Category: CBSE Sample Papers

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

CBSE Sample Papers for Class 12 Informatics Practices Term 2 Set 2 with Solutions

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.
Evaluate:

OR
The marginal revenue function for a item is given by MR = 10 + 4x – 3x² – 4x³. Find the demand function.
Answer:
Put, 1 + x² = t
or 2x dx = dt
or x dx = \(\frac{d t}{2}\)

OR

Given, MR = 10 + 4x – 3x² – 4x³
TR = ∫(10 + 4x – 3x² – 4x³) dx
TR = 10x + 2x² – x³ – x⁴ + C
When x = 0, TR = 0, so C = 0
TR = 10x + 2x² – x³ – x⁴
⇒ px = 10x + 2x² – x³ – x⁴
⇒ p = 10 + 2x – x² – x³,
which is the demand function.

Question 2.
Find the present value of perpetuity of ₹ 900 at end of each quarter if money is worth 6 % compounded quarterly.
Answer:
R = ₹ 900
i = \(\frac{0.06}{4}\)
= 0.015
Present value of perpetuity, P = \(\frac{R}{i}\)
⇒ P = \(\frac{900}{0.015}\)
⇒ ₹ 60,000

Question 3.
What effective rate is equivalent to a nominal rate of 7% per annum compounded semi annually? 0 [Use (1.035)² = 1.071225]
OR
Find the present value of an annuity of ₹ 5000 payable at the end of each year for 7 years if money is worth 7% compounded annually. [Given (1.07)^-7 = 0.6227]
Answer:
Since, r_eff = \(\left(1+\frac{r}{m}\right)^{m}\) – 1
∴ r_eff = \(\left(1+\frac{0.07}{2}\right)^{2}\) – 1
= (1.035)² – 1
= 1.071225 – 1
= 0.071225
or 7.12 %
So, effective rate is 7.12 % compounded annually

OR

Present value of ordinary annuity

Question 4.
Find the sample size for the given standard deviation 10 and the standard error with respect of sample mean is 3.
Answer:

= 11.11
≅ 11,
The required sample size is 11.

Question 5.
Find the trend values using 3 yearly moving average for the education loans sanctioned to students for higher studies in abroad.

Answer:

Question 6.
In which quadrant, the bounded region for inequations x + y ≤ 1 and x – y ≤ 1 is situated ?
Answer:

As shown in graph drawn for x + y = 1 and x – y = 1, the origin included in the area. Hence, the bounded region situated in all four quadrants.

Section – B (3 marks each)

Question 7.
If the supply function for a commodity is p = 4 + x, and 12 units of goods are sold, then find the producer’s surplus.
Answer:
Given, the supply function is
p = 4 + x ……. (i)
and the market demand
x₀ = 12.
At equilibrium,
P₀ = 4 + x₀
Substituting this value of x₀ in (i), we get
P₀= 4 + 12
⇒ P₀ = 16
So, producer’s surplus

= 192 – (48 + 72) + 0
= 192 – 120
= 72 units

Question 8.
Calculate four-yearly moving averages of number of students studying in a higher secondary school in a particular city from the following data.

Year	Number of Students
2011	124
2012	120
2013	135
2014	140
2015	145
2016	158
2017	162
2018	170
2019	175

OR
Below are given the figures of production (in m. tonnes) of a rice factory:

(i) Fit the straight line trend to these figures and calculate trend values.
(ii) Estimate the likely sales of the company during 2019.
Answer:
Computation of four-yearly moving averages:

OR

(i) Filling straight line trend

Now, a = \(\frac{\Sigma y}{n}=\frac{630}{7}\) = 90
b = \(\frac{\Sigma x y}{\Sigma x^{2}}=\frac{56}{28}\) = 2
Thus, trend equation is: y_t = 90 + 2x
Trend values are
y₂₀₁₂ = 90 + 2 (- 3) = 84
y₂₀₁₃ = 90 + 2 (- 2) = 86
y₂₀₁₄ = 90 + 2 (- 1) = 88
y₂₀₁₅ = 90 + 2 (0) = 90
y₂₀₁₆ = 90 + 2 (1) = 92
y₂₀₁₇ = 90 + 2 (2) = 94
y₂₀₁₈ = 90 + 2 (2) = 96

(ii) Now,
For 2019, x would be 4.
Putting x = 4 in trend equation, we get
y₂₀₁₉ = 90 + 2(4) = 98 %
∴ Production in 2019 is 98m. tonnes

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 α = 0.05 ? Assume that the populations are normally distributed.
Answer:
Hypothesis H₀: µ₁ = µ₂ and H₀: µ₁ ≠ µ₂ (claim) The test is two-tailed and a = 0.05, also variances are unequal, the degrees of freedom are the smaller of iq – 1 or n2 – 1. In this case, the degrees of freedom are 8 – 1 = 7.

Hence, from t-table F, the critical values are -2.365 and – 2.365.
t = \(\frac{\bar{x}_{1}-\bar{x}_{2}}{\sqrt{\frac{s_{1}}{n_{2}}+\frac{s_{2}}{n_{2}}}}\)
= \(\frac{191-199}{\sqrt{\frac{38^{2}}{8}+\frac{12^{2}}{10}}}\)
= – 0.57
Do not reject the null hypothesis, since – 0.57 > – 2.365.
There is not enough evidence to support the claim that the average size of the farms is different.

Question 10.
Riya invested ₹ 20,000 in a mutual fund in year 2016. The value of mutual fund increased to ₹ 32,000 in year 2021. Calculate the compound annual growth rate of her investment. [Given, log(1.6) = 0.2041, antilog (0.04082) = 1.098]
Answer:
Given beginning value of investment = ₹ 20,000
Final value of the investment = ₹ 32,000
No. of years = 5

Taking log both sides, we get
log x = \(\frac{1}{5}\)log (1.6)
⇒ log x = \(\frac{1}{5}\) × 0.2041
⇒ log x = 0.04082
⇒ x = antilog (0.04082)
= 1.098
C.A.G.R. = 1.098 – 1 = 0.098 = 9.8%

Section – C (4 marks each)

Question 11.
A company manufacturers two types of products A and B. Each unit of A requires 3 gram of nickel and 1 gram of chromium, while each unit of B requires 1 gram of nickel and 2 gram of chromium. The firm can produce 9 gram of nickel and 8 grams of chromium. The profit is ₹ 40 on each unit of product of type A and ₹ 50 on each unit of type B. How many units of each type should the company manufactures so as to earn maximum profit? Use linear programming to find the solution.
Answer:
Let x = Number of units of type A
y = Number of units of type B
Maximize Z = 40x + 50y
Subject to the constraints,
3 x + y < 9
x + 2y ≤ 8
and x ≥, 0, y ≥ 0
Consider the equation,
3x + y = 9

Shaded region in the diagram represents the feasible solution.
Now, we can determine the maximum value of Z by evaluating the value of Z at the four joints (vertices) as shown below.

Vertices	Z = 40x + 50y
(0,0)	Z = 40 × 0 + 50 × 0 = 0
(3,0)	Z = 40 × 3 + 50 × 0 = ₹ 120
(0,4)	Z = 40 × 0 + 50 × 4 = ₹ 200
(2,3)	Z = 40 × 2 + 50 × 3 = ₹ 230 Max

From graph,
Maximum profit, Z = ₹ 230
∴ Number of units of type A is 2 and number of units of type B is 3.

Commonly Made Error: In some cases, all the constraints were not used and hence coordinates of all feasible points were not obtained.

Answering Tip: Students should solve the constraints equations in pairs to obtain all feasible points leading to maximum or minimum value of desired function.

Question 12.
A machine costing ₹ 200000 has effective life 7 years and its scrap value is ₹ 30000. What amount should the company put into a sinking fund earning 5% per annum, so that it can replace the machine after its useful life ? Assume that a new machine will cost ₹ 300000 after 7 years. [Given log (1.05) = 0.0212 and antilog (0.1484) = 1.407]
Answer:
Cost of new machine = ₹ 3,00,000
Scrap value of old machine = ₹ 30,000
Hence,the money required for new machine after 7 years = ₹ 300000 – ₹ 30,000 = ₹ 2,70,000
So, we have A = ₹ 2,70,000

Now,let (1.05)⁷ = x
Taking log both sides, we get
7 log(1.05) = log x
⇒ log x = 7 × 0.0212 = 0.1484
⇒ x = antilog (0.1484) = 1.407
So, (1.05)⁷ = 1.407
Thus, P = \(\frac{270000 \times 0.05}{1.407-1}\)
⇒ P = \(\frac{13500}{0.407}\)
⇒ P = ₹ 33169.53
Hence, the company should deposit ₹ 33170 (Approx) at the end of each year for 7 years.

Question 13.
Calculate the monthly payment on ₹ 2,00,000 at a 6% annual interest rate that is amortized over 10 years. [Given (1.005)¹²⁰ = 1.819]
OR
Seema purchased bonds for 3 years of ₹ 1000 with period coupon rate 10%. The minimum per-annum yield that she would accept is 14%. Find the fair-value of bond. Find the bond value.
Answer:

OR

= 100 × 0.87719 + 100 × 0.769467 + 100 × 0.674972 + 1000 × 0.674972
= 87.719 + 76.947 + 67.479 + 674.97
= 907.125

Case Study

Question 14.
Riya a biologist runs an experiment in the laboratory involving a culture of bacteria. She is doing this experiment first time. She notices that the mass of the bacteria in culture increases exponentially. The bacteria count in the culture is 1,00,000. The rate of growth of bacteria is proportional to the number of bacteria present in the culture. Also, the number of bacteria is increased by 10% in 2 hours.

(i) If y be the number of bacteria at time t, then find the differential equation that models this scenario. Also, find the general solution of the differential equation. (2)
Answer:
If y be the number of bacteria at time t, then
\(\frac{d y}{d t}\) ∝ y
⇒ \(\frac{d y}{d t}\) = ky
\(\frac{d y}{d t}\) = ky
⇒ \(\frac{d y}{y}\) = k dt
Integrating both sides, we get
log y = kt + c

(ii) Find the particular solution of differential equation and in how many hours will the Count reach and in how many hours will the count reach 2,00,000. (2)
Answer:
Put t = 0 and y = 1,00,000 in equation
log y = kt + c, we get
log 1,00,000 = k × 0 + c
⇒ c = log 1,00,000
∴ Particular solution is log y = kt + log 1,00,000
Now, bacteria count after 2 hours i.e.,
When, t = 2 hours
y = 1,00,000 + 10% × 1,00,000
= 1,10,000
Now Putting t = 2, y = 1,10,000
log 1,10,000 = 2k + log 1,00,000
k = \(\frac{1}{2}\) log \(\frac{1,10,000}{1,00,000}\)
= \(\frac{1}{2}\) log \(\frac{11}{10}\)
∴ log y = \(\frac{1}{2}\) log \(\left(\frac{11}{10}\right)\)t + log 1, 00, 000
log 2,00,000 = \(\frac{1}{2}\) log \(\left(\frac{11}{10}\right)\)t + log 1, 00, 000
\(\frac{2 \log 2}{\log \left(\frac{11}{10}\right)}\) = t
= \(\frac{0.69315 \times 2}{0.09531}\)
= 14.55 years (Approx)

Students can access the CBSE Sample Papers for Class 12 Applied Mathematics with Solutions and marking scheme Term 2 Set 3 will help students in understanding the difficulty level of the exam.

CBSE Sample Papers for Class 12 Informatics Practices Term 2 Set 3 with Solutions

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)

Question 1.
Evaluate: ∫¹₀ e^x2 xdx
OR
Find the area of region bounded by the curve x = 2y + 3, Y-axis and the lines y = 1 and y = -1.
Answer:
Let, I = ∫₀¹ e^x2 xdx
Put, x² = t
or
xdx = \(\frac{1}{2}\)dt

Also when x = 0 or t = 0 and when x = 1 ⇒ t = 1
∴ I = \(\frac{1}{2}\) ∫₀¹ e^tdt
or
I = \(\frac{1}{2}\)[e^t]₀¹
= \(\frac{1}{2}\)(e – 1)

OR

From the figure, area of the shaded region,
A = ∫¹_-1 (2y + 3)dy
= [y² + 3y]¹_-1
= [1 + 3 – 1 + 3]
= 6 sq. units

Question 2.
A machine costing ₹ 50,000 has a useful life of 4 years. If the estimated scrap value is ₹ 10,000, then find the annual depreciation.
Answer:
Annual depreciation = \(\frac{\text { Original cost }-\text { Scrap value }}{\text { Useful life }}\)
= \(\frac{50,000-10,000}{4}\)
= \(\frac{40,000}{4}\)
= ₹ 10,000

Question 3.
What is the sum of money needed now, so as to get ₹ 6000 at the beginning of every month forever, when the money is worth 6% per annum compounded monthly?
OR
Assume that Shyam holds a perpetual bond that generates an annual payment of ₹ 500 each year. He believes that the borrower is creditworthy and that an 8% interest rate will be suitable for this bond. Find the present value of this perpetuity.
Answer:
Given R = ₹ 6000
r = \(\frac{6}{12}\)% = 0.5 per month

So, i = \(\frac{0.5}{100}\) = 0.005
So, P = R + \(\frac{R}{i}\)
= 6000 + \(\frac{6000}{0.005}\)
= 6000 + 1200000
= ₹ 1206000

OR

P.V. of perpetuity = \(\frac{\text { Annual Payment/Cash flow }}{\text { Interest rate/yield }}\)

= ₹ 6250

Question 4.
A wholesaler in apples claims that only 4% of the apples supplied by him are defective. A random sample of 600 apples contained 36 defective apples. Calculate the standard error concerning good apples.
Answer:
Sample Size = 600
No. of defective apples = 36
Sample proportion p = \(\frac{36}{600}\) = 0.06
Population proportion P = probability of defective apples = 4% = 0.04
Q = 1 – P
= 1 – 0.04 = 0.96
The S.E. for sample proportion is given by S.E.

Question 5.
Calculate three-yearly moving averages of number of students studying in a higher secondary school in a particular village from the following data.

Year	Number of Students
2011	332
2012	317
2013	357
2014	392
2015	402
2016	405
2017	410
2018	427
2019	435
2020	438

Answer:
Computation of three-yearly moving averages.

Question 6.
Draw the feasible region for given inequation system y ≤ 6, x + y ≤ 3, x ≥ 0, y ≥ 0.
Answer:

Thus, feasible region is OAB.

Section – B (3 marks each)

Question 7.
Find the producer’s surplus defined by the supply curve S(x) = 4x + 8 for the supply of 5 units.
Answer:
Here, S(x) = 4x + 8 and Q_e = 5
P_e = S(5) = 4(5) + 8 = 28

Question 8.
Given below are the data relating to the sales of a product in a district.
Fit a straight line trend by the method of least squares and tabulate the trend values.

OR
From the following data calculate the 4- yearly moving averages and determine the trend values.

Answer:
Computation of trend values by the method of least squares.
In case of EVEN number of years, let us consider
X = \(\frac{(x \text {-Arithimetic mean of two middle years })}{0.5}\)

a = \(\frac{\Sigma Y}{n}=\frac{47.8}{8}\) = 5.975
b = \(\frac{\Sigma X Y}{\Sigma X^{2}}=\frac{8.6}{168}\) = 0.05119
Therefore, the required equation of the straight line trend is given by
Y = a + bX; Y = 5.975 + 0.05119X.

When X = 2015, X= 1995, Y_t = 5.975 + 0.05119\(\left(\frac{2015-2018.5}{0.5}\right)\) = 5.6167
When X = 2016, X = 1996, Y_t = 5.975 + 0.05119\(\left(\frac{2016-2018.5}{0.5}\right)\) = 5.7190
Similarly, other values can be obtained.
OR
Calculation of Trend values by it four yearly Moving Averages:

Question 9.
Ten individuals are chosen at random from the population and their heights are found to be in inches 63, 63, 64, 65, 66, 69, 69, 70, 70, 71. Discuss the freedom value of student’s height and 5 % level of significance is 2.62.
Answer:

x̄ = mean
= \(\frac{\sum x}{n}\)
= \(\frac{670}{10}\) = 67

Now, compute the standard deviation using formula as,

The number of degree of freedom = n – 1 = 9
Given that the tabulated value for 9 d.f. at level of significance is 2.62.

Since calculated value of f is less than the tabulated value i.e., 2.02 < 2.62, the error has arisen due to fluctuations and we may conclude that the data are consistent with the assumption of mean of height in the universe of 65 inches.

Question 10.
A company ABC Ltd. 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 that (1.025)²⁰ = 1.6386.
Answer:
Sinking Fund, A = ₹ 1,000 x 1000 = ₹ 1,000,000, r = 5% or 0.05, No. of years, n = 10 years and No. of payments per year, m = 2 (Half Yearly)

Periodic Contribution,

Therefore, the company will be required to contribute a sum of ₹ 39,148 half-yearly in order to build the sinking fund to retire the zero-coupon bonds after 10 years.

Section – C (4 marks each)

Question 11.
A farmer has a supply of chemical fertilizer of type A which contains 10% nitrogen and 6% phosphoric acid and of type B which contains 5% nitrogen and 10% phosphoric acid. After soil test, it is found that atleast 7 kg of nitrogen and same quantity of phosphoric acid is required for a good crop. The fertilizer of type A costs ₹ 5.00 per kg and the type B costs ₹ 8.00 per kg. Using Linear Programming, find how many kilograms of each type of the fertilizer should be bought to meet the requirement and for the cost to be minimum. Find the feasible region in the graph.
Answer:
Let the fertilizer of type A be x kg and type 3 be ykg.

According to Question:

∴ Min.Z = 5x + 8y
Subject to constraints
2x + y ≥ 140
3x + 5y ≥ 350
x ≥ 0, y ≥ 0
2x + y = 140 ……..(i)

3x + 5y = 350 ……………(ii)

On solving eq (i) and (ii), we get x = 50 and y – 40

We draw the lines 2x + y = 140 and 3x + 5y = 350 and obtain the feasible region (unbounded and convex) as shown in figure. Thus, corner points are A(0, 140), B(50, 40),

The values of Z at these points are given in following table:

x	114	0
y	0	71.25

Since, there is no common point between the feasible region and 5x + 8y < 570
Hence, the cost will be minimum, if Fertilizer of type A used = 50 kg
Fertilizer of type B used = 40 kg
Minimum cost = ₹ 570

Commonly Made Error:
Many candidates made errors in calculating the conditions using constraints. Some candidates were unable to represent feasible region on the graph as such they may not have clarity of the concept of solving linear equations graphically. Some candidates solved it in terms of decimals and mostly went incorrect in simplifications.

Answering Tip:
The optimum function and all possible constraints in the form of inequations must be put down from what is stated in the problem.

Question 12.
Find the value of a ₹ 1,000 corporate bond with an annual interest rate of 5%, making semi-annual interest payments for 2 years, after which the bond matures and the principal must be repaid. Assume a Y.T.M. (yield to maturity) of 3%.
Answer:
Given, P = ₹ 1000 for corporate bond Annual Coupon Payment
= ₹ 1,000 × 5% = ₹ 50

Semi-annual Coupon Payment,
C = ₹ 50 – 2 = ₹ 25
N = 2 years × 2
= 4 periods for semi-annual coupon payments
r = Y.M.T.
= 3% = 0.03

Commonly Made Error
Sometimes there were calculation mistakes while simplification.

Answering Tip
When computing the bond value, you may have to round off the bond value up to nearest hundred to ensure that enough money in the given amount of time. Always recheck your solution.

Question 13.
The cost of a T.V depreciates by ₹ 800 during the second year and by ₹ 700 during the third year. Calculate:
(i) the rate of depreciation per annum.
(ii) the original cost of the T.V.
(ii) the value of the T.V at the end of third year.
OR
A machine costs ₹ 97,000 and its effective life is estimated to be 12 years. If scrap realises ₹ 2,000 only, what amount should be retained out of profits at the end of each year to accumulate at compound interest of 5% per annum in order to buy a new machine after 12 years ? (use 1.0512 = 1.769)
Answer:
(i) Let the original cost of the T.V. be ₹ P and the rate of depreciation be r% p.a. Then the value of T.V (in ₹) after 1 year, 2 years and 3 years are P(1 – i), P(1 – i)² and P(1 – i)³, respectively,
Where i = \(\frac{r}{100}\)

According to Question,
P(1 – i) – P(1 – i)² = 800
and P(1 – i)² – P(1 – i)3 = 700
⇒ P(1 – i)²[1 – (1 – i)] = 800
and P(1 – i)² [1 – (1 – i)] = 700
⇒ P(1 – i)i = 800 ……….(i)
and P(1 – i)²i = 700 …(ii)

On dividing eq. (ii) by eq (i), we get

Hence, the rate of depreciation = 12.5% p.a.

(ii) Putting i = \(\frac{1}{8}\) in eq. (i), we get

Hence, the original cost of TV is ₹ 7314.

(iii) The value of TV at the end of 3rd = P(1 – i)³

Thus, the value of TV at the end of the 3rd year is ₹ 4900.
OR
Cost of machine = ₹ 97,000
Value of scrap = ₹ 2,000
Required money = ₹ 95,000

Now, M = \(\frac{A}{r}\)95,000, n = 12, r = 5% = 0.05

Case Study

Question 14.
Mr. Lai was 75 year old businessman. He had two sons and three daughters. On the silver jubilee of his company, he gifted an equal amount of money among his children. One of his daughters buys jewellery from the gifted money whereas the other daughter invested his money in bank.
It is known that if the interest is compounded continuously, the principal changes at the rate equal to the product of the rate of bank interest per annum and principal.

(i) If P denotes the principal at time t and the rate of interest be r% per annum compounded continuously, then find the differential equation that models this scenario. Also, find the particular solution of the differential equation. (2)
Answer:
If P denotes the principal at time t and the rate of interest be r% per annum compounded continuously, then according to the law given in the problem, we get

Let P0 be the initial principal i.e. at t = 0, P = P₀.
Putting P = P₀ in eq. (i),
we get log P₀ = C
Putting C = log P₀ in eq. (i), we get
log P = \(\frac{r t}{100}\) + log P₀
⇒ log\(\left(\frac{P}{P_{0}}\right)\) = \(\frac{r t}{100}\)

(ii) At what rate of interest will X 100 double itself in 10 years ? (2)
(Use log_e2 = 0.6931)
Answer:
We have,
log\(\left(\frac{P}{P_{0}}\right)\) = \(\frac{r t}{100}\)
Put t = 10, P₀ = ₹ 100 and P = ₹ 200 = 2P₀ in the above equation, we get
log 2 = \(\frac{10r}{100}\)
⇒ r = 10 log_e2
= 10 × 0.6931
= 6.931% or 6.93% per annum (Approx)

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)

Students can access the CBSE Sample Papers for Class 12 Applied Mathematics with Solutions and marking scheme Term 2 Set 1 will help students in understanding the difficulty level of the exam.

CBSE Sample Papers for Class 12 Informatics Practices Term 2 Set 1 with Solutions

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)

Question 1.
The marginal revenue function for a commodity is given by MR = 9 + 2x – 6x². Find the demand function.
OR
The marginal cost of producing x pairs of tennis shoes is given by the following function:
MC = 50 + \(\frac{300}{x+1}\)
If the fixed cost is ₹ 2000, then find the total cost function.
Answer:
Given, MR = 9 + 2x – 6x²
Let the demand function be p,
We know that,
TR =f(x + 2x – 6x²)dx
TR = 9x + x² – 2x³+ C
where C is Integration Constant
When x = 0, TR = 0, so C = 0
TR = 9x + x² – 2x³ = px
⇒ px = 9x + x² – 2x³
⇒ p = 9 + x – 2x²
which is the required demand function.

OR

Let the total cost function be TC
TC = ∫(50 + \(\frac{300}{x+1}\))dx
TC = 50x + 300log|x + 1| + C
If x = 0, TC = ₹ 2000
So 2000 = 300(log 1) + C
⇒ C = 2000
So TC = 50x + 300log(x + 1) + 2000
Hence, the required total cost function
TC = 50x + 300log(x + 1) + 2000

Question 2.
Find the present value of a perpetuity of ₹ 600 at end of each quarter, if money is worth 8% compounded quarterly.
Answer:
Let the present value of perpetuity be P
Given, R = ₹ 600
i = \(\frac{0.08}{4}\) = 0.02
Present value of perpetuity = P = \(\frac{R}{i}\)
⇒ P = \(\frac{600}{0.02}\) = ₹ 30.000

Question 3.
What effective rate is equivalent to a nominal rate of 8% per annum compounded quarterly?
OR
Find the present value of an annuity of ₹ 1000, payable at the end of each year for 5 years, if money is worth 6%compounded annually. [Given (1.06)^-5 = 0.7473]
Answer:
Let the effective rate be r(subscript){eff}
Since, r_eff = (1 + \(\frac{r}{m}\))^m – 1
(Given r = 0.08)
= (1 + \(\frac{0.08}{4}\))⁴ – 1
= (1.02)⁴ – 1
= 0.0824 or 8.24%

So effective rate is 8.24% compounded quarterly.
OR
Present value of ordinary annuity

Question 4.
A sampling distribution of the sample means X̄ is formed from a population with mean weight μ = 60 kg and standard deviation σ = 9kg. What is the expected value and standard deviation of X̄ t if sample size is 36?
Answer:
Let the expected value be E, standard deviation be SD and sample size be n
√n = E(X̄) = 6kg
Standard deviation of
X̄ = SD(X̄)
= \(\frac{\sigma}{\sqrt{n}}=\frac{9}{6}\)
= 1.5kg

Question 5.
Find the trend values using 3 yearly moving average for the loans sanctioned to farmers by a particular branch of a bank in a village.
The data is given as follows:

Answer:

Question 6.
The feasible region of the L.PP Min Z = 3x + 2y subject to constraints
2x + y ≥ 6,x – y ≥ 0, x ≥ 0, y ≥ 0 is given below:

Determine the optimal solution Justify your answer.
Answer:
The feasible region is given as follows:

The values at the corner points P and Q can be shown as:

Corner points	Z = 3x + 2y
P (2, 2)	10
Q (3, 0)	9

The smallest value of Z is 9. Since, the feasible region is unbounded, we draw the graph of 3x + 2y < 9. The resulting open half plane has points common with feasible region, therefore Z = 9 is not the minimum value of Z. Hence, the optimal solution does not exist.

Section – B (3 marks each)

Question 7.
The supply function for a commodity is given by p = x² + 4x + 3, where x is the quantity supplied at the price p. Find the producer’s surplus, when the price of the commodity is ₹ 48.
Answer:
Substituting, p₀ = ₹ 48 in p = x² + 4x + 3 We get x₀ = 5

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

OR
Fit a straight line trend by the method of least squares and also estimate the trend for the year 2023.

Answer:

The trend value are given by 4 quarterly centered moving average.
OR

a = \(\frac{\Sigma Y}{n}=\frac{532}{7}\) = 76
b = \(\frac{\Sigma X Y}{\Sigma X^{2}}=\frac{672}{28}\) = 24

Y_C = a + bX, Y_C = 76 + 24X
Estimated sales = Y_C for 2023 = 76 + 24 × 6 = ₹ 220 lacs

Question 9.
A machine produces washers of thickness 0.50 mm. To determine whether the machine is in proper working order, a sample of 10 washers is chosen for which the mean thickness is 0.53 mm and the standard deviation is 0.03 mm. Test the hypothesis at 5% level of significance so that the machine is working in proper order. [Given critical value, t_0.025 = 2.262 at v(d.f) = 9]
Answer:
Define Null hypothesis H0 and alternate hypothesis Hx as follows:
H₀: μ = 0.50 mm
H₁ : μ ≠ 0.50 mm
Thus, a two-tailed test is applied under hypothesis H₀,

We have,
t = \(\frac{\bar{X}-\mu}{S} \sqrt{n-1}\)
= \(\frac{0.53-0.50}{0.03}\) × 3 = 3

Since, the calculated value of t = 3 does not lie in the interval – t_0.025 to t_0.025 i.e., – 2.262 to 2.262 for 10 – 1 = 9 degree of freedom. So we reject H₀ at 0.05 level. Hence, we conclude that machine is not working properly.

Question 10.
A person invested ₹ 15000 in a mutual fund and the value of investment at the time of redemption was ₹ 25000. If CAGR for this investment is 8.88%, then calculate the time period for which the given amount was invested? [Given log(1.667) = 0.2219 & log(1.089) = 0.037]
Answer:
We Know
CAGR = \(\left[\left(\frac{F V}{I V}\right)^{\frac{1}{n}}-1\right]\) × 100

where, IV = Initial value of investment
FV = Final value of investment
⇒ 8.88 = \(\left[\left(\frac{25000}{15000}\right)^{\frac{1}{n}}-1\right]\) × 100
⇒ 0.0888 = (\(\frac{5}{3}\))^1/n – 1
⇒ 1.089 = (1.667)^1/n
⇒ \(\frac{1}{n}\)log(1 .667) = log(1 .089)
⇒ n(0.037) = 0.2219
⇒ n = 5.99 ≈ 6 years

Section – C (4 marks each)

Question 11.
S & D chemicals produces two products, an alkaline solution and a base oil that are sold as raw material to the companies manufacturing soaps and detergents. On the basis of current inventory levels and estimated demand for the coming month, S & D’s management has decided that combined production of alkaline solution and the base oil must be at least 3500 gallons. S & D chemicals are also committed to supply 1250 gallons of alkaline solution to one of its major customer. The alkaline solution and base oil requires respectively 2 hours and 1 hour of processing time per gallon. The total processing time available for the coming month is 6000 hours. The production cost is ₹ 200 per gallon for the alkaline solution and ₹ 300 per gallon for the base oil.

Formulate the above as an L.EP and solve it by using graphical method, to help S & D chemicals for determining the minimum production cost.
Answer:
Let the company produces x and ij gallons of alkaline solution and base oil respectively, also
let C be the production cost.
Min C = 200x + 300y

Subject to constraints:
x + y ≥ 3500 …(1)
x ≥ 1250 …(2)
2x + y ≤ 6000 …(3)
x,y ≥ 0

Corner points	C = 200x + 300y
P (1250,2250)	₹ 9,25,000
Q (1250, 3500)	₹ 13,00,000
R (2500,1000)	₹ 8,00,000

Minimum cost is 8,00,000 when 2500 gallons of alkaline solutions & 1000 gallons of base oil are manufactured.

Question 12.
A machine costing ₹ 50,000 is to be replaced at the end of 10 years, when it will have a salvage value of ₹ 5000. In order to provide money at that time for a machine costing the same amount, a sinking fund is set up. If equal payments are placed in the fund at the end of each quarter and the fund earns at the rate of 8% compounded quarterly, then what should each payment be? [Given (1.02)⁴⁰ = 2.208]
Answer:
The amount of sinking fund S at any time is given by
S = R\(\left[\frac{(1+i)^{n}-1}{i}\right]\)
where R = periodic payment,
i = Interest per period,
n = number of payments
S = Cost of machine – Salvage value
= ₹ 50,000 – ₹ 5000 = ₹ 45,000
i = \(\frac{8 \%}{4}\) = 0.02
⇒ 45000 = R\(\left[\frac{(1+0.02)^{40}-1}{0.02}\right]\)
⇒ 45000 = R\(\left[\frac{2.208-1}{0.02}\right]\)
⇒ R = \(\frac{900}{1.208}\)
⇒ R = ₹ 745.03

Question 13.
A couple wishes to purchase a house for ₹ 15,00,000 with a down payment of ₹ 4,00,000. If they can amortize the balance at an interest rate of 9% per annum compounded monthly for 10 years, then find the monthly installment (EMI). Also find the total interest paid. [Given (1.0075)^-120 = 0.4079 ]
OR
A X 2000,8% bond is redeemable at the end of 10 years at ₹ 105. Find the purchase price to yield 10% effective rate. [Given (1.1)^-10 = 0.3855]
Answer:
P = Cost of house – Cash down payments
P = ₹ 15,00,000 – ₹ 4,00,000 = ₹ 11,00,000

Total interest paid = nR – R
= 13933.5 × 120 – 11,00,000 = ₹ 5,72,020

OR

Face value of bond, F = ₹ 2000
Redemption value C = 1.05 × 2000 = ₹ 2100

Nominal rate = 8%
R = C × id = 2000 × 0.08 = ₹ 160

Number of periods before redemption i.e., n = 10
Annual yield rate, i =10% or 0.1

Purchase price,
V = R\(\left[\frac{1-(1+i)^{-n}}{i}\right]\) + C(1 + i)^-n
= 160\(\left[\frac{1-(1+0.1)^{-10}}{0.1}\right]\) + 2100(1 + 0.1)^-10
= 160 × 6.14 + 2100 × 0.3855
= ₹ 1792
Thus, present value of the bond is ₹ 1792

Case Study

Question 14.
General anaesthesia is used for major operations to cure the patients and conduct pain free surgeries. Propofol is a commonly used anaesthetic injected for major operations such as knee replacement or open heart surgery. It also acts as a sedative and an analgesic.

A patient is rushed to operation theatre for a 2-hour cardiac surgery. A person is anaesthetized when its blood stream contains at least 3 mg of propofol per kg of body weight. The rate of change of propofol (x), in the body is proportional to the quantity of propofol present at that time. Based on the above information. Answer the following questions:
(i) Show that the propofol given intravenously is eliminated exponentially from the patients’ blood stream. (2)
(ii) What dose of propofol should be injected, to induce unconsciousness in a 50 kg adult for a two hours operation? (2)
(Given (2)^1/5 = 1.1487 & assume half-life of propofol = 5 hours )
Answer:
∵ \(\frac{d x}{d t}\) ∝ x,
\(\frac{d x}{d t}\) = -kx
⇒ ∫\(\frac{d x}{d t}\) = ∫-kdt
⇒ log x = -kt + C
⇒ x = e^-kt+c
⇒ x = λ^-kt where e⁰ = λ

(i) Let x = x₀ at t = 0
∵ x₀ = λ
x = x₀ e^-kt
where x₀ = original quantity

(ii) Let the half life be t and amount of propofol be x₀
x = x₀ e^-kt …(i)
Now, \(\) = x₀ e^-5k ( half life = 5 hours)
⇒ e_-5k = \(\frac{1}{2}\)
⇒ e^k = 2^1/5

The quantity of propofol needed in a 50 kg adult at the end of
2 hours = 50 × 3 = 150 mg
⇒ 150 = x₀e^-2k [using ..(i)]
⇒ x₀ = 150 (e^k)²
⇒ x₀ = 150(2^1/5)2 = 150 × 1.3195
x₀ = 197.93mg
So, 197.53 mg of propofol is needed.

Students can access the CBSE Sample Papers for Class 12 Biology with Solutions and marking scheme Term 2 Set 8 will help students in understanding the difficulty level of the exam.

CBSE Sample Papers for Class 12 Biology Standard Term 2 Set 8 with Solutions

Time Allowed: 2 Hours
Maximum Marks: 40

General Instructions:

• All questions are compulsory.
• The question paper has three sections and 13 questions. All questions are compulsory.
• Section-A has 6 questions of 2 marks each; Section-B has 6 questions of 3 marks each; and Section-C has a case-based question of 5 marks.
• There is no overall choice. However, internal choices have been provided in some questions. A student has to attempt only one of the alternatives in such questions.
• Wherever necessary, neat and properly labeled diagrams should be drawn.

Section – A

Question 1.
Humans have innate immunity for protection against pathogens that may enter the gut along with food.
What are the two barriers that protect the body from such pathogens? [2]
Answer:
Pathogens enter the body through food and water and the two barriers that protect the body from such pathogens are:

• Physical barriers: Mucus coating of the epithelium lining the gastrointestinal tract and respiratory tract helps in trapping microbes entering our body.
• Physiological barriers: Low pH in stomach and presence of bacteriolytic lysozyme in tears, saliva and other secretions of body prevent the growth of bacteria.

Question 2.
A patient admitted in ICLI was diagnosed to have suffered from myocardial infarction. The condition of coronary artery is depicted in the image below.
Name two bioactive agents and their mode of action that can improve this condition.

Substantiate by giving two reasons as to why a holistic understanding of the flora and fauna the cropland is required before introducing an appropriate biocontrol method. [2]
Answer:
Myocardial infarction commonly known as a heart attack, occurs when there is insufficient supply of blood to some parts of heart, causing damage to the heart muscle.
The two bioactive agents and their mode of action that can improve this condition are:
(1) Streptokinase (produced by the bacterium Streptococcus) is used as a ‘clot buster’ for removing clots from the blood vessels of patients who have undergone myocardial infarction.
(2) Statins (produced by the yeast Monascus purpureus) act as blood cholesterol lowering agents.
OR
Eradication of pests will disrupt predator-prey relationships, where beneficial predatory and parasitic insects which depend upon flora and fauna as food or hosts, may not be able to survive. Holistic approach ensures that various life forms that inhabit the field, their life cycles, patterns of feeding and the habitats that they prefer are extensively studied and considered.

Question 3.
Identify the compound having chemical structure as shown below. State any three of its physical properties. [2]

Answer:
The chemical structure is shown above is Morphine (alkaloid).
The physical properties are: It appears as a white, odourless, crystalline compound.

Question 4.
Water samples were collected at points A, B and C in a segment of a river near a sugar factory and tested for BOD level. The BOD levels of samples A, B and C were 400 mg/L, 480 mg/L and 8 mg/L respectively. What is this indicative of? Explain why the BOD level gets reduced considerably at the collection point C? [2]

Answer:
At collection points A and B, the BOD level is high due to high organic pollution caused by sugar factory and sewage discharge in the river. At the collection point C, the water was released after secondary treatment/ biological treatment (where vigorous growth of useful aerobic microbes into floes consume the major part of the organic matter present in the river water or effluent due to sugar factory and sewage discharge) which is clearly indicated by the reduced BOD.

Question 5.
An ecologist studied an area with population A, thriving on unlimited resources and showing exponential growth and introduced population B and C to the same area.
What will be the effect on the growth pattern of the population A, B and C when living together in the same habitat? [2]
Answer:
This interaction of the populations A, B and C will lead to competition for resources. Since the supply of resources is limited for the combined needs of all individuals, natural selection plays an important role and eventually the ‘fittest’ individuals will survive and reproduce.

Since the resources for growth will become finite and limiting, and population growth will no longer remain exponential but becomes realistic.

Question 6.
With the decline in the population of fig species it was noticed that the population of wasp species also started to decline. What is the relationship between the two and what could be the possible reason for decline of wasps?
OR
With the increase in the global temperature, the inhabitants of Antarctica are facing fluctuations in the temperature. Out of the regulators and the conformers, which of the two will have better chances of survival?
Give two adaptations that support them to survive in the ambient environment? Give one suitable example. [2]
Answer:
A mutualistic relationship is established between two organisms of different species that work together or live together and each is getting benefitted from the relationship. This is the example of the relationship between the plant and pollinator. Fig depends on wasp for pollination, and wasp depends on fig for obtaining food and shelter. With the decline in population of figs, wasp loses its source of food and shelter while if wasp is declined the fig will not get pollinated.
OR
Conformers: They are the organisms which cannot maintain a constant internal environment with respect to changing external environmental conditions like body temperature and osmotic concentration. Eg. fishes, etc.

Regulators: Organisms which can maintain homeostasis, maintain constant body temperature and osmotic concentration. Eg. mammals.

Hence regulators have better survival chances. The two adaptations that support them to survive in the ambient environment is thermoregulation (maintaining constant body temperature) and osmoregulation (maintaining osmotic pressure). E.g. Birds or mammals.

SECTION – B

Question 7.
How do normal cells get transformed into cancerous neoplastic cells? Elaborate giving three examples of inducing agent.
OR
A person is suffering from a high-grade fever. Which symptoms will help to identify if he/she is suffering from Typhoid, Pneumonia or Malaria? [3]
Answer:
Normal cells get transformed into cancerous neoplastic cells due to the effect of carcinogens.
Carcinogenic agents could be physical, chemical or biological agents that changes the normal sequence of DNA. They are:

• Ionising radiations like X-rays and gamma rays
• Non-ionizing radiations like UV.
• Chemical carcinogens present in tobacco smoke
• Cellular oncogenes (c-onc) or proto-oncogenes, when activated under certain conditions cause cancer. Viruses with oncogenes can transform normal cells to cancerous cells.

OR
Typhoid which is caused by Salmonella typhi and transmitted through contaminated food and water. The person suffering from typhoid has sustained high fever (39° to 40°C), weakness, stomach pain, constipation, headache and loss of appetite.

Pneumonia is caused by Diplococcus pneumonia and spread through droplet infection . The person suffering from pneumonia has fever, chills, cough and headache; and the lips and fingernails turn gray to bluish.

Malaria is transmitted by the bite of female anopheles mosquito and is caused by Plasmodium. The person suffering from malaria has chills and high fever recurring every three to four days.

Question 8.
Recognition of an antigenic protein of a pathogen or exposure to a pathogen occurs during many types of immune responses, including active immunity and induced active immunity.
Specify the types of responses elicited when human beings get encountered by a pathogen. [3]
Answer:
When our body encounters an antigenic protein or a pathogen for the first time it produces a response which is of low intensity and our body retains memory of the first encounter.

• The subsequent encounter with the same pathogen elicits a highly intensified response called as anamnestic response and is carried out with the help of two special types of lymphocytes present in our blood, B lymphocytes, and T-lymphocytes.
• The B-lymphocytes produce an army of proteins in response to these pathogens into our blood to fight with them. These proteins are called antibodies. Antibodies are specific against specific pathogens. The T-cells themselves do not secrete antibodies but help B-cells produce them and also play an important role in graft rejection.

Question 9.
In a pathological lab, a series of steps were undertaken for finding the gene of interest. Describe the steps, or make a flow chart showing the process of amplification of this gene of interest. [3]
Answer:
PCR is the technique of amplification of small fragment of DNA. The technique was developed by Kary
Mullis in the 1980s.
The flow chart shows the three steps involved in the process of PCR showing below:
(i) Denaturation: The DNA strands are treated with a temperature of 94°C (Heat) which results in denaturation of DNA caused due to break down of hydrogen bonds which separates double- stranded DNA to single-stranded DNA. The temperature at which 50% of the dsDNA is denatured is known as the melting temperature (Tm) and is determined by G + C content, the length of the sample, and the concentration of ions (Mg2+).

(ii) Annealing: The primers are added which get anneal to the complementary strands and provides the site of attachment of DNA polymerase. During this step, the sample is cooled to 40-60°C.

(iii) Extension: The DNA polymerase (Taq DNA polymerase) facilitates the extension of both the strands of DNA resulting in millions of copies in few cycles. This final step occurs at 70-75°C. At this temperature taq polymerase can synthesize and elongate the target DNA quickly and accurately.

Question 10.
‘The Evil Quartet’ describes the rates of species extinction due to human activities. Explain how the population of organisms is affected by fragmentation of the habitats.
Introduction of alien species has led to environmental damage and decline of indigenous species. Give any one example of how it has affected the indigenous species?
Could the extinction of Steller’s sea cow and passenger pigeon be saved by man? Give reasons to support your answer. [3]
Answer:
(a) The Evil Quartet is the concept which explains the reasons which cause a decrease in the number of species. The reasons are overexploitation, loss of habitat, introduction of the exotic species and co- extinction of species.
When a large habitat is broken into small fragments due to various activities, mammals and birds requiring large territories and certain animals with migratory habitats are badly affected, leading to population decline.

(b) Invasive alien species are plants, animals that are non-native to an ecosystem, may cause environmental harm or may pose an adverse impact upon biodiversity, including decline or elimination of native species – through competition, predation, or transmission of pathogens and the disruption of local ecosystems and ecosystem functions.
E.g., Nile perch introduced in lake Victoria eventually led to the extinction of an ecologically unique assemblage of more than 200 species of cichild fish. Parthenium/Lantana/water hyacinth caused environmental damage and threat to our native species. African catfish-C/arias gariepinus introduced for aquaculture purposes is posing a threat to the indigenous catfishes in our rivers.

(c) Yes; the extinction of Steller’s sea cow and passenger pigeon could be saved by man. Humans have overexploited natural resources for their ‘greed’ rather than ‘need’ leading to extinction of these animals. Sustainable harvesting could have prevented extinction of these species.

Question 11.
(a) The image shown below is of a sacred grove found in India. Explain how has human involvement helped in the preservation of these biodiversity rich regions.

(b) Value of Z (regression coefficient) is considered for measuring the species richness of an area. If the value of Z is 0.7 for area A ,and 0.15 for area B, which area has higher species richness and a steeper slope?
Answer:
(a) India’s history of religious and cultural traditions emphasized the protection of nature. In many cultures, tracts of forest are set aside, all the trees and wildlife within are venerated and given total protection. Sacred groves in many states are the last refuges for a large number of rare and threatened plants.
(b) The value of regression coefficient is 0.7 for area A and thus Area A will have more species richness and a steeper slope. [3]

Question 12.
The image below depicts the result of gel electrophoresis

If the ladder represents sequence length upto 3000 base pairs (bp),
(a) Which of the bands (I – IV) correspond to 2500 bp and 100 bp respectively?
(b) Explain the basis of this kind of separation and also mention the significance of this process. [3]
Answer:
(a) The maximum length of ladder of gel shown is 3000 bp. Band III corresponds to 2500 base pairs, and Band IV corresponds to 100bp.

(b) The fragments will resolve according to their size. The shorter sequence fragments would move farthest from well as seen in Band IV (100 bp) which is lighter as compared to Band III which is heavier being 2500 base pairs.
The significance of electrophoresis is to purify the DNA fragments for use in constructing recombinant DNA by joining them with cloning vectors.

SECTION – C

Question 13.
Some restriction enzymes break a phosphodiester bond on both the DNA strands, such that only one end of each molecule is cut and these ends have regions of single stranded DNA. BamHl is one such restriction enzyme which binds at the recognition sequence, 5′-GGATCC- 3′ and cleaves these sequences just after the 5′- guanine on each strand.
(a) What is the objective of this action?
(b) Explain how the gene of interest is introduced into a vector.
(c) You are given the DNA shown below.
5′ ATTTTGAGGATCCGTAATGTCCT 3′
3′ TAAAACTCCTAGGCATTACAGGA 5′
If this DNA was cut with BamHI, how many DNA fragments would you expect? Write the sequence of these double-stranded DNA fragments with their respective polarity.
(d) A gene M was introduced into E.coli cloning vector pBR322 at BamHI site. What will be its impact on the recombinant plasmids? Give a possible way by which you could differentiate non recombinant to recombinant plasmids. [5]
OR
GM crops especially Bt crops are known to have higher resistance to pest attacks. To substantiate this, an experimental study was conducted in 4 different farmlands growing Bt and non Bt-Cotton crops. The farm lands had the same dimensions, fertility and were under similar climatic conditions. The histogram below shows the usage of pesticides on Bt crops and non-Bt crops in these farm lands.

OR
(a) Which of the above 4 farm lands has successfully applied the concepts of biotechnology to show
better management practices and use of agrochemicals? If you had to ci±ltivate, which crop would
you prefer (Bt or non- Bt) and why?
(b) Cotton boliworms were introduced in another experimental study on the above farm lands wherein
no pesticide was used. Explain what effect would a Bt and non Bt crop have on the pest.
Answer:
(a) The two different DNA molecules will have compatible ends to recombine when cut with same
restriction enzymes.

(b) Restriction enzyme cuts the DNA of the vector and then ligates the gene of interest into the DNA o1
the vector.

(c) 2 fragments
5’ A1TFTGAG 3’5’GATCCGTAATGTCCT 3’
3’ TAAAACTCCTAG 5’.3’GCATI’ACAGGA 5’

(d) BamH1 site will affect tetracycline antibiotic resistance gene, hence the recombinant plasmids will
lose tetracycline resistance due to inactivation of the resistance gene.
Recombinants can be selected from non-recombinants by plating into a medium containing
tetracycline, as the recombinants will not grow in the medium because the tetracycline resistance
gene is cut and non-recombinants will grow in medium containing tetracycline.
OR
(a) Farm land II requires less application of pesticide.
I would prefer Bt crop as it involves no application of fertilizers and pesticides and shows more
yield when compared to non Bt crop.
Because the use of pesticides is highly reduced for Bt crop. Decrease of pesticide used is also more
significant for Bt crop.

(b) In Bt cotton a cry gene has been introduced from bacterium Bacillus thuringiensis (Bt) which causes
synthesis of a toxic protein. This protein becomes active in the alkaline gut of boliworm feeding
on cotton, punching holes in the lining of the gut, causing death of the insect. However; a non-Bt
crop will have no effect on the cotton bollworm and the yield of cotton will decrease, non-Bt will
succumb to pest attack.