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WHERE INSPIRATION FINDS YOU A PATH TO SUCCESS

MAHESH TUTORIALS COMMERCES.Y.J.C.

Preliminary Examination – IISubject : Mathematics and Statistics

Note : (i) All questions are compulsory.

(ii) Figure to the right indicates full marks.

(iii) Answer to every question must be written on a new page.

SECTION – I

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

(1) Express the truth of the following logical statement by Venn diagram:

(a) All rational numbers are real numbers.

(b) All natural numbers are real numbers and x is not a natural number.

(2) If A =2 1 3

3 2 1

, B =

1 2

3 4

2 3

Show that (AB)’ = B’⋅ A’

(3) Find dy

dx if y = (tan x)sin x

(4) Find dy

dx if y = sin – 1 (cos x)

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

(6) Evaluate1

1+cosx dx

(7) Evaluate 9

4

1

x dx.

(8) Find the area of the region bounded by the curve y = x4 between x = 1 and x = 5.

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

(1) By using truth table, show that p q = (p q) (q p)

(2) Solve the following equations by reduction method.

x + y + z = 3, 3x - 2y + 3z = 4 and 5x + 5y + z = 11.

(3) Find the inverse of the matrix A = 1 5

3 2

Date : 02/01/2014

Marks : 80

Duration : 3 Hrs.

Set No. : II

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(B) Attempt any TWO of the following : ( 4 Marks Each) (08) (1) If a function of ‘f’ is continuous at x = 0, where

f(x) =sin(3x)

5x + a, for x < 0

= x + 4 –b for x 0Find the value of a + b.

(2) Find dy

dx if y = (cos x)log x + (log x)x

(3) Evaluate 2

dx

3x 4x 2 .

Q.3. (A) Attempt any TWO of the following : ( 3 Marks Each) (06)(1) Write the converse, inverse and contrapositive of the following statement :

"If you are good in Mathematics then you are good in Logic".

(2) Evaluate sin(logx) cos(logx) dx

(3) Evaluate 2

8

(x 2)(x 4) dx

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

(1) The cost C of producing x chairs is given by C = ` x

1003

. The chairs are sold at

` x

103000

per chair. Find the number of chairs to be sold for maximum profit.

(2) Discuss the continuity at x = 1 for the function defined as

f(x) =2

1 cos x

(1 x)

, for x ≠ 1

= , for x = 1

(3) Evaluate /2

0

logsinx dx

.

SECTION – II

Q.4. (A) Attempt any SIX of the following : ( 2 Marks Each) (12) (1) A salesman is appointed on a fixed monthly salary of ` 1,500 together with a

commission at 5% on the sales over ` 10,000 during a month. If his monthly income is ` 2,050, find his sales during that month.

(2) A car worth ` 3,80,000 is insured for ` 2,50,000. In an accident it is damaged to the extent of ` 76,000. Find the amount of compensation that can be claimed under the policy.

(3) For an immediate annuity paid for 3 years with interest compounded at 10% p.a., the present value is 10,000. What is its accumulated value after 2 years.

(4) For a bivariate data,

( x - x ) (y - y ) = 121 and n = 10.

Calculate covariance between x and y.

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(5) Following are the two regression equations for a bivariate data. 8x-10y+ 66 = 0 and 40x-18y-214 = 0, find the mean values of x and y.

(6) Three balanced coins are tossed simultaneously. If X denotes the number of heads, find the probability distribution of X.

(7) If X has Poisson distribution with parameter m = 1; Find P(X 1), (Use e-1 = 0.3678794)

(8) Give the p.d.f. of a continuous random variable X as

f(x) =2x

3, - 1 < x < 2

= 0 , otherwiseDetermine the c.d.f. of X and hence find P(X < 1).

Q.5. (A) Attempt any TWO of the following : ( 3 Marks Each) (06)(1) Calculate CDR from the data given below :

Age group (years) No. of persons (in 000) No of deathsBelow 10 12 15010 – 30 20 1030 – 45 35 38045 – 70 24 210

Above 70 15 540

(2) The ratio of prices of two cycles was 16: 23. Two years late when the price of the first cycle has increased by 10% and that of second by ` 477, the ratio of prices becomes 11 : 20, find the original prices of two cycles.

(3) Solve the following minimal assignment problem.

A B C DI 3 4 6 5II 5 6 10 9III 1 2 3 2IV 4 10 6 4

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

(1) If for a bivariate data, u = X 70

10

and

v = Y 60

20

and u = 60, v = 40, u2 = 4260,

v2 = 1720, u v = 1150 and n = 10, find regression coefficient of x on y and on x. Also find the value of r.

(2) Complete the life table for the following data :

X 0 1 2 3 4 5

lx 30 26 18 10 4 0

(3) Two products A and B are available at the cost of ` 30 and ` 20 per pack respectively. Food A contains 80 and 9 units of proteins and vitamins respectively and Food B contains 40 and 5 units of proteins and vitamins respectively. Find how many packs of A and B must be purchased so as to meet the requirement of 600 units of proteins and 72 units of vitamins at the minimum cost.

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Q. 6. (A) Attempt any TWO of the following : ( 3 Marks Each) (06) (1) In a certain city 20% of person’s are vegetarians. If 5 persons from the city are

chosen at random, find the probability that(i) none is vegetarian(ii) at least one is vegetarian.

(2) On an average a company produce three defective bikes everyday. Find the probability that the company produces (i) no defective bikes(ii) at most 1 defective bike (iii) at least 2 defective bikes Given e-3 = 0.4979]

(3) How much should a company set aside at the end of each year if it has to buy a machine expected to cost ` 1,00,000 at the end of 4 years & interest rate is 5% p.a. compounded annually.

(B) Attempt any TWO of the following : ( 4 Marks Each) (08)(1) Solve the following L.P.P.

Minimize Z = 10x+ 15y

Subject to 2x+ 3y 12 2x+ y 6 x 0, y 0

(2) Taking population A as a standard population, find the standard death rate of the two districts. Which of the districts is healthier.

Age Group 0 – 10 10 – 55 Above 55

Population ADeath in A

400048

600018

300090

Population BDeath in B

300027

700028

400072

(3) Six jobs go first over the machine M and then over machine N, one at a time. The time schedule for the task is given below. Determine the sequences of the jobs, which will minimize the processing time. Also find the total elapsed time and the idle time for both the machines.

JobsMachines

A B C D E F

M 5 9 4 7 8 6

N 7 4 8 3 9 5