**CBSE Class 12 Applied Mathematics (Code No. 241)**

**(Sample Paper 2022-23)**

**Time Allowed: 3 hrs Maximum Marks: 80**

**General Instructions:**

- This question paper contains five sections A, B, C, D and E. Each section is compulsory.
- Section – A carries 20 marks weightage, Section – B carries 10 marks weightage, Section – C carries 18 marks weightage, Section – D carries 20 marks weightage and Section – E carries 3 case-based with total weightage of 12 marks.
**Section – A:**It comprises of**20 MCQs of 1 mark**each.**Section – B:**It comprises of**5 VSA type questions of 2 marks**each.**Section – C:**It comprises of**6 SA type of questions of 3 marks**each.**Section – D:**It comprises of**4 LA type of questions of 5 marks**each.**Section – E:**It has**3 case studies**. Each case study comprises of 3 case-based questions, where**2 VSA type questions are of 1 mark**each and**1 SA type question is of 2 marks**. Internal choice is provided in**2 marks**question in each case-study.- Internal choice is provided in
**2 questions in Section – B, 2 questions in Section – C, 2 questions in Section – D.**you have to attempt only one of the alternatives in all such questions.

**SECTION – A**

(All questions are compulsory. No internal choice is provided in this section)

- What is the least value of ‘x’ that satisfies x
- 27
- 30
- 31
- 35

27 (mod 4), when 27 < x 36? - Let p > 0 and q < 0 and p, q

p + q ________ p – q- >
- <

z, then choose the correct inequality from the given below options to complete the statement: - A machine makes car wheels and in a random sample of 26 wheels, the test statistic is found to be 3.07. As per the t-distribution test (of 5% level of significance), what can you say about the quality of wheels produced by the machine? (Use t
_{25}(0.05) = 2.06)- Superior quality
- Inferior quality
- Same quality
- Cannot say

- For the purpose of t-test of significance, a random sample of size (n) 34 is drawn from a normal population, then the degree of freedom (v) is-
- 33
- 34
- 35

- A person can row a boat along the stream of the river at 10 km/h and against the stream in 6 km/h. What is the speed of the stream flow?
- 1 km/h
- 2 km/h
- 4 km/h
- 5 km/h

- Standard deviation of a sample from a population is called a –
- Standard error
- Parameter
- Statistic
- Central limit

- Two water supplying trucks – A and B supply water to remote areas. Truck A is carrying 100 litres of water to a village 1.5 km away and truck B is delivering 80 litres of water to another village, 1 km away. Due to bad road conditions, each truck loses 20 ml water while travelling each metre distance. Which truck is able to deliver more water and by how much more?
- Truck A, 20 litres
- Truck B, 20 litres
- Truck A, 10 litres
- Truck B, 10 litres

- What is the face value of a sinking fund that yields a dividend of ₹1800 at 10% semi-annually?
- ₹ 3600
- ₹ 18000
- ₹ 24000
- ₹ 36000

- In the given figure, the area bounded by the curve x = f(y), y-axis and abscissa y = a and y = b is equal to –

- A factory production is delayed for three weeks due to breakdown of a machine and unavailability of spare parts. Under which trend oscillation does this situation fall under?
- Seasonal
- Cyclical
- Secular
- Irregular

- A newspaper printing machine costs ₹ 4,80,000 and estimated scrap value of ₹ 25,000 at the end of its useful life of 10 years. What is its annual depreciation as per linear method?
- ₹ 4,550
- ₹ 45,500
- ₹ 50,500
- ₹ 61,500

- In the given figure (I), what is the LPP shaded region known as?

- Feasible region
- Feasible solution
- Optimal region
- Objective region

- General solution of differential equation: y log y dx – xdy = 0 is-
- y = log |Cx|
- y = e
^{|Cx|} - y = e
^{-Cx} - log y = |C + x|

- An investment of ₹ 10,000 becomes ₹ 60,000 in 4 years, then the CAGR (compound annual growth rate) is given by-
- In what ratio shall I add water to the liquid detergent costing ₹480 per litre to get resulting mixture worth ₹300 per litre?
- 5 : 3
- 3 : 8
- 3 : 5
- 5 : 8

- A grain whole-seller visits the granary market. While going around to make a good purchase, he takes a handful of rice from random sacks of rice, in order to inspect the quality of farmers produce. The handful of rice taken from a sack of rice for quality inspection is a:
- statistic
- population
- parameter
- sample

- For predicting the straight-line trend in the sales of scooters (in thousands) on the basis of 6 consecutive years data, the company makes use of 4-year moving averages method. If the sales of scooters for respective years are a, b, c, d, e and f respectively, then which of the following average will not be computed?
- In a school, a random sample of 145 students is taken to check whether a student’s average calory intake is 1500 or not. The collected data of average calories intake of sample students is presented in a frequency distribution, which is called a:
- Statistics
- Sampling distribution
- Parameter
- Population sampling

**For questions 19 and 20, two statements are given – one labelled Assertion(A) and the other labelled Reason (R). Select the correct answer to these questions from the codes (i), (ii), (iii) and (iv) as given below: (i) Both A and R are true and R is the correct explanation of the assertion (ii) Both A and R are true but R is not the correct explanation of the assertion (iii) A is true, but R is false (iv) A is false, but R is true**

**Assertion (A):**Kuhu and Beena are two equally capable badminton players. Probability that Beena will beat Kuhu in 3 games out of 4 is 25%

**Reason (R):**The probability of r successes in n trials, denoted by P(x = r) is given by P(x = r) =^{n}C_{r}p^{r}q^{n-r}, r = 0, 1, … , n where p denotes success and q denotes failure in each trial.- (i)
- (ii)
- (iii)
- (iv)

**Assertion (A):**If the nominal rate of interest is 12.5% and the inflation is 2%, then the effective rate of interest is 10.5%

**Reason (R):**If the interest is calculated only at the end of an year, then the effective rate of interest is same as the nominal rate of interest.- (i)
- (ii)
- (iii)
- (iv)

**SECTION – B**

(All questions are compulsory. In case of internal choice, attempt any one question only)

- ₹ 2,50,000 cash is equivalent to a perpetuity of ₹ 7,500 payable at the end of each quarter. What is the rate of interest convertible quarterly?
- Find value of 2a + 3b − c, if
**OR**There are two real value(s) of x, for which the value of the determinant

is 86. Find the value(s) of x. is a skew-symmetric matrix - A book publisher sells a hard cover edition of a book for ₹ 72 and a paperback edition for ₹ 40. In addition to a fixed weekly cost of ₹ 9,600, the cost of printing hardcover and paperback editions are ₹ 56 and ₹ 28 per book respectively. Each edition requires 5 minutes on the printing machine whereas hardcover binding takes 10 minutes and paperback takes 2 minutes on the binding machine. The printing machine and the binding machine are available for 80 hours each week.

Formulate the linear programming problem to maximise the publisher’s profit. - A boatman takes half as much time in rowing his boat for a certain distance downstream than upstream. What is the ratio between his speed of rowing the boat in still water and speed of current?
**OR**In a 200-metre race, Anuj can beat Param by 5 metre or 3 seconds. How much time did Anuj take to complete the race?

- Mitul invested ₹ 3,50,000 in a fund. At the end of the year the value of the fund is ₹ 4,37,500. What is the nominal rate of interest, if the market price is same at the end of the year?

**SECTION – C**

(All questions are compulsory. In case of internal choice, attempt any one question only)

- Find the interval(s) in which the function , is strictly increasing and strictly decreasing.
- Two badminton teams A and B are staying in the same hotel. Team A has 2 male and 3 female players accompanied by 1 coach. Team B comprises of 1 male, 2 female players and 2 coaches. The daily diet requirement (calories and protein) for each person is as given below:
Calories Protein Male Player 2500 65 g Female player 1900 50 g Coach 2000 54 g Use matrix algebra to calculate the total diet requirement of calories and protein for each team.

- Evaluate
**OR**Evaluate

- Under the pure market competition scenario, the demand function p
_{d}and the supply function p_{s}for a certain commodity are given as p_{d}= and respectively, where p is the price and x is the quantity of the commodity. Using integrals, find the producer’s surplus.**OR**The demand function p for maximising a profit monopolist is given by p = 274 – x

^{2}while the marginal cost is 4 + 3x, for x units of the commodity. Using integrals, find the consumer surplus. - Surjeet purchased a new house, costing ₹ 40,00,000 and made a certain amount of down payment so that he can pay the balance by taking a home loan from xyZ Bank. If his equated monthly instalment is ₹ 30,000, at 9% interest compounded monthly (reducing balance method) and payable for 25 years, then what is the initial down payment made by him? [Use (1.0075)
^{-300}= 0.1062] - 10 years ago, Mr Mehra set up a sinking fund to save for his daughter’s higher studies. At the end of 10 years, he has received an amount of ₹ 10,21,760. What amount did he put in the sinking fund at the end of every 6 months for the tenure, which paid him 5% p.a. compounded semi-annually? [Use (1.025)
^{20}= 1.6386]

**SECTION – D**

(All questions are compulsory. In case of internal choice, attempt any one question only)

- It is known that 3% of plastic buckets manufactured in a factory are defective. Using the Poisson distribution on a sample of 100 buckets, find the probability of:
- Zero defective buckets
- At most one bucket is defective

[Use e^{-3}= 0.049]

**OR**In a math aptitude test, student scores are found to be normally distributed having mean as 45 and standard deviation 5. What percentage of students have scores-

- more than the mean score?
- between 30 and 50?

- An event management company charges ₹ 4,800 per guest, for a bulk booking for 100 guests. In addition, it offers a discount of ₹ 200 for each group of 10 guests over and above 100 guest booking. What is the number of guests that will maximise the amount of money the company receives on a booking? What is the maximum profit on such booking?
**OR**To manufacture ‘x’ number of dolls, a company’s total cost function C(x) is given by C (x) = 100 + 0.025x

^{2}and the total revenue function R (x) is described as R (x) = 5x. Given that C (x) and R (x) are in thousand rupees, what number of dolls shall be manufactured to maximise the profit of the company? What is the maximum profit? - Rahul is at the whole sale market to purchase folding tables and chairs, to later sell them at his furniture shop. He has only ₹ 5,760 to spend and his van has space to carry at the most 20 items. A table costs him ₹ 360 and a chair costs ₹ 240. Back at his shop, he plans to sell a table at a profit of ₹ 22 and a chair at a profit of ₹ 18. Given that he can sell all the items that he purchases, how many tables and chairs shall he purchase in order to maximise his profit?
- The equilibrium conditions for three competitive markets are described as given below, where p
_{1}, p_{2}and p_{3}are the equilibrium price for each market respectively.

p_{1}+ 2p_{2}+ 3p_{3}= 85

3p_{1}+ 2p_{2}+ 2p_{3}= 105

2p_{1}+ 3p_{2}+ 2p_{3}= 110

Using matrix method, find the values of respective equilibrium prices.

**SECTION – E**

(All questions are compulsory. In case of internal choice, attempt any one question only)

**CASE STUDY – I**

An overhead water tank has three pipes A, B and C attached to it (as shown in figure (II)). The inlet pipes A and B can fill the empty tank independently in 15 hours and 12 hours respectively. The outlet pipe C alone can empty a full tank in 20 hours.

Based on the above information, answer the following questions. Show steps to support your answers.- For a routine cleaning of the tank, the tank needs to be emptied. If pipes A and B are closed at the time when the tank is filled to two-fifth of its total capacity, how long will pipe C take to empty the tank completely?
- How long will it take for the empty tank to fill completely, if all the three pipes are opened simultaneously?
- On a given day, pipes A, B and C are opened (in order) at 5 am, 8 am and 9 am respectively, to fill the empty tank. In how many hours will the tank be filled completely?
**OR**Given that the tank is half-full, only pipe C is opened at 6 AM, to empty the tank. After closing the pipe C and an hour’s cleaning time, tank is filled completely by pipe A and B together. What is the total time taken in the whole process?

**CASE STUDY – II**

When observed over a long period of time, a time series data can predict trend that can forecast increase or decrease or stagnation of a variable under consideration. Such analytical studies can benefit a business for forecasting or prediction of future estimated sales or production.

Mathematically, for finding a line of best-fit to represent a trend, many methods are available. Methods like moving-averages and least-squares squares are some of the techniques to predict such trends.

Mrs. Shamita runs a bread factory and the record of her sales of bakery items for the period of 2015 – 2019 is as follows:Year 2015 2016 2017 2018 2019 Sales

(in ₹ thousands)35 42 46 41 48 Based on the above information, answer the following questions. Show steps to support your answers.

- By taking year 2017 as origin, use method of least-squares to find the best-fit trend line equation for Mrs. Shamita’s business. Show the steps of your working.
**OR**Demonstrate the technique to fit the best-suited straight-line trend by the method of 3-years moving averages. Also draw the trend line.

- What are the estimated sales for Mrs. Shamita’s business for year 2022?
- Mrs Shamita wishes to grow her business to yearly sale of ₹ 67400. In which year will she be able to reach her target?

- By taking year 2017 as origin, use method of least-squares to find the best-fit trend line equation for Mrs. Shamita’s business. Show the steps of your working.
**CASE STUDy – III**

According to an educational board survey, it was observed that class XII students apply at least one to four weeks ahead of colleges application deadline. Let X represent the week when an average student applies ahead of a college’s application deadline and the probability of student to get admission in the college P(X = x) is given as follows:

Where k is a real number.

Based on the above information, answer the following questions. Show steps to support your answers.- Find the value of k.
- What is the probability that Sonali will get admission in the college, given that she applied at least 2 weeks ahead of application deadline?
- Calculate the mathematical expectation of number of weeks taken by a student to apply ahead of a college’s application deadline.
**OR**To promote early admissions, the college is offering scholarships to the students for applying ahead of deadline as follows:

₹ 50000 for applying 4 weeks early, ₹ 20000 for applying 3 weeks early, ₹ 12000 for applying 2 weeks early, and ₹ 9600 for applying 1 week early

What is the expected scholarship offered by the college?

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