The Central Board of Secondary Education (CBSE) has made the Class 12 Sample Question Papers (SQPs) and Marking Schemes (MS) for the 2026-27 academic session available on its official academic website. Students appearing for the Class 12 board examinations in 2027 can use these papers to understand how questions may be presented in the examination.
Applied Mathematics is listed separately from Mathematics on the CBSE Class 12 sample paper page. Students should make sure they download the correct subject paper, as both subjects have separate question papers.
The sample paper is designed to help students understand the structure of the examination and practise applying mathematical concepts to different types of questions. Read this article to download the CBSE Class 12 Applied Mathematics Sample Paper 2026-27 PDF.
CBSE Class 12 Applied Mathematics Sample Paper 2026-27
General Instructions:
Read the following instructions very carefully and strictly follow them:
1. This Question paper contains 38 questions. All questions are compulsory.
2. This Question paper is divided into five Sections - A, B, C, D and E.
3. 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.
4. In Section B, Question number 21 to 25 are Very Short Answer (VSA)-type questions, carrying 2 marks each.
5. In Section C, Question number. 26 to 31 are Short Answer (SA)-type questions, carrying 3 marks each.
6. In Section D, Question number 32 to 35 are Long Answer (LA)-type questions, carrying 5 marks each.
7. In Section E, Question number 36 to 38 are case study-based questions carrying 4 marks each.
8. 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.
9. Draw neat and clean figures wherever required. Take Ο = 22 /7 wherever required, if not stated 10. Use of calculators is not allowed.
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SECTION β A (20 x 1 = 20)
This section comprises of 18 multiple choice questions and two assertion and reason type questions of 1 mark each.
6. Which of the following statements about the future value (FV) of a perpetuity is true? 1
(A) The future value can be calculated by compounding the present value to a specific future date.
(B) The future value is equal to twice the present value multiplied by number of payments.
(C) The future value is always equal to the total cash flows received up to the future date.
(D) The future value of perpetuity is undefined.
7. The order of the differential equation whose general solution is given by π¦ = (a + b)log(π₯ + c) β dπ π₯+k 1
8. The Marginal Revenue (MR) of a company is given by the function ππ
(π₯) = 20π 0.5π₯ where π₯ is the number of units sold. The additional revenue generated when sales increase from 2 units to 4 units is: 1
(A) 20 π 2
(B) 20(π 2 β π 1 )
(C) 40 π 2
(D) 40(π 2 β π 1 )
9. In the context of standard reducing-balance EMI, over the loan period, which of the following is correct? 1
(A) Principal and Interest components both remain constant.
(B) Principal component decreases, and Interest component increases.
(C) Principal component increases, and Interest component decreases.
(D) Principal component remains constant, and Interest component decreases.
10. If you are willing to pay βΉ 40,000 today for an investment that promises to pay βΉ 3,200 at the end of every year indefinitely, then the implied rate of return is. 1
(A) 6 %
(B) 8 %
(C) 10 %
(D) 12.5 %
11. A foreign research lab observes that the number of microscopic defects on a solarpanel surface follows a Poisson distribution with mean 2 defects per 100 ππ2 . If a scientist inspects an area of 250 ππΒ², what is the probability that exactly 1 defect is found? 1
(A) 6π β5
(B) 5π β5
(C) 4π β5
(D) 3π β2
12. Which of the following statement(s) is/are true: 1
I: The mean of a population is denoted by π₯Μ
.
II: The population mean is a statistic.
(A) I only
(B) II only
(C) Both I and II
(D) Neither I nor II
13. Two random variables X and Y have the same mean π = 50. If πππ(π) = 100 and πππ(π) = 25, which statement is TRUE? 1
(A) X is more consistent than Y
(B) Y is more consistent than X
(C) Both have equal consistency
(D) Cannot be determined with the given data
14. A trend line is considered the best fit when the sum of squares of residuals is. 1
(A) Maximum
(B) Minimum
(C) Positive
(D) Negative
15. A sample of 100 cereal boxes has mean = 505g and SD = 15g. The 99% confidence interval for the true mean ΞΌ is (501.125, 508.875) grams. Which of the following is the correct interpretation of this interval? 1
(A) 99% of repeated samples will have means in the interval (501.125,508.875) grams.
(B) The probability that the mean weight, π, is between 501.125 and 508.875 grams is 0.99.
(C) We are 99% confident that the calculated interval of (501.125, 508.875) grams contains the true mean weight π.
(D) Approximately 99% of all boxes have a weight between 501.125 and 508.875 grams.
16. Let X be a random variable following the binomial distribution π΅(6, p). If 2 P(X = 4) = 3 P(X = 3), then the value of p is: 1
(A) 1/3
(B) 1/2
(C) 2/3
(D) 3/4
17. Suppose a random variable π follows the binomial distribution with parameters π and π, where 0 < π < 1. If π(π=π) π(π=πβπ) is independent of π and π, then π equals (A) 1/2
(B) 1/3
(C) 1/5
(D) 1/7
18. A firm manufacturing two types of lubricants whose non-negative production quantities, represented by decision variables π₯ and π¦ satisfy the following constraints: 1
π₯ β₯ 2, π¦ β€ 3, π₯ + π¦ = 5
Then the feasible region of the above LPP is:
(A) A triangle
(B) A quadrilateral
(C) A line segment
(D) unbounded
Assertion-Reason Based Questions Directions:
Question numbers 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 options (A), (B), (C) and (D) as given below:
(A) Both Assertion (A) and Reason (R) are true and (R) is the correct explanation of Assertion (A).
(B) Both Assertion (A) and Reason (R) are true but (R) is not the correct explanation of Assertion (A).
(C) Assertion (A) is true but Reason (R) is false.
(D) Assertion (A) is false but Reason (R) is true.
19. Assertion (A): Let π΄ be a non-singular square matrix of order 3. If |π΄| = 4, then the value of |2. πππ (π΄)| is 128. Reason (R): For a matrix π΄, |πππ(π΄)| = |π΄| πβ1 and |π π΄| = π |π΄| 1
20. Assertion (A): If two probability distributions have the same mean, they must have the same variance. Reason (R): Variance measures the spread of data around the mean and is calculated as πππ(π) = πΈ(π 2 ) β [πΈ(π)] 2
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SECTION β B (5 x 2 = 10) This section comprises of 5 very short answer (VSA) type questions of 2 marks each.
21(A). If π₯ is an odd integer, show that π₯ 2 β‘ 1(πππ8) 2
OR
21(B). If 3 333 β‘ π₯ πππ11 , find π₯. 22. An asset was purchased for βΉ120,000 with an expected salvage value of βΉ10,000 and 10-year useful life. What is the asset's book value at the end of sixth year? 2
23. A man rows 12 km downstream and 12 km upstream, taking 4 hours in total. If the speed of the stream is 4 km/h, find the speed of the man in still water. 2
24(A). The number of customer complaints received by an e-commerce company follows a Poisson distribution. It is observed that the probability of receiving exactly 2 complaints in a day is equal to the probability of receiving exactly 3 complaints in a day. Find the mean and the variance of the distribution. 2
OR
24(B). A metro station observes that trains arrive at a particular platform at an average rate of 20 trains per hour during peak hours. [Use π β2 = 0.1353]
(i) What is the expected number of trains arriving in a 6-minute interval?
(ii) Find the probability that exactly 2 trains arrive in a 6-minute interval.
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SECTION β C (6 x 3 = 18) This section comprises of 6 short answer (SA) type questions of 3 marks each.
26(A). The price of a stock S (in dollars) t days after its IPO is modeled by π(π‘) = 4π‘ 2+1 π‘ 3
OR
26(B). Consider a simple market where quantity demanded is Qd = 15 β 2P and quantity supplied is QS = 3P β 5. The rate of change of price i.e., ππ ππ is half of the excess demand i.e. (Qd β QS ).
(i) Show that dP dt = 1 2 ( 20 β 5P).
(ii) If P = 2 when t = 0, then show that P = 2 ( 2 β e β2.5t )
27. A βΉ1000 bond with a 4% coupon rate is 5 years until maturity and pays quarterly interest. If the annual market yield to maturity is 6%, what is the purchase price of the bond? [Given: 1.015β20 = 0.742 ] 3
28(A) The overall increase in a company's market share was 33.1%. Management reports the growth rate as a CAGR of 8.5%. Assuming the CAGR is accurate, how many years did it take for the market share to grow this much? [Use log(1.331) = 0.124 and log(1.085) = 0.035] 3
28 (B) A company needs to accumulate βΉ 45,000,000 in π years to replace a fleet of vehicles. They decide to make annual deposits of βΉ 5,00,000 at the end of each year into a sinking fund. If the interest rate is 15% compounded annually, then in how many years will it take to reach the target? [Use log(14.5) = 1.161 and log(1.15) = 0.061].
29 For the matrix A = [ 1 2 2 1 ], show that A 3 β 7A β 6 I = O, where I is an identity matrix of order 2 and O is a zero matrix of order 2 Γ 2. Hence find A β1 .
30 In an examination, 2000 students appeared and the mean of the normal distribution of marks is 30 with standard deviation as 6.25. Find how many students are expected to score:
(i) between 20 and 40 marks?
(ii) less than 25 marks? [Use π(0 β€ π < 1.6) = 0.4452 and π(0 β€ π β€ 0.8) = 0.2881]
31 A nutritionist claims that the average daily calorie intake of Indian teenagers is 2200 calories. To verify this claim, a random sample of 25 teenagers was selected, and their daily calorie intake was recorded. The sample mean was found to be 2080 calories with a sample standard deviation of 280 calories. Test at the 5% level of significance whether the nutritionist's claim is valid. Assume that the calorie intake follows a normal distribution. [Use π‘24(0.025) = 2.064] 3
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SECTION β D (4 x 5= 20) This section comprises of 4 long answer (LA) type questions of 5 marks each.
32. Under monopoly, the quantity sold and market price are determined by demand function π = 274 β π₯ 2 . If the marginal cost for a profit maximising monopolist is equal to 4 + 3π₯, find the consumerβs surplus. 5
33(A). A textile factory manufactures three types of fabrics: Cotton (C), Silk (S), and Wool (W). These fabrics must pass through three processing units: Weaving (π1), Dyeing (π2), and Packaging (π3). The per hour processing speeds of the units are as follows: 5
Unit π1 (Weaving): 20 units of Cotton, 40 units of Silk, 20 units of Wool.
Unit π2 (Dyeing): 25 units of Cotton, 50 units of Silk, 50 units of Wool.
Unit π3 (Packaging): 20 units of Cotton, 20 units of Silk, 40 units of Wool.
The total available processing time for the units are 20 hours for Weaving (π1), 12 hours for Dyeing (π2), and 19 hours for Packaging (π3). Using Cramer's Rule, find the number of units of Cotton, Silk, and Wool that can be produced if the available time is fully utilized.
33 (B) "Cyber-Safe" has upgraded their encryption protocol. They now use a more complex Encoding Matrix (E) to encrypt the 3-letter keyword that unlocks their central server. The encryption follows the formula: C = E . P, where P is the column matrix of the letter positions (A = 1, B = 2, C = 3, β¦ , Z = 26). You are an interceptor who has received an encrypted message matrix C and you have managed to crack the Encoding Matrix E. The Encoding matrix E is given as E = [ 2 1 1 1 0 1 0 2 1 ] The intercepted encrypted matrix C is given as C = [ 52 36 35 ] Using matrix algebra, determine the original numerical matrix P. Also decode the numbers into letters to reveal the 3-letter secret code word.
34 A farmer can buy local goats at βΉ400 each and hybrid goats at βΉ600 each. A local goat produces milk worth βΉ190 per week, while a hybrid goat produces milk worth βΉ250 per week. The weekly maintenance cost of each goat is βΉ60. The farmer has at most βΉ12,000 to spend on buying goats and can maintain not more than 22 goats. Formulate the Linear Programming Problem and find how many local goats and hybrid goats the farmer should buy to maximize the weekly profit? Also, find the maximum profit. 5
35 (A) The following data represents the production of wheat in India (in Million Tonnes) over a period of 9 years. Due to climatic variations, the production fluctuates annually:
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Calculate the trend values using the 4-Year Centred Moving Average method for the given data.
35 (B) The "Digital India" initiative has led to a massive surge in digital payment transactions across the country. The following data represents the volume of digital transactions (UPI, NEFT, IMPS, debit card etc.) in India (in Crores) for 6 consecutive years. The growth has been significant due to the push for a digital economy.
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| Year | 2018 | 2019 | 2020 | 2021 | 2022 | 2023 |
|---|---|---|---|---|---|---|
| Volume (in crores) | 2071 | 3134 | 4572 | 5554 | 8839 | 13462 |
(i) Fit a straight-line trend to the above data using the method of Least Squares.
(ii) Estimate the likely volume of transactions for the year 2025, assuming the current trend continues.
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SECTION β E (3 x 4 = 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
36 Three pipes A, B and C can fill a tank in 12 hours, 15 hours and 20 hours respectively. Pipe A is kept open continuously. Pipes B and C in that order are open alternately for one hour each.
Based upon the above information, answer the following questions:
(i) Find the part of the tank filled in one hour when pipes A and C are open together. 1
(ii) Find the part of the tank filled in one hour when pipes A and B are open together. 1
(iii) (A) Find the part of tank filled by the three pipes in the first 6 hours. 2
OR
(iii) (B) Find the part of the unfilled tank at the end of 7 hours.
37 The cost of building an office block, π₯ floors high, consists of three components:
Based on the above information, answer the following questions:
(i) Find the total cost of building the office block π₯ floors high. 1
(ii) Find the average cost per floor in terms of π₯. 1
(iii) (A) Find, how many floors the office block should have if the average cost per floor is to be minimized? 2
OR
(iii) (B) Find, for how many floors the average cost keeps decreasing? [Use β10 = 3.2]
38 A couple wishes to purchase a house for βΉ15,00,000 with a down payment of βΉ4,00,000. If they amortize the balance at an interest rate of 9% per annum compounded monthly for 10 years.
(i) Find the monthly installment using the flat rate method. 2
(ii) Find the monthly installment using the reducing balance method. 2
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To get the full PDF, students can check the link below:
CBSE Class 12 Applied Mathematics Marking Scheme 2026-27
The CBSE Class 12 Applied Mathematics Marking Scheme for 2026β27 is available in the PDF below for students to download and review.
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