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Mathematics and Statistics 2013-2014 HSC Commerce (Marketing and Salesmanship) 12th Board Exam Question Paper Solution

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Mathematics and Statistics
Marks: 80Academic Year: 2013-2014
Date: October 2013

(1) All questions are compulsory.
(2) Answer to every question must be written on new page.
(3) Log table will be provided on demand.
(4) Answer to Section-I and Section-II should be written in two seperate answer books.
(5) Questions from Section-I attempted in the answer book of Section-II and vice versa will not be assessed/not given any credit.
(6) Graph paper is necessary for L.P.P.
(7) Figure to the right indicate full marks.


SECTION-I
[12]1 | Attempt any SIX
[2]1.A

Write the following statements in symbolic forms :
(a) Either 49 is a perfect square or 39 is divisible by 11.
(b) It is not true that if 'i is a real number, then '2' is an even prime number.

Concept: Normal Distribution
Chapter: [0.14] Random Variable and Probability Distribution
[2]1.B

If A = `[(1,2,3), (2,k,2), (5,7,3)]` is a singular matrix then find the value of 'k'.

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

If A = `[(7,1), (2,5)]` and B = `[(1,2), (3,-1)]` then verify that |AB| = |A|  |B|.

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

Discuss the continuity of the function at the point given. If the function is discontinuous, then remove the discontinuity.

f (x) = `(sin^2 5x)/x^2` for x ≠ 0 
= 5   for x = 0, at x = 0

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

Find the value of x for which the function `f(x) = x^3 - 3x^2 - 9x + 25` is increasing.

Concept: Maxima and Minima
Chapter: [0.05] Applications of Derivative
[2]1.F

Differentiate tan-1 (cot 2x) w.r.t.x.

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

Discuss the continuity of the function `f(x) = (3 - sqrt(2x + 7))/(x - 1)` for x ≠ 1

= `-1/3`   for x = 1, at x = 1

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

Evaluate : `int e^x [(x + 3)/(x + 4)^2] dx`

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

Without using the truth table, show that p ∧ [(∼p ∨ q) ∨ ∼q] ≡ p

Concept: Indefinite Integration - Definition of an Integral
Chapter: [0.06] Indefinite Integration
[3]2.A.ii

If Y = tan-1 `[(cos 2x - sin 2x)/(sin2x + cos 2x)]` then find `(dy)/(dx)`

Concept: Continuous Function of Point
Chapter: [0.03] Continuity
[3]2.A.iii

Evaluate = `int (tan x)/(sec x + tan x)` . dx

Concept: Properties of Definite Integrals
Chapter: [0.016] Definite Integration [0.07] Definite Integrals
[8]2.B | Attempt any Two
[4]2.B.i

If the function
f(x) = x2 + ax + b,         x < 2

      = 3x + 2,                 2≤ x ≤ 4

      = 2ax + 5b,             4 < x

is continuous at x = 2 and x = 4, then find the values of a and b

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

The total cost function of a firm is C = x2 + 75x + 1600 for output x. Find the  output for which the average cost ls minimum. Is CA= Cm at this output?  

Concept: Derivative - Derivative of Inverse Function
Chapter: [0.04] Differentiation
[4]2.B.iii

Find the area of the ellipse `x^2/4 + y^2/25 = 1`

Concept: Indefinite Integration - Integral of Standard Functions
Chapter: [0.06] Indefinite Integration
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[14]3
[6]3.A | Attempt any Two
[3]3.A.i

Examine whether the following statement (p ∧ q) ∨ (∼p ∨ ∼q) is a tautology or contradiction or neither of them.

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

Find `(dy)/(dx)` if x = a cosec θ, y = b cot θ at θ = `π/4`

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

Evaluate : `int 1/(x^2 + 8x + 20)` dx

Concept: Indefinite Integration - Integral of Standard Functions
Chapter: [0.06] Indefinite Integration
[8]3.B | Attempt any Two
[4]3.B.i

Express the following equations in matrix form and solve them by the method of inversion.
x + 2y + 3z = 8,
2x - y + z = 1,
3x + y - 4z = 1

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

The expenditure Ee of a person with income. x  ls given by Ec = (0.000035).x2 + (0.045)x~ Find the marginal propensity to consume and marginal propensity to save when x = 5000. Also find the average propensity to consume and average propensity to save.

Concept: Insurance and Annuity
Chapter: [0.1] Insurance and Annuity
[4]3.B.iii

Evaluate : `int_(π/6)^(π/3) 1/(1 + sqrt(cotx)` dx

Concept: Indefinite Integration - Integral of Standard Functions
Chapter: [0.06] Indefinite Integration
SECTION = II
[12]4 | Attempt any Six
[2]4.A

Obtain the Crude Death Rate (CDR) for city A from the data given below.

Age group
(in years) 
City A
  Population No. of deaths
Below 15 800 32
15 - 25 3000 12
25 - 65 4800 48
65 and above 1400 42
Concept: Uses of Vital Statistics in Demography
Chapter: [0.11] Demography
[2]4.B

Compute the product moment coefficient of correlation for the following data: 
n = 100, `bar x` = 62, `bary` = 53, `sigma_x` = 10, `sigma_y` = 12

`Sigma (x_i - bar x) (y_i - bary) = 8000`

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
[2]4.C

Find the age standard death rate (S.D.R.) for the following data:

Age 
(in years)
Population No. of deaths
0 - 20 40000 350
20 - 65 65000 650
65 and above 15000 x

Using the above information find x, if the CDR = 13.4 per thousand.

Concept: Inequations in Management Mathematics
Chapter: [0.15] Management Mathematics
[2]4.D

The ranks of the 10 students in Mathematics and Accountancy are as follows  : 
Two numbers given in the brackets denote the ranks of the students in Mathematics and Accountancy respectively.
(1. 1). (2. 10). (3. 3). (4. 4). (5. 5). (6. 7). (7. 2). (8. 6). (9. 8). (10. 9) 
Calculate the rank correlation coefficient

Concept: Rank Correlation
Chapter: [0.12] Bivariate Data and Correlation
[2]4.E

If X ∼ N (4,25), then find P(x ≤ 4)

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

Find common region for the following system of linear inequations :

x ≥ -3, 4x - 5y ≥ -20, 3x + 4y ≤ 12, y ≥ -2

Concept: Linear Programming Problem in Management Mathematics
Chapter: [0.15] Management Mathematics
[2]4.G

For the following problem, find the sequence that minimize total elapsed time (in hours) required to complete the following jobs on two machines M1 and M2 in the order M1 - M2

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

A company produces mixers and food processor. Profit on selling one mixer and one food processor is Rs 2,000 and Rs 3,000 respectively. Both the products are processed through three machines A, B, C. The time required in hours for each product and total time available in hours per week on each machine arc as follows:

Machine  Mixer Food Processor Available time
A 3 3 36
B 5 2 50
C 2 6 60

Formulate the problem as L.P.P. in order to maximize the profit.

Concept: Sequencing in Management Mathematics
Chapter: [0.15] Management Mathematics
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[14]5
[6]5.A | Attempt any Two
[3]5.A.i

The defects on a plywood sheet occur at random with an average of the defect per 50 sq. ft. What Is the probability that such sheet will have-

(a) No defects
(b) At least one defect 
[Use e-1 = 0.3678]

Concept: Random Variables and Its Probability Distributions
Chapter: [0.027999999999999997] Probability Distributions [0.14] Random Variable and Probability Distribution
[3]5.A.ii

A Banker's discount calculated for 1 year is 26 times his gain. Find the rate or interest.

Concept: Commission, Brokerage and Discount
Chapter: [0.09] Commission, Brokerage and Discount
[3]5.A.iii

For the two regression equations 4y = 9x + 15 and 25x = 6y + 7 find correlation coefficient r, `barx, bary`

Concept: Regression Coefficient of X on Y and Y on X
Chapter: [0.13] Regression Analysis Introduction
[8]5.B | Attempt any Two
[4]5.B.i

Incomes or Mr. Shah, Mr. Patel and Mr. Mehta are ln the ratio 1 : 2 : 3, while their expenditures are in the ratio 2 : 3 : 4. if Mr. Shah saves 20% or his lncome, find the ratio of their savings.

Concept: Ratio, Proportion and Partnership
Chapter: [0.08] Ratio, Proportion and Partnership
[4]5.B.ii

A sample of 5 items is taken from the production of a firm. Length and weight of the five items arc given below :

Length (cm) 3 4 6 7 10
Weight (gm) 9 11 14 15 16

Calculate Karl Pearson's coefficient of   correlation between the length and weight and interpret the result.

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

A departmental head has three jobs and four subordinates. The subordinates differ in their capabilities and the jobs differ in their work
contents. With the help of the performance matrix given below, find out which of the four subordinates should be assigned which jobs ?

Subordinates Jobs
I II III
A 7 3 5
B 2 7 4
C 6 5 3
D 3 4 7
Concept: Assignment Problem
Chapter: [0.027000000000000003] Assignment Problem and Sequencing [0.15] Management Mathematics
[14]6
[6]6.A | Attempt any Two
[3]6.A.i

Raghu, Madhu and Ramu started a business in partnership by investing Rs 60,000, Rs 40,000 and Rs 75,000 respectively. At the end of the year, they found that they have incurred a loss of Rs 24,500. Find how much loss each one had to bear.

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

A card is drawn at random and replaced four times from a well shuftled pack of 52 cards. Find the probability that -

(a) Two diamond cards are drawn.
(b) At least one diamond card is drawn.

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

Find tha agents commission at 15% on the first premium if he places insurance for Rs 2,00,000 on the life of a person, the premium being at the rate of Rs 35 per thousand, per annum, paid annually.

Concept: Commission, Brokerage and Discount
Chapter: [0.09] Commission, Brokerage and Discount
[8]6.B | Attempt any two
[4]6.B.i

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
[4]6.B.ii

Information on v:ehicles [in thousands) passing through seven different highways during a day (X) and number of accidents reported (Y) is given as follows :   

`Sigmax_i` = 105, `Sigmay_i` = 409, n = 7, `Sigmax_i^2` = 1681, `Sigmay_i^2` = 39350 `Sigmax_iy_i` = 8075

  Obtain the linear regression of Y on X.

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
[4]6.B.iii

In a factory there are six jobs to be performed each of which should go through two machines A and B in the order A - B. The processing timing (in hours) for the jobs arc given here. You are required to determine the sequence for performing the jobs that would minimize the total elapsed time T. What is the value of T? Also find the idle time for machines · A and B.

Jobs J1 J2 J3 J4 J5 J6
Machine A 1 3 8 5 6 3
MAchine B 5 6 3 2 2 10
Concept: Assignment Problem
Chapter: [0.027000000000000003] Assignment Problem and Sequencing [0.15] Management Mathematics
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Maharashtra State Board previous year question papers 12th Board Exam Mathematics and Statistics with solutions 2013 - 2014

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