Tamil Nadu 12th Std Model Question Paper Business Maths

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3317 (NS)
!3317NSBusiMatheandStati! £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 ©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

3317 (NS) 2

4 x
1. A = 13 ©Ø-Ö® Adj A = GÛÀ, x &ß ©v¨-£õ-Úx :
5 7

(A) 3 (B) 4 (C) 2 (D) −5

4 x
If A = 13 and Adj A = , then the value of x is :
5 7

(a) 3 (b) 4 (c) 2 (d) −5

2. ¤ß-Á-¸-Á-Ú-ÁØ-ÔÀ Gx J¸ AoU-PõÚ Ai¨-£øh E¸-©õØ-Ó® BPõx ?
(A) Ci→Ci+5Cj (B) Ri↔Rj
(C) Ri→2Ri+2Cj (D) Ri→2Ri−4Rj
Which of the following is not an elementary transformation ?

(a) Ci→Ci+5Cj (b) Ri↔Rj

(c) Ri→2Ri+2Cj (d) Ri→2Ri−4Rj

3. k ≠ __________ GÛÀ, x+y+z=2, 2x+y−z=3, 3x+2y+k=4 GßÓ ÷|›¯
\©ß£õm-kz öuõ-S¨-£õ-Úx, J÷µ J¸ wº-øÁ¨ ö£Ø-Ô-¸U-S®.
(A) 1 (B) 0 (C) −4 (D) 4
The system of linear equations x+y+z=2, 2x+y−z=3, 3x+2y+k=4 has unique
solution, if k is not equal to :

(a) 1 (b) 0 (c) −4 (d) 4

4. j(n+2)=90 j(n) ; (n > 0) GÛÀ, n &ß ©v¨-£õ-Úx :

(A) 9 (B) 8 (C) 10 (D) 7
If j(n+2)=90 j(n) ; (n > 0), then the value of ‘n’ is :

(a) 9 (b) 8 (c) 10 (d) 7

3 3317 (NS)

ex
5.
∫ 1 + ex dx &ß ©v¨-¦a \õº¦ :
(A) 2 1 + ex + C (B) ex 1 + ex + C

ex
(C) (D) +C
1 + ex + C
1 + ex

ex
∫ 1 + ex dx is :
(a) 2 1 + ex + C (b) ex 1 + e x + C

ex
(c) (d) +C
1 + ex + C
1 + ex

6. y= x GÝ® ÁøÍ-Áøµ, 0 &Â-¼¸
- ¢x 2 Áøµ HØ-£k
- z-x® Aµ[-Pz-vß £µ¨¦ :
(A) 4 \.A-»-S-PÒ (B) 1 \.A-»-S
(C) 3 \.A»-S-PÒ (D) 2 \.A»-S-PÒ
Area bounded by y= x between the limits 0 and 2 is :
(a) 4 sq.units (b) 1 sq.unit
(c) 3 sq.units (d) 2 sq.units

7. CÖ-v {ø» ö\»-Äa \õº-¦ MC=100 x , TC=0 ©Ø-Ö® öÁÎ-±k 0 GÛÀ \µõ\›a
\õº¦ AC BÚx :
200 200
(A) 1 (B) 200 x 1 2 (C) 200 x 3 2 (D) 3
3x 2 3 3 3x 2

The marginal cost function is MC=100 x . Find AC, given that TC=0 when the
output is zero :

200 200 1 2 200 3 2 200
(a) 1 (b) x (c) x (d) 3
3x 2 3 3 3x 2

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3317 (NS) 4

d2 y dy
8.
2
= +5 GßÓ ÁøPU- ö P- Ê a \©ß- £ õm- i ß Á›- ø \ ©Ø- Ö ® £i
dx dx
•øÓ÷¯ :
(A) 2 ©Ø-Ö® 3 (B) 2 ©Ø-Ö® 1 (C) 3 ©Ø-Ö® 2 (D) 2 ©Ø-Ö® 2

d2 y dy
The order and degree of the differential equation = + 5 are respectively :
2 dx
dx
(a) 2 and 3 (b) 2 and 1 (c) 3 and 2 (d) 2 and 2

9. (3D2+D−14)y=13e2x &ß ]Ó¨-¦z öuõøP :

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

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

10. ∆ 2y0 =
(A) y2+y1+2y 0 (B) y2−2y1+y0
(C) y2+2y1−y 0 (D) y2+2y1+y0
∆ 2y0 =
(a) y 2+y 1+2y 0 (b) y 2−2y1 +y 0
(c) y 2+2y 1−y 0 (d) y 2+2y1 +y 0

11. |õÒ Jß-ÖUS ö£õ-¸Ò-P-Îß ÷uøÁ-¯õ-Úx, ‰ßÖ |õÒ-P-ÐUS •øÓ-÷¯
21, 19, 22 A»SPÒ BS®. AÁØÔß {PÌuPÄPÒ •øÓ÷¯ 0.29, 0.40, 0.35
BS®. A»S Jß-ÖUS C»õ-£® 0.50 ø£\õU-PÒ GÛÀ, ‰ßÖ |õÒPÐUPõÚ
Gvº-£õºU-P¨-£mh C»õ-£® :
(A) 3.045, 3.8, 3.85 (B) 21, 19, 22
(C) 0.29, 0.40, 0.35 (D) 21.5, 19.5, 22.5
Demand of products per day for three days are 21, 19, 22 and their respective
probabilities are 0.29, 0.40, 0.35. Profit per unit is 0.50 paisa, then expected profits for
three days are :
(a) 3.045, 3.8, 3.85 (b) 21, 19, 22
(c) 0.29, 0.40, 0.35 (d) 21.5, 19.5, 22.5

5 3317 (NS)

12. J¸ \©-Áõ´¨¦ ©õ-Ô-°ß {PÌ-u-PÄ \õº¦ ¤ß-Á-¸-©õÖ Áøµ-¯-ÖU-P¨-£m-kÒÍx
X=x −1 −2 0 1 2
P(x ) k 2k 3k 4k 5k
GÛÀ, k &Cß ©v¨-£õ-Ú-x :
1 1
(A) 4 (B) 15 (C) §ä-¯® (D) -Jß-Ö
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) (c) zero (d) one
4 15

13. J¸ EØ-£z-v-¯õͺ u¯õ-›U-S® ªß-Âø\ ©õØ-ÖUSªÌ-P-ÎÀ (Switches) 2 \uÃu
u¯õ-›¨-¦-PÒ SøÓ-£õ-kÒÍøÁ GÚ AÔ-¯¨-£-k-Q-Óx. J¸ ÷£øÇ-°À C¸US®
50 ªß-Âø\ ©õØ-ÖU-S-ªÌ-P-ÎÀ Av-P-£m-\-©õP 2 SøÓ-£õ-k-PÒ C¸¨-£-uØ-PõÚ
{PÌ-u-P-Áõ-Úx :
(A) 1.5e−1 (B) 3e−1 (C) 2.5e−1 (D) 2e−1
A manufacturer produces switches and experiences that 2 percent switches are defective.
The probability that in a box of 50 switches, there are at the most two defective is :
(a) 1.5e −1 (b) 3e−1 (c) 2.5e −1 (d) 2e−1

14. H0 : µ=µ0 Gß-£-uØS HØ£ H1 : µ < µ0 G-Ý® J¸ ÷\õ-u-øÚ-°À, ªøP Põs
©v¨¦ α=0.01 BP C¸U-S® ÷£õx, Auß wº-©õ-ÛU-S® ©v¨¦ :
(A) −1.645 (B) −2.33 (C) 2.33 (D) 1.645
For testing H0 : µ=µ0 against H1 : µ < µ0 , what is the critical value at α=0.01 ?
(a) −1.645 (b) −2.33 (c) 2.33 (d) 1.645

15. N AÍ-ÄÒÍ J¸ •Ê-ø©z öuõ-S-v-°-¼-¸¢x \©-Áõ´¨¦ TöÓ-k¨¦ •øÓ°À
•uß-•øÓ J¸ EÖ¨¦ ÷uºÄ ö\´-²® ÷£õ-x Auß {PÌ-u-PÄ :
N n 1
(A) n (B) N (C) N (D) 1
In simple random sampling from a population of N units, the probability of drawing
any unit at the first draw is :
N n 1
(a) (b) (c) (d) 1
n N N

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3317 (NS) 6

16. Cµs-hõ-Áx ÁøP¨-¤øÇ Gß-£x :
(A) H0 uÁÖ GÛÀ HØ-£-x (B) H0 uÁÖ GÛÀ ©Ö¨-£-x
(C) H0 Esø© GÛÀ ©Ö¨-£-x (D) H0 Esø© GÛÀ HØ-£-x
Type II error is :
(a) Accept H0 when it is false (b) Reject H0 when it is false
(c) Reject H0 when it is true (d) Accept H0 when it is true

17. T, S, C ©Ø-Ö® I BQ-¯ TÖ-PøÍU öPõsh Põ-»®\õº öuõ-h-›ß ö£¸U-PÀ
Ái-Á-ø©¨-£õ-Úx :
(A) y=T+S×C×I (B) y=T×S×C×I
(C) y=T+S×C+I (D) y=T+S+C+I
The multiplicative model of the time series with the components T, S, C and I is :
(a) y=T+S×C×I (b) y=T×S×C×I
(c) y=T+S×C+I (d) y=T+S+C+I

18. EØ-£z-v¨ ö£õ-¸-Îß uµzøu £õ-vU-PU-T-i¯ ©õ-Ö-£õ-k-PÒ Gz-uøÚ ?
(A) 1 (B) 4 (C) 3 (D) 2
How many causes of variation will affect the quality of a product ?
(a) 1 (b) 4 (c) 3 (d) 2

19. JxU-Rk PnU-QÀ ÁÇ[-PÀ ©Ø-Ö® ÷\¸-ªh- ® \©-©õP CÀ-»õ-Âm-hõÀ AøÁ :
(A) \©a-^-µØ-Ó-x (B) \©-©õ-Ú-x
(C) \©a-^-µõ-Ú-x (D) -\-©-{-ø»-¯Ø-Ó-x
If number of sources is not equal to number of destinations, the assignment problem is
called __________.
(a) unsymmetric (b) balanced
(c) symmetric (d) unbalanced

20. ^µØÓ wº-ÂÀ JxU-Rmk AøÓ-P-Îß Gs-oUøP BÚ-x :
(A) m+n+1 &U-Sa \©-©Ø-Ó-x (B) m+n−1 &U-Sa \©®
(C) m+n+1 &U-Sa \©® (D) m+n−1 &U-Sa \©-©Ø-Ó-x
In a non-degenerate solution, number of allocations is :
(a) Not equal to m+n+1 (b) Equal to m+n−1
(c) Equal to m+n+1 (d) Not equal to m+n−1

7 3317 (NS)

£Sv & II / PART - II
SÔ¨¦ : H÷u- Ý ® 7 ÂÚõUPÐUS Âøh¯ÎUPÄ®. ÂÚõ Gs 30 &US
Pmhõ¯©õP Âøh¯ÎUPÄ®. 7x2=14
Note : Answer any 7 questions. Question no. 30 is compulsory.
21. 11 ö£ß- ] À- P Ò ©Ø- Ö ® 3 AÈ- ¨ £õß- P - Î ß ö©õzu Âø» ` 64. ÷©¾®
8 ö£ß]ÀPÒ ©Ø- Ö ® 3 AȨ- £ õß- P - Î ß ö©õzu Âø» ` 49. Q÷µ- © - › ß
Âvø¯¨ £¯ß-£-kzv J¸ ö£ß-]À ©Ø-Ö® 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.

cos 2x + 2 sin 2 x
22. ©v¨-¤-kP : ∫ dx .
cos 2 x

cos 2x + 2 sin 2 x
Evaluate :
∫ cos 2 x
dx .

23. y=4−x2 GßÓ £µ-Á-øÍ-¯®, x Aa_, x=0 ©Ø-Ö® x=2 GßÓ ÷Põ-k-P-Ð-hß
HØ£-kz-x® Aµ[-Pz-vß £µ¨-ø£U Põs-P.
Find the area of the region bounded by the parabola y=4−x2, x-axis and the lines
x=0, x=2.

24. Bv ÁÈa- ö \À- ¾ ® AøÚzx ÷|º- ÷ Põm- k z öuõ- S ¨- ¤ ß ÁøPU- ö P- Ê a
\©ß£õmøh Aø©U-P.
Find the differential equation of the family of all straight lines passing through the
origin.

25. öuõ-hºa-]-¯õÚ \©-Áõ´¨¦ ©õ-Ô-°ß £s-¦-PÒ ¯õøÁ ?
What are the properties of continuous random variable ?

26. D¸- Ö ¨- ¦ ¨ £µ- Á - ¼ ß \µõ- \ › ©v¨¦ 20 GÚ- Ä ®, ©õ- Ö - £ m- h ÍøÁ ©v¨¦
16 GÚÄ® öPõs-hõÀ, “p” ©Ø-Ö® “n” &Cß ©v¨-¦-PøÍU Põs-P.
The mean of Binomial distribution is 20 and variance is 16. Find “p” and “n”.

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3317 (NS) 8

27. ¦Ò-Î-°-¯À AÝ-©õ-Úz-vß Cµsk £S-v-PøÍ GÊ-x-P.
Mention two branches of Statistical Inference.

28. öPõ-kU-P¨-£mh ¦ÒÎÂÁ-µ[-PøÍU öPõsk, £Sva \µõ-\› •øÓ-°À J¸
÷£õUS ÷Põmøh ö£õ-¸z-xP.

Bsk 2000 2001 2002 2003 2004 2005 2006

EØ-£z-v 105 115 120 100 110 125 135

Fit a trend line by the method of semi-averages for the given data.

Year 2000 2001 2002 2003 2004 2005 2006
Production 105 115 120 100 110 125 135

29. öPõ-kU-P¨-£m-kÒÍ AoU-PõÚ EP¢u ³-Pzøu
(i) «a-]-Ö-Âß «¨-ö£¸ ©Ø-Ö®
(ii) «¨-ö£-¸-Âß «a-]Ö BQ-¯-ÁØøÓ £¯ß-£-kzv Põs-P.

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

Given the following pay-off matrix (in rupees) for three strategies and two states of
nature.

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

9 3317 (NS)

30. ©Ø-Ö® (2, 20) GßÓ ¦Ò-Î-PÒ ÁÈa-ö\À-¾® y=ax2+bx+c GßÓ
(0, 0), (1, 1)
£µÁ-øͯ Aø©¨-¦-øh¯ \©ß-£õm-iøÚ, ö»U-µõg-]-°ß Cøha-ö\-¸-P-ø»¨
£¯ß-£-kz-vU Põs-P.
Find an equation of the parabolic form y=ax2+bx+c passing through (0, 0), (1, 1)
and (2, 20) using Lagranges Interpolation.

£Sv & III / PART - III
SÔ¨¦ : H÷uÝ® 7 ÂÚõUPÐUS Âøh¯ÎUPÄ®. ÂÚõ Gs 40 &US
Pmhõ¯® Âøh¯ÎUP ÷Ás-k®. 7x3=21
Note : Answer any 7 questions. Question no. 40 is compulsory.

1 1 −1   1 −2 5
31. 
A =  2 −3  T 
4  ©Ø-Ö® B = −2 4 1  GÛÀ
 3 −2 3   3 −6 −1 
  

AB &°ß uµz-v-øÚU PõsP.

1 1 −1   1 −2 5

If A = 2 −3  T 
4  and B = −2 4 1  , then find the rank of AB.

 3 −2 3   3 −6 −1 
  

1
32. ©v¨-¤-kP : ∫ dx .
x 2+6 x+13

1
Evaluate :
∫ x2+6x+13 dx .

33. ` 2,000 GßÓ öuõ-øPUS öuõ-hºa] Tm-k-Ámi PnU-Q-h¨-£-k-Q-Óx. Ám-i
- à - u ® Bs- ö hõß- Ö US 5% C¸¨- ¤ ß, Az- ö uõøP Gz- u øÚ Bs- k - P - Î À
Bµ®-£z öuõ-øP-ø¯¨ ÷£õÀ C¸-©-h[-Põ-S® ? (loge2=0.6931)
The sum of ` 2,000 is compounded continuously, the nominal rate of interest being
5% per annum. In how many years, will the amount be double the original principal ?
(loge2=0.6931)

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3317 (NS) 10

34. RÌU-Psh ÂÁ-µ[-P-øÍU 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

35. J¸ ¯õ-£õµ •¯Ø-]-°À J¸-Áº ` 2,000 C»õ-£® Dm-k-Á-uØ-PõÚ {PÌ-u-PÄ
0.4 AÀ- » x ` 1,000 CǨø£ ö£Ö- Á - u Ø- P õÚ {PÌ- u - P Ä 0.6 GÛÀ, AÁ- µ x
Gvº-£õºz-uÀ, ©õ-Ö-£õk ©Ø-Ö® vm-h-Â-»U-P® C»õ-£® GßÚ ?
In a business venture, a man can make a profit of ` 2,000 with a probability of 0.4 or
have a loss of ` 1,000 with a probability of 0.6. What is his expected, variance and
standard deviation of profit ?

36. C¯À-{ø» {PÌ-u-PÄ ÁøÍ-Á-øµ-°ß H÷u-Ý® ‰ßÖ •uß-ø©¨ £s-¦-PøÍ
GÊ-x-P.
Write down any three chief characteristics of normal probability curve.

37. J¸ £Pøh 9000 •øÓ Ã\¨-£-k®-÷£õx Auß ÷©À EÒÍ Gs-PÒ 3 AÀ-»x
4 BP 3240 •øÓ QøhU- Q ß- Ó Ú. ¤øÇ- ¯ ØÓ £P- ø h- ° ß vm- h ¨- ¤ øÇ
ÂQuzøuU P-nU-Q-k-P.
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.

38. J¸ SÔ¨-¤mh |P-µz-vÀ EÒÍ E¯º-{-ø»¨-£Ò-Î-°À £iU-S® ©õ-n-Áº-P-Îß
Gs- o U- ø P- ø ¯ |õßS Á¸- h õ¢- v µ |P- ¸ ® \µõ- \ - › - ø ¯¨ ¤ß- Á - ¸ ®
uµÄPμ¸¢x PnU-Q-kP.
Bsk 2001 2002 2003 2004 2005 2006 2007 2008 2009

©õ-n-Áº-P-Îß Gs-oU-øP 124 120 135 140 145 158 162 170 175
Calculate four-yearly moving averages of number of students studying in a higher
secondary school in a particular city from the following data.
Year 2001 2002 2003 2004 2005 2006 2007 2008 2009
Number of students 124 120 135 140 145 158 162 170 175

11 3317 (NS)

39. JxU-Rk PnU-Qß Po-u Ái-Á® u¸-P.
Give mathematical form of Assignment Problem.

40. ¦v-¯ EØ-£zv ö£õ-¸-Îß Âø»a-\õº¦ f (x)=(100+2x2)ex, GßP. C[S x Gߣx
\¢-øu-°À A¨-ö£õ-¸Ò QøhU-S® |õÒ-P-Îß Gs-oUøP GßP. •uÀ |õßS
|õm-P-ÎÀ A¢u ö£õ-¸-Îß ö©õzu ÂØ-£-øÚ-ø¯U Põs-P. (e−4=0.018)
The rate of new product is given by f (x)=(100+2x2)ex, where x is the number of days
the product is on the market. Find the total sale during the first four days. (e−4=0.018)

£Sv & IV / PART - IV

SÔ¨¦ : AøÚzx ÂÚõUPÐUS® Âøh¯ÎUPÄ®. 7x5=35

Note : Answer all the questions.

41. (A) öPõ-kU-P¨-£mh CÖ-v-{-ø»a ö\»-Ä ©Ø-Ö® Á¸-Áõ´ \õº-¦-PÒ •øÓ-÷¯
x
C
(x ) = 50 + ©Ø-Ö® R9(x)=60 ©õ-Óõa ö\»-Ä ` 200 GÛÀ, «¨-ö£¸
50
C»õ-£z-øuU Põs-P.

AÀ»x

(B) ¤ß- Á - ¸ ® ÂÁ- µ [- P - Ð US, L¤- å º Âø»U SÔ- ± mk Gs- ø nU
Pm- h - ø ©UP- Ä ®. ÷©¾® Ax Põ- » - © õØ- Ö a ÷\õ- u øÚ, Põ- µ o ©õØ- Ö a
÷\õ-uøÚ BQ-¯-ÁØ-øÓ¨ §ºzv ö\´-²® GÚ {¹-¤U-P-Ä®.

³Ûm JßÖUS Âø» (`) A»SPÎß GsoUøP
ö£õ¸ÒPÒ
Ai¨£øh Bsk |h¨¦ Bsk Ai¨£øh Bsk |h¨¦ Bsk
A 6 10 50 56
B 2 2 100 120
C 4 6 60 60
D 10 12 50 24
E 8 12 40 36

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3317 (NS) 12

x
(a) The marginal cost C9(x) and marginal revenue R9(x) are given by C
(x ) = 50 +
50
and R9(x)=60. The fixed cost is ` 200. Determine the maximum profit.

OR

(b) Using the following data, construct Fisher’s Ideal Index and show how it satisfies
Factor Reversal Test and Time Reversal Test.

Price in Rupees per unit Number of units
Commodity
Base Year Current Year Base Year Current Year
A 6 10 50 56
B 2 2 100 120
C 4 6 60 60
D 10 12 50 24
E 8 12 40 36

42. (A) J¸ ÷uº- Â À 500 ©õ- n - Á º- P - Î ß \µõ- \ - › - ©- v ¨- ö £s 40 ©Ø- Ö ® vmh
»UP® 25 GÛÀ ©z-v¯ 60% ©õ-n-Áº-PÒ ö£Ö® ©v¨-ö£s-P-Îß
GÀ-ø»-P-øÍU Põs-P. [P(0 < z < 0.84)=0.30]
AÀ»x
(B) ¤ß-Á-¸®- Â-Á-µ[-P-Î-¼-¸¢-x f 9(x) &ß ©v¨ø£, {³m-h-Ûß •ß-÷|õUS
Cøha-ö\-¸-P-¼ß `z-v-µz-øu¨ £¯ß-£-kz-vU PõsP.

x 0 1 2 3
f (x) 2 4 8 20

(a) The mean score of 500 students for an examination is 40 and S.D is 25. Determine
the limit of the marks of the central 60% of the candidates. [P(0 < z < 0.84)=0.30]

OR

(b) Using Newton’s forward interpolation formula, find f 9(x) from the following table.

x 0 1 2 3
f (x) 2 4 8 20

13 3317 (NS)

43. (A) J¸ SÔ¨-¤mh Ak-©-øÚ-°À J¸ |õ-ÎÀ ÂØÖ •i¢u öµõmi x &ß
AÍ-Ä-PÒ (¡-Ö £Äs-k-P-ÎÀ) J¸ Gs \õº¢u \©-Áõ´¨¦ {PÌ-Áõ-PU
Ps-h-Ô-¯¨-£m-hx. Auß {PÌ-u-P-Áõ-Ú-x, f (x) GßÓ {PÌ-u-PÄ Ahºz-va
\õº-¤ß ‰»® öPõ-kU-P¨-£m-kÒÍx GÛÀ,
 Ax , 0 ≤ x < 10

f (x ) = A(20−x ), 10 ≤ x < 20
 0, ©ØöÓ[Q¾®

(i) A &Cß ©v¨-ø£U Põs-P.
(ii)©Ö-|õ-ÎÀ ÂØ-P¨-£-h-Â-¸U-S® öµõm-i-P-Îß Gs-oU-øP-°À
(A) 10 £Äs-k-P-ÐUSU Av-P-©õ-P
(B) 10 £Äs-k-P-ÐU-SU SøÓ-Áõ-P
(C) 5 ©Ø- Ö ® 15 £Äs- k - P - Ð US Cøh- ° À C¸¨- £ - u Ø- P õÚ
{PÌuPÂøÚU Põs-P.
AÀ»x
(B) ‘a’ ©Ø- Ö ® ‘b’ Cß G®- © - v ¨- ¦ - P - Ð US x+y+z=6, x+2y+3z=10,
x+2y+az=b GßÓ \©ß-£õ-k-PÒ :
(i) G¢u wº-Ä® ö£Ø-Ô-µõ-x
(ii) J÷µ J¸ wºøÁ ö£Ø-Ô-¸U-S®
(iii) Gs-oU-øP-¯ØÓ wº-Ä-Pøͨ ö£Ø-Ô-¸U-S® GÚ Bµõ´-P.
(a) The amount of bread (in hundreds of pounds) x that a certain bakery is able to
sell in a day is found to be a numerical valued random phenomenon, with a
probability function specified by the probability density function f (x) is given by,
 Ax , for 0 ≤ x < 10

f (x ) = A(20−x ), for 10 ≤ x < 20
 0, otherwise

(i) Find the value of A.
(ii) What is the probability that the number of pounds of bread that will be sold
tomorrow is :
(a) More than 10 pounds,
(b) Less than 10 pounds, and
(c) Between 5 and 15 pounds ?
OR
(b) Investigate for what values of ‘a’ and ‘b’ the following system of equations
x+y+z=6, x+2y+3z=10, x+2y+az=b have,
(i) no solution,
(ii) a unique solution,
(iii) an infinite number of solutions.

[ v¸¨¦P / Turn over

3317 (NS) 14

∫ 2x 4 − 3x2 − 2 dx .
44. (A) ©v¨-¤-kP : x

AÀ»x
(B) x Põ-»o
- P
- Ò u¯õ-›¨-£u
- Ø-PõÚ CÖ-v{
- ø
- »a ö\»Ä (3xy+y2) dx+(x2+xy) dy=0
©Ø-Ö® J¸ ÷áõi Põ-»-o-PÒ u¯õ-›¨-£-uØ-PõÚ ö©õzu ö\»Ä ` 12 GÛÀ,
ö©õzu ö\»-Äa \õº-ø£U Põs-P.

∫ 2x 4 − 3x2 − 2 dx .
x
(a) Evaluate :

OR
(b) If the marginal cost of producing x shoes is given by (3xy+y2) dx+(x2+xy) dy=0
and the total cost of producing a pair of shoes is given by ` 12, then find the total
cost function.

45. (A) -öPõ-kU-P¨-£m-kÒÍ ÷£õU-S-Á-µzx PnU-Qß Bµ®£ Ai¨-£-øhz wºøÁ
RÌU-Psh •øÓ-P-ÎÀ Põs-P.

I II III IV Aئ
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 21 25 17 17

(i) Áh ÷©ØS ‰ø» •øÓ
(ii) «a-]Ö ö\»Ä •øÓ
(iii) ÷Áõ-P-¼ß ÷uõ-µõ¯ •øÓ
AÀ»x
(B) - C ¯À- { ø» £µ- Á - ¼ À EÒÍ J¸ öuõ- È Ø- \ õø» FÈ- ¯ º- P - Î ß
Fv¯[PÎß ©õ-Ö-£m-hÍøÁ 25 GßP. 50 £-o-¯õͺ-PÒ öPõsh J¸
TÔÀ EÒÍ-Áº-P-Îß ö©õzu Fv-¯® ` 2,550 GßP. P¸-x-÷PõÒ, µ=52
Gß-£-øu-²® AuØS ©õ-ÓõÚ P¸-x-÷PõÒ µ=49 &ø¯-²® 1% ªøP-Põs
{ø»-°À ÷\õ-uøÚ ö\´-P.

15 3317 (NS)

(a) Find the initial basic feasible solution of the following transportation problem.
I II III IV Supply
A 5 1 3 3 34
B 3 3 5 4 15
C 6 4 4 3 12
D 4 1 4 5 19
Demand 21 25 17 17
Using (i) North west corner rule
(ii) Least cost method
(iii) Vogel’s approximation method
OR
(b) Wages of the factory workers are assumed to be normally distributed with variance
25. A random sample of 50 workers gives the total wages equal to
` 2,550. Test the hypothesis µ=52, against the alternative hypothesis µ=49 at
1% level of significance.

46. (A) Áøµ- ¯ - Ö zu öuõ- ø P- ± møh J¸ Tm- h - ¼ ß GÀø» GÚU öPõsk
2
3
∫ (2x −4)dx.
1
AÀ»x
(B) ªß ÂÍU-S-PÒ EØ-£zv ö\´-¯¨-£-k® ö\¯À-£õm-iÀ J¸ ©oUS J¸
ÂÍUS Ãu® 6 ªß ÂÍU-S-PÒ GkU-P¨-£-k-Qß-ÓÚ. C÷u-÷£õÀ 6 TÖ-PÒ
GkU- P ¨- £ - k - Q ß- Ó Ú. ¤ß- Á - ¸ ® ÂÁ- µ [- P Ò AÁØ- Ô ß £¯ß- £ õm- k U
−
Põ-»zøu (©-o-°À) SÔU-Qß-ÓÚ G-ÛÀ X ©Ø-Ö® R £h[-PÒ Áøµ¢x
Av-¼-¸¢x Eß •i-Ä-PøÍU SÔ¨-¤-k-P.
TÖ Gs -£-¯ß-£õmk Põ-»® (©-o-°À)
1 620 687 666 689 738 686
2 501 585 524 585 653 668
3 673 701 686 567 619 660
4 646 626 572 628 631 743
5 494 684 659 643 660 640
6 634 755 625 582 683 555
(n=6 GÛÀ, A2=0.483, D3=0, D4=2.004 GÚ öPõ-kU-P¨-£m-kÒÍ-x)
2
Evaluate the integral as the limit of a sum (2 x 3 −4)dx.
(a)
∫
1
OR

[ v¸¨¦P / Turn over

3317 (NS) 16

(b) The following data relate to the life (in hours) of 6 electric bulbs each drawn at an
−
interval of one hour from a production process. Draw the control chart for X
and R, and comment.
Sample No. Life time (in hours)
1 620 687 666 689 738 686
2 501 585 524 585 653 668
3 673 701 686 567 619 660
4 646 626 572 628 631 743
5 494 684 659 643 660 640
6 634 755 625 582 683 555
(Given for n=6, A2=0.483, D3=0, D4=2.004)

dp d2 p
47. (A) Qd = 29 − 2p − 5 + 2 ©Ø- Ö ® Q s=5+4p Gß- £ Ú •øÓ÷¯ J¸
dt dt
ö£õ¸- Î ß ÷uøÁ AÍÄ ©Ø- Ö ® AΨ¦ AÍÄ BQ- ¯ - Á Ø- ø ÓU
SÔU-Qß-ÓÚ. C[S ‘p’ Âø»-ø¯U SÔU-Q-Óx. \¢øu £›-©õØ-Óz-vÀ
\©ß-{ø» Âø»-ø¯U Põs-P.
AÀ»x
(B) J¸- {-Ö-Á-Úz-vØ-S Põø» 10.00 ©o-°À C¸¢x ©v-¯® 2.30 ©o Áøµ
Á¸® öuõ- ø »- ÷ £] AøǨ- ¦ - P - Î ß Gs- o UøP \µõ- \ - › - ¯ õP J¸
{ªhzvØS 2.5 BS®. J¸ SÔ¨-¤mh {ª-hz-vÀ
(i) AøǨ-¦-PÒ -CÀ-ø»
(ii) \›-¯õP 3 AøǨ-¦-PÒ ©m-k®
(iii) SøÓ¢-u-£m-\® 5 AøǨ¦PÒ Á¸-Á-uØ-PõÚ {PÌ-u-P-ÂøÚ PõsP.
(e−2.5=0.08208)

dp d2 p
(a) Suppose that the quantity demanded Qd = 29 − 2p − 5 + 2 and
dt dt
quantity supplied Qs=5+4p, where ‘p’ is the price. Find the equilibrium price
for market clearance.
OR
(b) The average number of phone calls per minute into the switch board of a company
between 10.00 a.m. and 2.30 p.m. is 2.5. Find the probability that during one
particular minute there will be,
(i) no phone at all,
(ii) exactly 3 calls,
(iii) at least 5 calls.
(e−2.5=0.08208)

-oOo-