CBSE Sample Papers for Class 12 Applied Mathematics Term 2 for Practice

Students can access the CBSE Sample Papers for Class 12 Applied Mathematics with Solutions and marking scheme Term 2 for Practice will help students in understanding the difficulty level of the exam.

CBSE Sample Papers for Class 12 Applied Mathematics Term 2 for Self Assessment

Class 12 Applied Mathematics Sample Paper Term 2 Set 1 for Self Assessment

Time: 2 Hours
Maximum Marks: 40

General Instructions:

• The question paper is divided into 3 sections -A, B and C
• Section A comprises of 6 questions of 2 marks each. Internal choice has been provided in two questions.
• Section B comprises of 4 questions of 3 marks each. Internal choice has been provided in one question.
• Section C comprises of 4 questions of 4 marks each. It contains one case study based question. Internal choice has been provided in one question.

Section – A (2 marks each)

Question 1.
The marginal cost function of manufacturing x shoes is 6 + 10x – 6x². The cost producing a pair of shoes is ? 12. Find the total and average cost function.
OR
Find the area of the region bounded below by y = x² + 1, bounded above by y = x and bounded on the sides by x = 0 and x = 1.

Question 2.
Find the present value of perpetuity of ₹ 800 at end of each quarter if money is worth 7% compounded quarterly.

Question 3.
What effective rate is equivalent to a nominal rate of 5% per annum compounded quarterly?
[Use (1.05)⁴ = 1.05094]
OR
Find the present value of an annuity of ₹ 2000 payable at the end of each year for 6 years if money is worth 6% compounded annually. [Given (1.06)^-6 = 0.70496]

Question 4.
A sampling distribution of the sample means X̄ is formed from a population with mean weight µ = 80 kg and standard deviation σ = 14 kg. What is the expected value and standard deviation of X̄, if sample size is 49 ?

Question 5.
The following figures relate to the profits of a commercial concern for 8 years.

Find the trend of profits by the method of three-yearly moving averages.

Question 6.
Draw the feasible region for the given L.PE
Minimize Z = 5x + 10y
Subject to x + 2y ≤ 120
Constraints x + y ≥ 60
x – 2y ≥ 0
and x, y ≥ 0

Section – B (3 Marks each)

Question 7.
Find the particular solution of the differential equation e^x\(\sqrt{1-y^{2}}\)dx + \(\left(\frac{y}{x}\right)\)dy = 0 given that y = 1 when x = 0.

Question 8.
Apply the method of least square to obtain the trend values from the following data:

OR

Calculate the 5-yearly and 7-yearly moving average for the following data of a number of commercial industrial failures in a country during 2003-2018.

Year	No. of failures
2003	23
2004	26
2005	28
2006	32
2007	20
2008	12 .
2009	12
2010	10
2011	09
2012	13
2013	11
2014	14
2015	12
2016	09
2017	03
2018	01

Question 9.
It is claimed that the average weight of a bag of biscuits is 250 grams with the standard deviation 20.5 grams. Would you agree to this claim if random sample of 50 bags of biscuits showed an average weight of 240 grams, using a 0.05 level of significance ?

Question 10.
An investment made by Mr. Roy has initial value ₹ 10,000 and it increases to ₹ 40,000 in 3 years what wiU be C.A.G.R. ? [Given, (4)^1/3 = 1.587]

Section – C (4 Marks each)

Question 11.
A company manufactures two types of toys A and B. A toy of type A requires 5 minutes for cutting and 10 minutes for assembling. A toy of type B requires 2 minutes for cutting and 8 minutes for assembling. There are 3 hours available for cutting and 4 hours available for assembling the toys in a day. The profit is ₹ 50 each on a toy of type A and ₹ 60 each on a toy of type B. How many toys of each type should the company manufacture in a day to maximize the profit ? Use linear programming to find the solution.

Question 12.
Rohan has completed his M.B.A. and now he wants to start a new business. So, he approaches to many banks. One bank is agreed to give loan to Rohan. So, Rohan has borrowed ₹ 5 lakhs from a bank on the interest rate of 12 per cent for 10 years.
(i) Calculate monthly installment using (1.01)¹²⁰ = 3.300
(ii) Find the amount of interest paid by Rohan.
OR
In 10 years, a machine costing ₹ 50,000 will have a salvage value of ₹ 5,000. A New Machine at that time is expected to sell for ₹ 55,000. In order to provide funds for the difference between the replacement cost and the salvage cost, a sinking fund is set up into which equal payments are placed at the end of each year. If the fund earns interest at the rate 6% compounded annually, how much should each payment be ? [Given that (1.06)¹⁰ = 1.7908]

Question 13.
An investor is considering purchasing a new issue of 5-year bonds of ₹ 1,000 value and an annual fixed coupon rate of 12%, while coupon payments are made semi-annually. The minimum semi-annual yield that the investor would accept is 6.75%. Find the fair value of the bond.
[Given that (1.0675)^-10 = 0.52038]

CASE STUDY

Question 14.
The demand function for a popular make of 12- speed bicycle is given by p = D(x) = – 0.001x² + 250, where p is the unit price in Rupees and x is the quantity demanded in units of a thousands. The supply function for the same product is given by p = S(x) = 0.0006x² + 0.02x + 100, where p is the unit price in Rupees and x is the quantity supplied in units of a thousands.

(i) If at equilibrium, curves of demand function and supply function intersect at (x, y), then find point of intersection. (2)
(ii) Find producer’s profit (in ₹).

Class 12 Applied Mathematics Sample Paper Term 2 Set 2 for Self Assessment

Section – A (2 marks each)

Question 1.
A company has determined that the marginal cost function for a product of a particular commodity is given by MC = 12, + 10x – \(\frac{x^{2}}{9}\) where x rupees is the cost of producing x units of the commodity. If the fixed cost is ₹ 250 what is the cost of producing 15 units.
OR
Evaluate:

Answer:

Question 2.
Assume that Reena holds a perpetual bond that generates an annual payment of ₹ 700 each year. She believes that the borrower is credit worthy and that an 6% interest rate will be suitable for this bond. Find the present value of the bond.
Answer:

Question 3.
The value of a car purchased 3 years back, depreciates at the annual rate of 7%. If its present value is ₹ 7,50,000, then find its value after 3 years. [Given that (0.93)³ = 0.80435]
OR
The present value of perpetuity of ₹ 1600 payable at end of every 3 month be ₹ 15,000. Find the rate of interest.
Answer:

Question 4.
The standard deviation of a sample of size 100 is 8.5. Determine the standard error whose population standard deviation is 16 ?
Answer:

Question 5.
From the following time series obtain trend value by 3 yearly moving averages.

Answer:

Question 6.
Draw the feasible reason for the following L.P.P.
Maximize
Z = 4 x + y
x + y ≤ 50,
3x + y ≤ 90,
x ≥ 10
x, y ≥ 0

Section – B (3 marks each)

Question 7.
If the demand function for a commodity is p = 35 – 2x – x², then find the consumer’s surplus at equilibrium price p₀ = 20.
Answer:

Question 8.
Calculate trend values from the following data assuming 7- yearly moving average.

OR

The following data shows the percentage of rural, urban and suburban Indians who have a high speed internet connection at home:

Find the straight line trend by the method of least square for the Rural Indians.

Question 9.
Country A has an average farm size of 191 acres, while Country B has an average farm size of 199 acres. Assume the data were attained from two samples with standard deviations of 38 and 12 acres and sample sizes of 8 and 10, respectively. Is it possible to infer that the average size of the farms in the two countries is different at a = 0.05 ? Assume that the populations are normally distributed.

Question 10.
Anna owns a produce truck, invested ₹ 700 in purchasing the truck, some other initial admin related and insurance expenses of ₹ 1500 to get the business going and has now a day to day expense of ₹ 500/p.m. Consider hypothetically that her everyday profit is ₹ 550/p.m. (ideally, it will be based on sales). At the end of 6 months, Anna takes up her accounts and calculates her rate of return.
Answer:

Section – C (4 marks each)

Question 11.
Two tailors P and Q earn ₹ 150 and ₹ 200 per day respectively. P can stitch 6 shirts and 4 trousers a day, while Q can stitch 10 shirts and 4 trousers per day. How many days should each work to produce at least 60 shirts and 32 trousers at minimum labour cost?
Answer:

Question 12.
An investor is considering purchasing a new issue of 5-year bonds of ₹ 7,000 par value and an annual fixed coupon rate of 10%. While coupon payments are made semi-annually. The minimum semi annual yield that the investor would accept is 7%. Find fair value of bond.
[Given (1.07)^-10 = 0.5083]
Answer:

Question 13.
A sinking fund with a monthly periodic contribution of ? 2100. The fund will be required to retire a newly taken debt (zero-coupon bonds) rose for the ongoing expansion project. Find the amount of sinking fund if the annualized rate of interest is 7% and the debt will be repaid in 6 years.
[Given that (1.00583)⁷² = 1.5197]
OR
Rohan takes a car loan of ₹ 15 lakhs at an 7.5% interest for 7 years loan tenure. What would be his E.M.I. ? [Given (1.00625)⁸⁴ = 1.6877]
Answer:

CASE STUDY

Question 14.
Doctors have shown certain drugs leave a person’s bloodstream at a rate that is proportional to the amount present. In an experiment a patient is injected with 450 mg of a substance. Seven hours later it is found that 50 mg of the substance remains. Assuming the proportional model is correct for the particular substance.
(i) Express the differential equation that models this scenario. Also, find general solution. (2)
(ii) Find the time constant, k.(log 9 = 2.197225). (2)

Class 12 Applied Mathematics Sample Paper Term 2 Set 3 for Self Assessment

Section – A (2 marks each)

Question 1.
The marginal cost(M.C.) is given by the function MC = 2 + 5e^x.
Find the cost(C) and the average cost(A.C.) function if C(0) = 100.
OR
Evaluate: ∫\(\frac{x^{3}+5 x^{2}+4 x+1}{x^{2}}\)

Question 2.
Find the present value of perpetuity of ₹ 1800 at end of each quarter if money is worth 7% compounded quarterly.

Question 3.
What effective rate is equivalent to a nominal rate of 4% per annum compounded semi annually?
OR
Find the present value of an annuity of ₹ 2500 payable at the end of each year for 4 years if money is worth 5.5% compounded annually. [Given (1.055)-4 = 0.8072]

Question 4.
Define the terms population and sample.

Question 5.
The salaries of the certain company workers are given as below. Using 3 yearly moving average indicate the trend in salaries.

Question 6.
Draw the feasible region for given L.P.P.
Minimize: Z = 6x + 3y

4x + y > 80 x + 5y >115 3x + 2y < 150 x > 0,y > 0

Section – B (3 marks each)

Question 7.
Evaluate the following integral: ∫\(\frac{\sqrt{x^{2}+1}\left\{\log \left(x^{2}+1\right)-2 \log x\right\}}{x^{4}}\)dx.

Question 8.
The following table shows the quarterly sales (in ₹ crore) of a MNC company. Compute the trend by quarterly moving averages.

OR

Fit a straight line trend for the following data and estimate the likely profit for the year 2022. Also calculate the trend values.

Question 9.
For a random sample of 10 pigs fed on diet A, the increases in weight in pounds in a certain period were 10, 6, 16, 17, 13, 12, 8, 14, 15, 9 lbs. For another random sample of 12 pigs fed on diet B, the increases in the same period were 7,13,22,15,12,14,18,8,21,23,10,17 lbs. It is given that mean of first sample is 12 pounds and mean of second sample is 15 pounds. Find the standard error for the given data.
Using the fact the 5% value of t for 20 degrees of freedom is 2.09. Whether diets A and B differ significantly as regarding the effect on increases in weight.

Question 10.
A person invested ₹ 10000 in a mutual fund and the value of investment at the time of redemption was ₹ 20000. If C.A.G.R. for this investment is 9%, Calculate the time period for which the amount was invested? [Given log(2) = 0.3010 &log(1.09) = 0.0374]

Section – C (4 marks each)

Question 11.
A manufacturer manufactures two types of tea-cups, A and B. Three machines are needed for manufacturing the tea cups. The time in minutes required for manufacturing each cup on the machines is given below:

Each machine is available for a maximum of six hours per day. If the profit on each cup of type A is ₹ 1.50 and that on each cup of type B is ₹ 1.00, find the number of cups of each type that should be manufactured in a day to get maximum profit.

Question 12.
Mr. Lai took a car loan of ₹ 10 lakhs at an 11% interest rate for a 15 years tenure. What would be his EMI ? [Given (1.0091)¹⁸⁰ = 5.1069.]

Question 13.
A new issue of 7-year bonds of ₹ 1,700 par value and an annual fixed coupon rate of 9 %, was purchased by Mr. Pandit. It is given that the coupon payments are made semi-annually. The minimum semi-annual yield that Mr. Pandit would accept is 8 %. Find the fair value of bond using the semi-annual coupon payment and the formula as required.
[Given that (1.08)^-14 = 0.3405]
OR
A ₹ 4000, 6% bond is redeemable at the end of 7 years at ₹ 210. Find the purchase price to yield 8% effective rate. [Given (1.08)^-7 = 0.5835]

CASE STUDY

Question 14.
A radioactive substance is unstable and produces dangerous kinds of radiation. It is unstable because the strong nuclear force that holds the nucleus of the atom together is not balanced with the electric force that wants to push it apart. Because it is unstable, the atoms will decay into more stable ones.

A radioactive substance has a half life of h days. Using the given information answer the following:
(i) Find a formula for its mass m in terms of the t, if the initial mass is m₀. (2)
(ii) What is the initial decay rate ? (2)

Class 12 Applied Mathematics Sample Paper Term 2 Set 4 for Self Assessment

Section – A (2 marks each)

Question 1.
The rate of a new product is given by f'(x) = 100 – 90 e^-x, where x is the number of days the product in on the market. Find the total sale during the first four days. [Given, e^-4 = 0.018]
OR
Evaluate: ∫\(\left[\sqrt{x}+\frac{1}{\sqrt{x}}\right]^{2}\) dx.

Question 2.
At what rate of interest will the present value of a perpetuity of ₹ 300 payable at the end of each quarter be ₹ 24,000 ?

Question 3.
What sum of money invested now could establish a scholarship of ₹ 5000 which is to be awarded at the end of every year forever, if money is worth 8% per annum.
OR
Define the terms ‘useful life’ and ‘scrap value’.

Question 4.
A sample of 1000 students whose mean weight is 119 lbs (pounds) from a school in Uttar Pradesh State was taken and their average weight was found to be 120 lbs with a standard deviation of 30 lbs. Calculate the standard error of the mean.

Question 5.
Find the trend values using 3 yearly moving average for the loans sanctioned to villagers for small startups by a particular branch of a bank in a village.

Question 6.
Draw the feasible region for given LPE
Minimize: Z = 3x + 9y
When: x + 3y ≤ 60
Subject to constraints x + y ≥ 10
x ≤ y
x ≥ 0, y ≥ 0

Section – B (3 marks each)

Question 7.
Evaluate: ∫\(\frac{3 x+1}{\left(x^{3}-x^{2}-x+1\right)}\)dx

Question 8.
Calculate 5-year moving averages for the following data:

OR

Fit a straight line trend to the following data by Least Square Method and estimate the sale for the year 2022.

Question 9.
A group of 7 weeks old chickens, reared on a high protein diet weight 12,15,11,16,14,14,16 ounces; a second group of 5 chickens, similarly treated except that they receive a low protein diet, weight 8,10, 14,10,13 ounces. Find the value of t.

Question 10.
₹ 5000 is invested in a Term Deposit Scheme that fetches interest 6% per annum compounded quarterly. What will be the interest after one year ?
[Given that (1.015)⁴ = 1.0613]

Section – C (4 marks each)

Question 11.
A company produces two types of items, P and Q. Manufacturing of both items requires the metals gold and copper. Each unit of item P requires 3 gms of gold and 1 gm of copper while that of item Q requires 1 gm of gold and 2 gm of copper. The company has 9 gm of gold and 8 gm of copper in its store. If each unit of item P makes a profit of ₹ 50 and each unit of item Q makes a profit of ₹ 60, determine the number of units of each item that the company should produce to maximize profit. Also, find the maximum profit.

Question 12.
A company ABC Ltd which has raised funds in the form of 1,000 zero-coupon bonds worth ₹ 1,000 each. The company wants to set up a sinking fund for repayment of the bonds, which will be after 10 years. Determine the amount of the periodic contribution if the annualized rate of interest is 5%, and the contribution will be done half-yearly. [Given (1.025)²⁰ = 1.6386]

Question 13.
Kirti borrowed money and returned it in 3 equal quarterly installments of ₹ 4630.50 each. What sum had she borrowed if the rate of interest was 20% p.a. compounded quarterly? Also find the total interest charged.
OR
Calculate the price of a Tata Corp. corporate bond which has a par value of ₹ 1000 and coupon payment is 6% and yield is 10%. The maturity period of the bond is 3 years.

Case Study

Question 14.
To model more realistic population growth, scientists developed the logistic growth model, which illustrates how a population may increase exponentially until it reaches the carrying capacity of its environment. When a population’s number reaches the carrying capacity, population growth slows down or stops altogether.

The population of a town grows at the rate of 10% per year.
(i) Find the general solution of the differential equation, related to given problem. (2)
(ii) Find how long will it take for the population to grow 4 times. (2)

Class 12 Applied Mathematics Sample Paper Term 2 Set 5 for Self Assessment

Section – A (2 marks each)

Question 1.
A company produces 50,000 units per week with 200 workers. The rate of change of production with respect to the change in the number of additional labour x is represented as 300 – 5x^2/3. If 64 additional labours are employed, find out the additional number of units, the company can produce.
OR
What is the area bounded by the curve y = log x, x-axis and the ordinates x = 1, x = 2 ?
(Use log 4 = 1.38629)

Question 2.
Find the effective interest rate corresponding to a nominal rate of interest of 7% per year compounded quarterly.

Question 3.
Find the present value of perpetuity of ₹ 2580 at end of each quarter if money is worth 6 % compounded semi-annually.
OR
At what rate of interest will the present value of a perpetuity of ₹ 500 payable at the end of each quarter be ₹ 35,000?

Question 4.
Define the terms ‘statistics’ and ‘parameter’.

Question 5.
What is the need for studying time series ?

Question 6.
Draw the graph of following L.EE
Maximize Z = 1000x + 600y
Subject to the constraints
x + y ≤ 200
x > 20
y – 4x ≥ 0
x, y ≥ 0

Section-B (3 marks each)

Question 7.
The demand and supply function of a commodity are P_d = 18 – 2x – x² and P_s = 2x – 3 respectively. Find the consumer’s surplus at equilibrium price.

Question 8.
Calculate the trend values by the method of least squares from the data given below:

Year	Sales of T.V sets (in lakhs)
2013 – 2014	12
2014-2015	18
2015-2016	20
2016 – 2017	23
2017-2018	27

OR
Compute 4-year moving averages centred for the following time series:

Question 9.
Find the f-test value for the following given two sets of values: 7,2, 9,8 and 1,2, 3,4.

Question 10.
Find the present value of an annuity of ₹ 1380 payable at the end of each year for 5 years if money is worth 6.5% compounded annually.
[Given (1.065)^-5 = 0.7299]

Section – C (4 marks each)

Question 11.
A new cereal, formed of a mixture of bran and rice, contains at least 88 grams of protein and at least 36 milligram of iron. Knowing that bran contains 80 gram of protein and 40 milligram of iron per kilogram, and that of rice contains 100 gram of protein and 30 milligrams of iron per kilogram, find the minimum cost of producing a kilogram of this new cereal if bran costs ₹ 28 per kilogram and rice costs ₹ 25 per kilogram.

Question 12.
Mrs. Lata took a home loan of ₹ 30 lakhs at an interest rate of 7.8% for 14 years loan tenure. What would be her E.M.I. ? [Given (1.0065)¹⁶⁸ = 2.969]

Question 13.
Mr. Sharma purchased a bond of par value ₹ 2200 for 10 years with annual fixed coupon rate of 7.5 %. It is given that the coupon payments are made quarterly. The minimum semi-annual yield that Mr. Sharma would accept is 8.5 %. Find the fair value of bond. [Given (1.085)^-40 = 0.0383]
OR
To raise the fund, an IT based company release 10,000 zero coupon bonds worth ₹ 1500 each. The company wants to set up a sinking fund for repayment of the bonds, which will be after 5 years. Determine the amount of the periodic contribution if the annualized rate of interest is 7%, and the contribution will be done semi-annually. [Given (1.035)¹⁰ = 1.4106]

Case Study

Question 14.
Populations change over time and space as individuals are born or immigrate (arrive from outside the population) into an area and others die or emigrate (depart from the population to another location). Populations grow and shrink and the age and gender composition also change through time and in response to changing environmental conditions. Some populations, for example trees in a mature forest, are relatively constant over time while others change rapidly.

Suppose the growth of a population is proportional to the number present.
(i) Find the particular solution of the differential equation in this scenario. (2)
(ii) If the population of a colony doubles in 25 days, in how many days will the population become triple ? (Given log 3 = 1.0986 log 2 = 0.6931) (2)

Class 12 Applied Mathematics Sample Paper Term 2 Set 6 for Self Assessment

Section – A (2 marks each)

Question 1.
The rate of change of sales of a company after an advertisement campaign is represented as, f(t) = 3000e^-0.3t where t represents the number of months after the advertisement. Find out the total cumulative sales after 4 months. [Given that e^-1.2 = 0.3012]
OR
Evaluate:

Question 2.
Find the effective interest rate corresponding to a nominal rate of interest of 8% per year compounded monthly. [Given that (1.0067)¹² = 1.0834]

Question 3.
At what rate of interest will the present value of a perpetuity of ₹ 850 payable at the end of each 6 months be ₹ 94,000?
OR
What sum of money invested now could establish a scholarship of ₹ 3000 which is to be awarded at the end of every year forever, if money is worth 7% per annum?

Question 4.
A random sample of 60 observations was drawn from a large population and its standard deviation was found to be 2.5. Calculate the suitable standard error if this sample is taken from a population with standard deviation 3 ?

Question 5.
Define secular trend and seasonal variations.

Question 6.
Draw the graph of following L.P.P.
Maximize Z = x₁ + x₂,
Subject to constraints
x₁ + x₂ > 1,
3x₁ + x₂ > 3
and x₁, x₂ ≥ 0

Section – B (3 marks each)

Question 7.
The demand and supply function of an article are D(q) = 1000 – 0.4q² and S(q) = 42q. Find the producer’s surplus at equilibrium price.

Question 8.
Fit a straight line trend on the following data using the Least Squares Method.

OR
Compute 5-year moving averages for the following data.

Question 9.
A machine which produces mica insulating washers of used in the electric devices is set to turn out washers having a thickness of 10 mils (1 mil = 0.001 inch). A sample of 10 washers has an average thickness of 9.52 mils with a standard deviation of 0.60 mil. Find out t.

Question 10.
Rohan invested ₹ 25,000 in a corporate bond and the value of investment at the time of redemption was ₹ 45,000. If C.A.G.R. for this investment is 7%, calculate the time period for which the amount was invested? [Given log 5 = 0.6989, log 9 = 0.9542 & log(1.07) = 0.0294]

Section – C (4 marks each)

Question 11.
A manufacturer produces two types of steel trunks. He has two machines, A and B. The first type of trunk requires 3 hours on machine A and 3 hours on machine B. The second type requires 3 hours on machine A and 2 hours on machine B. Machines A and B can work at most for 18 hours and 15 hours per day respectively. He earns a profit of ₹ 30 per trunk on the first type of trunk and ₹ 25 per trunk on the second type. Formulate a Linear Programming Problem to find out how many trunks of each type he must make each day to maximize his profit.

Question 12.
A textile company has raised funds in the form of 5,000 zero-coupon bonds worthing ₹ 1,100 each. The company wants to set up a sinking fund for repayment of the bonds, which will be after 7 years. Determine the amount of the periodic contribution if the annualized rate of interest is 6%, and the contribution will be done quarterly. [Given (1.015)²⁸ = 1.5172]

Question 13.
Sameeksha borrowed money for new startup from a small finance company and returned it in 5 equal quarterly installments of ₹ 7500 each. What sum had she borrowed if the rate of interest was 12% p.a. compounded quarterly?
OR
Calculate the price of a Reliance corporate bond which has a par value of ₹ 1825 and coupon payment is 7.6% and yield is 9.8%. The maturity period of the bond is 3 years.

CASE STUDY

Question 14.
Radium a chemical element, in the form of radium chloride, was discovered by Marie and Pierre Curie in 1898 from ore mined at Jachymov. They extracted the radium compound from Uraninite and published their discovery at the French Academy of Sciences five days later. Pure radium is silvery-white, but it readily reacts with nitrogen (rather than oxygen) on exposure to air, forming a black surface layer of radium nitride. It is also known as the alkaline earth metal.

It is given that radium decomposes at a rate proportional to the amount present.
(i) Find the particular solution of the differential equation in this scenario. (2)
(ii) If P % of the original amount of radium disappears in l years. What percentage of it will remain after 21 years ? (2)

Class 12 Applied Mathematics Sample Paper Term 2 Set 7 for Self Assessment

Section – A (2 marks each)

Question 1.
The price of a machine is 6,40,000 if the rate of cost saving is represented by the function f(t) = 20,000t. Find out the number of years required to recoup the cost of the function.
OR
Evaluate: ∫\(\frac{x^{2}}{x^{2}-4}\)dx.

Question 2.
Assuming that Shyam holds a perpetual bond that generates an annual payment of ₹ 500 each year. He believes that the borrower is credit-worthy and that an 8% interest rate will be suitable for this bond. Compute the present value(P.V.) for this perpetuity.

Question 3.
At what rate converted semi-annually will the present value of a perpetuity of ₹ 500 payable at the end of each 6 months be ₹ 25,000 ?
OR
On 1^st April, 2021, Rekha purchased a LED TV costing ₹ 50,000 and spent ₹ 5,000 on its erection. The estimated effective life of the LED TV is 5 years with a scrap value of ₹ 3,000. Calculate the depreciation using linear method with accounting year ending on 31^st March 2022.

Question 4.
A simple random sample consist of four observations 1, 3, 5, 7. What is the point estimate of population standard deviation ?

Question 5.
Define time series and its components.

Question 6.
The feasible solution for a L.EE is shown in given figure. Let Z = 3x-4y be the objective function. Find the minimum and maximum value of Z.

Section – B (3 marks each)

Question 7.
Solve: (x² – yx²) dy + (y² + xy²) dx = 0.

Question 8.
Calculate 5 year moving averages for the following data.

OR
The following data shows the percentage of rural, urban and sub-urban Indians who have a high speed internet connection at home:

Find a straight line trend by the method of least square for the Sub-urban Indians.

Question 9.
Let us consider the average rainfall in a given area be 8 inches. However, a local meteorologist claims that rainfall was above average from 2016-2020 and argues that average rainfall during this period was significantly different from overall average rainfall. The following is the average rainfall for the observed period of 2016-2020:

Analyse the argument of the average rainfall and find out the actual condition.

Question 10.
Rehaan invested ₹ 7000 in a Term Deposit Scheme that fetches interest 7.5% per annum compounded semi-annually. What will be the interest after 3 years ? [Given (1.0375)⁶ = 1.2472]

Section – C (4 marks each)

Question 11.
A dealer in rural area wishes to purchase a number of sewing machines. He has only ₹ 5760 to invest and has space for at most 20 items of storage. An electronic sewing machine cost him ₹ 360 and a manually operated sewing machine ₹ 240. He can sell an electronic sewing machine at a profit of ₹ 22 and a manually operated sewing machine at a profit of ₹ 18. Assuming that he can sell all the items that he can buy, how should he invest his money in order to maximize his profit ? Make it as a L.P.P. and solve it graphically.

Question 12.
A company XYZ borrowed ₹ 1.5 lakhs for reformation. The company plans to set up a sinking fund that will pay back the loan at the end of 3 years. Assuming a rate of 9% compounded quarterly, and the sinking fund of the ordinary annuity. Calculate the amount of the sinking fund. [Given that (1.0225)¹² = 1.3060]

Question 13.
Ajay is a service man. He lives in a joint family. There are 6 members in his family. He is planning to purchase a car so he is searching for a bank loan. He take a loan of ? 2,50,000 at the interest rate of 6% p.a. compounded monthly which is to be amortize by equal payment at the end of each month for 5 years. Find the size of each monthly payment.
[Given that (1.005)⁶⁰ = 1.3489, (1.005)²¹ = 1.1104]
OR
A bond of par value ₹ 1450 is purchased for 7 years with annual fixed coupon rate of 8.5%. It is given that the coupon payments are made semi-annually. The minimum semi-annual yield that would accept is 8.7%. Find the fair value of a bond. [Given that (1.087)^-14 = 0.3110]

CASE STUDY

Question 14.
Suppose the demand for the certain product is given by p = – 0.01x² – 0.1x + 6, where p is the unit price given in rupees and x is the quantity demanded per month given in the units of 1000. The unit market price for the product is ₹ 4 per unit.

(i) Find the quantity demanded at the given price. (2)
(ii) Find the consumer’s surplus if the market price for the product is ₹ 4 per unit. (2)

Class 12 Applied Mathematics Sample Paper Term 2 Set 8 for Self Assessment

Section – A (2 marks each)

Question 1.
For the marginal revenue function MR = 35 + 7x – 3x², find the revenue function and demand function.
OR
Evaluate:

Question 2.
Find the effective interest rate corresponding to a nominal rate of interest of 8.5% per year compounded monthly. [Given (1.00708)¹² = 1.0884]

Question 3.
The present value of a perpetuity of ₹ 885 payable at the end of each quarter be ₹ 90,250. Find the rate of interest.
OR
Define the terms ‘Bond’ and ‘Discount Rate’.

Question 4.
A server channel monitored for an hour was found a have an estimated mean of 20 transactions transmitted per minute. The variance is known to be 4. Find the standard error.

Question 5.
Assuming no trend, calculate seasonal variation indices for the following data.

Question 6.
Solve the given L.EE
Minimize Z = 3x + by
Subject to constraints:
x, y ≥ 0
x + 3y – 3 ≥ 0
x + y – 2 ≥ 0

Section – B (3 marks each)

Question 7.
Assume that a spherical raindrop evaporates at a rate proportional to its surface area. If its radius
originally is 3 mm and 1 hour later has been reduced to 2 mm, find an expression for the radius of the rain drop at any time.

Question 8.
Compute 7-year and 9-year moving averages for the following data.

OR
Fit a straight line trend equation by the method of least squares and estimate the trend values.

Question 9.
A sample of 900 members has a mean 3.4 cm and SD 2.61 cm. Is the sample taken from a large population with mean 3.25 cm. and SD 2.62 cm? (95% confidence limit)

Question 10.
Calculate C.A.G.R. of unit sales on the basis of given information:

[Given, (1.9811)^{\(\frac{1}{4}\)} = 0.18639]

Section – C (4 marks each)

Question 11.
A company produces soft drinks that have a contract which requires that a minimum of 80 units of chemical A and 60 units of the chemical B go into each bottle of the drink. The chemicals are available in prepared mix packets from two different suppliers. Supplier S had a packet of mix of 4 units of A and 2 units of B that costs ₹ 10. The supplier T has a packet of mix of 1 unit of A and 1 unit of B that costs ₹ 4. How many packets of mixes from S and T should the company purchase to honour the contract requirement and yet maintain the minimum cost ? Make a L.EE and solve graphically.

Question 12.
Find the purchase price of a ₹ 600,8% bond, dividends payable semi-annually redeemable at par in 5 years, if.the yield rate is to be 8% compounded semi-annually. [Given (1.04)^-10 = 0.6755]

Question 13.
In 10 years’, a machine costing ₹ 40,000 will have a salvage value of ₹ 4,000. A new machine at that time is expected to sell for ₹ 52,000. In order to provide funds for the difference between the replacement cost and the salvage cost, a sinking fund is set up into which equal payments are placed at the end of each year. If the fund earn interest at the rate 7% compounded annually, how much should each payment be ? [Given (1.07)¹⁰ = 1.9672]
OR
A couple wishes to purchase a house for ₹ 12,00,000 with a down payment of ₹ 2,50,000. If they can amortize the balance at 9% per annum compounded monthly for 20 years, (i) What is their monthly payment ₹ (ii) What is the total interest paid ? [Given(1.0075)²⁴⁰ = 6.00965]

Case Study

Question 14.
Consumer Surplus can be defined as the surplus that is retained with the consumer after he purchases a product for which he paid lesser than what he was able to. This is the difference between what the consumer pays and what he would have been willing to pay.

Producer Surplus can be defined as the surplus that is retained with the producer after he sells a product for which he accepted more than what he was expected to receive. This is the difference between the price a firm receives and the price it would be willing to sell it at.

Suppose the demand for a product is given by p = d(q) = – 0.8q + 150 and the supply for the same product is given by p = s(q) = 5.2q. For both functions, q is the quantity and p is the price, in ₹.
(i) Find the consumer’s surplus at the equilibrium price. (2)
(ii) Find the producer’s surplus at the equilibrium price. (2)

Class 12 Applied Mathematics Sample Paper Term 2 Set 9 for Self Assessment

Section – A (2 marks each)

Question 1.
A company receives a shipment of 200 cars every 30 days. From experience, it is known that the inventory on hand is related to the number of days. Since, the last shipment, I(x) = 200 – 0.2x. Find the daily holding cost for maintaining inventory for 30 days if the daily holding cost is ₹ 3.5.
OR
In year 2008, World Gold Production was 2547 metric tons and it was growing exponentially at the rate of 0.6% per year. If the growth continues at this rate, how many tons of gold will be produced from 2008 to 2021? [Given, e^0.078 = 1.0811]

Question 2.
How much money is needed to endure a series of lectures costing ₹ 5500 at the beginning of each year indefinitely, if money is worth 4% compounded annually?
OR
At 7.7% converted quarterly, find the present value of a perpetuity of ₹ 8540 payable at the end of each quarter.

Question 3.
Find the present value of an annuity of ₹ 1900 payable at the end of each year for 3 years if money is worth 5.6 % compounded annually. [Given (1.056)^-3 = 0.8492]

Question 4.
The standard deviation of a sample is 6.3. Determine the size of the sample if the standard error is 0.6 and population standard deviation is 6.

Question 5.
Compute the seasonal indices by 3-year moving averages from the given data of production of paper (in thousand tons)

Question 6.
Draw the graph of the following L.P.P.
Maximize: P = 7X + 5Y
Subject to constraints: 4X + 3Y ≤ 240
2X + Y ≤ 100
X ≥ 0, Y ≥ 0

Section – B (3 marks each)

Question 7.
The demand and supply function of a commodity are P_d = 18 – 2x – x² and P_s = 2x – 3 respectively.
Find the producer’s surplus at equilibrium price.

Question 8.
The annual production of a commodity is given as follows:

Fit a straight line trend by the method of least squares.
OR
Calculate 5 yearly and 7 yearly moving average for the following data of the numbers of commercial and industrial failure in a country during 1987 to 2002.

Year	No. of failures
1987	23
1988	26
1989	28
1990	32
1991	20
1992	12
1993	12
1994	10
1995	9
1996	13        –
1997	11
1998	14
1999	12
2000	9
2001	3
2002	1

Question 9.
Consider the following hypothesis test:
H₀ = µ = 15 ,
H₁ = µ =≠ 15
A sample of 50 provided a sample mean of 14.15. The population standard deviation is 3.
(i) Compute the value of the test statistics.
(ii) What is the p-value?
(iii) At α = 0.05, what is your conclusion?

Question 10.
Raveesh invested ?₹ 20,000 in a mutual fund in year 2015. The value of mutual fund increased to ₹ 32,000 in the year 2020. Calculate the annual growth rate of his investment.

Section – C (4 marks each)

Question 11.
A toy company manufactures two types of dolls A and B. Market test and available resources have indicated that the combined production level should not exceed 1200 dolls per week and the demands for the dolls of type B is almost half of that for dolls of type A. Further, the production level of type A can exceed three times the production of dolls of other type by at most 600 units. If the company makes profit of ₹ 12 and ₹ 16 per doll on dolls A and B respectively how many of each type of dolls should be produced weekly, in order to maximise the profit ?

Question 12.
Determine the monthly payment into the sinking fund, the amount is ₹ 10,000, and the interest per pay period is 9% divided by 12, because it is compounded monthly. The number of time periods over the ten years is 120. [Given, (1.0075)¹²⁰ = 2.4514]

Question 13.
A person amortizes a loan of ₹ 1500000 for renovation of his house by 8 years mortgage at the rate of 12% p.a. compounded monthly. Find the equated monthly installment. [Given (1.01)⁹⁶ = 2.5993]
OR
A bond matures in 5 years has coupon rate of 10% per annum and has face value of ₹ 15,000. Find the fair value of bond if the yield to maturity is 8%. [Given (1.08)^-5 = 0.6806]

Case Study

Question 14.
In a college hostel accommodating 1000 students, one of the hostellers came in carrying Corona virus, and the hostel was isolated. The rate at which the virus spread is assumed to be proportional to the product of number of infected students and remaining students. There are 25 infected students after 4 days.

(i) If N(f) is the number of students infected by Corona virus at any time f, then find maximum value of N(f). Also find N(4). Formulate the differential equation in this scenario. (2)
(ii) Find the solution of differential equation formed in the given situation. (2)

Class 12 Applied Mathematics Sample Paper Term 2 Set 10 for Self Assessment

Section – A (2 marks each)

Question 1.
A firm has the marginal revenue function given by MR = \(\frac{a}{(x+b)^{2}}\) – c where x is the output and a, b, c are constants. Show that the demand function is given by x = \(\frac{a}{b(p+c)}\) – b
OR
The marginal cost and marginal revenue with respect to commodity of a firm are given by C (x) = 8 + 6x and R'(x) = 24 respectively. Find the total profit given that the total cost at zero output is zero.

Question 2.
Find the present value of perpetuity of ₹ 1250 at end of each quarter if money is worth 6.6% compounded quarterly.

Question 3.
Find the effective interest rate corresponding to a nominal rate of interest of 9.7% per year compounded semi-annually.
OR
The present value of a perpetuity of ₹ 555 payable at the end of each quarter be ₹ 95,000. Find the rate of interest.

Question 4.
A sampling distribution of the sample means X̄ is formed from a population with mean weight µ = 50 kg and standard deviation σ = 6 kg. What is the expected value and standard deviation of X̄, if sample size is 25?

Question 5.
Assuming no trend, calculate seasonal variation indices for the following data.

Question 6.
A furniture manufacturer makes two products: chairs and tables. Processing of these products is done on two machines A and B. A chair requires 2 hours on machine A and 6 hours on machine B. A table requires 5 hours on machine A and no time on machine B. There are 16 hours per day available on machine A and 30 hours on machine B. Profit gained by the manufacturer from a chair and a table is ₹ 2 and ₹ 10, respectively. Formulate this problem as a linear programming problem to maximize the total profit of the manufacturer.

Section – B (3 marks each)

Question 7.
The demand and supply function of an article are D(q) = 1000 – 0.4q² and S(q) = 42q, respectively. Find the consumer’s surplus at equilibrium price.

Question 8.
Fit a straight line trend by the method of least squares for the following data:

OR

Question 9.
Consider the following hypothesis test:
H₀ = µ < 25
H₁ = µ > 15
A sample of 40 has a sample mean of 26.4. The population standard deviation is 6.
(i) Compute the value of the test statistics.
(ii) What is the p-value ?
(iii) At α = 0.01, what is your conclusion ?

Question 10.
An interviewer gives the following graph on a client’s sales in the last 7 years to candidate and said find the C.A.G.R. Compute the C.A.G.R. for the given data as an interviewee.

Section – C (4 marks each)

Question 11.
A mill owner buys two types of machines A and B for his mill. Machine A occupies 1,000 sq.m of area and requires 12 men to operate it; while machine B occupies 1,200 sq.m of area and requires 8 men to operate it. The owner has 7,600 sq.m of area available and 72 men to operate the machines. If machine A produces 50 units and machine B produces 40 units daily, how many machines of each type should he buy to maximize the daily output? Use Linear Programming to find the solution.

Question 12.
Find the purchase price of a ₹ 500, 5% bond, dividends payable semi-annually redeemable at par in 3 years, if the yield rate is to be 5% compounded semi-annually. [Given (1.025)^-6 = 0.8623]

Question 13.
Roshan needs ₹ 5,000 in three years. Flow much should he deposit each month in an account that pays 8% compounded monthly in order to achieve his goal? [Given, (1.0066)³⁶ = 1.2672]
OR
Mr. Kundan wants to purchase a car for ₹ 25,00,000 with down payment of ₹ 10,00,000. If he can amortize the balance at 8% per annum compounded monthly for 7 years.
(i) What is the monthly payment?
(ii) What is the total interest paid?
[Given (1.0066)⁸⁴ = 1.7474]

Case Study

Question 14.
A thermometer reading 80°F is taken outside. Five minutes later the thermometer reads 60° F. After another 5 minutes the thermometer reads 50° F. At any time t the thermometer reading be T°F and the outside temperature be S°F.

(i) If X is positive constant of proportionality, then find \(\frac{d T}{d t}\). Also, find value of T(5). (2)
(ii) Find the general solution of differential equation formed in given situation.
Also, find the value of constant of integration c in the solution of differential equation formed in given situation. (2)