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Question Paper Solutions - Mathematics and Statistics 2014 - 2015-H.S.C-12th Board Exam Maharashtra State Board (MSBSHSE)


Marks: 80
[12]1 | Attempt any SIX of the following
[2]1.1

Express the following statement in symbolic form and write its truth value.

"If 4 is an odd number, then 6 is divisible by 3 "

Chapter: [1] Mathematical Logic
Concept: Mathematical Logic - Statement Patterns and Logical Equivalence
[2]1.2
 

Find the values of x and y, if 

`2[[1,3],[0,x]]+[[y,0],[1,2]]=[[5,6],[1,8]]`

 

 
Chapter: [2] Matrices
Concept: Solution of Equations
[2]1.3

Find the value of 'k' if the function

`f(X)=(tan7x)/(2x) , `

`=k`,            for x=0

is continuos at x=0

 

Chapter: [3] Continuity
Concept: Continuous Function of Point
[2]1.4

Find dy/dx if `y=cos^-1(sqrt(x))`

Chapter: [4] Differentiation
Concept: Derivative - Derivative of Inverse Function
[2]1.5

Price P for demand D is given as P = 183 +120D - 3D2 Find D for which the price is increasing

Chapter: [5] Applications of Derivative
Concept: Increasing and Decreasing Functions
[2]1.6

Evaluate : `int1/(x(3+logx))dx`

 

Chapter: [6] Indefinite Integration
Concept: Indefinite Integration - Methods of Integration
[2]1.7

If   `a=[[2,1],[1,1]] " then show taht " A^2-3A+I=0`

 

Chapter: [2] Matrices
Concept: Solution of Equations
[2]1.8

evaluate : `int xcosxdx`

Chapter: [6] Indefinite Integration
Concept: Indefinite Integration - Integration by Parts
[14]2
[6]2.1 | Attempt any TWO of the following
[3]2.1.1

Prove that the following statement pattern is equivalent :

(p ∨ q)  r and (p → r) ∧ (q → r)

Chapter: [1] Mathematical Logic
Concept: Mathematical Logic - Statement Patterns and Logical Equivalence
[3]2.1.2

Examine the continuity of the following function : 

`f(x)=x^2-x+9` , for x ≤ 3

        `=4x+3`          for x > 3`

 

Chapter: [3] Continuity
Concept: Continuous Function of Point
[3]2.1.3

find dy/dx if `y=tan^-1((6x)/(1-5x^2))`

Chapter: [4] Differentiation
Concept: Derivative - Derivative of Inverse Function
[8]2.2 | Attempt any TWO of the following :
[4]2.2.1

 Find the inverse of the following matrix by elementary row transformations if it exists. `A=[[1,2,-2],[0,-2,1],[-1,3,0]]`

Chapter: [2] Matrices
Concept: Inverse of Matrix
[4]2.2.2

Find the area of elipse `x^2/a^2+y^2/b^2=1`

Chapter: [7] Definite Integrals
Concept: Applications of Definite Integrals
[4]2.2.3

The expenditure Ec of a person with income I is given by Ec = (0.000035) I2 + (0.045) I. Find marginal propensity to consume (MPC) and marginal propensity to save (MPS) when I = 5000. Also find A (average) PC and A (average)

PS.

Chapter: [5] Applications of Derivative
Concept: Maxima and Minima
[14]3
[6]3.1 | Attempt any TWO of the following :
[3]3.1.1

Express the truth of each of the following statements by Venn diagram:

(a) Some hardworking students are obedient.

(b) No circles are polygons.

(c) All teachers are scholars and scholars are teachers. 

 

Chapter: [12.01] Set
Concept: Venn Diagrams
[3]3.1.2

 If 'f' is continuous at x = 0, then find f(0).

`f(x)=(15^x-3^x-5^x+1)/(xtanx) , x!=0`

Chapter: [3] Continuity
Concept: Continuous Function of Point
[3]3.1.3

find dy/dx if x=e2t , y=`e^sqrtt`

Chapter: [4] Differentiation
Concept: Derivatives of Functions in Parametric Forms
[8]3.2 | Attempt any TWO of the following :
[4]3.2.1

Evaluate : `int (1+logx)/(x(2+logx)(3+logx))dx`

Chapter: [6] Indefinite Integration
Concept: Indefinite Integration - Methods of Integration
[4]3.2.2

Evaluate :`int_0^(pi/2)dx/(1+cotx)`

Chapter: [6] Indefinite Integration
Concept: Indefinite Integration - Integration by Parts
[4]3.2.3

A firm wants to maximize its profit. The total cost function is C = 370Q + 550 and revenue is R = 730Q-3Q2. Find the output for which profit is maximum and also find the profit amount at this output.

Chapter: [5] Applications of Derivative
Concept: Maxima and Minima
[12]4 | Attempt any SIX of the following :
[2]4.1

The ratio of number of boys and girls in a school is 3 : 2. If 20% of the boys and 30% of the girls are scholarship holders, find the percentage of students who are not scholarship holders.

Chapter: [8] Ratio, Proportion and Partnership
Concept: Ratio, Proportion and Partnership
[2]4.2

Obtain crude death rates (C. D. R.) for city A and city B from the data given below :

Age groups (in years)

City A

City B

Population

No. of deaths

Population

No. of deaths

Below 15

800

32

900

12

15-25

3000

12

1500

8

25-65

4800

48

4500

38

65 and above

1400

42

600

30

Chapter: [11] Demography
Concept: Measurements of Mortality
[2]4.3

Coefficient of rank correlation between x and y is 0.5 and `sumd_i^2`= 42. Assuming that no ranks are repeated, find the number of pairs of observations.

Chapter: [12] Bivariate Data and Correlation
Concept: Rank Correlation
[2]4.4

An agent charges 12% commission on the sales. What does he earn if the total sale amounts to Rs. 36,000? What does the seller get?     

Chapter: [9] Commission, Brokerage and Discount
Concept: Commission, Brokerage and Discount
[2]4.5

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

Age groups (in years)

Population (in '000)

No. of deaths

0-10

11

240

10-20

12

150

20-60

9

125

60 and above

2

90

 

Chapter: [11] Demography
Concept: Life Tables
[2]4.6

Following table gives the age of husbands and wives

Age of wives (in years)

Age of husbands (in years)

20-30

30-40

40-50

50-60

15-25

5

9

 -

25-35

10

25

2

35-45

1

12

2

45-55

4

16

55-65

4

 

Find:

(a)The marginal frequency distribution of the age of husbands.

(b)The conditional distribution of the age of husbands

 

Chapter: [11] Demography
Concept: Uses of Vital Statistics in Demography
[2]4.7

The present worth of the sum of Rs. 5,830, due 9 months hence, is Rs. 5,500. Find the rate of interest.

Chapter: [9] Commission, Brokerage and Discount
Concept: Commission, Brokerage and Discount
[2]4.8

For a binomial distribution mean is 6 and variance is 2. Find n and p.

Chapter: [14] Random Variable and Probability Distribution
Concept: Binomial Theorem and Binomial Distribution
[14]5
[6]5.1 | Attempt any TWO of the following :
[3]5.1.1

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

Jobs

A

B

C

D

E

Machine M1

5

1

9

3

10

Machine M2

2

6

7

8

4

Chapter: [15] Management Mathematics
Concept: Sequencing in Management Mathematics
[3]5.1.2

Natarajan and Mr. Gopalan are partners in the company having capitals in the ratio 4 : 5 and the profits received by them are in the ratio 5 :4. If Mr. Gopalan invested capital in the company for 16 months, how long was Mr. Natarajan’s investment in the company?

Chapter: [9] Commission, Brokerage and Discount
Concept: Commission, Brokerage and Discount
[3]5.1.3

From a lot of 25 bulbs of which 5 are defective a sample of 5 bulbs was drawn at random with replacement. Find the probability that the sample will contain -

(a) exactly 1 defective bulb.

(b) at least 1 defective bulb.

Chapter: [14] Random Variable and Probability Distribution
Concept: Random Variables and Its Probability Distributions
[8]5.2 | Attempt any TWO of the following :
[4]5.2.1

Given the following table which relates to the number of parrots at age x, complete the life table for parrots.

X

0

1

2

3

4

5

 lx

1000

940

780

590

25

0

Chapter: [11] Demography
Concept: Life Tables
[4]5.2.2

 You are given the following information about advertising expenditure and sales:

 

 

 

 

 

                          Advertisement

Expenditure (Rs. in lakh)

            (X)

Sales (Rs. in lakh)

           (Y)

Arithmetic mean

10

90

Standard deviation

3

12

 

Correlation coefficient between X and Y = 0.8.

(a) Obtain the two regression equations.

(b) What is the likely sales when the advertising budget is ? 15 lakh?

(c) What should be the advertising budget if the company

 

Chapter: [13] Regression Analysis Introduction
Concept: Lines of Regression of X on Y and Y on X Or Equation of Line of Regression
[4]5.2.3

Electro Corp. Co. manufactures two electrical products :Air conditioners and Fans. The assembly process for each is similar in which both require a certain amount of wiring and drilling. Each air conditioner takes 4 hours for wiring and 2 hours for drilling. Each fan also takes 2 hours for wiring and 1 hour for drilling. During the next production period, 240 hours of wiring time are available and upto 100 hours of drilling time may be used. Each air-conditioner assembled may be sold for Rs. 2,000 profit and each fan assembled may be sold for Rs. 1,000 profit. Formulate this problem as an L.P.P. in order to maximize the profit.

Chapter: [9] Commission, Brokerage and Discount
Concept: Commission, Brokerage and Discount
[14]6
[6]6.1 | Attempt any TWO of the following :
[3]6.1.1

The equations given of the two regression lines are :

2x + 3y-6 = 0 and 5x + 7y-12 = 0.

Find :

(a) Correlation coefficient

(b) `sigma_x/sigma_y`

Chapter: [13] Regression Analysis Introduction
Concept: Lines of Regression of X on Y and Y on X Or Equation of Line of Regression
[3]6.1.2

Find graphical solution for the following system of linear inequations:

2x + 3y ≥ 2, -x + y ≤ 3,  x ≤ 4, y ≥ 3

 

Chapter: [15] Management Mathematics
Concept: Linear Programming Problem in Management Mathematics
[3]6.1.3

The number of complaints which a bank manager receives per day is a Poisson random variable with parameter m = 4. Find the probability that the manager will receive -

(a) only two complaints on any given day.

(b) at most two complaints on any given day

[Use e-4 =0.0183]

Chapter: [14] Random Variable and Probability Distribution
Concept: Poisson Distribution
[8]6.2 | Attempt any TWO of the following :
[4]6.2.1

A warehouse valued at Rs. 10,000 contained goods worth Rs. 60,000. The warehouse was insured against fire for Rs. 4,000 and the goods to the extent of 90% of their value. A fire broke out and goods worth Rs. 20,000 were completely destroyed, while the remainder was damaged and reduced to 80% of its value. The damage to the warehouse was to the extent of Rs. 2,000. Find the total amount that can be claimed.

Chapter: [10] Insurance and Annuity
Concept: Insurance and Annuity
[4]6.2.2

In the following data, one of the values of Y is missing. Arithmetic means of X and Y series are 6 and 8

X

6

2

10

4

8

Y

9

11

?

8

7

 (a) Estimate the missing observation.

 (b) Calculate correlation coefficient.

Chapter: [18] Statistics
Concept: Statistics - Karl Pearson’s Coefficient of Correlation
[4]6.2.3

A job production unit has four jobs A, B, C, D which can be manufactured on each of the four machines P, Q, R and S. The processing cost of each job is given in the following table:

         Jobs

 

 

                          Machines

P

Q

R

S

                Processing Cost (Rs.)

 

A

31

25

33

29

B

25

24

23

21

C

19

21

23

24

D

38

36

34

40

 How should the jobs be assigned to the four machines so that the total processing cost is minimum?

Chapter: [15] Management Mathematics
Concept: Assignment Problem
S