1WHERE INSPIRATION FINDS YOU A PATH TO SUCCESS

MAHESH TUTORIALS COMMERCES.Y.J.C.

Preliminary Examination II Subject : Mathematics and Statistics

Note : (i) Write Section-I and Section-II on separate answer books.

(ii) All questions of Section-I and Section-II are compulsory.(iii) Figure to the right indicate full marks.

SECTION I

Q.1. Attempt any SIX of the following : ( 2 Marks Each) (12) (1) Write the negation of the following statements :

(a) Economic growth and per capita income is more in America. (b) All students have completed their homework and the teacher is present.

(2) Verify, whether D =

1 0 0

0 2 0

0 0 3

is singular or non singular matrix.

(3) Find dy

dx if y = ( 2 + x)x.

(4) Evaluate tan x

x dx.(5) If the average revenue RA is 50 and elasticity of demand is 5, find marginal revenue Rm.

(6) Evaluate /2

2

0

cos x dx

.

(7) If a

2

1

(3x 2x 1) dx = 11, find a(8) The revenue function is given by R = D2 -40D, where D is demand of the commodity.

For what values of D, the revenue is increasing?

Q.2. (A) Attempt any TWO of the following : ( 3 Marks Each) (06) (1) If the function f is continuous at x = 1, then find f(1), where

f (x) = 2

1 cos x

(1 x)

for x 1.

(2) Evaluate 2

1

3 4x 4x dx.(3) A wholesale merchant observe that demand function for chillies is p = 31e-q/5, where q

is quantity demand for chillies. For what demand the revenue is maximum?

(B) Attempt any TWO of the following : ( 4 Marks Each) (08) (1) In a city, there are two factories A and B. Each factory produces sports clothes for

boys and girls. There are three types of clothes produced in both the factories, type I,II and III.

Date : 02/01/2014

Marks : 80

Duration : 3 Hrs.

Set No. : I

2For boys, the number of units of type I, II and III are 80, 70 and 65 in factory A and 85, 65 and 72 in factory B. For girls, the number of units of type I, II and III are 80, 75 and 90 in factory A and 50, 55 and 80 in factory B. Express the information in terms of matrices and using matrix algebra, answer the following questions : (a) How many total units of type I are produced for boys? (b) What is the total production of each type for boys and for girls?

(2) If cosec-13 3

3 3

x - y

x + y

= tan k, then show that dy

dx =

y

x

(3) Find the volume of the solid obtained by the complete revolution of the ellipse2 2x y

36 25 = 1, about the X-axis.

Q.3. (A) Attempt any TWO of the following : ( 3 Marks Each) (06)(1) Show that (p q) (q p) is tautology.(2) Using rules of negation of compound statements write the negation of the following

symbolic statements :(1) p q (2) p q (3) p q

(3) If A =

1 1 1

2 1 3

1 1 1

Find the adjoint of the matrix A and hence find A-1.

(B) Attempt any TWO of the following : ( 4 Marks Each) (8)(1) In a firm the cost function for output x is given as

C = 3x

3-20x2 + 70x. Find the output of for which

(a) Marginal Cost (MC) is minimum,(b) Average Cost (AC) is minimum.

(2) If f (x) = 2

1 sinx

( 2x)

, for x 2

, is continuous at x =

2

, then find f

2

.

(3) By using integration, find the area of circle x2 + y2 = 25.

SECTION II

Q.4. (A) Attempt any SIX of the following : ( 2 Marks Each) (12)(1) If the numerator of a function is increased by 20% and its denominator be diminished

by 10%, the value, of the fraction is 16

21. Find the original fraction.

(2) An agent charges 8% commission plus a 2% delcredere. If he sells goods worth ` 12,400,find his total earnings.

(3) Obtain crude death rate (CDR) for the population given below :

Age group: Population Deaths

Below 10 900 12

10 20 1500 8

20 60 4500 36

Above 60 600 30

3(4) Calculate mean and variance of Poisson variable X, if P(X = 4) = P(X = 5).(5) Calculate Karl Pearsons Coefficient of correlation if variance of x is 50, variance of y

is 800 and covariance between x and y is 15.(6) Suppose X is a random variable with p.d.f.,

f(x) =k

x, 0 < x < 4

= 0, elsewherefind the value of k.

(7) Find the solution set represented by

5x + 4y 20 , x 6, y 4, x, y, 0,(8) If the present worth of a certain amount due 6 months at 4% p.a. is ` 625, find the sum

due and the true discount.

Q.5. (A) Attempt any TWO of the following : ( 3 Marks Each) (06)(1) For a population the following information is available :

Number of persons who attain the exact age of 71 years = 100Number of persons who attain the exact age of 72 years = 80.Number of persons living from age 71 years until all of them die = 250.Find :(a) The probability that a person of age 71 years will survive till his next birthday.(b) Number of persons living between the age71 years and 72 years.

(c) The average number of years a person of age72 years is expected to live under the prevailing conditions of mortality.

(2) Calculate Karl Pearsons coefficient of correlation for the following data.

X 10 12 14 18 20 16

Y 20 25 30 35 26 20

(3) The probability of defective bolts in a workshop is 40%. Find mean and variance of the defective bolts out of 10 randomly selected bolts out of 10 randomly selected bolts.

(B) Attempt any TWO of the following : ( 4 Marks Each) (08)(1) If 5x+ y = 21 and 4x+ 5y = 42 are two regression equations, find the mean values of

x and y . Also find the coefficient of correlation r.

(2) A chartered accountant's firm has accepted five new cases. The estimated number of days required by each of their five employees for each cases are given below, where _____ means that the particular employee cannot be assigned the particular case. Determine the optimal assignment of cases to the employees so that the total number of days required to complete these five cases will be minimum. Also find the minimum number of days.

Employees Cases

I II III IV V

E1 5 2 4 2 6

E2 3 4 - 5 7

E3 6 3 4 1 2

E4 4 2 2 3 5

E5 3 6 4 7 3

4(3) A firm manufactures two products A and B on which the profit per unit is ` 30 and ` 40 respectively. Each product is processed on two machines M1 and M2, which are available for use for at most 450 minutes and 600 minutes during any working day, respectively.

The product A requires 1 minute processing time on M1 and 2 minutes time on M2, per unit. The product B requires 3 minutes processing time on M1 and 1 minute processing time on M2, per unit. It is required to find the number of units of products A and B to be produced per day so as to get maximum profit. Formulate this problem as L.P.P. and show feasible region graphically and state the vertices of feasible region.

Q.6. (A) Attempt any TWO of the following : ( 3 Marks Each) (06)(1) Find the two regression lines for the following data and hence if a student obtained 18

marks in Mathematics estimate his marks in Statistics.

Mathematics Statistics

Mean 20 25

S.D. 2 3

Correlation coefficient r = 0.75

(2) If 2% of electric bulbs are defective, find the probability that in a sample of 200 bulbs more than two are defective. (e-4 = 0.0183)

(3) A salesman is paid commission at 5% on total sales, if his total sales are up to ` 15,000 and at 7% on total sales if total sales exceed ` 15,000. In two consecutive months period, his total sales increased by ` 12,000 and his commission also increased by ` 1,040. Find the sales in these two months

(B) Attempt any TWO of the following : ( 4 Marks Each) (08)(1) An article was sold for a net ` 6,664 after allowing a 15% trade discount and a further

2% for cash payment and the profit thus earned was 33.28% on cost. Find the list price and cost price.

(2) Find the rate of interest compounded per annum if an annuity immediate at ` 10,000 per year amounts to ` 1,30,000 in 3 years.

(3) Taking population of A as standard population, find the standardised death rate of the two districts. Which of the districts is healthier?

Age group: 0 10 10 55 above 55

Population in A 4000 6000 3000

Deaths in A 48 18 90

Population in B 3000 7000 4000

Deaths in B 27 28 72

WHERE INSPIRATION FINDS YOU A PATH TO SUCCESS

MAHESH TUTORIALS COMMERCE

S.Y.J.C.

Preliminary Examination II

Subject : Mathematics and Statistics

Note : (i)Write Section-I and Section-II on separate answer books.

(ii)All questions of Section-I and Section-II are compulsory.

(iii)Figure to the right indicate full marks.

SECTION I

Q.1. Attempt any SIX of the following : ( 2 Marks Each) (12)

(1)Write the negation of the following statements :

(a) Economic growth and per capita income is more in America.

(b) All students have completed their homework and the teacher is present.

(2)Verify, whether D =

is singular or non singular matrix.

(3)Find

if y = ( 2 + x)x.

(4)Evaluate

dx.

(5)If the average revenue RA is 50 and elasticity of demand ( is 5, find marginal revenue Rm.

(6)Evaluate

.

(7)If

dx = 11, find a

(8)

The revenue function is given by R = D2 -40D, where D is demand of the commodity. For what values of D, the revenue is increasing?

Q.2. (A)Attempt any TWO of the following : ( 3 Marks Each)(06)

(1)If the function f is continuous at x = 1, then find f(1), where

f (x) =

for x 1.

(2)Evaluate

dx.

(3)A wholesale merchant observe that demand function for chillies is p = 31e-q/5, where q is quantity demand for chillies. For what demand the revenue is maximum?

(B)Attempt any TWO of the following : ( 4 Marks Each)(08)

(1)In a city, there are two factories A and B. Each factory produces sports clothes for boys and girls. There are three types of clothes produced in both the factories, type I, II and III.

For boys, the number of units of type I, II and III are 80, 70 and 65 in factory A and 85, 65 and 72 in factory B. For girls, the number of units of type I, II and III are 80, 75 and 90 in factory A and 50, 55 and 80 in factory B. Express the information in terms of matrices and using matrix algebra, answer the following questions :

(a)How many total units of type I are produced for boys?

(b)What is the total production of each type for boys and for girls?

(2)If cosec-1

= tan k, then show that

=

(3)Find the volume of the solid obtained by the complete revolution of the ellipse

= 1, about the X-axis.

Q.3. (A)Attempt any TWO of the following : ( 3 Marks Each) (06)

(1)Show that (p ( q) ( (q ( p) is tautology.

(2)Using rules of negation of compound statements write the negation of the following symbolic statements :

(1) (p ( (q (2) p ( q (3) (p ( (q

(3)If A =

Find the adjoint of the matrix A and hence find A-1.

(B)Attempt any TWO of the following : ( 4 Marks Each) (8)

(1)In a firm the cost function for output x is given as

C =

-20x2 + 70x. Find the output of for which

(a)Marginal Cost (MC) is minimum,

(b)Average Cost (AC) is minimum.

(2)If f (x) =

, for x

, is continuous at x =

, then find f

.

(3)By using integration, find the area of circle x2 + y2 = 25.

SECTION II

Q.4. (A)Attempt any SIX of the following : ( 2 Marks Each) (12)

(1)If the numerator of a function is increased by 20% and its denominator be diminished by 10%, the value, of the fraction is

. Find the original fraction.

(2)An agent charges 8% commission plus a 2% delcredere. If he sells goods worth ` 12,400, find his total earnings.

(3)Obtain crude death rate (CDR) for the population given below :

Age group:

Population

Deaths

Below 10

900

12

10 20

1500

8

20 60

4500

36

Above 60

600

30

(4)Calculate mean and variance of Poisson variable X, if P(X = 4) = P(X = 5).

(5)Calculate Karl Pearsons Coefficient of correlation if variance of x is 50, variance of y is 800 and covariance between x and y is 15.

(6)Suppose X is a random variable with p.d.f.,

f(x)=

, 0 < x < 4

=0, elsewhere

find the value of k.

(7)Find the solution set represented by

5x + 4y ( 20 , x ( 6, y ( 4, x, y, ( 0,

(8)

If the present worth of a certain amount due 6 months at 4% p.a. is ` 625, find the sum due and the true discount.

Q.5. (A)Attempt any TWO of the following : ( 3 Marks Each) (06)

(1)For a population the following information is available :

Number of persons who attain the exact age of 71 years = 100

Number of persons who attain the exact age of 72 years = 80.

Number of persons living from age 71 years until all of them die = 250.

Find :

(a)The probability that a person of age 71 years will survive till his next birthday.

(b)Number of persons living between the age71 years and 72 years.

(c)The average number of years a person of age72 years is expected to live under the prevailing conditions of mortality.

(2)Calculate Karl Pearsons coefficient of correlation for the following data.

X

10

12

14

18

20

16

Y

20

25

30

35

26

20

(3)The probability of defective bolts in a workshop is 40%. Find mean and variance of the defective bolts out of 10 randomly selected bolts out of 10 randomly selected bolts.

(B)Attempt any TWO of the following : ( 4 Marks Each) (08)

(1)If 5x+ y = 21 and 4x+ 5y = 42 are two regression equations, find the mean values of

and

. Also find the coefficient of correlation r.

(2)A chartered accountant's firm has accepted five new cases. The estimated number of days required by each of their five employees for each cases are given below, where _____ means that the particular employee cannot be assigned the particular case. Determine the optimal assignment of cases to the employees so that the total number of days required to complete these five cases will be minimum. Also find the minimum number of days.

Employees

Cases

I

II

III

IV

V

E1

5

2

4

2

6

E2

3

4

-

5

7

E3

6

3

4

1

2

E4

4

2

2

3

5

E5

3

6

4

7

3

(3)A firm manufactures two products A and B on which the profit per unit is ` 30 and ` 40 respectively. Each product is processed on two machines M1 and M2, which are available for use for at most 450 minutes and 600 minutes during any working day, respectively.

The product A requires 1 minute processing time on M1 and 2 minutes time on M2, per unit. The product B requires 3 minutes processing time on M1 and 1 minute processing time on M2, per unit. It is required to find the number of units of products A and B to be produced per day so as to get maximum profit. Formulate this problem as L.P.P. and show feasible region graphically and state the vertices of feasible region.

Q.6.(A)Attempt any TWO of the following : ( 3 Marks Each) (06)

(1)

Find the two regression lines for the following data and hence if a student obtained 18 marks in Mathematics estimate his marks in Statistics.

Mathematics

Statistics

Mean

20

25

S.D.

2

3

Correlation coefficient r = 0.75

(2)If 2% of electric bulbs are defective, find the probability that in a sample of 200 bulbs more than two are defective. (e-4 = 0.0183)

(3)A salesman is paid commission at 5% on total sales, if his total sales are up to ` 15,000 and at 7% on total sales if total sales exceed ` 15,000. In two consecutive months period, his total sales increased by ` 12,000 and his commission also increased by ` 1,040. Find the sales in these two months

(B)Attempt any TWO of the following : ( 4 Marks Each) (08)

(1)An article was sold for a net ` 6,664 after allowing a 15% trade discount and a further 2% for cash payment and the profit thus earned was 33.28% on cost. Find the list price and cost price.

(2)Find the rate of interest compounded per annum if an annuity immediate at ` 10,000 per year amounts to ` 1,30,000 in 3 years.

(3)Taking population of A as standard population, find the standardised death rate of the two districts. Which of the districts is healthier?

Age group:

0 10

10 55

above 55

Population in A

4000

6000

3000

Deaths in A

48

18

90

Population in B

3000

7000

4000

Deaths in B

27

28

72

Duration : 3 Hrs.

Set No. : I

Date : 02/01/2014

Marks : 80

PAGE

4

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