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Code No. 75'(English Version )

lnstructiotis : i) ! The question paper consists o{Jive Parts - A. D. C - ; . D and E.Answer all the Paris.

ii) Part - A carries 10 marks. Part - B carries 20 marks.Part - C carries 40 marks. Part - D carries 20 marks andPart - E carries 10marks.

iii) Write the question numbers properly as indicated in thequestion paper.

PART-A

Answer all the ten questions :.; 10 x 1 = 10

1. If P and q are propositions with truth values T and Frespectively, find thetruth value of - p ~ - q.

2. In how many ways can a candidate select 5 questions out of10 questions?

3. If A = 11 2 31. B = [ : 1 find AB.4. Find the fourth proportional of 8. 12. 16.

5. Find the average of the first 5 prime numbers.

6. Find the Banker's discount on a bill of Rs. 415 due 9 months hence at- .. '"15% p.a.

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Code No. 75 107. Find the length of the lat.us rectum of the parabola 3x 2 + 4y = O.

~. Evaluate: Ltx .....x2+x-2x2 - 3x + 2 .9. If y = {4X+ 7. find ~.

10. Evaluate f (ex + x" ~ 2e e ) dx.PART .; B

Answer any ten questions : 10 x 2 = 2011. Write the converse and the contrapositive of "IfIwork hard then I get the

grade".12. How many words can be formed from the letters ofthe word DAUGHTER

so that the vowels do not appear together?13. Three students x, y, z write an examination. Their chances of passing are~.i and ~ respectively. Find the probability that at least one of them

passes the examination.14. Solve by Cramer's Rule:

x,+ y = 72x + Y = S.

15. Find the middle term in the expansion of ( x +~ ) 12.

16. A can finish a job in IS days and B can do the same job in 15 days. Bworked for 10 days and left the job. In how many days, A alone can finishthe remaining work?

17. Find A if the line 3x + y + A = 0 touches the circleX 2 + Y 2 - 2x - 4y - 5 = 0..

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)

1 1 Code No. 75

18. Evaluate: {X+3 - 2x-Itx-+l19. If s = at 3 + bi, find a and b given t = 3 seconds, velocity 0 and

acceleration 14units.

20. If y = ~ log x + ~ log x + ." J log x + ... to 00 show that( 2y _ 1) dy = 1dx x

21. Evaluate J 1 dx.{X-tf - . "Jx- 2f (x + log x) dx.2. EvaluatePART - C

I. Answer any three questions : 3 x 5 = 1523. Vertfy if the proposition [ - P 1\ ( P V q ) ) ~ q is a tautology,

contradiction or neither.

24. A committee of 12 is to be formed from 9 women and 8 men. In howmany ways can this be done if at least 5 women have to be includedin the committee ? In how many of these

i) women are in majority

il ) men are in majority ?1 + 2x25. Resolve into partial fractions (x + 2 ) 2 ( x-I) .

[ ~ 2 - 3 ]26. If A = 0 : then verify A . adjA = adjA .A = I A I I.- 1[Tum over

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Code No. 75II. Answer any two questions :.

122 x 5 = 10

27. a} A person mixes 80 kg sugar worth Rs. 675 per kg with 120 kgsugar worth Rs. 8 per kg. At what rate should he sell themixture to gain 20% ? 3

b) If a man walks at the rate of 5 kmZhr, he misses a train by7 min. However, if he walks at the rate of 6 km/hr, he reachesthe station 5 min before the arrival of the train. Find thedistance covered by him to reach the station. 2

28. a) A invests a sum of money in 55% stock at 90 and B, an equalsum in the 35%. IfA's income is 10% more than B, find theprice of the 35% stock. 3

b) How much money has to be invested in 115% stock at 73( including brokerage) to obtain an income of Rs. 150 after a taxdeduction at source of 20% ? 2

29. A company requires 200 hours to produce the first ten units atRs. 10 per hour. If the learning effect is 90%. find the total numberof hours required to produce 320 units and also the total cost toproduce 320 units.

30. Solve the L.P.P. graphically:Maximize Z = lOx + 30y

subject to the limitations:x + 2y ~ 20x + 5y~ 35x ; : : : 0, y ; : : : O.

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13 Code No. 75III. Answer; any three questions: 3 x 5 = 15

31. Find the equation of a circle passing through the points ( 1. 1 ).

(5. - 5 ) and (6. - 4 ).

3Q. a) Find the equation" of the parabola given its dtrectrtx ~ + 2= O.axis Y = 3 and length of the latus rectum = 6. 2

.dy f 1b) Find dx i x = log t, Y = t . 3(

33. If Y = e=: log x. then prove that xy 2 - ( 2x - 1 ) y 1 + ( x-I) Y = O.

34. Find the. area enclosed between the curves y = 8 - X 2 and y = x 2.

PART - D

Answer any two questions : 2 x 10 = 2035. a) Find the coefficient of x 8 in ( 3x 2 - 2~ ) 10. 5

b) Define the continuity of a function. Discuss the continuity of

I( x) = 4 if x = 13x 2 + 1 . if x < 1

2x + 2 . if x 1

at x = 1. 5[Tum over

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Code No. 75 1436. a) In a college. 35% of the students failed in Accountancy. 25% of the

students failed in Commerce and 20% of the students failed in both.A student is selected at random. Find the probability that

i) he failed in Accountancy if he had failed in Commerce

ti) he failed in Commerce if he had failed in Accountancy

iii) he failed in either of the two subjects. 5

b) The surface area of a spherical bubble is increasing at the rate of

08 sq.cm/sec. At what rate is its volume increasing when the radiusis 25 cm ? 5

37. a) Prove:1 1 1

= (a - b) ( b - c ) ( c - a). 5c ca abb s - c c-i- u a+ b

b) A bill for Rs. 1.460 was drawn on 1st April for 6 months after dateand was discounted at 5% p.a. for Rs. 1.451. On what date was thebill discounted? 5

~18. a) Find the vertex. focus and the length of the latus rectum of theparabola y 2 '- 12x - 6y + 21 = O. 5

-'

b) Evaluate J (x- 1 ) (3:_-2\ (x- 3) dx. 5=====~~--------

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15 Code No. 75PART - E

Answer anyone question : 1x 10 = 10

39. a) Using Binomial theorem, find the value of ( 101 ) 5 upto 4 places ofdecimals. 4

b) Banker's gain on a bill due 6 months at 4% p.a. is Rs. 24. Find thetrue discount, banker's discount, bill amount and discounted value ofthe bill. 4

( c) If the marginal cost of a firm is J ( x) = 12 + 6x - 6x 2 where x is the), output. find the total cost assuming that the fixed cost is Rs. 125. 2

40. a) Old hens can be bought at Rs..2 each and young ones at Rs. 5 each.You have only Rs. 80 to spend on hens. Each old hen fetches Rs. 10and each young hen fetches Rs. 25 on selling. How many of eachshould you sell to maximize the profits, assuring that you cannot

accommodate more than 20 hens? Formulate the L.P.P. 4b) A finn has its demand function P = 75 - 2x and average costx3function AC = 350x + 12x2 +4where x is the number of units.

P is price. Determine the level at which (i) profit is maximized,(ii) price at which profit is maximized and (iii) the maximum profit. 4

c) A worker works for 8 hours on Monday, 9 hours on Tuesday,7 ~ hours on Wednesday and 7! hours on Thursday. He is paid anhourly wage of Rs. 750. What is his average daily earnings? 2