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# Board Paper of Class 12-Humanities Term-II 2022 Applied Math Delhi(Set 4) - Solutions

General Instructions:
Read the following instructions very carefully and strictly follow them:
1. This question paper contains three Sections - A, B and C.
2. Each section is compulsory.
3. Section - A has 6 short-answer type-I questions of 2 marks each.
4. Section - B has 4 short answer type-II questions of 3 marks each.
5. Section - C has 4 long-answer type questions of 4 marks each.
6. There is an internal choice in some questions.
7. Question 14 is a case study based question with two subparts of 2 marks each.

• Question 1
Evaluate: $\underset{0}{\overset{1}{\int }}\frac{x{e}^{x}}{{\left(x+1\right)}^{2}}dx$

OR

Solve the following differential equation : $\frac{dy}{dx}={e}^{x+y}+{x}^{2}{e}^{y}$ VIEW SOLUTION

• Question 2
Find the present value of a perpetuity of ₹18,000 payable at the end of 6 months, if the money is worth 8% p.a. compounded semi-annually. VIEW SOLUTION

• Question 3
Find the effective rate which is equivalent to nominal rate of 10% p.a. compounded monthly.
[Given that : (1.00833)12 = 1.1047]

OR

Abhay bought a mobile phone for ₹30,000. The mobile phone is estimated to have a scrap value of ₹3,000 after a span of 3 years. Using the linear depreciation method, find the book value of the mobile phone at the end of 2 years. VIEW SOLUTION

• Question 4
Consider the following hypothesis:
H0 : μ = 35
H1 : μ ≠ 35
A sample of 81 items is taken whose mean is 37∙5 and the standard deviation is 5. Test the hypothesis at 5% level of significance.
[Given: Critical value of Z for a two-tailed test at 5% level of significance is 1∙96] VIEW SOLUTION

• Question 5
The following table shows the annual rainfall (in mm) recorded for Cherrapunji, Meghalaya:
 Year Rainfall (in mm) 2001 1∙2 2002 1∙9 2003 2 2004 1∙4 2005 2∙1 2006 1∙3 2007 1∙8 2008 1∙1 2009 1∙3

Determine the trend of rainfall by 3-year moving average.
VIEW SOLUTION

• Question 6
Maximize z = 3x + 4y, if possible,
Subject to the constraints:
x≤ –1
x + y  ≤ 0
x, y ≥ 0 VIEW SOLUTION

• Question 7
Find: $\int \frac{2{x}^{2}+1}{{x}^{2}–3x+2}dx$

OR

The supply function of a commodity is 100p = (x + 20)2. Find the Producer's Surplus (PS), when the market price is ₹25. VIEW SOLUTION

• Question 8
Fit a straight line trend by the method of least squares and find the trend value for the year 2008 for the following data:
 Year Production (in lakh tonnes) 2001 30 2002 35 2003 36 2004 32 2005 37 2006 40 2007 36

VIEW SOLUTION

• Question 9
Ten cartons are taken at random from an automatic packing machine. The mean net weight of the ten cartons is 11.8 kg and standard deviation is 0.15 kg. Does the sample mean differ significantly from the intended mean of 12 kg?
[Given that for d.f. = 9, ${t}_{0.05}$ = 2.26] VIEW SOLUTION

• Question 10
Madhu exchanged her old car valued at ₹1,50,000 with a new one priced at ₹6,50,000. She paid ₹x as down payment and the balance in 20 monthly equal instalments of ₹21,000 each. The rate of interest offered to her is 9% p.a. Find the value of x.
[Given that: (1·0075)–20 = 0.86118985] VIEW SOLUTION

• Question 11
In a certain culture of bacteria, the rate of increase of bacteria is proportional to the number present. It is found that there are 10,000 bacteria at the end of 3 hours and 40,000 bacteria at the end of 5 hours. Determine the number of bacteria present in the beginning. VIEW SOLUTION

• Question 12
Calculate the EMI under 'Flat Rate System' for a loan of ₹5,00,000 with 10% annual interest rate for 5 years.

OR

A machine costing ₹2,00,000 has effective life of 7 years and its scrap value is ₹30,000. What amount should the company put into a sinking fund earning 5% p.a., so that it can replace the machine after its usual life? Assume that a new machine will cost ₹3,00,000 after 7 years. [Given that: (1·05)7 = 1·407] VIEW SOLUTION

• Question 13
A start-up company invested ₹3,00,000 in shares for 5 years. The value of this investment was ₹3,50,000 at the end of second year, ₹3,80,000 at the end of third year and on maturity, the final value stood at ₹4,50,000. Calculate the Compound Annual Growth Rate (CAGR) on the investment.
[Given that: (1·5)1/5 = 1·084]
VIEW SOLUTION

• Question 14
A dietician wishes to mix two types of Foods F1 and F2 in such a way that the vitamin content of the mixture contains at least 8 units of vitamin A and 10 units of vitamin C. Food F1 contains 2 units/kg of vitamin A and 1 unit/kg of vitamin C, while Food F2 contains 1 unit/kg of vitamin A and 2 units/kg of vitamin C. It costs ₹5 per kg to purchase Food F1 and ₹7 per kg to purchase Food F2.

Based on the above information, answer the following questions:
(a) To find out the minimum cost of such a mixture, formulate the above problem as a LPP.
(b) Determine the minimum cost of the mixture. VIEW SOLUTION
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