TN 12th Question Paper 2023 Business Mathematics and Statistics

Tamil Nadu
State Board

2023
QUESTION
PAPER

No. of Printed Pages : 16
6667
!6667BusiMatheandStati! £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.

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1. JÆöÁõ¸ EÖ¨¦® 1 GÚU öPõsh m×n Á›ø\ Eøh¯ Ao°ß uµ® :
(A) m (B) 0 (C) n (D) 1
The rank of m×n matrix whose elements are unity is :
(a) m (b) 0 (c) n (d) 1

2. |An×n|=3 |adj A| = 243 GÛÀ ‘n’ &ß ©v¨¦ :
(A) 6 (B) 4 (C) 7 (D) 5
If |An×n|=3 and |adj A|=243 then the value of ‘n’ is :
(a) 6 (b) 4 (c) 7 (d) 5

sin2x
3.
∫ 2sinx dx &ß ©v¨¦ :
1 1
(A) cosx+c (B) sinx+c (C) 2 cosx+c (D) 2
sinx+c

sin2x
The value of
∫ 2sinx dx is :
1 1
(a) cosx+c (b) sinx+c (c) cosx+c (d) sinx+c
2 2

∫ e dx &ß ©v¨¦ :
x
4.

1 1
(A) 2 e x + c (B) (C) +c (D)
ex + c 2 e x 2 ex + c

The value of ∫ e x dx is :

1 x 1
(a) e +c (b) x (c) +c (d)
2 e +c 2 ex 2 ex + c

5. 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

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6. CÖv {ø»a \õº¦ MR=100 – 9x2 &ß ÷uøÁa \õº¦ :
(A) 100x−9x2 (B) 100−3x2 (C) 100+9x2 (D) 100x−3x2
The demand function for the marginal function MR=100−9x2 is :
(a) 100x−9x2 (b) 100−3x2 (c) 100+9x2 (d) 100x−3x2

7. y=e–2x (A cosx+B sinx) &À A ©ØÖ® B &ø¯ }USÁuß ‰»® Aø©UP¨£k®
ÁøPUöPÊa \©ß£õk :
(A) y2−4y1−5=0 (B) y2−4y1+5=0 (C) y2+4y1+5=0 (D) y2+4y−5=0
The differential equation formed by eliminating A and B from
y=e–2x(A cosx+B sinx) is :
(a) y2−4y1−5=0 (b) y2−4y1+5=0 (c) y2+4y1+5=0 (d) y2+4y−5=0

8. (3D2+ D −14)y=13e2x &ß ]Ó¨¦z öuõøP :

x2 2x x
(A) e (B) 2 e2 x (C) 13xe2x (D) xe2x
2
The Particular Integral of (3D2+D−14)y=13e2x is :

x2 2x x 2x
(a) e (b) e (c) 13xe2x (d) xe2x
2 2

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

10. E (Ey0)=
(A) y2 (B) y0 (C) y3 (D) y1
E (Ey0)=
(a) y2 (b) y0 (c) y3 (d) y1

11. E [X−E(X)] Gߣx :
(A) 0 (B) E(X) (C) E(X)−X (D) V(X)
E [X−E(X)] is equal to :

(a) 0 (b) E(X) (c) E(X)−X (d) V(X)

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1
12. p(x)= , x=10 GÛÀ, E(X) &ß ©v¨£õÚx :
10

6
(A) 1 (B) §ä¯® (C) –1 (D) 8

1
If p(x)= , x=10 then E(X) is :
10

6
(a) 1 (b) Zero (c) –1 (d)
8

13. C¯À{ø»¨ £µÁø»U Psk¤izuÁº :
(A) Põì (B) »õ¨÷»ì
(C) ÷á®ì ö£º÷Úõ¼ (D) j ©õ´Áº
Normal distribution was invented by :
(a) Gauss (b) Laplace
(c) James Bernoulli (d) De-Moivre

14. \µõ\›²® ©õÖ£õmhÍøÁ²® \©©õP C¸US® {PÌuPĨ £µÁ»õÚx :
(A) C¯À{ø» (B) £õ´\õß
(C) D¸Ö¨¦ (D) ÷©ØTÔ¯ AøÚzx®
In a parametric distribution the mean is equal to variance is :
(a) normal (b) poisson
(c) binomial (d) all of the above

15. J¸ •Êø©z öuõSv°ß •iÄÖ EmPnzøu __________ GÚ TÓ»õ®.
(A) •Êø© (B) TÖ
(C) •Êø©U Po¨¦ (D) •Êø©z öuõSv
A finite subset of statistical individuals in a population is called __________.
(a) universe (b) a sample

(c) census (d) a population

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16. ©v¨¥mk AÍøÁ¯õÚx ©õv› ¦Òΰ¯À AÍøÁ°ß _________ I ©v¨¤h
£¯ß£kQÓx.
(A) ©õv› AÍÄ (B) •Êø©z öuõSv £s£ÍøÁ
(C) •Êø©U Po¨¦ (D) ¤øǯõÚ ©v¨¥k
An estimator is sample statistics used to estimate a :
(a) sample size (b) population parameter
(c) census (d) biased estimate

17. L¤åº Âø» SÔ±mk Gs Gߣx »õ줯º ©ØÖ® £õ] Âø» SÔ±mk
GsPÐUS Cøh÷¯¯õÚ __________ BS®.
(A) Tmk \µõ\› (B) ö£¸US \µõ\›
(C) Cø\ \µõ\› (D) (A) ©ØÖ® (C)
Fisher’s price index number is the __________ between Laspeyre’s and Paasche’s
price index number.
(a) Arithmetic mean (b) Geometric mean
(c) Harmonic mean (d) both (a) and (c)

18. £¸ÁPõ» ©õÖ£õk Gߣx __________ HØ£k® ÷ÁÖ£õkPÒ BS®.
(A) J¸ ©õuzvØSÒÍõP (B) ]» BskPÐUSÒ
(C) J¸ ÁõµzvØSÒÍõP (D) J¸ BsiØSÒÍõP
The seasonal variation means the variations occurring within :
(a) a month (b) some years
(c) a week (d) a year

19. ÷£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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20. Áh÷©ØS ‰ø» GߣuøÚ SÔ¨£x _________.
(A) RÌ Á»x ‰ø» (B) ÷©À Chx ‰ø»
(C) RÌ Chx ‰ø» (D) ÷©À Á»x ‰ø»
North-West corner refers to _________.
(a) bottom right corner (b) top left corner
(c) bottom left corner (d) top right corner

£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.

0 −1 5 
21.  2 4 −6 GßÓ Ao°ß uµzvøÚU PõsP.
 
 1 1 5 

0 −1 5 
Find the rank of the matrix  2 4 −6
 
 1 1 5 

22. ©v¨¤kP : ∫ (3 + x ) (2 − 5x ) dx .

Evaluate : ∫ (3 + x ) (2 − 5x ) dx .

23. y=4x+3 GßÓ ÁøÍÁøµ, x - Aa_, x=1 ©ØÖ® x=4 BQ¯ÁØÖhß HØ£kzx®
£µ¨ø£U PõsP.
Find the area bounded by the curve y=4x+3 with x - axis between the lines x=1
and x=4

24. ©v¨¤kP : ∆( log ax )

Evaluate : ∆( log ax )

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25. ¤ßÁ¸® uPÁÀ öÁØÔPÎß {PÌuPÄ £µÁø»U SÔUQÓx GÛÀ, öÁØÔ°ß
Gvº£õºzuÀ GsoUøPø¯U Psk¤iUPÄ®.

öÁØÔPÎß GsoUøP X=x 0 1 2

6 9 1
{PÌuPÄ P( x ) 11 22 22

The following information is the probability distribution of successes.

No. of successes X=x 0 1 2

6 9 1
Probability P(x )
11 22 22

Determine the expected number of success.

26. D¸Ö¨¦¨ £µÁÀ – Áøµ¯ÖUPÄ®.
Define Binomial distribution.

27. Gί \©Áõ´¨¦ TöÓk¨¤ß |ßø©PÒ GøÁ÷¯Ý® CµsiøÚ GÊxP.
State any two merits of simple random sampling.

dy
28. wºUP :- = xy+x+y+1
dx
dy
Solve :- = xy+x+y+1
dx

29. JxURk PnUQß Pou ÁiÁ® u¸P.
Give mathematical form of Assignment problem.

30. ∑ p 0 q 0=1974, ∑ p1q 0= 3140, ∑ p1q 1= 2005 , GÛÀ ÁõÌUøPz uµUSÔ±mk
Gsøn, ö©õzu ö\»Ä •øÓø¯¨ £¯ß£kzvU PõsP.

For ∑ p 0 q 0=1974, ∑ p1q 0= 3140, ∑ p1q 1= 2005 find the Cost of Living Index by
Aggregate Expenditure Method.

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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. ¤ßÁ¸® \©ß£õkPÒ J¸[Pø©Ä Eøh¯x GÛÀ k &ß ©v¨ø£U PõsP.
x+2y−3z=−2, 3x−y−2z=1 ©ØÖ® 2x+3y−5z=k.
Find k, if the equations x+2y−3z=−2, 3x−y−2z=1, 2x+3y−5z=k are
consistent.

32. MR=20−5x+3x2 GÛÀ, ö©õzu Á¸Áõ´a \õº¦ PõsP.
If MR=20−5x+3x2, find total revenue function.

33. wºUP : 9y99−12y9+4y=0
Solve : 9y99−12y9+4y=0

34. 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

35. J¸ \©Áõ´¨¦ ©õÔ X &UPõÚ {PÌuPÄ Ahºzva \õº£õÚõx,
4x 3 , 0 < x < 1
f(x) = 
0, ©ØöÓ[Q¾®


GÛÀ E(X) ©ØÖ® V(X) Psk¤iUPÄ®.
Consider a random variable X with probability density function,

4x 3 , if 0 < x < 1
f(x) = 
 0, otherwise

Find E(X) and V(X).

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36. 520 £UP[PøÍU öPõsh ¦zuPzvÀ, 390 umha_¨ ¤øÇPÒ EÒÍÚ. £õ´\õß
ÁÈ°øÚ AÝ©õÛzx \©Áõ´¨¦ •øÓ°À ÷uº¢öukUP¨£mh 5 £UP[PÎÀ
¤øÇ÷¯ CÀ»õ©À C¸¨£uØPõÚ {PÌuPÄ PõsP.
In a book of 520 pages, 390 typo-graphical errors occur. Assuming Poisson law for
the number of errors per page, find the probability that a random sample of
5 pages will contain no error.

37. vmh »UP® 10 ©ØÖ® ©õv›ø¯¨ ö£õÖzx vmh¨¤øÇ 3 GÛÀ, ©õv›°ß
AÍøÁU PõsP.
Find the sample size for the given standard deviation 10 and the standard error
with respect of sample mean is 3.

38. 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.

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

Construct the cost of living index number for 2011 on the basis of 2007 from the
given data using Family Budget method.

Price
Commodities W eights
2007 2011
A 350 400 40
B 175 250 35
C 100 115 15
D . 75 105 20

E . 60 . 80 25

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39. öPõkUP¨£mh AÎzuÀ Ao°ß EP¢u wºøÁ (i) «a]ÖÂß «¨ö£¸ ©ØÖ®
(ii) «¨ö£¸Âß «a]Ö BQ¯ÁØøÓ¨ £¯ß£kzv PõsP.

`Ì{ø»PÎß {ø»¨£õkPÒ
ö\¯Ø£õ[S
S1 S2 S3 S4
A1 14 9 10 5
A2 11 10 8 7
A3 9 10 10 11
A4 8 10 11 13

From the following pay-off matrix, find the optimal decision under each of the
following rule (i) maxmin (ii) minimax.

States of nature
Act
S1 S2 S3 S4
A1 14 9 10 5
A2 11 10 8 7
A3 9 10 10 11
A4 8 10 11 13

1
40. ©v¨¤kP : ∫ x+2 − x+3
dx

1
Evaluate : ∫ dx
x+2 − x+3

£Sv & IV / PART - IV

SÔ¨¦ : AøÚzx ÂÚõUPÐUS® Âøh¯ÎUPÄ®. 7x5=35
Note : Answer all questions.

41. (A) Q÷µ©›ß Âvø¯¨ £¯ß£kzv wºUP :
x+y+z=4; 2x−y+3z=1; 3x+2y−z=1

AÀ»x

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(B) J¸ SÔ¨¤mh |Pµzvß ©UPÒ öuõøP R÷Ç öPõkUP¨£mkÒÍx.

Á¸h® : X 1941 1951 1961 1971 1981 1991

©UPÒ öuõøP : Y
20 24 29 36 46 51
(C»m\zvÀ)

Cøha ö\¸PÀ `zvµzøu¨ £¯ß£kzv 1946 &® BskUPõÚ ©UPÒ
öuõøPø¯U PõsP.
(a) Solve by Cramer’s rule, x+y+z=4; 2x−y+3z=1; 3x+2y−z=1

OR

(b) The population of a certain town is as follows.

Year : X 1941 1951 1961 1971 1981 1991

Population in lakhs : Y 20 24 29 36 46 51

Using appropriate interpolation formula, estimate the population during the
period 1946.

42. (A) Áøµ¯Özu öuõøP±møh J¸ Tmh¼ß GÀø» GÚU öPõsk
2
∫ (2x+1) dx &I ©v¨¤kP.
1

AÀ»x
(B) £»Áõ´¨¦ ÂÚõUPÒ öPõsh ÷uºÂÀ £zx ÂÚõUPÐUS BÖ \›¯õÚ
£vÀPøÍU Po¨£uØPõÚ {PÌuPÂøÚU PõsP.

2
(a) Evaluate the integral as the limit of a sum ∫ (2x+1) dx
1

OR

(b) Find the probability of guessing correctly atleast six of the ten answers in a
TRUE/FALSE objective test.

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43. (A) ÷uøÁa \õº¦ Pd =25−3x ©ØÖ® AΨ¦a \õº¦ P s=5+2x GÛÀ, \©ß
{ø»°À ~Pº÷Áõº E£› ©ØÖ® EØ£zv¯õͺ E£›ø¯U PõsP.
AÀ»x
(B) X GÝ® ©õÔ C¯À{ø»¨ £µÁ¼ß \µõ\› 12 ©ØÖ® vmh »UP® 4 GÛÀ
P(X ≤ 20) ©ØÖ® P(0 ≤ X ≤ 12) ©v¨¤øÚU PõsP.

[P(0 < Z < 2)= 0.4772]

(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) X is normally distributed with mean 12 and S.D 4.

Find P(X ≤ 20) and P(0 ≤ X ≤ 12)

[P(0 < Z < 2)= 0.4772]

44. (A) wºUP : (D2−2D+1)y=e2x+ex.
AÀ»x
(B) AÁ\µ ©¸zxÁ ]Qaø\ ÁõPÚ ÷\øÁ ÁÇ[S® J¸ {ÖÁÚ®, u[PÐUS
QøhUP¨ö£Ö® AÁ\µ AøǨ¤ß÷£õx \µõ\›¯õP 8.9 {ªh[PÎÀ
AøǨ¤hzøu ö\ßÓøhÁuõP TÖQÓx. AÁºPÎß TØøÓ ÷\õvUP,
GkUP¨£mh 50 AÁ\µ AøǨ¤ß ©õv› ÷uºÄPÎÀ Auß \µõ\›
9.3 {ªh[PÒ, vmh »UP® 1.6 {ªh[PÒ GÚ AÔ¯¨£kQÓx. 5%
ªøPPõs {ø»°À {ÖÁÚzvß TØÖ \›¯õÚuõ ?
(a) Solve : (D2−2D+1)y=e2x+ex.

OR

(b) An ambulance service claims that it takes on an average 8.9 minutes to
reach its destination in emergency calls. To check on this claim, the agency
which licenses ambulance services, has then timed on 50 emergency calls,

getting a mean of 9.3 minutes with a standard deviation of 1.6 minutes.
What can they conclude at 5% level of significance ?

13 6667

45. (A) öPõkUP¨£mkÒÍ AmhÁøn°¼¸¢x y(10) &ß ©v¨ø£ C»Uµõg]°ß
Cøhaö\¸PÀ `zvµzøu¨ £¯ß£kzv PõsP.

X 5 6 9 11
Y 12 13 14 16

AÀ»x

dy y
(B) wºUP : + = x3
dx x

(a) Using Lagrange’s interpolation formula find y(10) from the following table.

X 5 6 9 11
Y 12 13 14 16

OR

dy y
(b) Solve : + = x3
dx x

46. (A) 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

AÀ»x

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(B) J¸ \©Áõ´¨¦ ©õÔ X &ß {PÌuPÄ \õº¦ R÷Ç öPõkUP¨£mkÒÍx.
1
 4 , x =−2

1, x= 0

p(x ) =  4
1
 , x = 10
2

 0, ©ØöÓ[Q¾®

GÛÀ, ¤ßÁ¸® {PÌuPÄPøÍ ©v¨¤hÄ®.

(i) P(X ≤ 0) (ii) P(X<0) (iii) P(|X|≤ 2) (iv) P(0 ≤ X ≤ 10)

(a) Compute (i) Laspeyre’s (ii) Paasche’s (iii) Fisher’s Index numbers for the year
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

OR

(b) The probability function of a random variable X is given by
1
 4 , for x=−2

 1 , for x = 0

p(x ) = 4
1
 , for x = 10
2

 0, elsewhere

Evaluate the following probabilities

(i) P(X ≤ 0) (ii) P(X<0) (iii) P(|X|≤ 2) (iv) P(0 ≤ X ≤ 10)

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47. (A) 5 AÍÄ öPõsh 10 ©õv›PÎß \µõ\› ©ØÖ® Ãa_ AÍÃkPÒ R÷Ç
öPõkUP¨£mkÒÍÚ. \µõ\› Áµ®¦ Áøµ £h[PøÍ Áøµ¯Ä® ©ØÖ®
ö\¯À•øÓ Pmk¨£õmiß {ø» SÔzx E©x P¸zøu ÂÁ›UPÄ®.

TÖ 1 2 3 4 5 6 7 8 9 10
X 43 49 37 44 45 37 51 46 43 47
R 5 6 5 7 7 4 8 6 4 6

n=5, A2=0.58, D3=0 ©ØÖ® D4=2.115 GÚU öPõkUP¨£mkÒÍÚ.

AÀ»x
(B) ÷ÁõP¼ß ÷uõµõ¯ •øÓø¯ öPõsk RÌUPsh ÷£õUSÁµzx PnUQß
Ai¨£øh Bµ®£zwºøÁ PõsP.

Qh[SPÒ PøhPÒ C¸¨¦
↓ I II III IV (a i )
A 5 1 3 3 34
B 3 3 5 4 15
C 6 4 4 3 12
D 4 1 4 5 19
÷uøÁ (bj) 21 25 17 17

(a) Given below are the values of sample mean ( X ) and the range (R) for ten
samples of size 5 each. Draw mean chart and comment on the state of
control of the process.

Sample
1 2 3 4 5 6 7 8 9 10
Number
X 43 49 37 44 45 37 51 46 43 47
R 5 6 5 7 7 4 8 6 4 6

Given the following control chart constraint for : n=5, A2=0.58, D3=0 and
D4=2.115

OR

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(b) Obtain an initial basic feasible solution to the following transportation problem
using Vogel’s approximation method.
Warehouses Stores Availability
↓ I II III IV (ai )
A 5 1 3 3 34
B 3 3 5 4 15
C 6 4 4 3 12
D 4 1 4 5 19

Requirement 21 25 17 17
(bj)

-oOo-