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# Mathematics and Statistics 2017-2018 HSC Commerce 12th Board Exam Question Paper Solution

Mathematics and Statistics
Date: March 2018

SECTION - 1
[12]1 | Attempt any six of the following
[2]1.A
[2]1.A.i

Draw a Venn diagram for the truth of the following statement :

All rational number are real numbers.

Concept: Venn Diagrams
Chapter: [1] Mathematical Logic

Draw Venn diagram for the truth of the following statements :

Some rectangles are squares.

Concept: Venn Diagrams
Chapter: [1] Mathematical Logic
[2]1.A.ii
[2]1.B

Find the inverse of the matrix A=[[1,2],[1,3]] using elementry transformations.

Concept: Introduction of Matrices
Chapter: [2] Matrices
[2]1.C

Examine the continuity of f(x)=x^2-x+9  "for"  x<=3

=4x+3  "for"  x>3,  "at"  x=3

Concept: Continuous Function of Point
Chapter: [3] Continuity
[2]1.D

find dy/dx, if y= cos ^-1 (sin 5x)

Concept: Concept of Demography
Chapter: [11] Demography
[2]1.E

The price P for demand D is given as P = 183 + 120 D – 3D2.
Find D for which the price is increasing.

Concept: Increasing and Decreasing Functions
Chapter: [5] Applications of Derivative
[2]1.F

Evaluateint (1)/(x(3+log x))dx

Concept: Properties of Definite Integrals
Chapter: [7] Definite Integrals
[2]1.G

Find cofactors of the elements of the matrix A = [[-1,2],[-3,4]]

Concept: Introduction of Matrices
Chapter: [2] Matrices
[2]1.H

Evaluate : int 1/(9x^2+49) dx

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

Find k, if f(x) =log (1+3x)/(5x) for x ≠ 0

= k                    for x = 0

is continuous at x = 0.

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

Examine whether the following statement pattern is tautology, contradiction or contingency :
p ∨ – (p ∧ q)

Concept: Regression Coefficient of X on Y and Y on X
Chapter: [13] Regression Analysis Introduction
[3]2.A.iii

If x = cos2 θ and y = cot θ then find dy/dx  at  θ=pi/4

Concept: Increasing and Decreasing Functions
Chapter: [5] Applications of Derivative
[8]2.B | Attempt any two of the following
[4]2.B.i

The sum of three numbers is 6. If we multiply the third number by 3 and add it to the second number we get 11. By adding first and third numbers we get a number, which is double than the second number. Use this information and find a system of linear equations. Find these three numbers using matrices.

Concept: Inverse of Matrix
Chapter: [2] Matrices
[4]2.B.ii

Find the area of the region bounded by the parabola y2 = 16x and the line x = 4.

Concept: Inverse of Matrix
Chapter: [2] Matrices
[4]2.B.iii

The consumption expenditure Ec of a person with the income x. is given by Ec = 0.0006x2 + 0.003x. Find MPC, MPS, APC and APS when the income x = 200.

Concept: Increasing and Decreasing Functions
Chapter: [5] Applications of Derivative
[14]3
[6]3.A | Attempt any TWO of the following :
[3]3.A.i

Discuss continuity of f(x) =(x^3-64)/(sqrt(x^2+9)-5) For x ≠ 4

= 10 for x = 4  at x = 4

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

Find dy/dx,if e^x+e^y=e^(x-y)

Concept: Increasing and Decreasing Functions
Chapter: [5] Applications of Derivative
[3]3.A.iii

Using truth table show that ∼ (p → ∼ q) ≡ p ∧ q

Concept: Increasing and Decreasing Functions
Chapter: [5] Applications of Derivative
[8]3.B | Attempt any TWO of the following :
[4]3.B.i

Evaluate :int Sinx/(sqrt(cos^2 x-2 cos x-3)) dx

Concept: Continuous Function of Point
Chapter: [3] Continuity
[4]3.B.ii

The total cost function of a firm is C = x^2 + 75x + 1600 for output x. Find the output (x) for which average
cost is minimum. Is C_A = C_M at this output?

Concept: Maxima and Minima
Chapter: [5] Applications of Derivative
[4]3.B.iii

Evaluate : int_1^2 1/((x+1)(x+3)) dx

Concept: Maxima and Minima
Chapter: [5] Applications of Derivative
SECTION - II
[12]4 | Attempt any SIX of the following :
[2]4.A

A shop valued at ₹ 2,40,000 is insured for 75% of its value. If the rate of premium is 90 paise percent, find the premium paid by the owner of the shop.

Concept: Insurance and Annuity
Chapter: [10] Insurance and Annuity
[2]4.B

Find the Age-Specific Death Rate (Age-SDR) for the following data :

 Age groups(in years) Population(in '000) Number of deaths 1 - 10 11 240 10 - 20 12 150 20 -60 9 125 60 and above 2 90
Concept: Life Tables
Chapter: [11] Demography
[2]4.C

If Σd_i^2 = 25, n = 6 find rank correlation coefficient where di, is the difference between the ranks of ith values.

Concept: Statistics - Bivariate Frequency Distribution
Chapter: [12] Bivariate Data and Correlation
[2]4.D

The following table gives the ages 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 : (i) The marginal frequency distribution of the age of husbands.
(ii) The conditional frequency distribution of the age of husbands when the age of wives lies between
25 - 35.

Concept: Statistics - Bivariate Frequency Distribution
Chapter: [12] Bivariate Data and Correlation
[2]4.E

The regression equation of Y on X is y = 2/9 x and the regression equation of X on Y is x=y/2+7/6

Find:

1.  The correlation coefficient between X and Y.
2.  σ_y^2 if σ _x^2=4
Concept: Statistics - Bivariate Frequency Distribution
Chapter: [12] Bivariate Data and Correlation
[2]4.F

Identify the regression equations of X on Y and Y on X from the following equations :
2x + 3y = 6 and 5x + 7y – 12 = 0

Concept: Lines of Regression of X on Y and Y on X Or Equation of Line of Regression
Chapter: [13] Regression Analysis Introduction
[2]4.G

If X has Poisson distribution with parameter m = 1, find P[X ≤ 1]. (Use e–1 = 0.3679)

Concept: Regression Propertise
Chapter: [13] Regression Analysis Introduction
[2]4.H

Three fair coins are tossed simultaneously. If X denotes the number of heads, find the probability distribution of X.

Concept: Random Variables and Its Probability Distributions
Chapter: [14] Random Variable and Probability Distribution
[6]5
[6]5.A | Attempt any TWO of the following :
[3]5.A.i

Ramesh, Vivek, and Sunil started a business by investing capital in the ratio 4 : 5 : 6. After 3 months Vivek withdrew all his capital and after 6 months Sunil withdrew all his capital from the business. At the end of the year, Ramesh received ₹ 6,400 as profit. Find the profit earned by Vivek.

Concept: Commission, Brokerage and Discount
Chapter: [9] Commission, Brokerage and Discount
[13]5.A.ii

Solve the following minimal assignment problem and hence find the minimum value :

 I II III IV A 2 10 9 7 B 13 2 12 2 C 3 4 6 1 D 4 15 4 9

Concept: Assignment Problem
Chapter: [15] Management Mathematics
[3]5.A.iii

Calculate from e_0^0,e_1^0,e_2^0 from the following data :

 Age x 0 1 2 l_x 1000 900 700 T_x - - 11500
Concept: Statistics - Karl Pearson’s Coefficient of Correlation
Chapter: [12] Bivariate Data and Correlation
[8]5.B | Attempt any TWO of the following :
[4]5.B.i

A bill was drawn on 12th April for 3,500 and was discounted on 4th July at 5% p.a. If the banker paid 3,465 for the bill. Find period of the bill.

Concept: Commission, Brokerage and Discount
Chapter: [9] Commission, Brokerage and Discount
[4]5.B.ii

Find Karl Pearson's correlation coefficient for the following data :

 X 3 2 1 5 4 Y 8 4 10 2 6
Concept: Statistics - Karl Pearson’s Coefficient of Correlation
Chapter: [12] Bivariate Data and Correlation
[4]5.B.iii

Solve the following using graphical method :

Minimize :Z=3x+5y

2x+3x>=12

-x+y<=3

x<=4,y>=3,x>=0,y>=0

Concept: Linear Programming Problem in Management Mathematics
Chapter: [15] Management Mathematics
[14]6
[6]6.A | Attempt any TWO of the following :
[3]6.A.i

Given the following information :

 Age groups(in years) Population Number of deaths 0 - 20 40,000 350 20 - 65 65,000 650 65 and above 15,000 x

Find X, if the CDR = 13.4 per thousand.

Concept: Probability Distribution of a Discrete Random Variable
Chapter: [14] Random Variable and Probability Distribution
[3]6.A.ii

The manager of a company wants to find a measure which he can use to fix the monthly wages of persons applying for a job in the production department. As an experimental project, he collected data of 7 persons from that department referring to years of service and their monthly income :

 Years of service 11 7 9 5 8 6 10 Income (` in thousands) 10 8 6 5 9 7 11

Find regression equation of income on the years of service.

Concept: Sequencing in Management Mathematics
Chapter: [15] Management Mathematics
[3]6.A.iii

Solve the following inequation :
– 8 < – (3x – 5) < 13.

Concept: Inequations in Management Mathematics
Chapter: [15] Management Mathematics
[8]6.B
[4]6.B.i

Find the probability of guessing correctly at most three of the seven answers in a True or False objective test.

Concept: Probability Distribution of a Discrete Random Variable
Chapter: [14] Random Variable and Probability Distribution
[4]6.B.ii

A person bought a television set paying ₹ 20,000 in cash and promised to pay ₹ 1,000 at the end of every month for the next 2 years. If the money is worth 12% p.a. converted monthly, what is the cash price of the television set?
[(1.01)-24 = 0.7884]

Concept: Commission, Brokerage and Discount
Chapter: [9] Commission, Brokerage and Discount
[4]6.B.iii

There are found jobs to be completed. Each job must go through machines M1 , M2 , M3 in the order M1 - M2 - M3.  Processing time in hours is given below. Determine the optimal sequesnce and idle time for machine M1 .

 Jobs A B C D M1 5 8 7 3 M2 6 7 2 5 M3 7 8 10 9
Concept: Linear Programming Problem in Management Mathematics
Chapter: [15] Management Mathematics

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