Mathematics and Statistics Set 1 2018-2019 HSC Commerce (Marketing and Salesmanship) 12th Board Exam Question Paper Solution

Mathematics and Statistics [Set 1]
Date & Time: 2nd March 2019, 11:00 am
Duration: 3h
• All questions are compulsory.
• Answer to every question must be written on a new page.
• Log lable will be provided on demand.
• Answer to Section-I and Section-II should be written in two separate answer books.
• Questions from Section-I attempted in the answer book of Section-II and vice versa will not be assessed/not given any credit.
• Graph paper is necessary for L.P.P.
• Figure to the right indicate full marks.

SECTION 1
 1 | Attempt any Six
 1.A

Write converse and inverse of the following statement:
“If a man is a bachelor then he is unhappy.”

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

Discuss the continuity of f at x = 1
Where f(X) = [ 3 - sqrt ( 2x + 7 ) / ( x - 1 )]           For x ≠ 1
= -1/3                                                 For x = 1

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

Find k, if the function f is continuous at x = 0, where

f(x)=[(e^x - 1)(sinx)]/x^2,      for x ≠ 0

= k                             ,        for x = 0

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

Find the marginal revenue if the average revenue is 45 and elasticity of demand is 5.

Concept: Application of Derivatives to Economics
Chapter: [0.013999999999999999] Applications of Derivatives
 1.E

Find dy/dx if x^3 + y^2 + xy = 7

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

Find the area bounded by the curve y = x4, x-axis and lines x = 1 and x = 5.

Concept: Applications of Definite Integrals
Chapter: [0.07] Definite Integrals
 1.G

Evaluate :  int_-2^3 dx/(x+5)

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

Evaluate
int  1/(16 - 9x^2) dx

Concept: Logarithmic Differentiation
Chapter: [0.04] Differentiation
 2
 2.A | Attempt any Two of the Following:
 2.A.i

Prove that the following statement pattern is a tautology : ( q → p ) v ( p → q )

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

Find dy/dx if y = x+ 5x

Concept: Logarithmic Differentiation
Chapter: [0.04] Differentiation
 2.A.iii

Evaluate :
int xcos^-1x dx

Concept: Indefinite Integration - Rules of Integration
Chapter: [0.06] Indefinite Integration
 2.B | Attempt any Two of the following :
 2.B.i

Find the inverse of the matrix [ (1, 2, 3), (1, 1, 5), (2, 4, 7)] by using the adjoint method.


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

If f is continuous at x = 0 then find f(0) where f(x) = [5^x + 5^-x - 2]/x^2, x ≠ 0

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

A manufacturer can sell x items (x > 0) at a price of Rs.(280 - x) each. The cost of producing x items is Rs. (x2 + 40x + 35). Find the number of items to be sold so that the manufacturer can make maximum profit.

Concept: Commission, Brokerage and Discount
Chapter: [0.09] Commission, Brokerage and Discount
 3
 3.A | Attempt any Two of the following :
 3.A.i

If p and q are true statements and r and s are false statements, find the truth value of the following :
( p ∧  ∼ r ) ∧ ( ∼ q ∧ s )

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

Differentiate e4x + 5 w.r..t.e3x

Concept: Derivatives of Implicit Functions
Chapter: [0.013000000000000001] Differentiation [0.04] Differentiation
 3.A.iii

Evaluate :
int [e^x ( 1 + x )]/[ cos^2 (xe^x)] dx

Concept: Indefinite Integration - Rules of Integration
Chapter: [0.06] Indefinite Integration
 3.B | Attempt any TWO of the following :
 3.B.i

If A = [(2, 3), (1, 2)], B = [(1, 0),(3, 1)], Find (AB)-1

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

For manufacturing x units, labour cost is 150 - 4x and processing cost is x2. Price of each unit is p = 10,800 - 4x2. Find the values of x for which :
(a) Total cost is decreasing
(b) Revenue is increasing

Concept: Algebra of Statements
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
 3.B.iii

Evaluate : int_3^9 [root(3)(12-x)]/[ root(3)(x) + root(3)(12 - x)]

Concept: Applications of Definite Integrals
Chapter: [0.07] Definite Integrals
SECTION - II
 4 | Attempt any SIX of the following:
 4.A

Two fair coins are tossed simultaneously. If X denotes the number of heads, find the probability distribution of X. Also find E(X).

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

If the correlation coefficient between X and Y is 0.8, what is the correlation
coefficient between.
(a) X and 3Y
(b) X - 5 and Y - 3

Concept: Statistics - Bivariate Frequency Distribution
Chapter: [0.12] Bivariate Data and Correlation
 4.C

Find the premium on property worth Rs.12,50,000 at 3% if the property is insured to the extent of 80% of its value.

Concept: Insurance and Annuity
Chapter: [0.1] Insurance and Annuity
 4.D

If the sum of squares of differences of ranks for 10 pairs of observations is 66, find the rank correlation coefficient.

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

If the present worth of a bill due 6 months hence is Rs. 2,500 at 10% per annum, what is the true discount ?

Concept: Commission, Brokerage and Discount
Chapter: [0.09] Commission, Brokerage and Discount
 4.F

From the following table find q0 :

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

Compute CDR using the information given below :

 Age group ( Years ) 0-15 15-35 35-65 65 and above Population 9000 25000 32000 9000

Total number of deaths in a year is given to be 900.

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

What must be subtracted from each of the numbers 5, 7 and 10, so that the
resulting numbers are in continued proportion?

Concept: Ratio, Proportion and Partnership
Chapter: [0.08] Ratio, Proportion and Partnership
 5
 5.A | Attempt any TWO of the following :
 5.A.i

An article is marked at Rs. 1500. A trader allows a discount at 3% and still gains 20% on the cost. Find the cost price of the article.

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

For a Binomial distribution n = 6 and p = 0.3, find the probability of getting
exactly 3 successes.

Concept: Binomial Theorem and Binomial Distribution
Chapter: [0.14] Random Variable and Probability Distribution
 5.A.iii

Diet for a sick person must contain at least 4000 units of vitamins, 50 units of minerals and 1500 calories Two foods F1 and F2 cost Rs. 50 and Rs. 75 per unit respectively. Each unit of food F1 contains 200 units of vitamins, 1 unit of minerals and 40 calories, whereas each unit of food F2 contains 100 units of  vitamins, 2 units of minerals and 30 calories. Formulate the above problem as L.P.P. to satisfy the sick person's requirements at minimum cost.

Concept: Linear Programming Problem in Management Mathematics
Chapter: [0.15] Management Mathematics
 5.B | Attempt any TWO of the following :
 5.B.i

Two samples from bivariate populations have 15 observations each. The sample mean of X and Y are 25 and 18 respectively. The corresponding sum of squares of deviations from means are 136 and 148. The sum of product of deviations from respective means is 122. Obtain the equation of line of regression of X on Y.

Concept: Regression Coefficient of X on Y and Y on X
Chapter: [0.13] Regression Analysis Introduction
 5.B.ii

Suggest optimum solution to the following assignment. Problem, also find the total minimum service time.
Service Time ( in hrs.)

 Counters Salesmen A B C D W 41 72 39 52 X 22 29 49 65 Y 27 39 60 51 Z 45 50 48 52
Concept: Assignment Problem
Chapter: [0.027000000000000003] Assignment Problem and Sequencing [0.15] Management Mathematics
 5.B.iii

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
 6
 6.A | Attempt any TWO of the following :
 6.A.i

For 50 students of a class, the regression equation of marks in statistics (X) on the marks in accountancy (Y) is 3y - 5x + 180 = 0. The mean marks in accountancy is 44 and the variance of marks in statisstics is (9/16)th of the variance of marks in accountancy. Find the mean marks in statistics and the
correlation coefficient between marks in the two subjects.

Concept: Regression Coefficient of X on Y and Y on X
Chapter: [0.13] Regression Analysis Introduction
 6.A.ii

The p.d.f. of a random variable X is given by :
f(x) = 2x,                  0 ≤ x ≤ 1
= 0,                    otherwise
Find P(1/3 < x < 1/2)`

Concept: Probability Distribution of a Discrete Random Variable
Chapter: [0.14] Random Variable and Probability Distribution
 6.A.iii

Find the sequence that minimizes the total elapsed time required to complete the following task. The table below gives the processing time in hours. Also, find the minimum elapsed time and idle times for both machines.

 Jobs 1 2 3 4 5 M1 3 7 4 5 7 M2 6 2 7 3 4
Concept: Sequencing in Management Mathematics
Chapter: [0.15] Management Mathematics
 6.B | Attempt any TWO of the following :
 6.B.i

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
 6.B.ii

The following data gives the marks of 20 students in mathematics (X) and statistics (Y) each out of 10, expressed as (x, y). construct ungrouped frequency distribution considering single number as a class.
Also prepare marginal distributions :

(2, 7) (3, 8) (4, 9) (2, 8) (2, 8) (5, 6) (5 , 7) (4, 9) (3, 8) (4, 8) (2, 9) (3, 8) (4, 8) (5, 6) (4, 7) (4, 7) (4, 6 ) (5, 6) (5, 7 ) (4, 6 )

Concept: Probability Distribution - Distribution Function of a Continuous Random Variable
Chapter: [0.14] Random Variable and Probability Distribution
 6.B.iii

Find the feasible solution for the following system of linear inequations:
0 ≤ x ≤ 3, 0 ≤ y ≤ 3, x + y ≤ 5, 2x + y ≥ 4

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

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