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

Mathematics and Statistics
Date & Time: 28th February 2015, 11:00 am
Duration: 3h

 1 | Attempt any SIX of the following
 1.1

Express the following statement in symbolic form and write its truth value.

"If 4 is an odd number, then 6 is divisible by 3 "

Concept: Statement Patterns and Logical Equivalence
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
 1.2

Find the values of x and y, if

2[[1,3],[0,x]]+[[y,0],[1,2]]=[[5,6],[1,8]]

Concept: Solution of Equations
Chapter: [0.02] Matrices
 1.3

Find the value of 'k' if the function

f(X)=(tan7x)/(2x) ,  "for " x != 0

=k,            for x=0

is continuos at x=0

Concept: Continuous Function of Point
Chapter: [0.03] Continuity
 1.4

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

Concept: The Concept of Derivative - Derivative of Inverse Function
Chapter: [0.04] Differentiation
 1.5

Price P for demand D is given as P = 183 +120D - 3D2 Find D for which the price is increasing

Concept: Increasing and Decreasing Functions
Chapter: [0.013999999999999999] Applications of Derivatives [0.05] Applications of Derivative
 1.6

Evaluate : int1/(x(3+logx))dx

Concept: Indefinite Integration - Methods of Integration
Chapter: [0.06] Indefinite Integration
 1.7

If   "A"=[[2,1],[1,1]] " then show taht " A^2-3A+I=0

Concept: Solution of Equations
Chapter: [0.02] Matrices
 1.8

evaluate : int xcosxdx

Concept: Indefinite Integration - Integration by Parts
Chapter: [0.06] Indefinite Integration
 2
 2.1 | Attempt any TWO of the following
 2.1.1

Prove that the following statement pattern is equivalent :

(p ∨ q) →  r and (p → r) ∧ (q → r)

Concept: Statement Patterns and Logical Equivalence
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
 2.1.2

Examine the continuity of the following function :

{:(,,f(x)= x^2 -x+9,"for",x≤3),(,,=4x+3,"for",x>3):}}"at "x=3

Concept: Continuous Function of Point
Chapter: [0.03] Continuity
 2.1.3

find dy/dx if y=tan^-1((6x)/(1-5x^2))

Concept: The Concept of Derivative - Derivative of Inverse Function
Chapter: [0.04] Differentiation
 2.2 | Attempt any TWO of the following :
 2.2.1

Find the inverse of the following matrix by elementary row transformations if it exists. A=[[1,2,-2],[0,-2,1],[-1,3,0]]

Concept: Inverse of Matrix
Chapter: [0.012] Matrices [0.02] Matrices
 2.2.2

Find the area of elipse x^2/a^2+y^2/b^2=1

Concept: Applications of Definite Integrals
Chapter: [0.07] Definite Integrals
 2.2.3

The expenditure Ec of a person with income I is given by Ec = (0.000035) I2 + (0.045) I. Find marginal propensity to consume (MPC) and marginal propensity to save (MPS) when I = 5000. Also find A (average) PC and A (average)

PS.

Concept: Maxima and Minima
Chapter: [0.05] Applications of Derivative
 3
 3.1 | Attempt any TWO of the following :
 3.1.1

Express the truth of each of the following statements by Venn diagram:

(a) Some hardworking students are obedient.

(b) No circles are polygons.

(c) All teachers are scholars and scholars are teachers.

Concept: Venn Diagrams
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
 3.1.2

If 'f' is continuous at x = 0, then find f(0).

f(x)=(15^x-3^x-5^x+1)/(xtanx) , x!=0

Concept: Continuous Function of Point
Chapter: [0.03] Continuity
 3.1.3

find dy/dx if x=e2t , y=e^sqrtt

Concept: Derivatives of Functions in Parametric Forms
Chapter: [0.04] Differentiation
 3.2 | Attempt any TWO of the following :
 3.2.1

Evaluate : int (1+logx)/(x(2+logx)(3+logx))dx

Concept: Indefinite Integration - Methods of Integration
Chapter: [0.06] Indefinite Integration
 3.2.2

Evaluate :int_0^(pi/2)dx/(1+cotx)

Concept: Indefinite Integration - Integration by Parts
Chapter: [0.06] Indefinite Integration
 3.2.3

A firm wants to maximize its profit. The total cost function is C = 370Q + 550 and revenue is R = 730Q-3Q2. Find the output for which profit is maximum and also find the profit amount at this output.

Concept: Maxima and Minima
Chapter: [0.05] Applications of Derivative
 4 | Attempt any SIX of the following :
 4.1

The ratio of number of boys and girls in a school is 3 : 2. If 20% of the boys and 30% of the girls are scholarship holders, find the percentage of students who are not scholarship holders.

Concept: Ratio, Proportion and Partnership
Chapter: [0.08] Ratio, Proportion and Partnership
 4.2

Obtain crude death rates (C. D. R.) for city A and city B from the data given below :

 Age groups (in years) City A City B Population No. of deaths Population No. of deaths Below 15 800 32 900 12 15-25 3000 12 1500 8 25-65 4800 48 4500 38 65 and above 1400 42 600 30
Concept: Measurements of Mortality
Chapter: [0.11] Demography
 4.3

Coefficient of rank correlation between x and y is 0.5 and sumd_i^2= 42. Assuming that no ranks are repeated, find the number of pairs of observations.

Concept: Rank Correlation
Chapter: [0.12] Bivariate Data and Correlation
 4.4

An agent charges 12% commission on the sales. What does he earn if the total sale amounts to Rs. 36,000? What does the seller get?

Concept: Commission, Brokerage and Discount
Chapter: [0.09] Commission, Brokerage and Discount
 4.5

Find the age standard death rate (S. D. R.) for the following  data:

 Age groups (in years) Population (in '000) No. of deaths 0-10 11 240 10-20 12 150 20-60 9 125 60 and above 2 90
Concept: Life Tables
Chapter: [0.11] Demography
 4.6

Following table gives the age of husbands and wives

 Age of wives (in years) Age of husbands (in years) 20-30 30-40 40-50 50-60 15-25 5 9 3 — 25-35 — 10 25 2 35-45 — 1 12 2 45-55 — — 4 16 55-65 — — — 4

Find:

(a)The marginal frequency distribution of the age of husbands.

(b)The conditional distribution of the age of husbands when the age of wives lies between 25 - 35

Concept: Uses of Vital Statistics in Demography
Chapter: [0.11] Demography
 4.7

The present worth of the sum of Rs. 5,830, due 9 months hence, is Rs. 5,500. Find the rate of interest.

Concept: Commission, Brokerage and Discount
Chapter: [0.09] Commission, Brokerage and Discount
 4.8

For a binomial distribution mean is 6 and variance is 2. Find n and p.

Concept: Binomial Theorem and Binomial Distribution
Chapter: [0.14] Random Variable and Probability Distribution
 5
 5.1 | Attempt any TWO of the following :
 5.1.1

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

 Jobs A B C D E Machine M1 5 1 9 3 10 Machine M2 2 6 7 8 4
Concept: Sequencing in Management Mathematics
Chapter: [0.15] Management Mathematics
 5.1.2

Natarajan and Mr. Gopalan are partners in the company having capitals in the ratio 4 : 5 and the profits received by them are in the ratio 5 :4. If Mr. Gopalan invested capital in the company for 16 months, how long was Mr. Natarajan’s investment in the company?

Concept: Commission, Brokerage and Discount
Chapter: [0.09] Commission, Brokerage and Discount
 5.1.3

From a lot of 25 bulbs of which 5 are defective a sample of 5 bulbs was drawn at random with replacement. Find the probability that the sample will contain -

(a) exactly 1 defective bulb.

(b) at least 1 defective bulb.

Concept: Random Variables and Its Probability Distributions
Chapter: [0.027999999999999997] Probability Distributions [0.14] Random Variable and Probability Distribution
 5.2 | Attempt any TWO of the following :
 5.2.1

Given the following table which relates to the number of parrots at age x, complete the life table for parrots.

 X 0 1 2 3 4 5 lx 1000 940 780 590 25 0
Concept: Life Tables
Chapter: [0.11] Demography
 5.2.2

 Advertisement Expenditure (Rs. in lakh)             (X) Sales (Rs. in lakh)            (Y) Arithmetic mean 10 90 Standard deviation 3 12

Correlation coefficient between X and Y = 0.8.

(a) Obtain the two regression equations.

(b) What is the likely sales when the advertising budget is ? 15 lakh?

(c) What should be the advertising budget if the company wants to attain sales target of ₹ 120 lakh?

Concept: Lines of Regression of X on Y and Y on X Or Equation of Line of Regression
Chapter: [0.023] Linear Regression [0.13] Regression Analysis Introduction
 5.2.3

Electro Corp. Co. manufactures two electrical products :Air conditioners and Fans. The assembly process for each is similar in which both require a certain amount of wiring and drilling. Each air conditioner takes 4 hours for wiring and 2 hours for drilling. Each fan also takes 2 hours for wiring and 1 hour for drilling. During the next production period, 240 hours of wiring time are available and upto 100 hours of drilling time may be used. Each air-conditioner assembled may be sold for Rs. 2,000 profit and each fan assembled may be sold for Rs. 1,000 profit. Formulate this problem as an L.P.P. in order to maximize the profit.

Concept: Linear Programming Problem in Management Mathematics
Chapter: [0.15] Management Mathematics
 6
 6.1 | Attempt any TWO of the following :
 6.1.1

The equations given of the two regression lines are 2x + 3y - 6 = 0 and 5x + 7y - 12 = 0.

Find:

(a) Correlation coefficient

(b) sigma_x/sigma_y

Concept: Lines of Regression of X on Y and Y on X Or Equation of Line of Regression
Chapter: [0.023] Linear Regression [0.13] Regression Analysis Introduction
 6.1.2

Find graphical solution for the following system of linear inequations:

2x + 3y ≥ 2, -x + y ≤ 3,  x ≤ 4, y ≥ 3

Concept: Linear Programming Problem in Management Mathematics
Chapter: [0.15] Management Mathematics
 6.1.3

The number of complaints which a bank manager receives per day is a Poisson random variable with parameter m = 4. Find the probability that the manager will receive -

(a) only two complaints on any given day.

(b) at most two complaints on any given day

[Use e-4 =0.0183]

Concept: Poisson Distribution
Chapter: [0.027999999999999997] Probability Distributions [0.14] Random Variable and Probability Distribution
 6.2 | Attempt any TWO of the following :
 6.2.1

A warehouse valued at Rs. 10,000 contained goods worth Rs. 60,000. The warehouse was insured against fire for Rs. 4,000 and the goods to the extent of 90% of their value. A fire broke out and goods worth Rs. 20,000 were completely destroyed, while the remainder was damaged and reduced to 80% of its value. The damage to the warehouse was to the extent of Rs. 2,000. Find the total amount that can be claimed.

Concept: Insurance and Annuity
Chapter: [0.1] Insurance and Annuity
 6.2.2

In the following data, one of the values of Y is missing. Arithmetic means of X and Y series are 6 and 8

 X 6 2 10 4 8 Y 9 11 ? 8 7

(a) Estimate the missing observation.

(b) Calculate correlation coefficient.

Concept: Statistics - Karl Pearson’s Coefficient of Correlation
Chapter: [0.12] Bivariate Data and Correlation
 6.2.3

A job production unit has four jobs A, B, C, D which can be manufactured on each of the four machines P, Q, R and S. The processing cost of each job is given in the following table:

 Jobs Machines P Q R S Processing Cost (Rs.) A 31 25 33 29 B 25 24 23 21 C 19 21 23 24 D 38 36 34 40

How should the jobs be assigned to the four machines so that the total processing cost is minimum?

Concept: Assignment Problem
Chapter: [0.027000000000000003] Assignment Problem and Sequencing [0.15] Management Mathematics

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