears on the die, else loses half the value that appears on it. The expected gain of the player
is
𝟏 𝟏
(A) − 𝟐 (B) 𝟎 (C) 𝟐 (D) 𝟏

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Q.8. The original cost of a machine is ₹ 1200000 and the scarp value of the machine after a useful
life of 3 years is ₹ 300000 , then the book value of the machine at the end 2 years is
(A) ₹ 100000 (B) ₹ 250000 (C) ₹ 600000 (D) ₹ 800000
Q.9. A fish jumps out of the water surface and follows the parabolic path y = 6 x − x 2 − 8; 2  x  4 .

The fish reaches the highest height in its path at ( 3,1) . The slope of the path of the fish at

( 3,1) is

(A) 0 (B) 1 (C) 2 (D) 3

Q.10. In a large consignment of electric bulbs 5% of a batch of batteries are defective. A random

sample of 80 is taken for inspection with replacement. Then the Variance of the number of

defectives in the sample, is

18 19
(A) (B) (C) 4.555 (D) 8
5 5

Q.11. If it is currently 6: 00 pm in 12 hours clock then what will be the time after 375 hours?

(A) 6 am (B) 6 pm (C) 9 am (D) 9 pm
1
Q.12. The values of 𝑥 for the given values of 𝑥 ∈ (−1,3) − {0} is

1 1 1 1
(A) (−1, 3) ∪ (3, ∞) (B) (−∞, −1) ∪ (3 , ∞) (C) (− 3 , 1) (D) (− 3 , −1)

Q.13. The component of a time series attached to long term variations is termed as

(A) Seasonal variations (B) Irregular variations

(C) Secular trend variations (D) Cyclic variations

Q.14. The present value of a sequence of payments of ₹ 800 made at the end of every 6 month

and continuing forever. If money is worth 4% per annum compounded semi-annually, then the

present value of the sequence is:

(A) ₹ 20000 (B) ₹ 40000 (C) ₹ 60000 (D) ₹ 80000

Q.15. Shown below is a curve.

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L1 is the tangent to any point ( x , y ) on the curve.

L2 is the line that connects the point ( x , y ) to the origin.

The slope of L1 is one third of the slope of L2 .

Then the differential equation, using the given conditions is:
𝑑𝑦 𝑦 𝑑𝑦 𝑦 𝑑𝑦 𝑥 𝑑𝑦 3𝑦
(A) 𝑑𝑥 = 3𝑥 (B) 𝑑𝑥 = 𝑥 (C) 𝑑𝑥 = 3𝑦 (D) 𝑑𝑥 = 𝑥

Q.16. For a 3 × 3 matrix if adj A = 2A−1, find |3AAT |
(A) 108 (B) 12 (C) 54 (D) 8
 3 4
 −1 2 1 
Q.17. For two matrices P =  −1 2  & QT =   ; ( where QT is the transposeof thematrix Q )
 0 1   1 2 3

, P − Q is:
2 3 4 3 4 3 2 3 
(A)  −3 0 
 (B)  −3 0 
 (C)  0 −3 
 (D)  0 −3 
 0 −3   −1 −2   −1 −2   0 −3 
d 2 y  dy 
4
1
+ + 5
= 0; respectively, are
dx 2  dx 
Q.18. The order and degree of a differential equation x

(A) 2 and 4 (B) 2 and 1
(C) 2 and 3 (D) 3 and 3

ASSERTION-REASON BASED QUESTIONS
(Questions number 19 and 20 are Assertion and Reason based questions carrying 1 mark
each. Two statements are given, one labelled Assertion (A) and the other labelled Reason
(R). Select the correct answer from the codes (A), (B), (C) and (D) as given below.)
1  2 = 2 

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

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Q.19. Assertion (A): The effective rate of interest equivalent to a nominal rate of 6% when

compounded continuously is equal to e 0.06 − 1 = 6.18% .

Reason (R): The relation between effective rate reff ( ) of interest and nominal rate ( r ) of
interest: reff = e r − 1; where ' e' - Euler’s number (approximate value is 2.71828 ), when

compounded continuously.
Q.20. Assertion(A): 𝐴 = [𝑎𝑖𝑗 ] = 𝑚; 𝑖 = 𝑗
0; 𝑖 ≠ 𝑗
where 𝑚 is a scalar, is an identity matrix if 𝑚 = 1
Reason (R): Every identity matrix is not a scalar matrix

SECTION B  2  5 = 10
(This section comprises of 5 very short answer (VSA) type questions of 2 marks each.)

Q.21. (a) In what ratio water must be added in milk costing ₹ 60 per litre, so that the resulting
mixture would be of worth ₹ 50 per litre?
OR
7
Q.21. (b) A pump can fill a tank with water in 2 hours. Because of leakage, it took hrs to fill the
3

tank. How much time will it take for the leakage to drain all the water in the full tank?

Q.22. In a 200 𝑚 race, A can give a start of 18 𝑚 to B and a start of 31 𝑚 to C. In a race of 350
𝑚, how much start can B give to C?
Q.23. A boat takes thrice as long to go upstream to a point as to return downstream to the starting
point. If the speed of the stream is 5𝑘𝑚/ℎ, find the speed of the boat in still water.
Q.24. (a) The incidence of occupational disease in an industry is such that the workers have a 20%
chance of suffering from it. What is the probability that out of six workers 4 or more will catch
the disease?
OR

Q.24. (b) The lifetime of an item produced by a machine has a normal distribution with mean 12
months and standard deviation of 2 months. Find the probability of an item produced by this
machine will last
(i) less than 7 months
(ii) between 7 and 14 months.

 5
(Given P  Z   = 0.9938 and P ( Z  1) = 0.8413 )
 2 

 0  1 0
Q.25. If A =   and B =   , then find the value of  (if exists) for which A2 = B .
 1 1 5 1

Page 5 of 11

SECTION C
 3  6 = 18
(This section comprises of 6 short answer (SA) type questions of 3 marks each.)

Q.26. Find the remainder when 561 is divided by 7.

Q.27. (a) Two batches of the same product are tested for their mean life. Assuming that, the lives

of the product follow a normal distribution with an unknown variance; test the hypothesis that

the mean life is the same for both the branches, given the following information:

Standard Deviation
Batch Sample Size Mean life (in hours) (in hours)

Batch I 10 750 12

Batch II 8 820 14

Given 4.4444 = 2.1081 and t16 ( 0.05 ) = 2.120 
 
OR

Q.27. (b) The manufacturer of electrical items makes bulbs and claims that these bulbs have a

mean life of 25 months. The life in months of a random sample of 6 such bulbs are given to be

24, 26, 30, 20, 20 and 18. Test the validity of the manufacturer’s claim at 1% level of significance.

[Given 𝑡5 (0.01) = 4.032]

Q.28. A traffic engineer records the number of bicycle riders that use a particular cycle track. He

records that an average of 3.2 bicycle riders use the cycle track every hour. Given that the

number of bicycles that use the cycle track follow a Poisson distribution, what is the probability

that 2 or less bicycle riders will use the cycle track within an hour? Also find the mean

expectation and variance for the random variable. (Given 𝑒 −3.2 = 0.041)

Q.29. Mr Rohit invested ₹ 5000 in a fund at the beginning of year 2021 and by the end of year 2021

his investment was worth ₹ 9000. Next year market crashed and he lost ₹ 3000 and ending up

with ₹ 6000 at the end of year 2022. Next year i.e. 2023 he gained ₹ 4500 and ending up with

₹ 10500 at the end of the year. Find CAGR (Compounded Annual Growth Rate) of his

investment. (Use ( 2.1)
1/ 3
= 1.2805 )

Q.30.A small firm manufactures necklaces and bracelets. The total number of necklaces and

bracelets that it can handle per day is at most 25. It takes one hour to make a bracelet and half

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an hour to make a necklace. The maximum number of hours available per day is 14. If the profit

on a necklace is ₹ 100 and that on a bracelet is ₹ 300, formulate an L.P.P. for finding how many

of each should be produced daily to maximize the profit? It is being given that at least one of

each must be produced.

(Note: No need to find the feasible region and optimal solution)

Q.31.(a) An octagonal prism is a three-dimensional polyhedron bounded by two octagonal bases

and eight rectangular side faces. It has 24 edges and 16 vertices.

The prism is rolled along the rectangular faces and number on the bottom face (touching the

ground) is noted. Let X denotes the number obtained on the bottom face and the following table

gives the probability distribution of X .

X: 1 2 3 4 5 6 7 8

P ( X ): p 2p 2p p 2p p2 2 p2 7 p2 + p

On the above context, answer the following questions.

(i) Find the value of p .

(ii) Find the mean, E ( X ) .

OR
Q.31.(b) If the probability of success in a single trial is 0.01 , how many minimum number of Bernoulli
1
trials must be performed in order that the probability of at least one success is or more?
2
( Use log 2 = 0.3010 and log 99 = 1.9956 )
10 10

SECTION D [𝟓 × 𝟒 = 𝟐𝟎]
(This section comprises of 4 long answer (LA) type questions of 5 marks each)

Page 7 of 11

Q.32. (a) Fit a straight-line trend by using the method of least squares for the following data
and calculate the trend values.

Year Production (in tonnes)

1962 2

1963 4

1964 3

1965 4

1966 4

1967 2

1968 4

1969 9

1970 7

1971 10

1972 8

OR

Q.32. (b) The quarterly profits of a small-scale industry (₹ in thousands) are as follows.

Year Quarter 1 Quarter 2 Quarter 3 Quarter 4

2020 39 47 20 56

2021 68 59 66 72

2022 88 60 60 67

Calculate 4-quarterly moving averages.

Q.33. (a) An owl was sitting at ( 0, k ) ; k  0 . Then it starts flying along the path whose equation

is given by y = ax 2 + bx + c , where a  − 0 , b, c  . It passes through the points

( 1,2 ) , ( 2,1) and ( 4,5 ) . Using Cramer’s Rule, find the values of 𝑎, 𝑏, 𝑐 and hence 𝑘
OR
Q.33. (b) A toy rocket is fired, from a platform, vertically into the air, its height above the ground
after t seconds is given by s ( t ) = at 2 + bt + c , where a ,b,c  ; a  0 and s ( t ) is measured in

Page 8 of 11

metres. After 10 second, the rocket is 16 m above the ground; after 20 seconds, 22 m; after
30 seconds, 25 m.
(i) Write down a system of three linear equations in terms of a , b and c .
(ii) Hence find the values of a , b and c , using matrix method.
Q.34. Supply and demand curves of a tyre manufacturer company is given below:

The above graph showing the demand and supply curves of a tyre manufacturer company which
are linear. ‘ABC’ tyre manufacturer sold 25 units every month when the price of a tyre was ₹ 20000
per units and ‘ABC’ tyre manufacturer sold 125 units every month when the price dropped to ₹
15000 per unit. When the price was ₹ 25000 per unit, 180 tyres were available per month for sale
and when the price was only ₹ 15000 per unit, 80 tyres remained. Find the demand function. Also
find the consumer surplus if the supply function is given to be 𝑺(𝒙) = 𝟏𝟎𝟎 𝒙 + 𝟕𝟎𝟎𝟎
Q.35. In 4 years, a mobile costing ₹ 36,000 will have a salvage value of ₹ 7200.
The following graph shows the depreciation of a mobile’s value over 4 years.

A new mobile at that time (i.e., after 4 years) is expected to cost for ₹ 55,200. In order to
provide funds for the difference between the replacement cost and the salvage cost, a sinking

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fund is set up into which equal payments are placed at the end of each year. If the fund earns
interest at the rate 7% compounded annually, how much should each payment be? Also find
the amount of Annual Depreciation of the mobile’s value over 4 years and find the rate of
depreciation (under straight line method). Use (1.07)4 = 1.3107.
SECTION- E  4  3 = 12
(This section comprises of 3 case-study/passage-based questions of 4 marks each with sub
parts. The first two case study questions have three sub parts (i), (ii), (iii) of marks 1, 1, 2
respectively. The third case study question has two sub parts of 2 marks each)

Case Study-1
Q.36. A student Shivam is running on a playground along the curve given by y = x 2 + 7. Another

student Manita standing at point ( 3, 7 ) on playground wants to hit Shivam by paper ball when

Shivam is nearest to Manita.
Based on above information, answer the following questions:
(i) Let at any instant while running along the curve y = x 2 + 7 , Shivam’s position be ( x , y ) . Find

the expression for the distance ( D ) between Shivam and Manita in terms of ' x ' . [𝟏]

(ii) Find the critical point(s) of the distance function. [𝟏]
(iii) (a) What is the distance between Shivam and Manita when they are at least distance
from each other. [𝟐]
OR
(iii) (b) Find the position of Shivam, when he is closest to Manita. [𝟐]

Case Study-2
Q.37. EQUATED MONTHLY INSTALMENTS (EMI): -

Each instalment can be considered as consisting of two parts:

(i) Interest on the outstanding loan (ii) Repayment of part of the loan.

Methods of calculation of EMI or Instalment: -

EMI or Installment can be calculated by two methods:

1. Flat Rate Method

2. Reducing-balance method or Amortization of Loan

Page 10 of 11

Rajesh purchased a house from a company for ₹ 2500000 and made a down payment of ₹ 500000
He repays the balance in 25 years by monthly instalments at the rate of 9% per annum

(
compounded monthly. Given ( 1.0075 )
−300
= 0.1062 )
Based on the above information, answer the following questions:
(i) Find the number of payments and find the rate of interest per month. [𝟏]
(ii) (a) What are the monthly payments of instalments using reducing balance method?
[𝟐]
OR
(ii) (b) What are the monthly payments of instalments using flat rate method? [𝟐]
(iii) What is the total interest payment made in the process applied to calculate EMI in the

above part ( 37 ( ii ) ) ? [𝟏]

Case Study- 3

Q.38. A company has two factories located at P and Q and has three depots situated at A, B and
C. The weekly requirement of the depots at A, B and C is respectively 5, 5 and 4 units, while
the production capacity of the factories P and. Q are respectively 8 and 6 units. The cost (in
₹) of transportation per unit is given below.

Based on the above information, answer the following questions:

(i) Formulate the objective function and the constraints of the above Linear programming

problem. [𝟐]

(ii) How many units should be transported from each factory to each depot in order that

the transportation cost is minimum? [𝟐]

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