# Mathematics and Statistics 2014-2015 HSC Commerce (Marketing and Salesmanship) 12th Board Exam Question Paper Solution

Mathematics and Statistics
Date & Time: 8th October 2015, 4:00 pm
Duration: 3h

(1) All questions are compulsory.
(2) Answer to every question must be written on a new page.
(3) Log table will be provided on demand.
(4) Answer to Section-I and Section-II should be written in two separate answer books.
(5) Questions from Section-I attempted in the answer book of Section-II and vice versa will not be assessed/not given any credit.
(6) Graph paper ls necessary for L.P.P.
(7) Figure to the right Indicate full marks.

[12]1 | Attempt any Six of the following:
[2]1.A

Find x and y if x + y = [(7,0),(2,5)] , x - y[(3,0),(0,3)]

Concept: Algebra of Matrices
Chapter: [0.012] Matrices [0.02] Matrices
[2]1.B

Find (dy)/(dx) if y = sin^-1(sqrt(1-x^2))

Concept: Derivatives of Implicit Functions
Chapter: [0.013000000000000001] Differentiation [0.04] Differentiation
[2]1.C

Use the quantifiers to convert the following open sentence defined on N into true statement
5x - 3 < 10

Concept: Mathematical Logic - Statement Patterns and Logical Equivalence
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic

Use the quantifiers to convert the following open sentence defined on N into true statement:
x2 ≥ 1

Concept: Mathematical Logic - Statement Patterns and Logical Equivalence
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
[2]1.D

Examine the continuity of the following function :

{:(,f(x),=(x^2-16)/(x-4),",","for "x!=4),(,,=8,",","for "x=4):}} " at " x=4

Concept: Continuous Function of Point
Chapter: [0.03] Continuity
[2]1.E

Find the adjoint of the matrix "A" = [(2,-3),(3,5)]

Concept: Introduction of Matrices
Chapter: [0.02] Matrices
[2]1.F

Find the elasticity of demand if the marginal revenue is ₹ 50 and price is ₹ 75.

Concept: Indefinite Integration - Methods of Integration
Chapter: [0.06] Indefinite Integration
[2]1.G

Evaluate:int(tansqrtx)/sqrtxdx

Concept: Indefinite Integration - Methods of Integration
Chapter: [0.06] Indefinite Integration
[2]1.H

Evaluate : int  1/("x"^2 + 8"x" + 20) dx

Concept: Indefinite Integration - Integral of Standard Functions
Chapter: [0.06] Indefinite Integration
[14]2
[6]2.A | Attempt any TWO of the followlng :
[3]2.A.i

Write converse, inverse contrapositive of the statement "If two triangles are not congruent then their areas are not equal.

Concept: Mathematical Logic - Logical Connectives
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
[3]2.A.ii

Examine the continuity of the followin function :

{:(,f(x),=x^2cos(1/x),",","for "x!=0),(,,=0,",","for "x=0):}}" at "x=0

Concept: Continuous Function of Point
Chapter: [0.03] Continuity
[3]2.A.iii

Find "dy"/"dx" if y = xx + 5x

Concept: Logarithmic Differentiation
Chapter: [0.04] Differentiation
[8]2.B |  Attempt any TWO of the following:
[4]2.B.i

Solve the following equations by reduction method :

x + 2y + z = 8

2x+ 3y - z = 11

3x - y - 2z = 5

Concept: Algebra of Matrices
Chapter: [0.012] Matrices [0.02] Matrices
[4]2.B.ii

Find the volume of the solid obtained by revolving about the X-axis, the region bounded by the curve "x"^2/4 - "y"^2/9 = 1 and the lines x = 2 , x = 4.

Concept: Indefinite Integration - Methods of Integration
Chapter: [0.06] Indefinite Integration
[4]2.B.iii

If the demand function is D = 50 - 3p - p2, find the elasticity of demand at (a) p = 5 (b) p = 2 ,  Interpret your result.

Concept: Increasing and Decreasing Functions
Chapter: [0.013999999999999999] Applications of Derivatives [0.05] Applications of Derivative
[14]3
[6]3.A |  Attempt any TWO of the following:
[3]3.A.i

By constructing the truth table, determine whether the following statement pattern ls a tautology , contradiction or . contingency.  (p →  q) ∧  (p ∧ ~ q ).

Concept: Mathematical Logic - Statement Patterns and Logical Equivalence
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
[3]3.A.ii

If f (x) = (1 - "sin x")/(pi - "2x")^2 , for x ≠ pi/2 is continuous at x = pi/4 , then find "f"(pi/2) .

Concept: Continuous Function of Point
Chapter: [0.03] Continuity
[3]3.A.iii

If "x"^(5/3) . "y"^(2/3) = ("x + y")^(7/3) , the show that "dy"/"dx" = "y"/"x"

Concept: Logarithmic Differentiation
Chapter: [0.04] Differentiation
[8]3.B
[4]3.B.i

Cost of assembling x wallclocks is ( x^3/3 - 40x^2) and labour charges are 500x. Find the number of wall clocks to be manufactured for which average cost and marginal cost attain their respective minimum.

Concept: Maxima and Minima
Chapter: [0.05] Applications of Derivative
[4]3.B.ii

Evaluate : int _0^1 ("x" . ("sin"^-1 "x")^2)/sqrt (1 - "x"^2) dx

Concept: Indefinite Integration - Methods of Integration
Chapter: [0.06] Indefinite Integration
[4]3.B.iii

Evaluate :  ∫ log (1 + x2) dx

Concept: Properties of Definite Integrals
Chapter: [0.016] Definite Integration [0.07] Definite Integrals
Section - II
[12]4 | Attempt any SIX of the following.
[2]4.A

Alex spends 20% of his income on food items and 12% on conveyance. If for the month of June 2010, he spent ~ 900 on conveyance, find his expenditure on food items during he same month.

Concept: Ratio, Proportion and Partnership
Chapter: [0.08] Ratio, Proportion and Partnership
[2]4.B

Find the age · specific Death Rates for the following data :

 Age group (in years) Population number of deaths 0 - 20 7000 140 20 - 45 20000 180 45 - 65 10000 120 65 and above 4000 160
Concept: Measurements of Mortality
Chapter: [0.11] Demography
[2]4.C

Compute the coefficient of correlation for the following data :

n = 100 ; bar x = 62 ,  bar y = 53 . σx = 10 ; σ y = 12 ,

sum ("x"_1 - bar x) ("y"_1 - bar y) = 8000

Concept: Rank Correlation
Chapter: [0.12] Bivariate Data and Correlation
[2]4.D

A building is insured for 80% of its value. The annual premium at 70 paise percent amounts to ₹ 2,800.  Fire damaged the building to the extent of 60% of its value. How much amount for damage can: be claimed under the policy?

Concept: Insurance and Annuity
Chapter: [0.1] Insurance and Annuity
[2]4.E

A random variable X has the following probability distribution :

 X = x -2 -1 0 1 2 3 P(x) 0.1 k 0.2 2k 0.3 k

Find the value of k and calculate mean.

Concept: Random Variables and Its Probability Distributions
Chapter: [0.027999999999999997] Probability Distributions [0.14] Random Variable and Probability Distribution
[2]4.F

A, B and C are in partnership. A's capital was  ₹ 65,000, C's capital was  ₹ 50,000. The total profit is ₹ 38,000, out of which B's profit is ₹ 15,000. What was B's capital ?

Concept: Ratio, Proportion and Partnership
Chapter: [0.08] Ratio, Proportion and Partnership
[2]4.G

If the Crude Death Rate (CDR) for the following data is 13.4 per thousand, find x:

 Age groups(in years) Population Number of deaths 0 - 20 40,000 350 20 - 65 65,000 650 65 and above 15,000 x
Concept: Commission, Brokerage and Discount
Chapter: [0.09] Commission, Brokerage and Discount
[2]4.H

Draw scatter diagram for the following data and identify the type of correlation.

 capital (₹ in crore) 2 3 4 5 6 8 9 Profit (₹ in lakh) 6 5 7 7 8 12 11
Concept: Rank Correlation
Chapter: [0.12] Bivariate Data and Correlation
[14]5
[6]5.A | Attempt any TWO of the following:
[3]5.A.i

Mrs. Menon plans to save for her daughter's marriage. She wants to accumulate a sum of ₹ 4,00,000 at the end of 4 years. How much should she invest at the end of each year from now, if she can get interest compounded at 10% p.a. ?

[Given (1.1)4 = 1.4641]

Concept: Insurance and Annuity
Chapter: [0.1] Insurance and Annuity
[3]5.A.ii

A fair coin is tossed 12 times. Find the probability of getting exactly 7 heads .

Concept: Random Variables and Its Probability Distributions
Chapter: [0.027999999999999997] Probability Distributions [0.14] Random Variable and Probability Distribution

A fair coin is tossed 12 times. Find the probability of getting  at least 2 heads .

Concept: Random Variables and Its Probability Distributions
Chapter: [0.027999999999999997] Probability Distributions [0.14] Random Variable and Probability Distribution
[3]5.A.iii

For the following problem find the sequence that minimizes total elapsed time (in hrs) required to complete the jobs on 2 machines M1 and M2 in the order M1 - M2 · Also find the minimum elapsed time T.

 Job A B C D E F M1 4 8 3 6 7 5 M2 6 3 7 2 8 4
Concept: Sequencing in Management Mathematics
Chapter: [0.15] Management Mathematics
[8]5.B | Attempt any TWO of the following:
[4]5.B.i

A bill of ₹ 2,000 drawn on 15th February 2003 for 10 months was discounted on 13th May 2003 at  3 3/4 % p.a.  Calculate the banker's discount.

Concept: Commission, Brokerage and Discount
Chapter: [0.09] Commission, Brokerage and Discount
[4]5.B.ii

From the following table which relates to the number of animals of a certain
species at age x. complete the life table :

 x 0 1 2 3 4 5 lx 1000 850 760 360 25 0
Concept: Life Tables
Chapter: [0.11] Demography
[4]5.B.iii

A person makes two types of gift items A and B requiring the services of a cutter and a finisher.  Gift item A requires 4 hours of the cutter's time and 2 hours of finisher's time. Fifth item B requires 2 hours of the cutter's time and 4 hours of finisher's time. The cutter and finisher have 208 hours and 152 hours available time respectively every month. The profit on one gift item of type A is ₹ 75 and on one gift item of type, B is ₹ 125. Assuming that the person can sell all the gift items produced, determine how many gift items of each type should he make every month to obtain the best returns?

Concept: Linear Programming Problem in Management Mathematics
Chapter: [0.15] Management Mathematics
[14]6
[6]6.A | Attempt any TWO of the following :
[3]6.A.i

In a partially destroyed laboratory record of an analysis of regression data, the following data are legible :

Variable of X = 9

Regression equations : 8x - 10g + 66 = 0 and 40x- 18g = 214

Find Mean values of X and Y on the basis of the above information .

Concept: Rank Correlation
Chapter: [0.12] Bivariate Data and Correlation

In a partially destroyed laboratory record of an analysis of regression data, the following data are legible :

Variable of X = 9

Regression equations : 8x - 10g + 66 = 0 and  40x- 18g = 214

Find Correlation coefficient between X and Y on the basis of the above information.

Concept: Rank Correlation
Chapter: [0.12] Bivariate Data and Correlation
[3]6.A.ii

Mr. Rajesh has ₹ 1800 to spend on fruits for a meeting. Grapes cost ₹ 160/kg llnd peaches ₹ 200/kg. Let x and g represent the number of kilogrames of grapes and peaches he can buy. Write the graph of an inequation to model the amounts of grapes and peaches he can buy within his budget.

Concept: Linear Programming Problem in Management Mathematics
Chapter: [0.15] Management Mathematics
[3]6.A.iii

If a random variable X follows Poisson distribution sucli that P(X = 1) = P(X = 2), find:
(a) the mean and
(b) P(X = 0). [Given: e-2 = 0.1353)

Concept: Probability Distribution - Distribution Function of a Continuous Random Variable
Chapter: [0.14] Random Variable and Probability Distribution
[8]6.B | Attempt any TWO of the following:
[4]6.B.i

Ranking of 8 trainees at the beginning (X) and at the end (Y), of a certain course are given below :

 Trainees A B C D E F G H X 1 2 4 5 6 8 3 7 Y 2 4 3 7 8 1 5 6

Calculate Spearman's rank correlation coefficient.

Concept: Rank Correlation
Chapter: [0.12] Bivariate Data and Correlation
[4]6.B.ii

A computers centre has four expert programmers . The centre needs four application programmes to be developed. The head of the computer centre , after stying carefully the programmes to be developed , estimates the computer time in minutes required by the respective experts to develop the application programmes as follows :

 Programmes Programmes 1 2 3 4 (Times in minutes) A 120 100 80 90 B 80 90 110 70 C 110 140 120 100 D 90 90 80 90

How should the head of the computer centre assign the programmes to the programmers so that the total time required is minimum ?

Concept: Algebra of Matrices
Chapter: [0.012] Matrices [0.02] Matrices
[4]6.B.iii

From the data 7 pairs of observations on X and Y following results are obtained :

∑(xi - 70) = - 38 ; ∑ (yi - 60) = - 5 ;

∑ (xi - 70)2 = 2990 ; ∑ (yi - 60)2 = 475 ;

∑ (xi - 70) (yi - 60) = 1063

Obtain :

(a) The line of regression of Y on X.

(b) The line of regression of X on Y.

Concept: Regression Coefficient of X on Y and Y on X
Chapter: [0.13] Regression Analysis Introduction

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