# Mathematics and Statistics 2013-2014 HSC Commerce 12th Board Exam Question Paper Solution

Mathematics and Statistics
Date: March 2014
1. All questions are compulsory.
2. Answer to every question must be written on a new page.
3. L.P.P. problem should be solved on a graph paper
4. Log table will be provided on demand.
5. Write answers of Section - I and Section - II in one answer book.

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

If "A" = [(1,2,-3),(5,4,0)] , "B" = [(1,4,3),(-2,5,0)], then find 2A + 3B.

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

If the function f is continuous at x = I, then find f(1), where f(x) = (x^2 - 3x + 2)/(x - 1), for x ≠ 1

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

If x = tan-1t and y = t3 , find (dy)/(dx).

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

Evaluate : ∫sin^2x dx

Concept: Indefinite Integration - Definition of an Integral
Chapter: [0.06] Indefinite Integration
[2]1.A.v

Write the negation of the following statements :

(a) Chetan has black hair and blue eyes.
(b) ∃ x ∈ R such that x2 + 3 > 0.

Concept: Random Variables and Its Probability Distributions
Chapter: [0.027999999999999997] Probability Distributions [0.14] Random Variable and Probability Distribution
[2]1.A.vi

If A = [(1,1),(2,2)] , "B" = [(1,2),(3,4)] then find |AB|.

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

Evaluate : ∫dx/(4 - 9x^2)

Concept: Indefinite Integration - Definition of an Integral
Chapter: [0.06] Indefinite Integration
[2]1.A.viii

If the function f is continuous at x = 2, then find 'k' where

f(x) = (x^2 + 5)/(x - 1), for  1< x ≤ 2
= kx + 1 , for x > 2

Concept: Continuous Function of Point
Chapter: [0.03] Continuity
[6]2
[3]2.A | Attempt any Two of the following
[3]2.A.i

If x^y = e^(x - y) , show that (dy)/(dx) = logx/(1 + logx)^2

Concept: Derivative - Derivative of Inverse Function
Chapter: [0.04] Differentiation
[3]2.A.ii

If sin y = xsin(a + y) prove that (dy)/(dx) = sin^2(a + y)/sin a

Concept: Concept of Differentiability
Chapter:
[3]2.A.iii

Discuss extreme values of the function f(x) = x.logx

Concept: Derivatives of Implicit Functions
Chapter: [0.013000000000000001] Differentiation [0.04] Differentiation
[8]2.B
[4]2.B.i

Discuss the continuity of the function f at x = 0, where
f(x) = (5^x + 5^-x - 2)/(cos2x - cos6x), for x ≠ 0
= 1/8(log 5)^2,  for x = 0

Concept: Continuous Function of Point
Chapter: [0.03] Continuity
[4]2.B.ii

The expenditure Ec of a person with income I is given by E= (0.000035) I2 + (0. 045) I. Find marginal propensity to consume (MPC) and average propensity to consume (APC) when I = 5000.

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

Evaluate : ∫x  cot^-1x dx

Concept: Indefinite Integration - Definition of an Integral
Chapter: [0.06] Indefinite Integration
[6]3
[3]3.A | Attempt any Two of the following
[3]3.A.i

If p : It is a day time , q : It is warm
Give the verbal statements for the following symbolic statements :
(a) p ∧ ∼ q (b) p v q (c) p ↔ q

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

Using the truth table statement, examine whether the statement pattern (p → q) ↔ (∼ p v q) is a tautology, a contradiction or a contingency.

Concept: Introduction of Matrices
Chapter: [0.02] Matrices
[3]3.A.iii

The cost C of producing x articles is given as C = x3-16x2 + 47x.  For what values of x, with the average cost is decreasing'?

Concept: Derivatives of Functions in Parametric Forms
Chapter: [0.04] Differentiation
[8]3.B | Attempt any Two of the following
[4]3.B.i

If A = [(1,0,0),(2,1,0),(3,3,1)] then find A-1 by using elementary transformation .

Concept: Introduction of Matrices
Chapter: [0.02] Matrices
[4]3.B.ii

Evaluate : ∫_0^3dx/(x + sqrt(9 - x^2))

Concept: Indefinite Integration - Definition of an Integral
Chapter: [0.06] Indefinite Integration
[4]3.B.iii

Find the volume of a solid obtained by the complete revolution of the ellipse x^2/36 + y^2/25 = 1 about X-axis.

Concept: Applications of Definite Integrals
Chapter: [0.07] Definite Integrals
Section - II
[12]4
[2]4.A
[2]4.A.i

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 the same month.

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

Find the premium on a property worth ₹12,50,000 at 3% if the property is fully insured.

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

The following table gives the age of the husbands and of the 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 the marginal frequency distribution of the age of husbands.

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

For a bivariate data,

bar x = 53 , bar y = 28 , "b"_"xy" = - 0.2 , "b"_"yx" = -1.5 Find

Estimate of Y , When X = 50.

Concept: Regression Coefficient of X on Y and Y on X
Chapter: [0.13] Regression Analysis Introduction
[2]4.A.v

Values of two regression coefficients between the variables X and Y are b_"yx" = - 0.4 and b_"xy" = - 2.025 respectively. Obtain the value of correlation coefficient.

Concept: Regression Coefficient of X on Y and Y on X
Chapter: [0.13] Regression Analysis Introduction
[2]4.A.vi

Verify whether the following function can be regarded as probability mass function (p.m.f.) for the given values of X :

 X -1 0 1 P(X = x) -0.2 1 0.2
Concept: Random Variables and Its Probability Distributions
Chapter: [0.027999999999999997] Probability Distributions [0.14] Random Variable and Probability Distribution
[2]4.A.vii

The p.m.f. of a random variable X is
"P"(x) = 1/5 , for x = I, 2, 3, 4, 5
= 0 , otherwise.
Find E(X).

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

The time (in hours) required to perform the printing and binding operations (in that order) for each book is given in the following table :

 Books I II III IV V Printing Machine M1 3 7 4 5 7 Binding Machine M2 6 2 7 3 4

Find the sequence that minimizes the total elapsed time (in hours) to complete the work .

Concept: Sequencing in Management Mathematics
Chapter: [0.15] Management Mathematics
[6]5
[3]5.A | Attempt any Two of the following
[3]5.A.i

Find the present value of an annuity immediate.of ₹18,000. p.a. for 3 years at 9% p.a. compounded annually. [Given (1.09)^-3 = 0.7722]

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

Compute rank correlation coefficient for the following data :

 Rx 1 2 3 4 5 6 Ry 6 3 2 1 4 5
Concept: Statistics - Karl Pearson’s Coefficient of Correlation
Chapter: [0.12] Bivariate Data and Correlation
[3]5.A.iii

If the rank correlation coefficient is 2/3 and Σ"d"_1^2 = 55 , then find the number of  pairs of observations. (Assume that no rank is repeated.)

Concept: Statistics - Karl Pearson’s Coefficient of Correlation
Chapter: [0.12] Bivariate Data and Correlation
[8]5.B | Attempt any Two of the following
[4]5.B.i

From the following data, find the crude death rates (C.D.R.) for Town I and Town II, and comment on the results :

 Age Group (in years) Town I Town II Population No. of deaths Population No. of deaths 0-10 1500 45 6000 150 10-25 5000 30 6000 40 25 - 45 3000 15 5000 20 45 & above 500 22 3000 54
Concept: Random Variables and Its Probability Distributions
Chapter: [0.027999999999999997] Probability Distributions [0.14] Random Variable and Probability Distribution
[4]5.B.ii

Calculate quantities indicated by'?' for the following part of a life table :

 x lx dx qx Lx Tx e_x^0 4 9100 60 ? . ? 510000 ? 5 ? 45
Concept: Life Tables
Chapter: [0.11] Demography
[4]5.B.iii

The probability that a bomb dropped from an aeroplane will strike a target is 1/5`, If four bombs are dropped, find the probability that :

(a) exactly two will strike the target,
(b) at least one will strike the target.

Concept: Random Variables and Its Probability Distributions
Chapter: [0.027999999999999997] Probability Distributions [0.14] Random Variable and Probability Distribution
[6]6
[3]6.A | Attempt any Two of the following
[3]6.A.i

Amit and Rohit started a business by investing ₹20,000 each. After 3 months Amit withdrew ₹5,000 and Rohit put in ₹5,000 additionally. How should a profit of ₹12,800 be divided between them at the end of the year?

Concept: Random Variables and Its Probability Distributions
Chapter: [0.027999999999999997] Probability Distributions [0.14] Random Variable and Probability Distribution
[3]6.A.ii

A bill of Rs.7,500 was discounted for Rs. 7,290 at a bank on 28th October 2006. If the rate of interest was 14% p.a., what is the legal due date ?

Concept: Commission, Brokerage and Discount
Chapter: [0.09] Commission, Brokerage and Discount
[3]6.A.iii

Let X be the number of matches played by the player and Y he the number of matches in which he scored more thun 50 runs. The following data is obtained for 5 players :

 No. of Matches Played (X) Data of matches of 5 players 21 25 26 24 19 Scored more than 50 in a match (Y) 19 20 24 21 16

Find the regression line of X on Y.

Concept: Regression Coefficient of X on Y and Y on X
Chapter: [0.13] Regression Analysis Introduction
[8]6.B | Attempt any Two of the following
[4]6.B.i

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

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

Solve the following L.P.P. :
Minimize : Z = 4x+ 2y
Subject to: 3x+ y ≥ 27 ,  x + y ≥ 21,
x + 2y ≥ 30, x ≥ 0 , y ≥ 0

Concept: Linear Programming Problem in Management Mathematics
Chapter: [0.15] Management Mathematics
[4]6.B.iii

Solve the following L.P.P. :
Minimize : Z = 4x + 10y,
Subject to : 2x + 5y ≤10 , 5x + 3y ≤ 15,
x + 2y  ≥ 30, x ≥ 0, y ≥ 0.

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

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