Tamil Nadu 11th Std Model Question Paper Business Maths

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No. of Printed Pages : 15
3467 (NS)
!3467NSIstYearBusinessMathematics! £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

3467 (NS) 2

1 2 3 3 1 2
1. ∆= 3 1 2 GÛÀ, 1 2 3 &ß ©v¨¦ :
2 3 1 2 3 1

(A) −3∆ (B) ∆ (C) −∆ (D) 3∆

1 2 3 3 1 2
If ∆= 3 1 2 then 1 2 3 is :
2 3 1 2 3 1

(a) −3∆ (b) ∆ (c) −∆ (d) 3∆

x x+2
2. GÛÀ x &ß ©v¨¦ :
x−2 x

(A) x 2 (B) +4 (C) 0 (D) 1

x x+2
The value of x if is :
x−2 x

(a) x2 (b) +4 (c) 0 (d) 1

3. öÁÆ÷ÁÖ C»UP[PøÍ Eøh¯ 9 C»UP GsPÎß ö©õzu GsoUøP :
(A) 10×10! (B) 10! (C) 9! (D) 9×9!
The total number of 9 digit numbers which have all different digits is :

(a) 10×10! (b) 10! (c) 9! (d) 9×9!

4. S¯® ÁÈa ö\À¾® CµmøhU Szuõ¯® Gߣx :
(A) Aa_ (B) S¯|õs (C) ö\ÆÁP»® (D) C¯USÁøµ
The double ordinate passing through the focus is :

(a) axis (b) focal chord (c) latus rectum (d) directrix

3 3467 (NS)

5. 6x−2y+5=0 GßÓ ÷Põmiß \õ´Ä :

1
(A) 3 (B) 3 (C) −3 (D) 0

The slope of the line 6x−2y+5=0 :

1
(a) (b) 3 (c) −3 (d) 0
3

6. sinA+cosA=1 GÛÀ sin 2A =

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

If sinA+cosA=1 then sin 2A is equal to :

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

1−x
7. f ( x )= , 1+x ≠ 0 GÛÀ, f (−x) Cß ©v¨¦ :
1+x

1 1
(A) f (x) (B) −f (x) (C) f (x) (D) − f ( x )

1−x
If f ( x )= , 1+x ≠ 0, then f (−x) is equal to :
1+x

1 1
(a) f (x) (b) −f (x) (c) f (x) (d) −
f (x)

8. y=e2x GÛÀ y2 Cß ©v¨¦ :

(A) e−2x (B) 2e2x (C) e2x (D) 4e2x
If y=e2x then y2=?
(a) e−2x (b) 2e2x (c) e2x (d) 4e2x

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3467 (NS) 4

1 dy
9. y=x ©ØÖ® z = GÛÀ, =
x dz

1
(A) − (B) x 2 (C) 1 (D) −x 2
x2

1 dy
If y=x and z = , then =
x dz

1
(a) − (b) x2 (c) 1 (d) −x 2
x2

2 ∂u
10. u=ex GÛÀ, &ß ©v¨¦ :
∂x

(A) 0 (B) 2xex2 (C) ex2 (D) 2ex2

2 ∂u
If u=ex then is equal to :
∂x
2 2 2
(a) 0 (b) 2xex (c) ex (d) 2ex

∂2u
11. u(x, y) Gߣx x ©ØÖ® y &À öuõhºa]¯õÚ \õº¦ GÛÀ =
∂y ∂x

∂2u ∂2u ∂2u
(A) ∂x ∂y (B) (C) ∂y 2 (D) 0
∂x 2

∂2u
If u(x, y) is a continuous function of x and y, then is equal to :
∂y ∂x

∂2u ∂ 2u ∂2u
(a) (b) (c) (d) 0
∂x ∂y ∂x 2 ∂y 2

5 3467 (NS)

1 1
12. ` 100 •P©v¨¦øh¯ J¸ £[S 9 % PÈÄ Âø»US, % uµS ÃuzvÀ
2 2
QøhUS® GÛÀ, A¢u £[Qß Áõ[Q¯ Âø» :

(A) ` 95 (B) ` 89 (C) ` 90 (D) ` 91

1
Purchasing price of one share of face value 100 available at a discount of 9 % with
2
1
brokerage % is :
2

(a) ` 95 (b) ` 89 (c) ` 90 (d) ` 91

13. DÄ öuõøP Gߣx GÆÁõÖ öÁΨ£kzu¨£kQÓx ?

(A) £[SPÎß GsoUøP (B) •u½k

(C) \uÃu® (A) CÁØÔÀ HxªÀø»

Dividend is expressed as :

(a) No. of shares (b) Investment

(c) Percentage (d) None of these

14. Cøh{ø»=45 ©ØÖ® Auß \µõ\› »UP öPÊ=0.25 GÛÀ, Cøh{ø»ø¯
ö£õÖzu \µõ\› »UP® :

(A) 45 (B) 11.25 (C) 180 (D) 0.0056

If median=45 and its co-efficient is 0.25, then the mean deviation about median is :

(a) 45 (b) 11.25 (c) 180 (d) 0.0056

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3467 (NS) 6

n
15. J¸ ÷\õuøÚ°ß TÖöÁÎ S={E1, E2, ....... En} GÛÀ, ∑ P(Ei ) =
i=1

1 1
(A) 3 (B) 0 (C) 1 (D) 2

n
Let a sample space of an experiment be S={E1, E2, ....... En} then ∑ P(Ei ) is equal to :
i=1

1 1
(a) (b) 0 (c) 1 (d)
3 2

2 1
16. A, B Gß£Ú \õµõ {PÌÄPÒ GÛÀ, P(A)= , P(B)= GßÓõÀ P(A ∩ B) &ß
5 5
©v¨¦ :

3 1 2
(A) 0 (B) 25 (C) 25 (D) 25

2 1
If A and B are independent events P(A)= and P(B)= then P(A ∩ B) :
5 5

3 1 2
(a) 0 (b) (c) (d)
25 25 25

17. Cµsk ©õÔPÒ CÓ[S vø\°À |PºQÓx GÛÀ JmkÓÄU öPÊÁõÚx :

(A) Gv›øh (B) ÷|›øh

(C) •Êø©¯õÚ Gv›øh (D) JmkÓÄ Cßø©
If two variables move in decreasing direction then the correlation is :

(a) negative (b) positive

(c) perfect negative (d) no correlation

7 3467 (NS)

18. ¤ßÁ¸® TØÔÀ Gx \› ?
(A) X ©ØÖ® Y Gß£Ú C¸ ©õÔPÒ GÛÀ, AvP£m\©õP C¸¨£x
£» öuõhº¦¨ ÷£õUSU ÷PõkPÒ.
(B) JmkÓÄU öPÊ −1 &¼¸¢x +1 &US Cøh÷¯ Kº ©v¨ø£¨
ö£ØÔ¸US®.
(C) ]uÓÀ ÂÍUP¨£hzvÀ SÔUP¨£mh ¦ÒÎPÒ RÌ÷|õUQ¯¨
÷£õUQøÚU öPõsi¸¢uõÀ, ©õÔPÐUQøh÷¯ ÷|›øh JmkÓÄ
EÒÍx GÚ»õ®.
(D) C¸ öuõhº¦¨ ÷£õUSU öPÊUPЮ ©v¨¦ 1&I Âh¨ ö£›¯x AÀ».
Which statement is correct ?
(a) X and Y are two variables, there can be at the most more regression.
(b) Co-efficient of correlation lies between −1 and +1.
(c) In scatter diagram, if the plotted points show a downward trend, the correlation
will be positive.
(d) Both the regression co-efficients cannot be greater than one.

19. Áø»¯ø©¨¦ Áøµ£hzvÀ }sh öuõhºa]¯õÚ \[Q¼ ÷£õßÓ
ö\¯ÀPøÍ GÆÁõÖ SÔ¨£x ?
(A) wºÄPÒ (B) ªP }sh Põ»®
(C) wºÄUS EP¢u¨ £Sv (D) wºÄUSP¢u¨ £õøu
The longest path connected by the activities in the network is called :
(a) Solution (b) Longest duration
(c) Feasible region (d) Critical path

20. (i, j) GßÓ ö\¯»õÚx wºÄUS EP¢u £õøu°À C¸¨£uØPõÚ {£¢uøÚPÎÀ
JßÖ :
(A) Ej−Ei=Lj−Li ≠ tij (B) Ej−Ei=Lj−Li=tij
(C) Ei−Ej=Lj−Li=tij (D) Ej−Ei=Li−Lj=tij
One of the conditions for the activity (i, j) to lie on the critical path is :
(a) Ej−Ei=Lj−Li ≠ tij (b) Ej−Ei=Lj−Li=tij
(c) Ei−Ej=Lj−Li=tij (d) Ej−Ei=Li−Lj=tij

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3467 (NS) 8

£Sv & II/PART - II
SÔ¨¦ : GøÁ÷¯Ý® 7 ÂÚõUPÐUS Âøh¯ÎUPÄ®. ÂÚõ Gs 30 &US
Pmhõ¯©õP Âøh¯ÎUPÄ®. 7x2=14
Note : Answer any seven questions. Question No. 30 is compulsory.

x y z
21. 2 x+2a 2 y+2b 2 z+2c =0 GÚU PõmkP.
a b c

x y z
Show that 2 x+2a 2 y+2b 2 z+2c =0 .
a b c

4
22. 2 &I £Sv ¤ßÚ[PÍõP ©õØÖP.
x −1

4
Resolve into partial fraction : 2 .
x −1

23. GßÓ ÁmhzvØS
x 2 +y 2 +8x+4y+8=0 (2, 3) GßÓ ¦Òΰ¼¸¢x
Áøµ¯¨£k® öuõk÷Põmiß }Í® PõsP.
Find the length of the tangent from the point (2, 3) to the circle x2+y2+8x+4y+8=0.

sin 2θ
24. =tanθ GÚ {ÖÄP.
1+cos 2θ

sin 2θ
Show that =tanθ .
1+cos 2θ

25. ©v¨¤kP : lim x tan  1 
x→∞ x

Evaluate : lim x tan  
1
x→∞ x

9 3467 (NS)

26. p=3 &À x=2p2+5 GßÓ AΨ¦ \õº¤ß AΨ¦ ö|QÌa]ø¯U PõsP.

Find the elasticity of supply for the supply function x=2p2+5 when p=3.

27. RÌUPsh ÂÁµ[PÐUS Cø\a \µõ\›ø¯U PõsP.
1, 45, 10, 4, 11.2

Calculate the Harmonic mean for the following values :

1, 45, 10, 4, 11.2

28. ¤ßÁ¸® ÂÁµ[Pμ¸¢x X &ß «x Y &ß öuõhº¦¨ ÷£õUSU ÷PõkPøÍ
PõsP.
N=25, ΣX=125, ΣY=100, ΣX2=650, ΣY2=436, ΣXY=520.

Find the Regression lines X on Y from the following data :

N=25, ΣX=125, ΣY=100, ΣX2=650, ΣY2=436, ΣXY=520.

29. RÌUPsh {PÌÄPøÍ öPõsh vmhzvß Áø»¯ø©¨ø£ ÁøµP.
{PÌÄPÒ : 1 2 3 4 5 6 7

EhÚi •¢øu¯ {PÌÄ : - 1 1 2, 3 3 4, 5 5, 6

Draw the event oriented network for the following data :

Events : 1 2 3 4 5 6 7

Immediate Predecessors : - 1 1 2, 3 3 4, 5 5, 6

2+x − 2−x
30. ©v¨¤kP : lim
x→0 2x

2+x − 2−x
Evaluate : lim
x→0 2x

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3467 (NS) 10

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

 4 2 1
 5 − 5 − 5
2 2 1  
A= 1 3 1 
1 3 1
31.   ©ØÖ® B=− −  GßÓ AoPÒ
5 5 5
 1 2 2   1 2 4
− − 
 5 5 5

JßÖUöPõßÖ ÷|º©õÖ BS® GÚU PõmkP.
 4 2 1
 5 − 5 − 5
2 2 1  
  
Show that the matrices A= 1 3 1  and B= −
1 3
−
1
 5 5 5
 1 2 2   1 2 4
− − 
 5 5 5
are inverses of each other.

 5 −7 
 GÛÀ, (A ) =A GÚU PõmkP.
32. A= −1 −1
 3 −4 

 5 −7  −1 −1
If A=  then prove that (A ) =A.
 3 −4 

33. (n+2) Cn=45 GÛÀ, n &ß ©v¨ø£ PõsP.
If (n+2) Cn=45, find n.

34. 3x+4y=13; 2x−7y=−1 ©ØÖ® ax−y−14=0 Gß£Ú J¸ ¦ÒÎ ÁÈU
÷PõkPÒ GÛÀ ‘a’ &ß ©v¨¦U PõsP.
Find the value of ‘a’ for which the straight lines 3x+4y=13; 2x−7y=−1 and
ax−y−14=0 are concurrent.

1 π
35. tan α= ©ØÖ® tan β= 1 GÛÀ, ( 2α+β )= GÚ {ÖÄP.
3 7 4
1 1 π
If tan α= and tan β= then prove that ( 2 α+β )= .
3 7 4

11 3467 (NS)

x 2 +x+1
36. ÁøPUöPÊ PõsP : x 2 −x+1

x 2 +x+1
Differentiate :
x 2 −x+1

∂2 f ∂2 f
37. f=x3y+y4z−z3x2y GÛÀ ©ØÖ® &ß ©v¨ø£U PõsP.
∂x 2 ∂y 2

∂2 f ∂2 f
If f=x3y+y4z−z3x2y, find and .
∂x 2 ∂y 2

38. BsiØS 10% Ámi ÂQuzvÀ \õuõµn uÁøn £[Rmkz öuõøP
` 3,200 &US 12 BskPÐUPõÚ öuõøP°øÚU PõsP. [(1.1)12=3.1384]

Find the amount of an ordinary annuity of ` 3,200 per annum for 12 years at the rate
of interest of 10% per year. [(1.1)12=3.1384]

39. ¤ßÁ¸® ÂÁµ[Pμ¸¢x JmkÓÄU öPÊøÁU PnUQkP.

N=9, ΣX=45, ΣY=108, ΣX2=285, ΣY2=1356, ΣXY=597.

Calculate the correlation co-efficient from the following data :

N=9, ΣX=45, ΣY=108, ΣX2=285, ΣY2=1356, ΣXY=597.

40. 2x 2 +7xy+3y 2 +5x+5y+2=0 Gߣx Cµmøh ÷|ºU÷PõkPøÍU SÔUS®
GÚU PõmkP. ÷©¾® CU÷PõkPÐUS Cøh¨£mh ÷Põn® PõsP.
Show that the equation 2x2+7xy+3y2+5x+5y+2=0 represents a pair of straight
lines. Also find the angle between them.

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3467 (NS) 12

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

1 a a2
41. (A) 1 b b 2 =(a−b)(b−c)(c−a) GÚ {ÖÄP.
1 c c2

AÀ»x
(B) 2 3n−1 Gߣx "" 7 &BÀ ÁS£k®'' (AøÚzx n ∈ N ) GÚ Pouz
öuõSzuÔuÀ Âv¨£i {¹¤.

1 a a2
2
(a) Evaluate 1 b b =(a−b)(b−c)(c−a) .
1 c c2

OR
(b) Show by the principle of mathematical induction that 23n−1 is divisible by 7, for
all n ∈ N.

42. (A) sin 6008 cos 3908+cos 4808 sin 1508=−1 GÚ {ÖÄP.
AÀ»x
(B) RÌUPsh ÷|›¯À vmhªhÀ PnUSPøÍ Áøµ£h® ‰»® wºUP.
36x 1 +6x 2 /108, 3x 1 +12x 2 /36, 20x 1 +10x 2 /100 ©ØÖ® x 1 , x 2 /0 GßÓ
Pmk¨£õkPÐUQn[P Z = 20x1+40x2 &ß «a]Ö ©v¨ø£U PõsP.
(a) Prove that sin 6008 cos 3908+cos 4808 sin 1508=−1.
OR
(b) Solve the following linear programming problem by graphical method :
Minimize Z = 20x1+40x2 subject to the constraints 36x1+6x2/108, 3x1+12x2/36,
20x1+10x2/100 and x1, x2/0.

13 3467 (NS)

3
43. (A) J¸ {ÖÁÚzvß ö©õzua ö\»Äa \õº£õÚx C( x ) = x − 5x 2 + 28x + 10 ,
3
C[S x BÚx EØ£zv BS®. EØ£zv°ß JÆöÁõ¸ A»QØS® ` 2
Ãu® ÂvUP¨£mh Á›ø¯ EØ£zv¯õͺ uß ö\»÷Áõk CønzxU
öPõÒQÓõº. ¯õ£õµa \¢øuUPõÚ ÷uøÁa \õº¦ p=2530−5x, GÚ
öPõkUP¨£mhõÀ, ö£¸© C»õ£® AøhÁuØPõÚ EØ£zv°ß
AÍøÁ²®, Âø»ø¯²® PõsP. C[S p Gߣx EØ£zv°ß
JÆöÁõ¸ A»Qß Âø»ø¯U SÔUQÓx.
AÀ»x
dy y
(B) xm ⋅ yn=(x+y)m+n GÛÀ, = GÚU PõmkP.
dx x
x3
(a) The total cost function of a firm is C( x ) = − 5x 2 + 28x + 10 , where x is the
3
output. A tax at the rate of ` 2 per unit of output is imposed and the producer
adds it to his cost. If the market demand function is given by p=2530−5x, where
p is the price per unit of output, find the profit maximizing the output and price.
OR
dy y
(b) If xm ⋅ yn=(x+y)m+n, then show that = .
dx x
44. (A) J¸ {ÇØ£hU Pø»bº J¸ ¦øP¨£hU P¸Âø¯ uÁøn•øÓ°À
Áõ[SQÓõº. Áõ[Q¯ ÷uv°¼¸¢x JÆöÁõ¸ uÁønUS® ` 36,000
Á¸hõ¢vµ uÁønPÎÀ 7 uÁønPÒ ö\¾zu ÷Ásk®. Ámi¯õÚx
16% Tmk Ámi GÛÀ, ¦øP¨£hU P¸Â°ß A\À Âø»ø¯U PõsP.
[(1.16) 7=2.828]
AÀ»x
(B) ¤ßÁ¸® ÂÁµ[PÐUS PõºÀ ¤¯º\Ûß JmkÓÄU öPÊÂøÚU
PnUQkP.
X 6 8 12 15 18 20 24 28 31
Y 10 12 15 15 18 25 22 26 28
(a) A photographer purchases a camera on instalments. He has to pay 7 annual
instalments each of ` 36,000 right from the date of purchase. If the rate of
compound interest is 16% then find the cost price (present value) of the camera.
[(1.16) 7=2.828]
OR
(b) Calculate Karl Pearson’s co-efficient of correlation from the following data :
X 6 8 12 15 18 20 24 28 31
Y 10 12 15 15 18 25 22 26 28

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3467 (NS) 14

45. (A) (1+x) 2n &ß Â›ÂÀ |k EÖ¨¦ 1.3.5...........(2n−1)2 n x n GÚU
n!
Põs¤UPÄ®.
AÀ»x
(B) ¤ßÁ¸® ÂÁµ[PÐUS Cøh{ø»ø¯¨ ö£õÖzx \µõ\› »UPzøuU
PõsP.

Á¯x (Á¸h[PÎÀ) 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80

|£ºPÎß GsoUøP 8 12 16 20 37 25 19 13

n n
(a) Show that the middle term in the expansion of (1+x)2n is 1.3.5..........(2n−1)2 x .
n!
OR
(b) Find out the co-efficient of mean deviation about median in the following series :

Age in years 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80
No. of persons 8 12 16 20 37 25 19 13

46. (A) A GßÓ ö£õ¸Îß ÷uøÁ q = 250 − P12 + 3P2 − P1P2 GÛÀ P 1 =2
Eq Eq
©ØÖ® P 2 =1 GßÓ ©v¨¦PÐUS ©ØÖ® GßÓ £Sv
EP1 EP2
ö|QÌa]PøÍU PõsP.
AÀ»x
(B) y 2 −8y−8x+24=0 GßÓ £µÁøͯzvß •øÚ, S¯®, Aa_,
C¯USÁøµ ©ØÖ® ö\ÆÁP»zvß }Í® BQ¯ÁØøÓ PõsP.
(a) The demand for a commodity A is q = 250 − P12 + 3P2 − P1P2 . Find the partial
Eq Eq
elasticities and when P1=2, P2=1.
EP1 EP2

OR
(b) Find the vertex, focus, axis, directrix and the length of latus rectum of the parabola
y 2 −8y−8x+24=0.

15 3467 (NS)

47. (A) X, Y, Z GßQÓ ©õnÁºPÎh® J¸ PnUS uµ¨£kQÓx. AUPnUøP
1 1 2
AÁºPÒ wº¨£uØPõÚ {PÌuPÄ •øÓ÷¯ 2 , 3 , 5 GÛÀ, AUPnUøP
wº¨£uØPõÚ {PÌuPÄ GßÚ ?
AÀ»x
(B) J¸ vmhzvß ö\¯ÀPЮ AÁØÖUPõÚ Põ» AÍÄPЮ (|õmPÎÀ)
¤ßÁ¸® AmhÁøn°À öPõkUP¨£mkÒÍx.
ö\¯À 1-2 1-6 2-3 2-4 3-5 4-5 6-7 5-8 7-8

Põ»AÍÄ 7 6 14 5 11 7 11 4 18

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1 1
(a) A problem is given to 3 students X, Y and Z whose chances of solving it are ,
2 3
2
and respectively. What is the probability that the problem is solved ?
5
OR
(b) A project has the following time schedule :
Activity 1-2 1-6 2-3 2-4 3-5 4-5 6-7 5-8 7-8
Duration (in days) 7 6 14 5 11 7 11 4 18
Construct the network and calculate the earliest start time, earliest finish time,
latest start time and latest finish time of each activity and determine the critical
path of the project and duration to complete the project.

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