TN 12th Model Question Paper Business Maths

No. of Printed Pages : 15
1367 (NP)
!1367NPBusinessMathematics! £vÄ Gs
Register Number

PART - III
ÁoPU Pou®/BUSINESS MATHEMATICS
( uªÌ ©ØÖ® B[Q» ÁÈ / Tamil & English Version )

Põ» AÍÄ : 2.30 ©o ÷|µ® ] [ ö©õzu ©v¨ö£sPÒ : 90
Time Allowed : 2.30 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®,
Ai÷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

1367 (NP) 2

A B
A  0.7 0.3 
1. T= Gߣx ©õÖuÀ {PÌuPÄ Ao GÛÀ, x=
B  x 0.8 

(A) 0.8 (B) 0.7 (C) 0.3 (D) 0.2
A B
A  0.7 0.3 
If T = is a transition probability matrix, then the value of x is :
B  x 0.8 
(a) 0.8 (b) 0.7 (c) 0.3 (d) 0.2

2. ©v¨¤h ÷Ási¯ ‰ßÖ ©õÔPÎÀ Aø©¢u ‰ßÖ ÷|›¯ A\©£izuõÚ
\©ß£õmkz öuõS¨¤À ρ(A, B)=ρ(A)<3 GÚ C¸¨¤ß, \©ß£õkPÒ
__________ .
(A) J¨¦ø©zußø©²hß J÷µ J¸ wºÄ öPõsi¸US®
(B) J¨¦ø©zußø©²hß GsnØÓ wºÄPÒ ö£ØÔ¸US®
(C) J¨¦ø©zußø© öPõshÚ
(D) J¨¦ø©zußø© AØÓøÁ
In a system of 3 linear non-homogeneous equations with 3 unknowns, if
ρ(A, B)=ρ(A) < 3 then the equations are :
(a) Consistent and it has unique solution
(b) Consistent and it has infinitely many solutions
(c) Consistent
(d) Inconsistent

3. y2=−4ax &Cß S¯® :
(A) (0, −a) (B) (−a, 0) (C) (a, 0) (D) (0, a)
Focus of y2=−4ax is :
(a) (0, −a) (b) (−a, 0) (c) (a, 0) (d) (0, a)

4. ö\ÆÁP»zvß }Í®, xøn¯a]ß }ÍzvÀ |õßQÀ J¸ £[S GÚU
öPõskÒÍ Av£µÁøͯzvß ø©¯z öuõø»z uPÄ.
15 17 5 17
(A) (B) (C) (D)
4 4 2 2
The eccentricity of the hyperbola whose latus rectum is equal to one fourth of its conjugate
axis is :
15 17 5 17
(a) (b) (c) (d)
4 4 2 2

3 1367 (NP)

5. y=3x+2 GßÓ \õº¦US x&BÚx 1.5 &¼¸¢x 1.6 &US AvP›US® ÷£õx y&ß
\µõ\› ©õÖ Ãu©õÚx :
(A) 0.6 (B) 3 (C) 1 (D) 0.5
For the function y=3x+2 the average rate of change of y when x increases from
1.5 to 1.6 is :
(a) 0.6 (b) 3 (c) 1 (d) 0.5

6. y=2x 2 −x+1 GßÓ ÁøÍÁøµUS (1, 2) GßÓ ¦ÒΰÀ Áøµ¯¨£mh
öuõk÷Põk, G¢u ÷PõmiØS Cøn¯õP C¸US® ?
(A) 2x+y+7=0 (B) y=5x−7 (C) y=3x (D) y=2x+4
The tangent to the curve y=2x2−x+1 at (1, 2) is parallel to the line :
(a) 2x+y+7=0 (b) y=5x−7 (c) y=3x (d) y=2x+4

∂u
7. u=xy (x>0) GÛÀ ∂y BÚx :
(A) yx log x (B) log yx (C) xy log x (D) log x
∂u
If u=xy (x>0) then ∂y is equal to :

(a) yx log x (b) log yx (c) xy log x (d) log x

8. y=4−2x−x2 GÝ® ÁøÍÁøµ¯õÚx :
(A) ÷©À÷|õUQ SÈÁõÚx (B) RÌ÷|õUQ SÈÁõÚx
(C) J¸ ÷|º÷Põk (D) CÁØÔÀ HxªÀø»
The curve y=4−2x−x2 is :
(a) Concave upward (b) Concave downward
(c) Straight line (d) None of these

π 5
2
cos 3 x
9. ∫ 5 5
dx =
0 cos 3 x + sin 3 x

π π
(A) 0 (B) π (C) 2
(D) 4
π 5
2
cos 3 x
∫ 5 5
dx is :
0 cos 3 x + sin 3 x

π π
(a) 0 (b) π (c) (d)
2 4

[ v¸¨¦P / Turn over

1367 (NP) 4

1
10. CÖv{ø» Á¸Áõ´ \õº¦ R9(x)= GÛÀ Á¸Áõ´a \õº¦ :
x+1

1 −1
1
(A) ( x+1)2 (B) log (C) log ?x+1?+k (D) x+1
x+1

1
The marginal revenue R9(x)= then the revenue function :
x+1

1 1 −1
(a) 2 (b) log (c) log ?x+1?+k (d)
( x+1) x+1 x+1

11. (D2+D+1)y=0 GßÓ \©ß£õmiß wºÄ :

3
− x  x x
(A) (Ax+B) e−x/2 (B) e 2  A cos + B sin 
 2 2

1
− x  3 3 
(C) e 2  A cos x + B sin x (D) Ae−x+Be−2x
 2 2 
The solution of (D2+D+1)y=0 is :

3
− x  x x
(a) (Ax+B) e−x/2 (b) e 2  A cos + B sin 
 2 2

1
− x  3 3 
(c) e 2  A cos x + B sin x  (d) Ae−x+Be−2x
 2 2 

d2 y dy
12. 2
−6 =0 GßÓ ÁøPUöPÊ \©ß£õmiß Á›ø\ ©ØÖ® £i
dx dx
•øÓ÷¯ :
(A) 2 ©ØÖ® 2 (B) 1 ©ØÖ® 1 (C) 2 ©ØÖ® 1 (D) 1 ©ØÖ® 2
d2 y dy
The order and degree of the differential equation 2
−6 = 0 are :
dx dx
(a) 2 and 2 (b) 1 and 1 (c) 2 and 1 (d) 1 and 2

5 1367 (NP)

13. ªP¨ ö£õ¸zu©õÚ ÷|ºU÷PõhõÚ y=5.8(x−1994)+41.6 &À x=1997 GÛÀ,
y&ß ©v¨¦ :
(A) 59 (B) 60 (C) 50 (D) 54
In a line of best fit y=5.8(x−1994)+41.6 the value of y when x=1997 is :
(a) 59 (b) 60 (c) 50 (d) 54

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

15. J¸ £õ´\õß ©õÔ°ß vmh»UP® 2 GÛÀ, Auß \µõ\› :
1
(A) 2 (B) 2
(C) 2 (D) 4
If the standard deviation of a Poisson variate is 2, then its mean is :

1
(a) 2 (b) (c) 2 (d) 4
2

16. J¸ uÛzu \©Áõ´¨¦ ©õÔ°ß {PÌuPÄ vs©a \õº¦
x 0 1 2 3
p(x ) k 2k 3k 5k
GÛÀ, k &ß ©v¨¦ :

3 4 1 2
(A) 11 (B) 11 (C) 11 (D) 11
If a discrete random variable has the probability mass function

x 0 1 2 3
then the value of k is :
p(x ) k 2k 3k 5k

3 4 1 2
(a) (b) (c) (d)
11 11 11 11

[ v¸¨¦P / Turn over

1367 (NP) 6

17. •Êø©z öuõSv AÍøÁø¯ ©v¨¥k ö\´²® ö£õÊx 80% |®£P
CøhöÁÎø¯¨ ö£Ó £¯ß£kzu¨£k® Z &ß ©v¨¦ :
(A) 1.33 (B) 2.58 (C) 1.28 (D) 2.33
The Z-value that is used to establish a 80% confidence interval for the estimation of a
population parameter is :

(a) 1.33 (b) 2.58 (c) 1.28 (d) 2.33

18. 10 ~Pº÷ÁõºPμ¸¢x 2 ~Pº÷ÁõºPøÍz öu›Ä ö\´²® ÁÈPÎß
GsoUøP :
(A) 45 (B) 50 (C) 90 (D) 60
The number of ways in which one can select 2 customers out of 10 customers is :

(a) 45 (b) 50 (c) 90 (d) 60

19. ^µØÓ ©õÖ£õmøh __________ GÚ AøÇUP»õ®.
(A) Gvº£õµõ ©õÖ£õkPÒ (B) }ÒPõ»¨ ÷£õUS
(C) _ÇÀ ©õÖ£õkPÒ (D) £¸ÁPõ» ©õÖ£õkPÒ
Irregular variation is also known as :

(a) Erratic variation (b) Secular trend

(c) Cyclic variation (d) Seasonal variation

20. ö£¸®£õßø©¯õP £¯ß£kzu¨£k® SÔ±mk GsPÒ :
(A) £µÁÀ SÔ±mk Gs (B) Âø» SÔ±mk Gs
(C) ©v¨¦ SÔ±mk Gs (D) CÁØÔÀ HxªÀø»
The most commonly used index numbers are :

(a) Diffusion index number (b) Price index number

(c) Value index number (d) None of these

7 1367 (NP)

£Sv & II / PART - II
H÷uÝ® 7 ÂÚõUPÐUS Âøh¯ÎUPÄ®. AÁØÔÀ ÂÚõ Gs 30 &US
Psi¨£õP Âøh¯ÎUPÄ®. 7x2=14
Answer any 7 questions in which question no. 30 is compulsory.

 2 4 5 

A=  4 8 10  GßÓ Ao°ß uµ® PõsP.
21.
−6 −12 −15 
 

 2 4 5 
 10  .
Find the rank of the matrix A =  4 8
−6 −12 −15 
 

22. 9x 2 −16y 2 =144 GßÓ Av£µÁøͯzvß SÖUPa_ ©ØÖ® xøn¯a]ß
}Í[PøÍU PõsP.
Find the length of transverse and conjugate axes of the hyperbola 9x2−16y2=144.

23. y=3x 2−4x+5 GßÓ ÁøÍÁøµUS® x Aa_ ©ØÖ® ÷|º÷Põk x=1, x=2
CÁØÔØQøh÷¯ Aø©²® £Sv°ß £µ¨ø£U PõsP.
Find the area of the region bounded by the curve y=3x2−4x+5, the x-axis and the
lines x=1 and x=2.

∂u ∂u ∂u
24. u=x3+y3+z3−3xyz GÛÀ x +y +z = 3u GÚ {ÖÄP.
∂x ∂y ∂z

∂u ∂u ∂u
If u=x3+y3+z3−3xyz, then prove that x +y +z = 3u .
∂x ∂y ∂z

25. ©ØÖ® b &PøÍ H÷uÝ® ©õÔ¼PÍõPU öPõsh
y=a cos(mx+b), a
ÁøÍÁøµU Sk®£zvß ÁøPUöPÊa \©ß£õmøhU PõsP.
Find the differential equation of a family of curves given by y=a cos(mx+b), a and b
being arbitrary constants.

26. Σx=10, Σy=25, Σx2=30, Σxy=90 ©ØÖ® n=5 GÛÀ, J¸ ªP¨ö£õ¸zu©õÚ
÷|ºU÷PõmiøÚ¨ ö£õ¸zv Auß \õ´Ä ©ØÖ® y Aa]ß öÁmkzxsk
BQ¯ÁØøÓU PõsP.
In a line of best fit, find the slope and the y intercept if Σx=10, Σy=25, Σx2=30,
Σxy=90 and n=5.

[ v¸¨¦P / Turn over

1367 (NP) 8

27. J¸ D¸Ö¨¦¨ £µÁ¼À P(X=3)=P(X=4) ©ØÖ® n=5 GÛÀ p ©ØÖ® q&Âß
©v¨¦PøÍU PõsP.
In a Binomial distribution, if P(X=3)=P(X=4) and n=5 then find p and q.

28. C¯¢vµ® JßÔÚõÀ J¸ ÁõµzvÀ u¯õ›UP¨£mh C¸®¦
E¸søhPμ¸¢x 200 E¸søhPøÍU öPõsh J¸ TÖ
GkUP¨£kQÓx. AUTÔ¾ÒÍ E¸søhPÎß \µõ\› Gøh 0.824 {³mhß
©ØÖ® vmh»UP® 0.042 {³mhßPÒ GÛÀ 95% {ø»°À, E¸søhPÎß
\µõ\› GøhUS |®¤UøP GÀø»PøÍU PõsP.
Measurements of the weights of a random sample of 200 ball bearings made by a certain
machine during one week showed mean of 0.824 Newtons and a standard deviation
of 0.042 Newtons. Find the 95% confidence limits for the mean weight of all the ball
bearings.

29. £Sva\µõ\› •øÓ°ß ‰»® ÷£õUS ©v¨¦PøÍU PõsP.

Bsk 1987 1988 1989 1990 1991 1992 1993

EØ£zv (hßÛÀ) 90 110 130 150 100 150 200

Obtain the trend values by the method of Semi-Average.
Year 1987 1988 1989 1990 1991 1992 1993
Production (In tonnes) 90 110 130 150 100 150 200

30. y2=20x GßÓ £µÁøͯzvß J¸ öuõk÷Põk x Aa_hß 608 ÷Põnzøu
HØ£kzxQÓx GÛÀ, Auß \©ß£õk PõsP.
Find the equation of the tangent to the parabola, y2=20x which forms an angle 608
with the x-axis.

£Sv & III / PART - III
H÷uÝ® HÊ ÂÚõUPÐUS Âøh¯ÎUPÄ®. AÁØÔÀ ÂÚõ Gs 40 &US
Pmhõ¯® Âøh¯ÎUP ÷Ásk®. 7x3=21
Answer any 7 questions in which question no. 40 is compulsory.

31. x+2y+kz=0, 3x+4y+6z=0, x+y+z=0 GßÓ \©ß£õkPÐUS §a]¯z
wºÄPÒ ©mk® EsöhÛÀ k &ß ©v¨ø£U PõsP.
Find k if the equations x+2y+kz=0, 3x+4y+6z=0, x+y+z=0 have only the trivial
solution.

9 1367 (NP)

32. Kº E÷»õPzøu u¯õ›US® {ÖÁÚzvß ©õuõ¢vµ EØ£zv x Q÷»õ
Qµõ®PÎß \µõ\› Âø» y &I `  
1 2
x − 3x + 50  Gߣx öPõkUQÓx. \µõ\›
 10 
Âø»°ß ÁøÍÁøµ J¸ £µÁøͯ® GÚU PõmkP. ÁøÍÁøµ°ß
•øÚ°À \µõ\› Âø» ©ØÖ® EØ£zvø¯U PõsP.
The average cost y of a monthly output x kgs. of a firm producing a metal is
 1 2 
`  x − 3x + 50  . Show that the average variable cost curve is parabola. Find the
 10 
output and average cost at the vertex of the parabola.

500
33. y=
x
&CÀ x &I ö£õÖzx x &BÚx 20 &¼¸¢x 20.5 &US Tk®ö£õÊx,
y &ß \µõ\› ©õÖÃu® PõsP. ÷©¾® x=20 &US, y &ß EhÚi ©õÖÃu®
¯õx ?
500
If y = find the average rate of change of y with respect to x, when x increases
x
from 20 to 20.5 units. Find also the instantaneous rate of change of y at x=20.

34. f (x)=x3−27x+108 GßÓ \õº¤ß ö£¸© ©v¨£õÚx Auß ]Ö© ©v¨ø£ Âh
108 Tku»õP EÒÍx GÚ {ÖÄP.
Show that the maximum value of the function f (x)=x3−27x+108 is 108 more than
the minimum value.

35. J¸ {ÖÁÚzvÀ J¸ ö£õ¸Îß CÖv{ø» ö\»Äa \õº¦ ©ØÖ® CÖv{ø»
Á¸Áõ´ •øÓ÷¯ C9(x)=4+0.08x ©ØÖ® R9(x)=12 EØ£zv Hx® CÀ»õuuõÀ
ö©õzu ö\»Ä §a]¯® GÛÀ ö©õzu C»õ£® PõsP.
The marginal cost and marginal revenue with respect to a commodity of a firm are
given by C9(x)=4+0.08x and R9(x)=12. Find the total profit, given that the total cost
at zero output is zero.

36. ¤ßÁ¸® AmhÁønø¯U öPõsk C»Uµõg]°ß `zvµzøu¨
£¯ß£kzv y (11) &ß ©v¨ø£U PõsP.
x 6 7 10 12
y 13 14 15 17
Using Lagrange’s formula find y (11) from the following table.

x 6 7 10 12
y 13 14 15 17

[ v¸¨¦P / Turn over

1367 (NP) 10

37. J¸ öuõhº \©Áõ´¨¦ ©õÔ X RÌUPsh {PÌuPÄ Ahºzva \õºø£¨
ö£ØÖÒÍx.
 1 , −1 < x < 1
f (x) =  2
0 , ©ØÓ£i
 GÛÀ
(i) E(X)
(ii) E(X2)
(iii) Var(X) BQ¯ÁØÔß ©v¨¦PøÍU PõsP.
 1 for −1 < x < 1
Let X be a continuous random variable with p.d.f. f ( x ) =  2

0 otherwise
Find (i) E(X)
(ii) E(X2)
(iii) Var(X)
38. öuõø»UPõm] {PÌa]Pøͨ £õº¨÷£õºPÎÀ 1000 ÷£›À, 320 ÷£ºPÒ J¸
SÔ¨¤mh {PÌa]ø¯¨ £õºzuÚº. öuõø»UPõm] {PÌa]PøÍU Põs÷£õº
AøÚÁøµ²® öPõsh •Êø©z öuõSv°¼¸¢x A¢u {PÌa]ø¯¨
£õºzuÁºPÎß GsoUøPUPõÚ 95% |®¤UøP GÀø»PøÍU PõsP.
Out of 1000 TV viewers, 320 watched a particular programme. Find 95% confidence
limits for TV viewers who watched this programme.
39. ¤ßÁ¸® ÂÁµ[PÐUS Âø»U SÔ±mk Gsøn »õ줯º •øÓ°À
PõsP.
ö£õ¸Ò 1999 1998

Âø» AÍÄ Âø» AÍÄ
A 4 6 2 8
B 6 5 5 10
C 5 10 4 14
D 2 13 2 19
Construct the price index number from the following data by applying Laspeyre’s
method.
Commodity 1999 1998
Price Quantity Price Quantity
A 4 6 2 8
B 6 5 5 10
C 5 10 4 14
D 2 13 2 19

11 1367 (NP)

dy y y2
40. wºUP : dx x
+ =
x2

dy y y2
Solve : + = 2 .
dx x x

£Sv & IV / PART - IV
AøÚzx ÂÚõUPÐUS® Âøh¯ÎUPÄ®. 7x5=35
Answer all the questions.

41. (A) uØ÷£õx P ©ØÖ® Q GßÓ C¸ ÂØ£øÚ¨ ö£õ¸ÒPÎß \¢øu ÂØ£øÚ
•øÓ÷¯ 70% ©ØÖ® 30% BP EÒÍx. JÆöÁõ¸ Áõµ•® ]»
~Pº÷Áõ›ß ¸¨£[PÒ ©õÖQßÓÚ. ö\ßÓ Áõµ® P &Áõ[Q¯ÁºPÎÀ
80% ÷£º «sk® Aøu Áõ[SQßÓÚº. 20% ÷£º Q &US ©õÔ
ÂkQßÓÚº. ö\ßÓ Áõµ® Q Áõ[Q¯ÁºPÎÀ 40% ÷£º «sk® Aøu
Áõ[SQßÓÚº. 60% ÷£º P &US ©õÔ ÂkQßÓÚº. Cµsk
Áõµ[PÐUS¨ ¤ÓS AÁºPÎß \¢øu¨ £[RkPøÍU PõsP. C¢u
÷£õUS öuõh¸©õÚõÀ G¨÷£õx \©{ø» Gmh¨£k® ?
AÀ»x
(B) Põº £zvÀ J¸Áº u® C¸ PsPøͲ® CÇ¢uõÀ CǨ¤øÚ Dk
ö\´ÁuØPõP J¸ Põ¨¥mk {ÖÁÚ® 4,000 |£ºPÐUS Põ¨¥k
ö\´xÒÍx. •¢øu¯ ÂÁµ[PÎߣi, 1,00,000 ÷£›À \µõ\› 10 ÷£º
JÆöÁõ¸ Á¸h•® Põº £zvÀ C®©õv› FÚ® HØ£kÁuõP
øÁzxU öPõsk Põ¨¥mk öuõøP Ãu[PÒ PnURk
ö\´¯¨£mkÒÍx. 3 ÷£¸US ÷©À FÚ•ØÓÁºPÒ J¸ Á¸hzvÀ
Põ¨¥mk öuõøPø¯ ö£ÖÁuØPõÚ {PÌuPÄ ¯õx ? (e−0.4=0.6703)
(a) Two products P and Q share the market currently with shares 70% and 30%
each respectively. Each week some brand switching takes place. Of those who
bought P the previous week, 80% buy it again whereas 20% switch over to Q. Of
those who bought Q the previous week, 40% buy it again whereas 60% switch
over to P. Find their shares after two weeks. If the price war continues, when
will the equilibrium be reached ?
OR
(b) An insurance company insures 4,000 people against loss of both eyes in car
accidents. Based on previous data, the rates were computed on the assumption
that on the average 10 persons in 1,00,000 will have car accidents each year that
result in this type of injury. What is the probability that more than 3 of the
injured will collect on their policy in a given year ? (e−0.4=0.6703)

[ v¸¨¦P / Turn over

1367 (NP) 12

42. (A) JÆöÁõßÖ® AÍÄ 5 EÒÍ £zx TÖPÎß \µõ\› X ©ØÖ® Ãa_PÒ R
£ØÔ¯ ÂÁµ[PÒ R÷Ç öPõkUP¨£mkÒÍÚ. \µõ\› ©ØÖ® Ãa_
£h[PÐUPõÚ ©zv¯U÷Põk ©ØÖ® Pmk¨£õmkU ÷PõkPÎß
GÀø»PøÍU Psk ö\¯À£õk Pmk¨£õmiÀ EÒÍuõ GßÖ
Psk¤iUPÄ®.

TÖPÒ : 1 2 3 4 5 6 7 8 9 10
\µõ\› : ( X ) 11.2 11.8 10.8 11.6 11.0 9.6 10.4 9.6 10.6 10.0

Ãa_ (R) 7 4 8 5 7 4 8 4 7 9

(n=5 GÛÀ, A2=0.577, D3=0, D4=2.115 GÚ öPõkUP¨£mkÒÍx)
AÀ»x
(B) A ©ØÖ® B GßÓ C¸ Ch[PÐUS Cøh¨£mh öuõø»Ä 100 Q.«.
A &CÀ J¸ ö£õ¸Îß Kµ»S EØ£zva ö\»Ä B &CÀ A÷u ö£õ¸Îß
EØ£zva ö\»øÁ Âh ` 12 SøÓÁõP EÒÍx. EØ£zv ö\´¯¨£mh
ö£õ¸ÒPÒ ÷|º÷Põmk¨ £õøu°À Aݨ£¨£mk AÎUP¨£kQßÓÚ
GßÖ®, Aݨ¦® ö\»Ä J¸ A»SUS J¸ Q÷»õ «mh¸US 20 ø£\õ
GßÖ® öPõÒP. G¢öu¢u Ch[PÐUS A &CÀ C¸¢x Aݨ¤
øÁzuõ¾®, B &CÀ C¸¢x Aݨ¤ øÁzuõ¾® ö©õzu ö\»Ä \©©õP
C¸US÷©õ AÆÂh[PøÍU SÔUS® ¦ÒÎPÒ Aø©²®
ÁøÍÁøµø¯U PõsP.
(a) The following data shows the value of sample mean X and the range R for ten
samples of size 5 each. Calculate the values for central line and control limits for
mean chart and range chart and determine whether the process is in control.

Sample No. 1 2 3 4 5 6 7 8 9 10
Mean ( X ) 11.2 11.8 10.8 11.6 11.0 9.6 10.4 9.6 10.6 10.0
Range (R) 7 4 8 5 7 4 8 4 7 9

[Given for n=5, A2=0.577, D3=0, D4=2.115]
OR
(b) The cost of production of a commodity is ` 12 less per unit at a place A than it is
at a place B and the distance between A and B is 100 km. Assuming that the
route of delivery of the commodity is along a straight line and that the delivery
cost is 20 paise per unit per km, find the curve, at any point of which the
commodity can be supplied from either A or B at the same total cost.

13 1367 (NP)

43. (A) x 2 +4y 2 −8x−16y−68=0 GÝ® T®¦ öÁmiUS G¨¦ÒΰÀ
öuõk÷Põk Aø©¢uõÀ Ax
(i) x &Aa_US
(ii) y &Aa_US
ö\[SzuõP C¸US® ?
AÀ»x
π
4
(B) ©v¨¤kP : ∫ log (1 + tan x ) dx .
0
(a) At what points on the conic x 2 +4y 2 −8x−16y−68=0 the tangent is
perpendicular to
(i) x-axis
(ii) y-axis.
OR
π
4
(b) Evaluate : ∫ log (1 + tan x ) dx .
0

44. (A) P ©ØÖ® Q GßÓ C¸ öuõÈØ\õø»PÎß EØ£zvPÐUPõÚ öuõhº¦
÷Põi ¹£õ´PÎÀ ¤ßÁ¸® AmhÁøn°À öPõkUP¨£mkÒÍx.
E£÷¯õQ¨÷£õº
EØ£zv¯õͺ ö©õzu EØ£zv
P Q
P 300 800 2,400
Q 600 200 4,000
P &UPõÚ ©ØÖ® Q &UPõÚ CÖvz ÷uøÁPÒ •øÓ÷¯ 5,000 ©ØÖ® 4,000
BP C¸US® ÷£õx A¢u öuõÈØ\õø»PÎß EØ£zvPøÍU PõsP.
AÀ»x
5
(B) wºUP : (D2+10D+25) y = 2 + e−5x
(a) The inter-relationship between the production of two industries P and Q in crores
of rupees is given below.
User
Producer Total Output
P Q
P 300 800 2,400
Q 600 200 4,000
If the level of final demand for the output of the two industries is 5,000 for P and
4,000 for Q, then find the output of the two industries.
OR
5
(b) Solve : (D2+10D+25) y = + e−5x .
2
[ v¸¨¦P / Turn over

1367 (NP) 14

45. (A) B°»›ß ÷uØÓzøu¨ £¯ß£kzv
 x− y ∂u ∂u 1
u = sin−1  3  GÛÀ x +y = tan u GÚ {ÖÄP.
 x +3y  ∂x ∂y 6
 

AÀ»x
(B) J¸ |Pµzvß ©UPÒ öuõøP ÂÁµ® R÷Ç öPõkUP¨£mkÒÍx.
Q›÷Põ› & {³mhß `zvµzøu¨ £¯ß£kzv 2005 &B® Bsiß
©UPÒ öuõøPø¯U PõsP.

Bsk x : 1971 1981 1991 2001 2011

©UPÒ öuõøP (B°µ[PÎÀ) y : 56 76 91 103 111

(a) Using Euler's theorem, prove that

∂u ∂u 1  x− y
x +y = tan u if u = sin−1  3 .
∂x ∂y 6  x +3y 
 
OR
(b) Using Gregory - Newton's formula estimate the population of town for the year
2005.

Year x : 1971 1981 1991 2001 2011
Population (in thousands) y : 56 76 91 103 111

46. (A) ö£õ¸mPÎß ÷Põ¸uÀ AÍÄ q AvP›US® ö£õÊx. ÷Põ¸uÀ ©ØÖ®
AÍøÁPøÍ C¸¨¦ øÁ¨£uØS©õÚ ö\»Ä C &ß AvP›US® Ãu®
dC C 2 + 2Cq
= GÝ® ÁøPUöPÊa \©ß£õmiÚõÀ uµ¨£mkÒÍx. C
dq q2
©ØÖ® q &US Cøh÷¯ EÒÍ öuõhºø£ C = 1 ©ØÖ® q = 1 GÝ®
{ø»°À PõsP.
AÀ»x
2
(B) ‘A’ GßÓ ö£õ¸Îß ÷uøÁ q 1=240−p1+6p 2−p1p 2 GÛÀ
Eq 1 Eq1
©ØÖ® GßÓ £Sv ö|QÌa]PøÍ p1=5, p2=4 GÝ®ö£õÊx
Ep1 Ep2
PõsP.

15 1367 (NP)

(a) The rate of increase in the cost C of ordering and holding as the size q of the order
C 2 + 2Cq
increases is given by the differential equation dC = . Find the
dq q2
relationship between C and q if C=1 when q=1.
OR
2
(b) The demand for a commodity A is q 1=240−p1+6p 2−p1p 2 . Find the partial
Eq 1 Eq1
Elasticities and when p1=5 and p2=4.
Ep1 Ep2

47. (A) 1600 ]ÖÁºPøÍU öPõsh TÖ JßÔ¼¸¢x AÁºPÎß \µõ\›
~snÔÄ DÄ (I.Q.) 99 BS®. \µõ\› ~snÔÄ DÄ 100 ©ØÖ®
vmh»UP® 15 GÚÄ® öPõsh •Êø©z öuõSv°¼¸¢x AUTÖ
GkUP¨£mhuõ GÚ ÷\õvUPÄ®. (5% •UQ¯zxÁ ©mhzvÀ)
AÀ»x
(B) J¸ C¯À{ø»¨ £µÁ¼À 20 ÂÊUPõk E¸¨£iPÒ 100 &US
SøÓÁõPÄ®, 30 ÂÊUPõk E¸¨£iPÒ 200 &US® ÷©÷» EÒÍx GÛÀ
A¨£µÁ¼ß \µõ\› ©ØÖ® vmh »UPzøu PõsP.
Z 0.84 0.525
£µ¨¦ 0.3 0.2

(a) The mean I.Q. of a sample of 1600 children was 99. Is it likely that this was a
random sample from a population with mean I.Q. 100 and standard deviation
15 ? (Test at 5% level of significance).
OR
(b) In a normal distribution 20% of the items are less than 100 and 30% are over 200.
Find the mean and S.D. of the distribution.

Z 0.84 0.525
Area 0.3 0.2

-o0o-

[ v¸¨¦P / Turn over