TN 12th Question Paper 2022 Business Mathematics & Statistics

No. of Printed Pages : 16
5967
!5967BusiMatheandStati! £vÄ Gs
Register Number

PART - III
ÁoPU Pou® ©ØÖ® ¦Òΰ¯À
BUSINESS MATHEMATICS AND STATISTICS
( uªÌ ©ØÖ® B[Q» ÁÈ / Tamil & English Version)

Põ» AÍÄ : 3.00 ©o ÷|µ® ] [ ö©õzu ©v¨ö£sPÒ : 90
Time Allowed : 3.00 Hours ] [Maximum Marks : 90

AÔÄøµPÒ : (1) AøÚzx ÂÚõUPЮ \›¯õP £vÁõQ EÒÍuõ GߣuøÚ \›£õºzxU
öPõÒÍÄ®. Aa_¨£vÂÀ SøÓ°¸¨¤ß AøÓU PsPõo¨£õÍ›h®
EhÚi¯õPz öu›ÂUPÄ®.
(2) }»® AÀ»x P¸¨¦ ø©°øÚ ©mk÷© GÊxÁuØS®,
AiU÷PõikÁuØS® £¯ß£kzu ÷Ásk®. £h[PÒ ÁøµÁuØS
ö£ß]À £¯ß£kzuÄ®.
Instructions : (1) Check the question paper for fairness of printing. If there is any lack of
fairness, inform the Hall Supervisor immediately.

(2) Use Blue or Black ink to write and underline and pencil to draw
diagrams.

£Sv & I/PART - I

SÔ¨¦ : (i) AøÚzx ÂÚõUPÐUS® Âøh¯ÎUPÄ®. 20x1=20

(ii) öPõkUP¨£mkÒÍ ©õØÖ ÂøhPÎÀ ªPÄ® Hئøh¯ Âøhø¯z
÷uº¢öukzxU SÔ±mkhß Âøh°øÚ²® ÷\ºzx GÊuÄ®.
Note : (i) Answer all the questions.

(ii) Choose the most appropriate answer from the given four alternatives

and write the option code and the corresponding answer.

[ v¸¨¦P / Turn over

5967 2

2 0
1. A=   GÛÀ, ρ(A) = __________.
0 8

(A) 2 (B) 0 (C) n (D) 1
2 0
If A =   , then ρ(A) is __________.
0 8
(a) 2 (b) 0 (c) n (d) 1

2. ρ(A)=ρ(A, B)= ©õÔPÎß GsoUøP GÛÀ öuõS¨£õÚx :
(A) J¸[Pø©Ä AØÓx
(B) J¸[Pø©Ä Eøh¯x ©ØÖ® GsoUøP¯ØÓ wºÄPÒ ö£ØÖÒÍx
(C) J¸[Pø©Ä Eøh¯x
(D) J¸[Pø©Ä Eøh¯x ©ØÖ® J÷µ J¸ wºÄ ö£ØÖÒÍx
If ρ(A)=ρ(A, B)= the number of unknowns, then the system is __________.
(a) inconsistent
(b) consistent and has infinitely many solutions
(c) consistent
(d) consistent and has a unique solution

∞
∫e
−2x
3. dx &ß ©v¨¦ __________.
0

(A) 2 (B) 0 (C) 12 (D) 1

∞
∫e
−2x
dx is __________.
0

(a) 2 (b) 0 (c) 1 (d) 1
2

4. C»õ£a \õº¦ p(x) BÚx ö£¸©©øhÁx :
(A) MR=0 (B) MC−MR=0 (C) MC+MR=0 (D) MC=0
The profit of a function p(x) is maximum when :
(a) MR=0 (b) MC−MR=0 (c) MC+MR=0 (d) MC=0

3 5967

5. ÷uøÁa \õº¦ p &US ∫ dp = k ∫ dx GÛÀ k =
p x

1 1
(A) − η (B) ηd (C) η (D) −ηd
d d

dp dx
For a demand function p, if
∫ p = k ∫ x , then k is equal to :
1 1
(a) −
ηd (b) ηd (c) ηd (d) −ηd

6. J¸ ö£õ¸Îß AΨ¦a \õº¦ p=3+x ©ØÖ® x0=3 GÛÀ, EØ£zv¯õͺ E£› :
3 5 7 9
(A) (B) (C) (D)
2 2 2 2
The producer’s surplus when the supply function for a commodity is p=3+x and
x0=3 is __________.
3 5 7 9
(a) (b) (c) (d)
2 2 2 2

3
 d2y  2  dy 
7.  2 −   −4 = 0 GßÓ ÁøPUöPÊa \©ß£õmiß Á›ø\ ©ØÖ® £i
 dx   dx 
•øÓ÷¯ :
(A) 1 ©ØÖ® 4 (B) 2 ©ØÖ® 6 (C) 2 ©ØÖ® 4 (D) 3 ©ØÖ® 6
3
 d2y  2  dy 
The order and degree of the differential equation   −   −4 = 0 are
2
 dx   dx 
respectively.
(a) 1 and 4 (b) 2 and 6 (c) 2 and 4 (d) 3 and 6

dx x 
8. = f   GßÓ ÁiÂÀ EÒÍ \©£izuõÚ ÁøPUöPÊa \©ß£õk wºUP¨£h
dy y 
£¯ß£kzu¨£k® ¤µv°hÀ :
(A) y=v (B) x=vy (C) x=v (D) y=vx
A homogeneous differential equation of the form dx = f  x  can be solved by
dy y 
 
making substitution.
(a) y=v (b) x=vy (c) x=v (d) y=vx

[ v¸¨¦P / Turn over

5967 4

9. ∆f (x)=__________.
(A) f (x+h)−f (x) (B) f (x+h) (C) f (x)−f (x−h) (D) f (x)−f (x+h)
∆f (x)=__________.
(a) f (x+h)−f (x) (b) f (x+h) (c) f (x)−f (x−h) (d) f (x)−f (x+h)

10. ∇f (a)=__________.
(A) f (a)−f (a−h) (B) f (a)+f (a−h) (C) f (a) (D) f (a)−f (a+h)
∇f (a)=__________.
(a) f (a)−f (a−h) (b) f (a)+f (a−h) (c) f (a) (d) f (a)−f (a+h)

11. J¸ öuõhºa]¯õÚ {PÌuPÄ £µÁ¼À c Gߣx J¸ ©õÔ¼ GßÓõÀ p(x=c)
G¨÷£õx® GuØS \©©õP C¸US® ?
(A) Gvº©øÓ (B) §ä⯮ (C) Põn C¯»õx (D) JßÖ
If c is a constant in a continuous probability distribution, then p(x=c) is always
equal to __________.
(a) negative (b) zero (c) does not exist (d) one

12. J¸ \©Áõ´¨¦ ©õÔ°ß {PÌuPÄ \õº¦ ¤ßÁ¸©õÖ Áøµ¯ÖUP¨£mkÒÍx.

X=x −1 −2 0 1 2

P( x ) k 2k 3k 4k 5k

GÛÀ k &Cß ©v¨£õÚx :
1 1
(A) (B) §ä¯® (C) JßÖ (D)
15 4
The probability function of a random variable is defined as :

X=x −1 −2 0 1 2

P( x ) k 2k 3k 4k 5k

Then k is equal to :

1 1
(a) (b) zero (c) one (d)
15 4

5 5967

13. J¸ EØ£zv¯õͺ u¯õ›US® ªß Âø\ ©õØÖUSªÌPÎÀ (switches) 2 \uÃu
u¯õ›¨¦PÒ SøÓ£õkÒÍøÁ GßÖ AÔ¯¨£kQÓx. J¸ ÷£øÇ°À C¸US®
50 ªßÂø\ ©õØÖUSªÌPÎÀ AvP£m\©õP 2 SøÓ£õkPÒ C¸¨£uØPõÚ
{PÌuPÁõÚx :
(A) e−1 (B) 2 e−1 (C) 2.5 e−1 (D) CøÁ HxªÀø»
A manufacturer produces switches and experiences that 2 percent switches are
defective. The probability that in a box of 50 switches, there are atmost two
defective is __________.
(a) e −1 (b) 2 e−1 (c) 2.5 e−1 (d) none of the above

14. D¸Ö¨¦¨ £µÁ¼À öÁØÔUPõÚ {PÌuPÁõÚx ÷uõÀÂUPõÚ {PÌuPøÁ¨ ÷£õÀ
C¸©h[S GÛÀ |õßS •¯Ø]PÎÀ §ä⯠öÁØÔ ö£ÖÁuØPõÚ {PÌuPÄ :
2 16 1 1
(A) (B) (C) (D)
27 81 81 16
In a binomial distribution, the probability of success is twice as that of failure.
Then out of 4 trials, the probability of no success is __________.

2 16 1 1
(a) (b) (c) (d)
27 81 81 16

15. J¸ •Êø©z öuõSv°ß ___________ TÖ GÚ AøÇUP¨£kQÓx.
(A) •iÄÖ Pn® (B) •iÄÓõ Pn®
(C) •Êø© Pn® (D) •iÄÖ EmPn®
A ___________ of statistical individuals in a population is called a sample.
(a) finite set (b) infinite set
(c) entire set (d) finite subset

16. __________ Gߣx •Êø©z öuõSv°¾ÒÍ JÆöÁõ¸ EÖ¨¦®
÷uº¢öukUP¨£kÁuØS J¸ \©©õÚ Áõ´¨ø£ AÎUS® JßÓõS®.
(A) TÖ £s£ÍøÁ (B) £s£ÍøÁ
(C) •Êø©z öuõSv (D) \©Áõ´¨¦ TÖ
A __________ is one where each item in the universe has an equal chance of
known opportunity of being selected.
(a) statistic (b) parameter
(c) entire data (d) random sample

[ v¸¨¦P / Turn over

5967 6

17. TÖ\µõ\›°ß vmh¨¤øǯõÚx :
σ σ σ2 σ
(A) n
(B) 2n
(C) (D)
n n
The standard error of sample mean is :

σ σ σ2 σ
(a) (b) (c) (d)
n 2n n n

18. y=a+bx GßÓ ÷£õUS ÷PõmiÀ ‘b’ &Cß ©v¨£õÚx :
(A) ªøP AÀ»x SøÓ (B) G¨÷£õx® ªøP
(C) §ä⯮ (D) G¨÷£õx® SøÓ
The value of ‘b’ in the trend line y=a+bx is :
(a) Either positive or negative (b) Always positive
(c) Zero (d) Always negative

19. RÌUPsh G¢u SÔ±mk Gs Põ» ©õØÖ ÷\õuøÚø¯ {øÓÄ ö\´QÓx ?
(A) £õ]°ß SÔ±mk Gs (B) L¤åº SÔ±mk Gs
(C) »õ줯º SÔ±mk Gs (D) ÷©ØTÔ¯ AøÚzx®
Which of the following Index number satisfy the time reversal test ?
(a) Paasche’s Index number (b) Fisher Index number
(c) Laspeyre’s Index number (d) All of them

20. wº©õÚ ÷Põm£õk Guß öuõhº¦øh¯x ?
(A) |®£Pzußø© öPõsh wº©õÚzøu AÍÃk ö\´Áx
(B) Á›ø\z öuõhº ¤µa]øÚPÐUS EP¢u wº©õÚ[PøÍ ÷uº¢öuk¨£x
(C) QøhUPUTi¯ uPÁÀPÎß AÍÄ
(D) ÷©ØTÔ¯ AøÚzx®
Decision theory is concerned with :
(a) decision making under certainty
(b) selecting optimal decisions in sequential problem
(c) analysis of information that is available

(d) all of the above

7 5967

£Sv - II / PART - II

SÔ¨¦ : GøÁ÷¯Ý® HÊ ÂÚõUPÐUS Âøh¯ÎUPÄ®. ÂÚõ Gs 30 &US
Pmhõ¯©õP Âøh¯ÎUPÄ®. 7x2=14

Note : Answer any seven questions. Question No. 30 is compulsory.

 1 2 3
 
21. A =  2 3 4  GßÓ Ao°ß uµzvøÚU PõsP.
3 5 7
 

 1 2 3
Find the rank of the matrix A =  2 3 4  .
3 5 7
 

22. f ' (x)=8x3−2x ©ØÖ® f(2)=8 GÛÀ, f(x) &I PõsP.

If f ' (x)=8x3−2x and f(2)=8, then find f(x).

23. Bv ÁÈaö\À¾® AøÚzx ÷|º÷Põmkz öuõS¨¤ß ÁøPUöPÊa \©ß£õmøh
Aø©UP.
Find the differential equation of the family of all straight lines passing through
the origin.

24. h=1 GÛÀ, (E−1∆)x3=3x2−3x+1 GÚ {ÖÄP.

If h=1 then prove that (E−1∆)x3=3x2−3x+1.

25. Cµsk £PøhPÒ J÷µ \©¯zvÀ Ã\¨£kQÓx. CvÀ ÷©÷» v¸¨£¨£mh
•P[PÎß TkuÀ \©Áõ´¨¦ ©õÔ¯õPU P¸u¨£kQÓx GÛÀ, Auß {PÌuPÄ
{øÓ \õºø£ E¸ÁõUPÄ®.
Two unbiased dice are thrown simultaneously and sum of the upturned faces
considered as random variable. Construct a probability mass function.

[ v¸¨¦P / Turn over

5967 8

26. ©õnÁºPÒ A uµ{ø»ø¯ ö£ÖÁuØPõÚ GsoUøPø¯ Áøµ¯ÖUS®
\©Áõ´¨¦ ©õÔ¯õP X C¸UPmk®. öPõkUP¨£mh AmhÁøn°¼¸¢x X &Cß
Gvº£õºzuÀ ©v¨ø£U PshԯĮ.

X=x 0 1 2 3
P(X=x ) 0.2 0.1 0.4 0.3

Let X be a random variable defining number of students getting A grade. Find the
expected value of X from the given table.

X=x 0 1 2 3
P(X=x ) 0.2 0.1 0.4 0.3

27. D¸Ö¨¦¨ £µÁ¼ß \µõ\› ©v¨¦ 20 GÚÄ® vmh»UPzvß ©v¨£õÚx
4 GÚÄ® öPõshõÀ, ‘n’ &Cß ©v¨¤øÚU PõsP.

If the mean of binomials distribution is 20 and standard deviation is 4, then find
the value of ‘n’.

28. J¸ TÔß AÍÄ 50 Eøh¯ J¸ ©õv›°ß vmh »UP® 6.3. AuØS›¯ •Êø©z
öuõøP°ß vmh »UP® 6 GÛÀ ©õv›°ß vmh¨¤øÇ PõsP.
The standard deviation of a sample of size 50 is 6.3. Determine the standard
error whose population standard deviation is 6.

29. SÔ±mk Gsoß £¯ß£õmøh ÂÁ›UPÄ®.
State the uses of Index number.

30. x=1, 2, 3, 4, 5 GÛÀ, y=f (x)=x3+3x GßÓ \õº¦US •ß÷|õUS ÷ÁÖ£õmiß
AmhÁønø¯ ÁiÁø©UPÄ®.
Construct a forward difference table for y=f (x)=x3+3x for x=1, 2, 3, 4, 5.

9 5967

£Sv & III / PART - III
SÔ¨¦ : GøÁ÷¯Ý® HÊ ÂÚõUPÐUS Âøh¯ÎUPÄ®. ÂÚõ Gs 40 &US
Pmhõ¯©õP Âøh¯ÎUPÄ®. 7x3=21

Note : Answer any seven questions. Question No. 40 is compulsory.
31. \¢øu°À EÒÍ A ©ØÖ® B BQ¯ uµ Aøh¯õÍ® öPõsh ö£õ¸ÒPÐUPõÚ
A B

©õÖuÀ {PÌuPÄ Ao A  0.9 0.1 
 GÛÀ, \©{ø»°À uµ Aøh¯õÍ®
B  0.3 0.7 
öPõsh JÆöÁõ¸ ö£õ¸ÒPÐUPõÚ \¢øu £[RkPøÍU PõsP.
Consider the matrix of transition probabilities of a product available in the market
in two brands A and B.

A B
A  0.9 0.1 
 
B  0.3 0.7 

Determine the market share of each brand in equilibrium position.

1
32. Cµshõ® Ai¨£øhz ÷uØÓzøu¨ £¯ß£kzv ∫ 2 2x + 3 dx &ß ©v¨¦
x + 3x + 7 −1
PõsP.
1
2x + 3
Using second fundamental theorem, evaluate ∫ x 2 + 3x + 7 dx .
−1

0
33. y=?x+3? GßÓ ÁøÍÁøµø¯ ÁøµP. ÷©¾® ∫
x + 3
dx &Cß ©v¨ø£U PõsP.
−6

0
Sketch the graph y=?x+3? and evaluate ∫
x +3
dx.
−6

34. wºUP : (1−x)dy−(1+y)dx=0
Solve : (1−x)dy−(1+y)dx=0

[ v¸¨¦P / Turn over

5967 10

35. öPõkUP¨£mkÒÍ AmhÁønø¯¨ £¯ß£kzv Âk£mh EÖ¨ø£U PõsP.
x 0 1 2 3 4

yx 1 3 9 - 81
Find the missing entry in the following table.

x 0 1 2 3 4

yx 1 3 9 - 81

36. PnUQ¯À Gvº£õºzu¼ß £s¦PÒ ¯õøÁ ?
What are the properties of Mathematical expectation ?

37. £õ´\õß £µÁ»õÚx D¸Ö¨¦¨ £µÁ¼ß GÀø»¯õP Aø©ÁuØPõÚ
Pmk¨£õkPøÍ GÊxP.
Write the conditions for which the poisson distribution is a limiting case of binomial
distribution.

38. Sk®£ ÁµÄ ö\»Äzvmh •øÓø¯¨ £¯ß£kzv ¤ßÁ¸® ÂÁµ[PÐUS
2012 &B® Bsøh Ai¨£øh BshõPU öPõsk 2015 &UPõÚ ÁõÌUøP
SÔ±mk GsønU Pmhø©UPÄ®.

ö£õ¸ÒPÒ Âø» {øÓPÒ
2012 2015
A›] 250 280 10

÷Põxø© 70 85 5

÷\õÍ® 150 170 6

Gsön´ 25 35 4

£¸¨¦ 85 90 3

Construct the cost of living Index number for 2015 on the basis of 2012 from the
following data using family budget method.
Price
Commodity W eights
2012 2015
Rice 250 280 10
Wheat 70 85 5
Corn 150 170 6
Oil 25 35 4
Dhal 85 90 3

11 5967

39. öPõkUP¨£mkÒÍ AoUPõÚ EP¢u ³Pzøu (i) «a]ÖÂß «¨ö£¸ ©ØÖ®
(ii) «¨ö£¸Âß «a]Ö BQ¯ÁØøÓ £¯ß£kzv PõsP.

`Ì{ø»PÎß {ø»¨£õkPÒ
³P®
E1 E2
S1 40 60
S2 10 −20
S3 −40 150

States - of - nature
Strategy
E1 E2
S1 40 60
S2 10 −20
S3 −40 150

Select a strategy using each of the following rule.
(i) Maximin (ii) Minimax

40. J¸ |£º J¸ |õn¯zøua _skQÓõº, uø» GÛÀ ` 6 &I¨ ö£ÖQÓõº ©ØÖ®
§ GÛÀ ` 3 &I ö\¾zxQÓõº. AÁµx C»õ£zvß Gvº£õº¨¦ ©ØÖ® ©õÖ£õmk
AÍøÁø¯U PshԯĮ.
A person tosses a coin and is to receive ` 6 for a head and is to pay ` 3 for a tail.
Find the expectation and variance of his gains.

£Sv - IV / PART - IV
SÔ¨¦ : AøÚzx ÂÚõUPÐUS® Âøh¯ÎUPÄ®. 7x5=35
Note : Answer all questions.

41. (A) J¸ öuõøP ` 5,000 BÚx BsiØS 6%, 7% ©ØÖ® 8% uµUTi¯ ‰ßÖ
£[SPÎÀ ¤›zx •u½k ö\´¯¨£mk, Bsk ö©õzu Á¸©õÚ©õP ` 358
ö£Ó¨£kQÓx. •uÀ Cµsk •u½kPμ¸¢x QøhUS® Á¸©õÚ®,
‰ßÓõÁx •u½mi¼¸¢x QøhUS® Á¸©õÚzøu Âh ` 70 AvP® GÛÀ,
A®‰ßÖ £[SPÎÀ ö\¾zu¨£k® •u½kPøÍ uµ•øÓ°À PõsP.
AÀ»x

[ v¸¨¦P / Turn over

5967 12

(B) J¸ öuõhºa]¯õÚ \©Áõ´¨¦ ©õÔ X BÚx ¤ßÁ¸® {PÌuPÄa \õºø£¨
ö£ØÖÒÍx GÛÀ,
Value of
0 1 2 3 4 5 6 7
X=x
2 2 2
P( x ) 0 k 2k 2k 3k k 2k 7k +k
(i) k &ß ©v¨ø£U PõsP.
(ii) P(x < 6), P(x / 6) ©ØÖ® P(0 < x < 5) &IU PõsP.
1
(iii) P(X £ x) > &UPõÚ x &Cß SøÓ¢u£m\ ©v¨ø£U Psk¤iUPÄ®.
2
(a) An amount of ` 5,000 is to be deposited in three different bonds bearing 6%,
7% and 8% per year respectively. Total annual income is ` 358. If the
income from first two investments is ` 70 more than the income from the
third, then find the amount of investment in each bond by rank method.
OR
(b) A continuous random variable X has the following probability function.

Value of
0 1 2 3 4 5 6 7
X=x
2 2 2
P( x ) 0 k 2k 2k 3k k 2k 7k +k
(i) Find k.
(ii) Evaluate P(x < 6), P(x / 6) and P(0 < x < 5).
1
(iii) If P(X £ x) > , then find the minimum value of x.
2
42. (A) x &I¨ ö£õÖzx x3e3x &Iz öuõøP°kP.
AÀ»x
(B) J¸ SÔ¨¤mh C¯¢vµ® Áõ°»õP EØ£zv ö\´¯¨£k® v¸S©øÓ°À
EÒÍ SøÓ£õkPÒ 18 \uÂQu® GÛÀ \©Áõ´¨¦ •øÓ°À
÷uº¢öukUP¨£k® |õßS v¸S©øÓ°À :
(i) \›¯õP J¸ SøÓ£õkÒÍ v¸S©øÓ
(ii) SøÓ£õk CÀ»õ v¸S©øÓ
(iii) AvP£m\® 2 SøÓ£õkPÒ EÒÍ v¸S©øÓ C¸¨£uØPõÚ {PÌuPÄ
PõsP.
(a) Integrate x3e3x with respect to x.
OR
(b) If 18% of the bolts produced by a machine are defective, determine the
probability that out of the 4 bolts chosen at random
(i) exactly one will be defective bolt
(ii) none of the bolt will be defective

(iii) atmost 2 will be defective bolts

13 5967

p + 2p2
43. (A) J¸ ö£õ¸Îß Âø» p &I ö£õ¸zx ÷uøÁ ö|QÌa] ηd =
100 − p − p2

GÛÀ Âø» 5 ©ØÖ® ÷uøÁ 70 GÝ® ö£õÊx Auß ÷uøÁ \õº¦ ©ØÖ®
Á¸Áõ´a \õºø£U PõsP.

AÀ»x

(B) J¸ C¯¢vµ® u¯õ›US® EØ£zv ö£õ¸Îß Ev›£õP[PÎß vmh»UP®
1.6 ö\.«. \©Áõ´¨¦ •øÓ°À ÷uº¢öukUP¨£mh 64 ©õv›PÎß \µõ\›
E¯µ® 90 ö\.«. BS®. Ev›£õP[PÎß E¯µ® 88 ö\.«mh¸US
SøÓÁõP÷Áõ AÀ»x 92 ö\.«US AvP©õP÷Áõ C¸US® ÷£õx
A¨£õP[PøÍ ÁõiUøP¯õͺ {µõP›UQÓõº. EØ£zv ö\´¯¨£mh \µõ\›
E¯µ® öPõsh Ev›£õP[PÒ, 95% |®¤UøP CøhöÁΰÀ Aø©²® GÚ
ÁõiUøP¯õ͸US EÖv£kzu •i²©õ ?

1% 2% 5% 10%

Z α
=2.58
Ζα
=2.33
Ζα
=1.96
Ζα
=1.645

(a) The elasticity of demand with respect to price p for a commodity is
p + 2p2
ηd = . Find demand function where price is ` 5 and the demand
100 − p − p2
is 70.
OR

(b) A machine produces a component of a product with a standard deviation of
1.6 cm in length. A random sample of 64 components was selected from the
output and this sample has a mean length of 90 cm. The customer will
reject the part if it is either less than 88 cm or more than 92 cm. Does the
95% confidence interval for the true mean length of all the components
produced ensure acceptance by the customer ?

1% 2% 5% 10%

Z α
=2.58
Ζα
=2.33
Ζα
=1.96
Ζα
=1.645

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5967 14

44. (A) J¸ ÁøÍÁøµ°À EÒÍ H÷uÝ® J¸ ¦ÒÎ (x, y) Chzx Aø©¯UTi¯
öuõk÷Põmiß \õ´Ä (y3−2yx2)dx+(2xy2−x3)dy=0 BS®. ÷©¾® C¢u
ÁøÍÁøµ¯õÚx (1, 2) ¦ÒÎ ÁÈa ö\ÀQÓx GÛÀ, ÁøÍÁøµ°ß
\©ß£õmøhU PõsP.

AÀ»x

(B) Gί \µõ\› •øÓø¯¨ £¯ß£kzv J¸ ö£õ¸Îß EØ£zv°ß Põ»õsk
£¸ÁPõ»U SÔ±møh PnUQkP.

Bsk I Põ»õsk II Põ»õsk III Põ»õsk IV Põ»õsk
2005 255 351 425 400
2006 269 310 396 410
2007 291 332 358 395
2008 198 289 310 357
2009 200 290 331 359
2010 250 300 350 400

(a) The slope of the tangent to a curve at any point (x, y) on it is given by
(y3−2yx2)dx+(2xy2−x3)dy=0 and the curve passes through (1, 2). Find the
equation of the curve.

OR

(b) Calculate the seasonal index for the quarterly production of a product using
the method of simple averages.

Year I Quarter II Quarter III Quarter IV Quarter
2005 255 351 425 400
2006 269 310 396 410
2007 291 332 358 395
2008 198 289 310 357
2009 200 290 331 359
2010 250 300 350 400

15 5967

45. (A) \¢øu°À EÒÍ A ©ØÖ® B C¸ÁøP¯õÚ ÷\õ¨¦PÎß uØ÷£õøu¯ \¢øu¨
£[Rk 15% ©ØÖ® 85% BS®. ö\ßÓ Bsk A Áõ[Q¯ÁºPÎÀ 65% ÷£º
«sk® Aøu C¢u Bsk® Áõ[SQÓõºPÒ. 35% ÷£º B &US
©õÔÂkQßÓÚº. ö\ßÓ Bsk B Áõ[Q¯ÁºPÎÀ 55% ÷£º C¢u Bsk®
«sk® Aøu Áõ[SQÓõºPÒ. 45% ÷£º A &US ©õÔ ÂkQÓõºPÒ. J¸
BskUS¨ ¤ÓS AÁØÔß \¢øu¨ £[RkPøÍU PõsP. ÷©¾® \¢øu°À
\©{ø» G¨÷£õx Gmh¨£k® ?
AÀ»x
(B) 10 Á¸h[PÐUS J¸•øÓ GkUP¨£k® J¸ |Pµzvß ©UPÒ öuõøP
PnUöPk¨¤ß ÂÁµ[PÒ R÷Ç öPõkUP¨£mkÒÍÚ. 1955 &B® Á¸hzvß
©UPÒ öuõøPø¯ ©v¨¤kP.
Á¸h® 1951 1961 1971 1981

©UPÒ öuõøP
35 42 58 84
(C»m\zvÀ)
(a) Two types of soaps A and B are in the market. Their present market shares
are 15% for A and 85% for B. Of those who bought A the previous year, 65%
continue to buy it again while 35% switch over to B. Of those who bought B
the previous year, 55% buy it again and 45% switch over to A. Find their
market shares after one year and when is the equilibrium reached.
OR
(b) The population of a city in a censes taken once in 10 years is given below.
Estimate the population in the year 1955.

Year 1951 1961 1971 1981

Population in lakhs 35 42 58 84

3x 2 + 6x + 1
46. (A) ©v¨¤kP : ∫ dx
(x + 3) (x 2 + 1)
AÀ»x
(B) Áh÷©ØS ‰ø» •øÓø¯¨ £¯ß£kzv ¤ßÁ¸® ÷£õUSÁµzx PnUQß
Bµ®£ Ai¨£øh \õzv¯©õÚz wºøÁ PõsP.
D1 D2 D3 D4 Aئ
O1 5 3 6 2 19
O2 4 7 9 1 37
O3 3 4 7 5 34
÷uøÁ 16 18 31 25

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5967 16

3x 2 + 6x + 1
(a) Evaluate :
∫ (x + 3) (x 2 + 1) dx
OR
(b) Find an initial basic feasible solution of following problem using north west
corner rule.
D1 D2 D3 D4 Supply
O1 5 3 6 2 19
O2 4 7 9 1 37
O3 3 4 7 5 34
Demand 16 18 31 25

47. (A) ÷uøÁa \õº¦ pd=25−3x ©ØÖ® AΨ¦a \õº¦ ps=5+2x GÛÀ, \©{ø»°À
~Pº÷Áõº E£› ©ØÖ® EØ£zv¯õͺ E£›ø¯U PõsP.
AÀ»x
(B) 2010 &B® BsiØS (i) »õ줯º (ii) £õ] (iii) L¤åº Âø»U SÔ±mk
GsPøÍ ¤ßÁ¸® ¦ÒÎ ÂÁµ[PÐUSU PnUQkP.
Âø» AÍÄ
ö£õ¸ÒPÒ
2000 2010 2000 2010
A 12 14 18 16
B 15 16 20 15
C 14 15 24 20
D 12 12 29 23
(a) Find the consumer’s surplus and producer’s surplus for the demand function
pd=25−3x and supply function ps=5+2x.
OR
(b) Compute (i) Laspeyre’s (ii) Paasche’s (iii) Fisher’s Index numbers for the
2010 from the following data.
Price Quantity
Commodity
2000 2010 2000 2010
A 12 14 18 16
B 15 16 20 15
C 14 15 24 20
D 12 12 29 23

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