# Mathematics and Statistics 2015-2016 HSC Commerce (Marketing and Salesmanship) 12th Board Exam Question Paper Solution

Mathematics and Statistics
Date & Time: 26th February 2016, 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 I
[12]1 | Attempt any SIX of the following :
[2]1.A

If y = ( sin x )x , Find dy/dx

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

If A = [(1, 3), (3, 1)], Show that A2 - 2A is a scalar matrix.

Concept: Inverse of Matrix
Chapter: [0.012] Matrices [0.02] Matrices
[2]1.C

Write the negation of the Following Statement :
∀ y ∈  N, y2 + 3 ≤ 7

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

Write the negation of the following statement :
If the lines are parallel then their slopes are equal.

Concept: Mathematical Logic - Statement Patterns and Logical Equivalence
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
[2]1.D

The total revenue R = 720 - 3x2 where x is number of items sold. Find x for which total  revenue R is increasing.

Concept: Properties of Definite Integrals
Chapter: [0.016] Definite Integration [0.07] Definite Integrals
[2]1.E

Evaluate : int  (sec^2 x)/(tan^2 x + 4) dx

Concept: Derivatives of Functions in Parametric Forms
Chapter: [0.04] Differentiation
[2]1.F

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

Concept: Properties of Definite Integrals
Chapter: [0.016] Definite Integration [0.07] Definite Integrals
[2]1.G

Discuss the continuity of function f at x = 0.
Where f(X) = [ [sqrt ( 4 + x ) - 2 ]/ ( 3x )], For x ≠ 0
= 1/12,                      For x = 0

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

State if the following sentence is a statement. In case of a statement, write down the truth value :
Every quadratic equation has only real roots.

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

State if the following sentence is a statement. In case of a statement, write down the truth value :
√-4 is a rational number.

Concept: Mathematical Logic - Statement Patterns and Logical Equivalence
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
[14]2
[6]2.A | Attempt any TWO of the following :
[3]2.A.i

Solve the following equations by the inversion method :
2x + 3y = - 5 and 3x + y = 3.

Concept: Inverse of Matrix
Chapter: [0.012] Matrices [0.02] Matrices
[3]2.A.ii

Find x and y, if
{ 3[(1, 2, 0), (0, -1, 3)] - [(1, 5, -2), (-3, -4, 4)]}[(1), (2), (1)] = [(x), (y)]

Concept: Solution of Equations
Chapter: [0.02] Matrices
[3]2.A.iii

Evaluate : int tan^-1x dx

Concept: Solution of Equations
Chapter: [0.02] Matrices
[8]2.B | Attempt any TWO of the following :
[4]2.B.i

Express the truth of each of the following statements using Venn diagram.
(1) All teachers are scholars and scholars are teachers.
(2) If a quadrilateral is a rhombus then it is a parallelogram..

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

Write converse and inverse of the following statement :
"If Ravi is good in logic then Ravi is good in Mathematics."

Concept: Mathematical Logic - Statement Patterns and Logical Equivalence
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
[4]2.B.ii

Find the area of the region bounded by the lines 2y + x = 8, x = 2 and x = 4.

Concept: Applications of Definite Integrals
Chapter: [0.07] Definite Integrals
[4]2.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
[14]3
[6]3.A | Attempt any TWO of the following:
[3]3.A.i

If f(x) = ("e"^(2"x") - 1)/"ax"   , for x < 0 , a ≠ 0

= 1                                    for x = 0

= ("log" (1 + 7"x"))/"bx" , for x > 0 , b ≠ 0

is continuous at x = 0, then find a and b.

Concept: Applications of Definite Integrals
Chapter: [0.07] Definite Integrals
[3]3.A.ii

If the function f is continuous at x = 0 then find f(0),
where f(x) =  [ cos 3x - cos x ]/x^2, x!=0

Concept: Continuous Function of Point
Chapter: [0.03] Continuity
[3]3.A.iii

If f '( x ) = 4x3 - 3x2 + 2x + k and f(0) = 1, f(1) = 4, find f(x).

Concept: Indefinite Integration - Definition of an Integral
Chapter: [0.06] Indefinite Integration
[8]3.B | Attempt any TWO of the following :
[4]3.B.i

Find MPC ( Marginal propensity to Consume ) and APC ( Average Propensity to Consume ) if the expenditure Ec of a person with income I is given as Ec = ( 0.0003 ) I2 + ( 0.075 ) I when I = 1000.

Concept: Increasing and Decreasing Functions
Chapter: [0.013999999999999999] Applications of Derivatives [0.05] Applications of Derivative
[4]3.B.ii

Cost of assembling x wallclocks is ( x^3/3 - 40x^2) and labour charges are 500x. Find the number of wall clocks to be manufactured for which average cost and marginal cost attain their respective minimum.

Concept: Maxima and Minima
Chapter: [0.05] Applications of Derivative
[4]3.B.iii

If cos^-1[(x^2 - y^2)/(x^2 + y^2)] = 2k    "show that y" dy/dx = xtan^2k

Concept: Indefinite Integration - Definition of an Integral
Chapter: [0.06] Indefinite Integration
SECTION II
[12]4
[12]4.A | Attempt any SIX of the following :
[2]4.A.i

Anandi and Rutuja invested Rs. 10,000 each in a business. Anandi withdrew her capital after 7 months. Rutuja continued for the year. After one year, the profit earned by them was Rs. 5,700. Find the profit earned by each person.

Concept: Ratio, Proportion and Partnership
Chapter: [0.08] Ratio, Proportion and Partnership
[2]4.A.ii

Calculate age specific death (A-SDR) rates for the following data :

 Age group ( in years ) Population (in '000) Number of Deaths Below 10 25 50 10 - 30 30 90 30 - 45 40 160 45 - 70 20 100

Concept: Life Tables
Chapter: [0.11] Demography
[2]4.A.iii

For a bivariate data byx = -1.2 and bxy = -0.3. Find the correlation coefficient between x and y.

Concept: Statistics - Bivariate Frequency Distribution
Chapter: [0.12] Bivariate Data and Correlation
[2]4.A.iv

A random variable x has the following probability distribution :

 x 0 1 2 3 4 5 6 P( X = x) k 3k 5k 7k 9k 11k 13k

Find k.

Concept: Probability Distribution - Distribution Function of a Continuous Random Variable
Chapter: [0.14] Random Variable and Probability Distribution
[2]4.A.v

The probability distribution function of continuous random variable X is given by
f( x ) = x/4,  0 < x < 2
= 0,       Otherwise
Find P( x ≤ 1)

Concept: Continuous Function of Point
Chapter: [0.03] Continuity
[2]4.A.vi

From the two regression equations y = 4x - 5 and 3x = 2y + 5, find barx and bary

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

Draw scatter diagram for the following data and interpret it :

 x 10 20 30 40 50 60 70 y 32 20 24 36 40 28 38
Concept: Life Tables
Chapter: [0.11] Demography
[2]4.A.viii

If Σd2 = 66 and n = 10 then find the rank correlation coefficient.

Concept: Statistics - Karl Pearson’s Coefficient of Correlation
Chapter: [0.12] Bivariate Data and Correlation
[12]5
[6]5.A | Attempt any TWO of the following :
[3]5.A.i

Determine l_92 and l_93, "given that"  l_91 = 97, d_91 = 38 and q_92 = 27/59

Concept: Assignment Problem
Chapter: [0.027000000000000003] Assignment Problem and Sequencing [0.15] Management Mathematics
[3]5.A.ii

Calculate CDR for districts A and B and compare them. Also state which district is more healthy..

 Age group( in years ) District A District B No. of persons ('000) No of Deaths No. of persons ('000) No. of Deaths 0 - 15 1 20 2 50 15 - 60 3 30 7 70 60 and above 2 40 1 25
Concept: Measurements of Mortality
Chapter: [0.11] Demography
[3]5.A.iii

If for a bivariate data barx = 10,  bary = 12, V(X) = 9, σ_y = 4 and r = 0.6, estimate y  when x = 5.

Concept: Statistics - Bivariate Frequency Distribution
Chapter: [0.12] Bivariate Data and Correlation
[8]5.B | Attempt any TWO of the following :
[4]5.B.i

Calculate the coefficient of correlation between X and Y series from the following data :

n = 15 ,bar x = 25, bary = 18, σ_x = 3.01, σ_y = 3.03,

sum ("x"_i - bar x) ("y"_i - bar y) = 122

Concept: Statistics - Karl Pearson’s Coefficient of Correlation
Chapter: [0.12] Bivariate Data and Correlation
[4]5.B.ii

Solve the following minimal assignment problem and hence find minimum time where  '- ' indicates that job cannot be assigned to the machine :

 Machines Processing time in hours A B C D E M1 9 11 15 10 11 M2 12 9 - 10 9 M3 - 11 14 11 7 M4 14 8 12 7 8
Concept: Assignment Problem
Chapter: [0.027000000000000003] Assignment Problem and Sequencing [0.15] Management Mathematics
[4]5.B.iii

Solve the following maximal assignment problem :

 Branch Manager Monthly Business ( Rs. lakh) A B C D P 11 11 9 9 Q 13 16 11 10 R 12 17 13 8 S 16 14 16 12

Concept: Assignment Problem
Chapter: [0.027000000000000003] Assignment Problem and Sequencing [0.15] Management Mathematics
[12]6
[6]6.A | Attempt any TWO of the following :
[3]6.A.i

Find the true discount, banker's discount and banker's gain on a bill of Rs. 36,600 due 4 months hence at 5% p.a.

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

Mr. Anil wants to invest at most Rs. 60,000 in Fixed Deposit (F.D.) and Public Provident Fund ( P.P.F. ). He wants to invest at least Rs. 20,000 in F.D. and at least Rs. 15,000 in P.P.F. The rate of interest on F.D. is 8% p.a. and that on P.P.F. is 10% p.a. Formulate the above problem as L.P.P. to determine maximum yearly income.

Concept: Insurance and Annuity
Chapter: [0.1] Insurance and Annuity
[3]6.A.iii

Find graphical solution for following system of linear inequations :
3x + 2y ≤ 180; x+ 2y ≤ 120, x ≥ 0, y ≥ 0
Hence find co-ordinates of corner points of the common region.

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
[8]6.B | Attempt any TWO of the following :
[4]6.B.i

Mrs. Menon plans to save for her daughter's marriage. She wants to accumulate a sum of ₹ 4,00,000 at the end of 4 years. How much should she invest at the end of each year from now, if she can get interest compounded at 10% p.a. ?

[Given (1.1)4 = 1.4641]

Concept: Insurance and Annuity
Chapter: [0.1] Insurance and Annuity
[4]6.B.ii

A car valued at Rs. 4,00,000 is insured for Rs. 2,50,000. The rate of premium is 5% less 20%. How much loss does the owner bear including the premium if value of the car is reduced to 60% of its original value ?

Concept: Ratio, Proportion and Partnership
Chapter: [0.08] Ratio, Proportion and Partnership
[4]6.B.iii

If random variable X has probability distribution function.
f(x) = c/x, 1 < x < 3, c > 0, find c, E(x) and Var(X)

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

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