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

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SubjectMathematics and Statistics
Year2014 - 2015 (March)

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

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

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

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Q: 1.2[2]

Find the values of x and y, if

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

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Q: 1.3[2]

Find the value of 'k' if the function

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

=k,            for x=0

is continuos at x=0

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Q: 1.4[2]

Find dy/dx if y=cos^-1(sqrt(x))

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Q: 1.5[2]

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

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Q: 1.6[2]

Evaluate : int1/(x(3+logx))dx

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Q: 1.7[2]

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

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Q: 1.8[2]

evaluate : int xcosxdx

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Q: 2[14]
Q: 2.1 | Attempt any TWO of the following[6]
Q: 2.1.1[3]

Prove that the following statement pattern is equivalent :

(p ∨ q)  r and (p → r) ∧ (q → r)

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Q: 2.1.2[3]

Examine the continuity of the following function : 

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Q: 2.1.3[3]

find dy/dx if y=tan^-1((6x)/(1-5x^2))

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Q: 2.2 | Attempt any TWO of the following :[8]
Q: 2.2.1[4]

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

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Q: 2.2.2[4]

Find the area of elipse 

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Q: 2.2.3[4]

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.

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Q: 3[14]
Q: 3.1 | Attempt any TWO of the following :[6]
Q: 3.1.1[3]

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.

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Q: 3.1.2[3]

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

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

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Q: 3.1.3[3]

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

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Q: 3.2 | Attempt any TWO of the following :[8]
Q: 3.2.1[4]

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

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Q: 3.2.2[4]

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

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Q: 3.2.3[4]

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.

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Q: 4 | Attempt any SIX of the following :[12]
Q: 4.1[2]

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.

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Q: 4.2[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
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Q: 4.3[2]

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.

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Q: 4.4[2]

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?

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Q: 4.5[2]

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

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Q: 4.6[2]

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

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Q: 4.7[2]

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

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Q: 4.8[2]

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

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Q: 5[14]
Q: 5.1 | Attempt any TWO of the following :[6]
Q: 5.1.1[3]

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
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Q: 5.1.2[3]

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?

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Q: 5.1.3[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.

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Q: 5.2 | Attempt any TWO of the following :[8]
Q: 5.2.1[4]

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
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Q: 5.2.2[4]

 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

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Q: 5.2.3[4]

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.

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Q: 6[14]
Q: 6.1 | Attempt any TWO of the following :[6]
Q: 6.1.1[3]

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

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Q: 6.1.2[3]

Find graphical solution for the following system of linear inequations:

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

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Q: 6.1.3[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]

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Q: 6.2 | Attempt any TWO of the following :[8]
Q: 6.2.1[4]

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.

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Q: 6.2.2[4]

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.

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Q: 6.2.3[4]

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?

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