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

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Mathematics and Statistics
2017-2018 March
Marks: 80

[12]1 | Attempt any six of the following
[2]1.1
[2]1.1.1

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

All rational number are real numbers. 

Concept: Venn Diagrams
Chapter: [1] Mathematical Logic
[2]1.1.2

Draw Venn diagram for the truth of the following statements : 

Some rectangles are squares. 

Concept: Venn Diagrams
Chapter: [1] Mathematical Logic
[2]1.2

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

Concept: Introduction of Matrices
Chapter: [2] Matrices
[2]1.3

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]1.4

find `dy/dx, if y= cos ^-1 (sin 5x)` 

Concept: Concept of Demography
Chapter: [11] Demography
[2]1.5

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]1.6

Evaluate`int (1)/(x(3+log x))dx` 

Concept: Properties of Definite Integrals
Chapter: [7] Definite Integrals
[2]1.7

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

Concept: Introduction of Matrices
Chapter: [2] Matrices
[2]1.8

Evaluate : `int 1/(9x^2+49) dx` 

Concept: Indefinite Integration - Integral of Standard Functions
Chapter: [6] Indefinite Integration
[14]2
[6]2.1 | Attempt any two of the following
[3]2.1.1

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]2.1.2

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]2.1.3

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]2.2 | Attempt any two of the following
[4]2.2.1

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]2.2.2

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]2.2.3

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]3.1 | Attempt any TWO of the following :
[3]3.1.1

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

= 10 for x = 4  at x = 4 

 

Concept: Continuous Function of Point
Chapter: [3] Continuity
[3]3.1.2

Find dy/dx,if `e^x+e^y=e^(x-y)` 

Concept: Increasing and Decreasing Functions
Chapter: [5] Applications of Derivative
[3]3.1.3

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

Concept: Increasing and Decreasing Functions
Chapter: [5] Applications of Derivative
[8]3.2 | Attempt any TWO of the following :
[4]3.2.1

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

Concept: Continuous Function of Point
Chapter: [3] Continuity
[4]3.2.2

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]3.2.3

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

Concept: Maxima and Minima
Chapter: [5] Applications of Derivative
[12]4 | Attempt any SIX of the following :
[2]4.1

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]4.2

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: Insurance and Annuity
Chapter: [10] Insurance and Annuity
[2]4.3

 If `Σdi^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]4.4

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]4.5

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

Find : (i) Correlation coefficient between X and Y. 

(ii) `σ_y^2 if σ _x^2=4`

Concept: Statistics - Bivariate Frequency Distribution
Chapter: [12] Bivariate Data and Correlation
[2]4.6

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]4.7

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]4.8

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]5.1 | Attempt any TWO of the following :
[3]5.1.1

Ramesh, Vivek and Sunil started a business by investing capitals 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]5.1.2

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]5.1.3

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]5.2 | Attempt any TWO of the following :
[4]5.2.1

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]5.2.2

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]5.2.3

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]6.1 | Attempt any TWO of the following :
[3]6.1.1

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]6.1.2

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]6.1.3

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

Concept: Inequations in Management Mathematics
Chapter: [15] Management Mathematics
[8]6.2
[4]6.2.1

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]6.2.2

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? 

Concept: Commission, Brokerage and Discount
Chapter: [9] Commission, Brokerage and Discount
[4]6.2.3

The given problem is of n jobs and three machines. We change the problem in of n jobs and two machines.
For this either Min M1 ≤ M3 × M2 or min M3 ≤ max M2 Here min M3 = 7 = M1 × M2 Hence we write M1+ M2 = G and M2 + M3 = H the problem will be as follows

Concept: Linear Programming Problem in Management Mathematics
Chapter: [15] Management Mathematics

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