Tamil Nadu 12th Std Model Question Paper 2026 Business Mathematics and Statistics

Tamil Nadu Board Model Question Paper

No. of Printed Pages : 15
8367
!8367BusiMatheandStati! £vÄ Gs
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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 |õßS ©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.

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8367 2

log x
1. ∫ x dx, (x > 0) &ß ©v¨¦a \õº¦ :
2 1 2
(A) 2
+c (B) 2 ( log x ) + c
x
2 1 2
(C) − 2
+c (D) − 2 ( log x ) + c
x
log x
∫ x dx, (x > 0) is :
2 1
(a) 2
+c (b) ( log x )2+ c
x 2

2 1
(c) − 2 +c (d) − ( log x )2 + c
x 2

4
 1 
2. ∫  x + x  dx &ß ©v¨¦ :
0

28 20 1 21
(A) 3
(B) 3 (C) 3 (D) 3
4
 1 
∫  x + x  dx is :
0

28 20 1 21
(a) (b) (c) (d)
3 3 3 3

 1
 
3. A =  2  GÛÀ AAT &ß uµ® :
 3
 

(A) 2 (B) 0 (C) 3 (D) 1
 1
 
If A = 2  , then the rank of AAT is :
 3
 
(a) 2 (b) 0 (c) 3 (d) 1

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4. A Gߣx n×n Á›ø\ Eøh¯ Ao GÛÀ adj A &ß ©v¨¦ :

(A) A n−1 (B) A n (C) A 1+n (D) A

If A is a matrix of order n×n, the value of adj A is :

(a) A n−1 (b) An (c) A 1+n (d) A

5. J¸ {ÖÁÚzvß CÖv{ø» Á¸Áõ´ ©õÔ¼ GÛÀ, Auß ÷uøÁa \õº¦ :
(A) C(x) (B) MR (C) AC (D) MC
If the marginal revenue of a firm is a constant, then the demand function is :
(a) C(x) (b) MR (c) AC (d) MC

6. y = x GÝ® ÁøÍÁøµ, 0 &¼¸¢x 2 Áøµ HØ£kzx® Aµ[Pzvß £µ¨¦ :

(A) 2 \.A»SPÒ (B) 1 \.A»S (C) 4 \.A»SPÒ (D) 3 \.A»SPÒ
Area bounded by the curve y = x between the limits 0 and 2 is :
(a) 2 sq. units (b) 1 sq. unit (c) 4 sq. units (d) 3 sq. units

3
 dx  1
7.   +2 y 2 = x GßÓ ÁøPUöPÊa \©ß£õk :
 dy 

(A) Á›ø\ 1 ©ØÖ® £i 6 Eøh¯x
(B) Á›ø\ 2 ©ØÖ® £i 1 Eøh¯x
(C) Á›ø\ 1 ©ØÖ® £i 2 Eøh¯x
(D) Á›ø\ 1 ©ØÖ® £i 3 Eøh¯x
3
 dx  1
The differential equation   +2 y 2 = x is :
 dy 
(a) of order 1 and degree 6
(b) of order 2 and degree 1
(c) of order 1 and degree 2
(d) of order 1 and degree 3

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dy
8. x − y = x 2 &Cß öuõøP±mkU Põµo :
dx
−1 1
(A) log x (B) x (C) x (D) x
dy
The integrating factor of x − y = x 2 is :
dx
−1 1
(a) log x (b) (c) x (d)
x x

9. C»Uµõg]°ß Cøhaö\¸P¼ß `zvµ® G¨ö£õÊx £¯ß£kzu¨£k® ?
(A) \©©ØÓ CøhöÁÎPÐUS ©mk®
(B) \© ©ØÖ® \©©ØÓ CøhöÁÎPÐUS
(C) \©©õÚ CøhöÁÎPÐUS ©mk®
(D) CÁØÖÒ Hx® Qøh¯õx
Lagrange’s interpolation formula can be used for :
(a) unequal intervals only
(b) both equal and unequal intervals
(c) equal intervals only
(d) none of these

10. ∇≡
(A) 1−E−1 (B) 1+E (C) 1+E−1 (D) 1−E
∇≡
(a) 1−E−1 (b) 1+E (c) 1+E−1 (d) 1−E

11. x &I ÂÁ›US® {PÌuPÄ SÔ¨¤mh ©v¨ø£ Âh \©©õP÷Áõ AÀ»x
SøÓÁõP÷Áõ EÒÍ {PÌuPÄ :
(A) Âή¦ {PÌuPÄ (B) uÛzu {PÌuPÄ
(C) öuõhºa]¯õÚ {PÌuPÄ (D) vµÒ {PÌuPÄ
Probability which explains x is equal to or less than particular value is classified as :
(a) marginal probability (b) discrete probability
(c) continuous probability (d) cumulative probability

12. E[X−E(X)]2 Gߣx :
(A) V(X) (B) E(X) (C) S.D(X) (D) E(X2)
E[X−E(X)]2 is :
(a) V(X) (b) E(X) (c) S.D(X) (d) E(X2)

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13. D¸Ö¨¦¨ £µÁ¼ß £s£ÍøÁPÍõÚ B(n, p) &US \µõ\›°ß ©v¨¦ 4 ©ØÖ®
©õÖ£õk 4 GÛÀ {PÌuPÄ P(X/5) &Cß ©v¨£õÚx :
3

6 6 6 5
1 2 2 2 1
(A)  3  (B)  3  (C) 4  3  (D)    
      3 3

4
If the parameters of a binomial distribution B(n, p) mean=4 and variance= , the probability,
3
P(X/5) is equal to :
6 6 6 5
1 2 2 2 1
(a)   (b)   (c) 4  (d)    
3 3 3  3 3

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) 27 (B) 81 (C) 81 (D) 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. __________Gߣx •Êø©z öuõSv°¾ÒÍ JÆöÁõ¸ EÖ¨¦®
÷uº¢öukUP¨£kÁuØS J¸ \©©õÚ Áõ´¨ø£ AÎUS® JßÓõS®.
(A) ¦Òΰ¯À AÍøÁ (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

16. TöÓk¨¤À EÒÍ ¤øÇPÒ __________.
(A) |õßS ÁøP (B) C¸ ÁøP (C) I¢x ÁøP (D) ‰ßÖ ÁøP
Errors in sampling are of :
(a) four types (b) two types (c) five types (d) three types

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17. J¸ Põ»®\õº öuõh›ß uµÄz öuõS¨¦ ÂÁµ[PøÍ £vÄ ö\´¯¨£k®
CøhöÁÎ :
(A) Áõµ® J¸•øÓ (B) öuõhºa]¯õÚ Põ» ¦ÒÎPÒ
(C) \©Põ» CøhöÁÎ (D) ÷©ØPsh AøÚzx®
A time series is a set of data recorded :
(a) Weekly (b) Successive points of time
(c) Periodically (d) all the above

18. __________ Âø» SÔ±mk Gs Põ»©õØÖa ÷\õuøÚ ©ØÖ® Põµo ©õØÖa
÷\õuøÚ BQ¯ Cµsk ÷\õuøÚPøͲ® {øÓÄ ö\´²®.
(A) £õ] (B) L¤åº
(C) »õ줯º (D) CøÁ Hx® CÀø»
__________ price index number satisfies both the Time Reversal and Factor Reversal test.
(a) Paache’s (b) Fisher’s
(c) Laspeyre’s (d) None of these

19. ]» ÷|µ[PÎÀ __________ •øÓ¯õÚx ÷£õUSÁµzx PnUQß EP¢u wºÁõP
Aø©²®.
(A) ÷ÁõP¼ß ÷uõµõ¯ •øÓ (B) Áh÷©ØS ‰ø» •øÓ
(C) {øµ°ß ]Ö© •øÓ (D) «a]Ö ©v¨¦ •øÓ
Solution for transportation problem using __________ method is nearer to an optimal solution.
(a) VAM (b) NWCM
(c) Row Minima (d) LCM

20. ÷£õUSÁµzx PnUS G¨ö£õÊx \©{ø»¯ØÓx ?
(A) m=n
(B) ö©õzu ÁÇ[PÀ ≠ ö©õzu ÷uøÁ
(C) m+n−1
(D) ö©õzu ÁÇ[PÀ = ö©õzu ÷uøÁ
The transportation problem is said to be unbalanced if __________.
(a) m=n
(b) Total supply ≠ Total demand
(c) m+n−1
(d) Total supply=Total demand

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£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.

3 1 −5 −1 
21. ¤ßÁ¸® Ao°ß uµ® PõsP :  1 −2 1 −5 
 1 5 −7 2 

 3 1 −5 −1 
Find the rank of the given matrix  1 −2 1 −5 
 
 1 5 −7 2 

1
22. ©v¨¤kP : ∫ dx
sin x cos2 x
2

1
Evaluate : ∫ dx
sin x cos2 x
2

23. ÂØ£øÚ ö£õ¸ÒPÎß CÖv{ø» Á¸Áõ´ \õº¦ MR=9−4x2 GÛÀ, ÷uøÁa
\õºø£U PõsP.
If the marginal revenue function for a commodity is MR=9−4x2, find the demand function.

24. J¸ uÛzu \©Áõ´¨¦ ©õÔ X &ß {PÌuPÄ £µÁÀ \õº¦ :

2k, x=1
 3k, x=3

f ( x )= 
 4k, x=5
 0, ©ØöÓ[Q¾®
C[S k J¸ ©õÔ¼ GÛÀ k &ß ©v¨ø£U PõsP.
The probability distribution function of a discrete random Variable X is :

2k, x=1
3k,
 x=3
f ( x )= 
4k, x=5

 0, otherwise
where k is some constant, find k.

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25. JxURk PnUQß Pou ÁiÁ® GÊxP.
Give mathematical form of Assignment problem.

26. y=x3−x2+x−1 GÛÀ x=0, 1, 2, 3, 4, 5 GߣÚÁØÖUS y&ß ©v¨¦PøÍU PnUQmk
•ß÷|õUS ÷ÁÖ£õmk AmhÁønø¯ Aø©UP.
If y=x3−x2+x−1, calculate the values of y for x=0, 1, 2, 3, 4, 5 and form the forward
differences table.

27. D¸Ö¨¦¨ £µÁ¼ß \µõ\› ©v¨¦ 20 GÚÄ®, vmh»UPzvß ©v¨£õÚx
4 GÚÄ® öPõshõÀ, £µÁ¼ß AøÚzx £s£ÍøÁPøͲ® PõsP.
The mean of Binomial distribution is 20 and Standard deviation is 4. Find the parameters of
the distribution.

28. J¸ Qµõ©zvÀ, 400 |£ºPøÍU öPõsh J¸ TÔÀ ø\Á EnÄ Es£ÁºPÒ
230 |£ºPÒ, ©ØÓÁºPÒ Aø\Á EnÄ Es£ÁºPÒ GßP. A¢u Qµõ©zvÀ ø\Á
©ØÖ® Aø\Á EnÄPÒ Es£ÁºPÎß GsoUøP \©® GÛÀ vmh¤øÇø¯U
PõsP.
In a sample of 400 population from a village, 230 are found to be eaters of vegetarian items
and the rest non-vegetarian items. Compute the standard error assuming that both vegetarian
and non-vegetarian foods are equally popular in that village.

29. öPõkUP¨£mkÒÍ ÂÁµ[PÐUS Áøµ£h •øÓ°ß ‰»® ÷£õUSU ÷Põmøh¨
ö£õ¸zxP.

Bsk 2000 2001 2002 2003 2004 2005 2006 2007

ÂØ£øÚ
30 46 25 59 40 60 38 65
(hßPÎÀ)
Fit a trend line by the method of freehand method for the given data.

Year 2000 2001 2002 2003 2004 2005 2006 2007
Sales
30 46 25 59 40 60 38 65
(Tons)

30. wºUP : (D2+2D+2)y=0
Solve : (D2+2D+2)y=0

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£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. 11 ö£ß]ÀPÒ ©ØÖ® 3 AȨ£õßPÎß ö©õzu Âø» ` 64. ÷©¾® 8 ö£ß]ÀPÒ
©ØÖ® 3 AȨ£õßPÎß ö©õzu Âø» ` 49. Q÷µ©›ß Âvø¯¨ £¯ß£kzv
J¸ ö£ß]À ©ØÖ® J¸ AȨ£õß Âø»ø¯U PõsP.
The total cost of 11 pencils and 3 erasers is ` 64 and the total cost of 8 pencils and 3 erasers is
` 49. Find the cost of each pencil and each eraser by Cramer’s Rule.

32. ©v¨¤kP : ∫ x log x dx
Evaluate : ∫ x log x dx

33. y2=4ax GßÓ £µÁøͯ® Auß ö\ÆÁP»zxhß HØ£kzx® £µ¨ø£U PõsP.
Calculate the area bounded by the parabola y2=4ax and its latus rectum.

a
34. y = + b GßÓ ÁøÍÁøµU Sk®£zvß ÁøPUöPÊa \©ß£õmøhU PõsP. C[S
x
‘a’ ©ØÖ® ‘b’ Gß£Ú ©õÓzuUP ©õÔ¼PÒ.
a
Find the differential equation of the family of curves y = + b where a and b are arbitrary
x
constants.
35. RÌPsh ÂÁµ[PøÍU öPõsk Âk£mh EÖ¨ø£U PõsP.
x 2 3 4 5 6
f(x) 45.0 49.2 54.1 _ 67.4
From the following table, find the missing value.
x 2 3 4 5 6
f(x) 45.0 49.2 54.1 _ 67.4

36. uÛzu \©Áõ´¨¦ ©õÔ°ß {PÌuPÄ {øÓ \õº£õÚx :
X=x 0 1 2 3
p(x ) 0.2 0.1 0.4 0.3
GÛÀ, E(3X+2X2) Cß ©v¨ø£U PõsP.
Suppose the probability mass function of the discrete random variable is :
X=x 0 1 2 3
p(x ) 0.2 0.1 0.4 0.3
What is the value of E(3X+2X2) ?

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37. ¤øǯØÓ J¸ |õn¯® 7 •øÓ _sh¨£kQßÓx. AÁØÔÀ \›¯õP 2 uø»PÒ
Qøh¨£uØPõÚ {PÌuPÄ PõsP.
A fair coin is tossed 7 times. Find the probability that exactly 2 heads occur.

38. J¸ £Pøh 9000 •øÓ Ã\¨£k® ÷£õx Auß÷©À EÒÍ GsPÒ 3 AÀ»x 4 BP
3240 •øÓ QøhUQßÓÚ. ¤øǯØÓ £Pøh°ß vmh¨¤øÇ ÂQuzøuU
PnUQkP.
A die is thrown 9000 times and a throw of 3 or 4 is observed 3240 times. Find the standard
error of the proportion for an unbiased die.

39. î¢xìuõß {ÖÁÚzvß Bµõ´a] xøÓ ‰ßÖ ÁøP¯õÚ åõ®¦PøÍ
AÔ•P¨£kzu \¢øu¨£kzx® xøÓUS {v JxUP £›¢xøµUQÓx. R÷Ç
öPõkUP¨£mkÒÍ öÁÆ÷ÁÓõÚ ÂØ£øÚ {ø»°À Gvº£õºUP¨£k®
AÎzuÀPÐUS HØ£ åõ®¦PøÍ \¢øu¨£kzxQÓx.
©v¨¤h¨£mh ÂØ£øÚ (A»SPÎÀ)
åõ®¦PÎß ÁøPPÒ
15000 10000 5000
•møh åõ®¦ 30 10 10

QÎÛU åõ®¦ 40 15 5

j»Uì åõ®¦ 55 20 3

\¢øu¨£kzx® ÷©»õÍ›ß •iÄ GßÚ Gߣøu
(i) «a]ÖÂß «¨ö£¸ ©ØÖ®
(ii) «¨ö£¸Âß «a]Ö BQ¯ÁØøÓ £¯ß£kzv PõsP.
The research department of Hindustan Ltd., has recommended to pay marketing department
to launch a shampoo of three different types. The marketing types of shampoo to be launched
under the following estimated pay-offs for various level of sales.
Estimated Sales (in units)
Types of shampoo
15000 10000 5000
Egg shampoo 30 10 10
Clinic shampoo 40 15 5
Deluxe shampoo 55 20 3
What will be the marketing manager’s decision if
(i) Maximin and
(ii) Minimax Principle applied ?

40. öuõøP°hø»¨ £¯ß£kzv Bvø¯ ø©¯©õPU öPõsk 5 A»S Bµ® Eøh¯
Ámhzvß £µ¨ø£U PõsP.
Using integration find the area of the circle whose centre is at the origin and the radius is
5 units.

11 8367

£Sv & IV / PART - IV
SÔ¨¦ : AøÚzx ÂÚõUPÐUS® Âøh¯ÎUPÄ®. 7x5=35
Note : Answer all the questions.
41. (A) ¤ßÁ¸® \©ß£õmk öuõS¨¤øÚ uµ•øÓ°À wºUP.
x+y+z=9, 2x+5y+7z=52 ©ØÖ® 2x+y−z=0
AÀ»x
2
(B) ©v¨¤kP : ∫ 3x −2 x2+5 dx
( x−1) ( x +5 )
(a) Solve the following system of equations by rank method :
x+y+z=9, 2x+5y+7z=52, 2x+y−z=0
OR

3x 2−2 x+5
(b) Evaluate : ∫ dx
( x−1) ( x 2+5 )
dy π
42. (A) + 2 y tan x = sin x ©ØÖ® x= GÛÀ y=0 GÝ® {ø»°À y &I x &ß
dx 3
Áõ°»õP GÊxP.
AÀ»x
(B) Áh÷©ØS ‰ø» •øÓø¯ £¯ß£kzv ¤ßÁ¸® ÷£õUSÁµzx PnUQß
Bµ®£ Ai¨£øh \õzv¯©õÚz wºøÁ PõsP.

D E F G Aئ
A 11 13 17 14 250
B 16 18 14 10 300
C 21 24 13 10 400

÷uøÁ 200 225 275 250

dy π
(a) If +2 y tan x = sin x and if y=0 when x= , express y in terms of x.
dx 3
OR
(b) Obtain an initial basic feasible solution to the following transportation problem by
North-west Corner method.

D E F G Available
A 11 13 17 14 250
B 16 18 14 10 300
C 21 24 13 10 400
Required 200 225 275 250

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43. (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¯ÁºPÎß 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.
Kº BsiØS ¤ÓS AÁØÔß \¢øu¨ £[RkPøÍU PõsP. ÷©¾® \¢øu°À
\©{ø» G¨÷£õx Gmh¨£k® ?
AÀ»x
(B) R÷Ç öPõkUP¨£mkÒÍ AmhÁøn°¼¸¢x x=7.5 GÝ®÷£õx
y &ß ©v¨ø£U PnUQkP.

x 1 2 3 4 5 6 7 8
y 1 8 27 64 125 216 343 512

(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) Calculate the value of y, when x=7.5 from the table below :

x 1 2 3 4 5 6 7 8
y 1 8 27 64 125 216 343 512

dy x 2+ y 2
44. (A) ÁøPUöPÊ \©ß£õmøhz wºUP : =
dx xy
AÀ»x
(B) |õßS SÇ¢øuPÒ öPõsh 750 Sk®£[PÎÀ,
(i) SøÓ¢u£m\® Kº Bs SÇ¢øu
(ii) AvP£m\® Cµsk ö£s SÇ¢øuPÒ ©ØÖ®
(iii) C¸ £õ¼Ú SÇ¢øuPЮ C¸¨£uØPõÚ {PÌuPÂøÚ PõsP ?
(Bs ©ØÖ® ö£s SÇ¢øuPÎß ¤Ó¨¦ \©©õÚ {PÌuPÁõP Gkzx
öPõÒP)

13 8367

(a) Solve the following differential equation.
dy x 2+ y 2
=
dx xy
OR
(b) Out of 750 families with 4 children each, how many families would be expected to
have
(i) atleast one boy
(ii) atmost two girls and
(iii) children of both sexes ?
(Assume equal probabilities for boys and girls)

45. (A) Gί \µõ\› •øÓø¯¨ £¯ß£kzv J¸ ö£õ¸Îß ©õuõ¢vµ ÂØ£øÚUS,
£¸ÁPõ» SÔ±møhU PõsP.
©õu[PÒ Bsk
2001 2002 2003
áÚÁ› 15 20 18
¤¨µÁ› 41 21 16
©õºa 25 27 20
H¨µÀ 31 19 28
֩ 29 17 24
áüß 47 25 25
áüø» 41 29 30
BPìm 19 31 34
ö\¨h®£º 35 35 30
AU÷hõ£º 38 39 38
|Á®£º 40 30 37
i\®£º 30 44 39
AÀ»x
(B) X GßÓ öuõhº \©Áõ´¨¦ ©õÔ°ß {PÌuPÄ Ahºzva \õº£õÚx
2e−2x , x > 0
f (x) = 

0 , ©ØöÓ[Q¾®

GÛÀ E(X) ©ØÖ® V(X) &IU PõsP.

[ v¸¨¦P / Turn over

8367 14

(a) Calculate the seasonal index for the monthly sales of a product using the method of
simple averages.

Months Year
2001 2002 2003
Jan 15 20 18
Feb 41 21 16
Mar 25 27 20
Apr 31 19 28
May 29 17 24
June 47 25 25
July 41 29 30
Aug 19 31 34
Sep 35 35 30
Oct 38 39 38
Nov 40 30 37
Dec 30 44 39

OR
(b) Consider a continuous random variable X with probability density function.

2e−2x , x > 0
f (x) = 
0 , otherwise

Find E(X) and V(X)

a
46. (A) J¸ {ÖÁÚzvß CÖv{ø» Á¸Áõ´ \õº¦ MR = − c . C[S
( x+b )2
x Gߣx ö£õ¸ÒPÎß EØ£zv ©ØÖ® a, b, c Gß£Ú ©õÔ¼PÒ GÛÀ,
a
÷uøÁa \õº¦ x = b ( p+c ) − b GÚ {ÖÄP.

AÀ»x
(B) £¢x •øÚ ÷£Úõ u¯õ›US® {ÖÁÚ©õÚx, uõß u¯õ›US® ÷£ÚõÂß
(GÊx®) B²Ò, \µõ\›¯õP 400 £UP[PÍõPÄ®, vmh»UP® 20 £UP[PÒ
GÚU TÖQÓx. J¸ •PÁº 100 ÷£ÚõUPøÍU öPõÒ•uÀ ö\´x ÷\õuøÚUS
Em£kzxQßÓõº. Auß \µõ\› (GÊx®) B²Ò 390 £UP[PÒ GÚU
PshÔQÓõº. öPõÒ•uÀ •PÁº {ÖÁÚzvß TØøÓ 1% ªøPPõs
{ø»°À {µõP›UP»õ©õ ?

15 8367

a
(a) A firm has the marginal revenue function given by MR = − c . Where x is the
( x+b )2
output and a, b, c are constants. Show that the demand function is given by
a
x= −b
b ( p+c )
OR
(b) A manufacturer of ball pens claims that a certain pen he manufactures has a mean
writing life of 400 pages with a standard deviation of 20 pages. A purchasing agent
selects a sample of 100 pens and puts them for test. The mean writing life for the
sample was 390 pages. Should the purchasing agent reject the manufacturer’s claim at
1% level ?

 5x+12 
47. (A) h=1 GÛÀ ∆  2  &I ©v¨¤kP.
 x +5x+6 
AÀ»x
(B) 2007 B® Bsiß Ai¨£øh°À 2011 B® BsiØPõÚ ÁõÌUøP SÔ±mk
GsønU öPõkUP¨£mh ÂÁµ[PÐUS Sk®£ ÁµÄ ö\»Ä •øÓø¯¨
£¯ß£kzvU PnUQkP.

ö£õ¸ÒPÒ Âø» {øÓPÒ
2007 2011
A 350 400 40
B 175 250 35
C 100 115 15
D 75 105 20
E 60 80 25

 5x+12 
(a) If h=1, Evaluate ∆  2 
 x +5x+6 
OR
(b) Construct the cost of living index number for 2011 on the basis of 2007 from the given
data using family budget method.
Price
Commodities Weights
2007 2011
A 350 400 40
B 175 250 35
C 100 115 15
D 75 105 20
E 60 80 25

-oOo-
[ v¸¨¦P / Turn over