Tamil Nadu 11th Std Model Question Paper Maths

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3412 (NS)
!3412NSIstYearMathematics! £vÄ Gs
Register Number

PART - III
Pou® / MATHEMATICS
( 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 GߣuøÚa
\›£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

3412 (NS) 2

1. A={(x, y) : y=sinx, x ∈R} ©ØÖ® B={(x, y) : y=cosx, x ∈R} GÛÀ, A∩B À :
(A) wº©õÛUP C¯»õx
(B) EÖ¨¦PÎÀø»
(C) Gso»h[Põ EÖ¨¦PÒ EÒÍÚ
(D) J÷µ J¸ EÖ¨¦ EÒÍx
If A={(x, y) : y=sinx, x ∈R} and B={(x, y) : y=cosx, x ∈R}, then A∩B contains :
(a) cannot be determined
(b) no element
(c) infinitely many elements
(d) only one element

2. f : [−3, 3] → S GßÓ \õº¦ f (x)=x2 GÚ Áøµ¯ÖUP¨£mk ÷©Ø÷PõºzuÀ GÛÀ,
S Gߣx :
(A) [0, 9] (B) [−9, 9] (C) R (D) [−3, 3]
If the function f : [−3, 3] → S defined by f (x)=x2 is onto, then S is :
(a) [0, 9] (b) [−9, 9] (c) R (d) [−3, 3]

1 3
3. − =
cos 80 sin 80
(A) 4 (B) 2 (C) 3 (D) 2
1 3
− =
cos 80 sin 80
(a) 4 (b) 2 (c) 3 (d) 2

cos6 x + 6 cos 4 x + 15cos 2 x + 10
4. =
cos 5x + 5 cos 3x + 10 cos x
(A) 2cosx (B) cos2x (C) cosx (D) cos3x
cos6 x + 6 cos 4 x + 15 cos 2 x + 10
is equal to :
cos 5x + 5 cos 3x + 10 cos x
(a) 2cosx (b) cos2x (c) cosx (d) cos3x

5. 2n C
3:
n C =11 : 1
3 GÛÀ n-&ß ©v¨¦ :
(A) 7 (B) 5 (C) 6 (D) 11
If 2n C3 : n C3=11 : 1, then n is :
(a) 7 (b) 5 (c) 6 (d) 11

3 3412 (NS)

1 1 1
6. + + +... -&ß ©v¨¦ :
2! 4! 6!

e 2−1 e 2+1 (e+1)2 (e−1)2
(A) (B) (C) (D)
2e 2e 2e 2e

1 1 1
The value of + + +... is :
2! 4! 6!

e 2−1 e 2+1 (e+1)2 (e−1)2
(a) (b) (c) (d)
2e 2e 2e 2e

7. 3x−y=−5 GßÓ ÷Põmkhß 458 ÷Põn® HØ£kzx® ÷Põmiß \õ´ÄPÒ :

1 1 1
(A) 2, − 2 (B) 1, −1 (C) 2 , −2 (D) 1,
2

The slope of the line which makes an angle 458 with the line 3x−y=−5 are :

1 1 1
(a) 2, − (b) 1, −1 (c) , −2 (d) 1,
2 2 2

8. y=−x GßÓ ÷PõmiØS (2, 3) GßÓ ¦Ò롧 ¤®£¨ ¦ÒÎ :
(A) (3, 2) (B) (−3, −2) (C) (−3, 2) (D) (−2, −3)
The image of the point (2, 3) in the line y=−x is :

(a) (3, 2) (b) (−3, −2) (c) (−3, 2) (d) (−2, −3)

9. A Gߣx J¸ \xµ Ao GÛÀ, ¤ßÁ¸ÁÚÁØÖÒ Gx \©a^µÀ»?
(A) A−AT (B) A+AT (C) AA T (D) ATA
If A is a square matrix, then which of the following is not symmetric ?

(a) A−AT (b) A+AT (c) AA T (d) AT A

[ v¸¨¦P / Turn over

3412 (NS) 4

3 −2
10. A+I=   GÛÀ (A+I)(A−I) &ß ©v¨¦ :
4 1 

− 5 − 4  − 5 − 4  − 5 4 5 4
(A) −8 −9  (B)  8 − 9
 (C) − 8 
9 (D)  
    8 9

 3 −2 
If A+I=   , then (A+I)(A−I) is equal to :
4 1 

− 5 − 4  − 5 − 4  − 5 4 5 4
(a)   (b)   (c)   (d)  
− 8 −9   8 − 9 − 8 9 8 9

→ →
11. a &US® b &US® Cøh¨£mh ÷Põn® 1208, ÷©¾® AÁØÔß GsnÍÄPÒ
→ →
•øÓ÷¯ 2, 3 GÛÀ a ⋅ b BÚx :

(A) − 3 (B) 3 (C) − 3 (D) 2
2

→ → → →
If a and b include an angle 1208 and their magnitudes are 2 and 3 , then a ⋅ b is
equal to :

− 3
(a) (b) 3 (c) − 3 (d) 2
2

→ → → → → →
12. a + b = 60 , a − b = 40 ©ØÖ® b = 46 GÛÀ a ß ©v¨¦ :

(A) 32 (B) 42 (C) 12 (D) 22

→ → → → → →
If a + b = 60 , a − b = 40 and b = 46 , then a is :

(a) 32 (b) 42 (c) 12 (d) 22

5 3412 (NS)

13. f (x)=|x|+|x−1|GßÓ \õº¦ :
(A) x=0, 1 GßÓ ¦ÒÎPÎÀ öuõhºa]¯ØÓx
(B) x=0 GßÓ ¦ÒΰÀ ©mk÷© öuõhºa]¯õÚx
(C) x=1 GßÓ ¦ÒΰÀ ©mk÷© öuõhºa]¯õÚx
(D) x=0, x=1 GßÓ ¦ÒÎPÎÀ öuõhºa]¯õÚx
f (x)=|x|+|x−1| is :
(a) discontinuous at x=0, 1
(b) continuous at x=0 only
(c) continuous at x=1 only
(d) continuous at both x=0 and x=1

a x − bx
14. lim =
x→0 x

a a b
(A) b (B) log ab (C) log  b  (D) log  
a

a x − bx
lim =
x→0 x

a a b
(a) (b) log ab (c) log   (d) log  
b b a

dy
15. x=at2, y=2at GÛÀ =
dx
1 1
(A) −t (B) t (C) − t (D) t
dy
If x=at2, y=2at, then =
dx
1 1
(a) −t (b) (c) − (d) t
t t

1 dz
16. y=
a−z
GÛÀ, dy &ß ©v¨¦ :
(A) −(z+a)2 (B) (a−z)2 (C) −(z−a)2 (D) (z+a)2
1 dz
If y = , then dy is :
a−z
(a) −(z+a) 2 (b) (a−z)2 (c) −(z−a) 2 (d) (z+a) 2

[ v¸¨¦P / Turn over

3412 (NS) 6

 x− 1
17. ∫  x + 1  dx =
1 x−1 2
(A) x+2log(x+1)+c (B) 2  x + 1  + c
 

( x − 1) 2
(C) x−2log(x+1)+c (D) log( x + 1) + c
2

∫ x + 1 dx =
 x− 1
 
 

1  x − 1 2
(a) x+2log(x+1)+c (b) +c
2  x + 1 

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

∫2
18. 3x + 5
dx =

2 3x + 5
(A) 3 log 2 + c (B)
(
3 2 3x + 5 ) +c
log 2

2 3x + 5 2 3x + 5
(C) 2 log (3x + 5) + c (D) 2 log 3
+c

∫2
3x + 5
dx is :

(a)
2 3x + 5
+c (b)
(
3 2 3x + 5 ) +c
3 log 2 log 2

2 3x + 5 2 3x + 5
(c) +c (d) +c
2 log (3x + 5) 2 log 3

dx
19. ∫ ex − 1 =
(A) log(ex+1)−log(ex)+c (B) log(ex)−log(ex−1)+c
(C) log(ex)+log(ex−1)+c (D) log(ex−1)−log(ex)+c
dx
∫ ex − 1 is :
(a) log(ex+1)−log(ex)+c (b) log(ex)−log(ex−1)+c
(c) log(ex)+log(ex−1)+c (d) log(ex−1)−log(ex)+c

7 3412 (NS)

20. £zx |õn¯[PøÍa _sk®÷£õx SøÓ¢ux 8 uø»PÒ Qøh¨£uØPõÚ
{PÌÄ :
7 7 7 7
(A) 128 (B) 64 (C) 32 (D) 16
Ten coins are tossed. The probability of getting at least 8 heads is :
7 7 7 7
(a) (b) (c) (d)
128 64 32 16

£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.
1
21. £Sv ¤ßÚ[PÍõP¨ ¤›UPÄ® :
x − a2
2

1
Resolve the rational expression into partial fractions.
x − a2
2

1 1 A
22.
7!
+ =
8! 9!
GÛÀ A-ß ©v¨¦ GßÚ ?

1 1 A
If + = , then find the value of A.
7! 8! 9!

17
 1 
23. x + 3  &ß Â›ÁõUPzvÀ x5 &ß SnPzøuU PõsP.
 x 

17
 1 
Find the coefficient of x5 in the expansion of x + 3  .
 x 

24. J÷µ ÷|ºU÷PõmiØS Bv°¼¸¢x Áøµ¯¨£k® ö\[SzxU ÷Põmiß }Í®
6 A»SPÒ. Aaö\[SzxU÷Põk x-Aa_hß HØ£kzx® ÷Põn® 120o GÛÀ,
A¢u ÷|ºU÷Põmiß \©ß£õmøhU PõsP.
Find the equation of the straight line, if the perpendicular from the origin makes an
angle of 1208 with x-axis and the length of the perpendicular from the origin is 6 units.

[ v¸¨¦P / Turn over

3412 (NS) 8

25. f ( x )= x , x/0 GÛÀ, lim f ( x ) QøhUP¨ö£Ö©õ GÚU PõsP.
x→0

Consider the function f ( x )= x , x/0. Does lim f ( x ) exist ?
x→0

 x 2 − 5x + 6 
26. lim   &ß ©v¨ø£U PnUQkP.
x → 3  x−3 


 x 2 − 5x + 6 
Calculate x →3 
lim
x−3
.

 

27. x &I¨ ö£õÖzx öuõøP°kP : (1+x2)−1
Integrate (1+x2)−1 with respect to x.

∫
1
28. ©v¨¤kP : dx
sin x cos2 x
2

∫
1
Evaluate : dx
sin x cos2 x
2

29. Cµsk |õn¯[PÒ J÷µ \©¯zvÀ _sh¨£kQßÓÚ GÛÀ AvP£m\©õP
C¸ § Qøh¨£uØPõÚ {PÌuPÄPøÍU PõsP.
If two coins are tossed simultaneously, then find the probability of getting at the most
two tails.

30. x -I¨ ö£õÖzx ÁøP°kP : xx
Differentiate xx with respect to x.

£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.

1
31. f (x) = &ß Ãa\P® PõsP.
1−3cosx

1
Find the range of f ( x ) =
1−3cosx

9 3412 (NS)

32. 5 ©õnÁºPÒ ©ØÖ® 4 ©õnÂPÒ J÷µ Á›ø\°À G¢u C¸ ©õnÂPЮ
Akzukzx Áµõ©À GzuøÚ ÁÈPÎÀ A©µøÁUP»õ®.
In how many ways 5 boys and 4 girls can be seated in a row, so that no two girls are
together ?

33. a1, a2, a3, ....., an&Gß£Ú ö£¸USz öuõhº•øÓ°À C¸US©õÚõÀ, ak(k>1)
Auß •ßÛ¯õÚ ak−1 &US®, öuõh›¯õÚ ak+1 &US® ö£¸USa \µõ\›¯õP
C¸US® GÚ {¹¤UPÄ®.
If a1, a2, a3, ....., an is a geometric progression, then prove that every term ak(k>1) is the
geometric mean of its immediate predecessor ak−1 and immediate successor ak+1.

34. J¸ ÷Põk B¯ Aa_PÐhß HØ£kzx® •U÷Põnzvß £µ¨¦ 36 \xµ A»S
©ØÖ® Bv°¼¸¢x AU÷PõmiØS Áøµ¯¨£k® ö\[Szx ÷Põk ªøP
x-Aa_hß HØ£kzx® ÷Põn® 458 GÛÀ, ÷|ºU÷Põmiß \©ß£õmøhU
PõsP.
Area of the triangle formed by a line with the coordinate axes is 36 square units. Find
the equation of the line if the perpendicular drawn from the origin to the line makes an
angle of 458 with positive of the x axis.

log 3 64 log 4 3 log 2 3 log 8 3
35. × GßÓ ö£¸UP¼ß ©v¨ø£U PõsP.
log 3 8 log 4 9 log 3 4 log 3 4
Find the value of the product ;
log 3 64 log 4 3 log 2 3 log 8 3
×
log 3 8 log 4 9 log 3 4 log 3 4

36. log(logx) GßÓ \õº¤ß Cµshõ® Á›ø\ ÁøPUöPÊøÁ x &I¨ ö£õÖzx
PõsP.
Find the second derivative of log(logx) with respect to x.

x 15
37. x &I¨ ö£õÖzx öuõøPU PõsP : .
1 + x 32

x15
Integrate with respect to x.
1 + x 32

→ → → → → → → → → →
38. a × ( b + c )+ b × ( c + a )+ c × ( a + b )= 0 GÚU PõmkP.
→ → → → → → → → → →
Show that a × ( b + c ) + b × ( c + a ) + c × ( a + b ) = 0 .

[ v¸¨¦P / Turn over

3412 (NS) 10

sin x
39. x→0 GÝ®÷£õx \õº¦ x US GÀø» ©v¨¦ EÒÍuõ GÚU PõsP.
ÂøhUPõÚ Põµn® TÖP.
sin x
Does the limit of the function x exist when x→0 ? State reasons for your answer.

2 c−b
40. a sin 2 θ + b cos2 θ = c GÛÀ, tan θ = a − c GÚ {ÖÄP.

2 c−b
If a sin 2 θ + b cos2 θ = c , show that tan θ = .
a−c

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

41. (A) f, g : R→R GßÓ \õº¦PÒ f (x)=|x|+x , g (x)=|x|−x GÚ Áøµ¯ÖUP¨£iß
fog ©ØÖ® gof BQ¯ÁØøÓU PõsP.
AÀ»x
(B) Z GßÓ PnzvÀ, m−n Gߣx 12&ß ©h[PõP C¸¢uõÀ, öuõhº¦ mRn
GÚ Áøµ¯ÖUP¨£kQÓx GÛÀ, R J¸ \©õÚz öuõhº¦ GÚ {¹¤UP.
(a) If f, g : R→R are defined by f (x)=|x|+x and g (x)=|x|−x, find gof and fog.
OR
(b) In the set Z of integers, define mRn if m−n is a multiple of 12. Prove that R is an
equivalence relation.

2x − 3
42. (A) (x − 2)(x − 4) < 0 GßÓ A\©ß£õmøh {øÓÄ ö\´²® x &ß AøÚzx
©v¨¦PøͲ® PõsP.
AÀ»x
π
(B) A+ B + C = 2 GÛÀ, cos 2A+cos 2B+cos 2C=1+4 sinA sinB sinC GÚ
{ÖÄP.
2x − 3
(a) Find all the values of x that satisfy the inequality ( x − 2)( x − 4) < 0.

OR
π
(b) If A+ B + C = , prove that cos 2A+cos 2B+cos 2C=1+4 sinA sinB sinC.
2

11 3412 (NS)

 B− C  b+ c A
43. (A) •U÷Põn® ABC -À, cos  2 
=
a
sin
2 GÚ {ÖÄP.

AÀ»x
(B) 3 126 &ß ©v¨ø£ 2 u\©ìuõÚ[PÐUS v¸zu©õPU PõsP.
 B− C  b+ c A
(a) In any triangle ABC, prove that cos  = sin .
 2  a 2
OR

(b) Find the value of 3 126 correct to two decimal places.

44. (A) Pouz öuõSzuÔuÀ •øÓ°À n/1 &US
n(2n − 1)(2n + 1)
1 2 + 32 + 52 + .... + (2n − 1)2 = GÚ {¹¤UP.
3

AÀ»x
(B) 9x 2 −24xy+16y 2 −12x+16y−12=0 Gߣx Cøn¯õÚ Cµmøh
÷|ºU÷PõkPÒ GÚ {ÖÄP. ÷©¾® CƸ ÷PõkPÐUS Cøh¨£mh
yµzøuU PõsP.
(a) By the principle of mathematical induction, prove that, for n/1

n(2n − 1)(2n + 1)
1 2 + 32 + 52 + .... + (2n − 1)2 = .
3
OR
(b) Show that the equation 9x2−24xy+16y2−12x+16y−12=0 represents a pair
of parallel lines. Find the distance between them.

2bc−a2 c2 b2 a b c
2

45. (A) c2 2ca−b2 a 2
=b c a GÚ {ÖÄP.
b2 a2 2ab−c2 c a b

AÀ»x
(B) ABCD GßÓ |õØPµzvÀ AC, BD - ß |k¨¦ÒÎPÒ E ©ØÖ® F - BP
→ → → → →
C¸¨¤ß AB + AD + CB + CD = 4 EF GÚ {ÖÄP.

[ v¸¨¦P / Turn over

3412 (NS) 12

2bc−a2 c2 b2 a b c
2

(a) Show that c2 2ca−b2 a2 =b c a
b2 a2 2ab−c2 c a b

OR
(b) If ABCD is a quadrilateral and E and F are the midpoints of AC and BD,
→ → → → →
respectively, prove that AB + AD + CB + CD = 4 EF

 4x + 5 ; x≤3
46. (A) f (x ) =  GßÓ \õº¦USz öuõhºa]z ußø©ø¯U
 4x − 5 ; x>3
öPõkUPõu ¦ÒÎPøÍU PõsP.
AÀ»x
sin−1 x
(B) y = 2
GÛÀ, (1−x2)y2−3xy1−y=0 GÚU PõmkP.
1−x
(a) Find the points of discontinuity of the function f, where
 4x + 5 ; x≤3
f (x) = 
 4x − 5 ; x > 3
OR
sin−1 x
(b) If y = , show that (1−x2)y2−3xy1−y=0.
2
1−x

5x − 2
47. (A) x -I¨ ö£õÖzx öuõøP°kP : 2 + 2x + x2 .
AÀ»x
(B) Jzu C¸ áõiPÎÀ, JßÔÀ 6 P¸¨¦ ©ØÖ® 4 ]Á¨¦ {Ó¨ £¢xPÒ
EÒÍÚ. ©ØöÓõ¸ áõi°À 2 P¸¨¦ ©ØÖ® 2 ]Á¨¦ {Ó¨ £¢xPÒ
EÒÍÚ. \©Áõ´¨¦ •øÓ°À J¸ áõi ÷uº¢öukUP¨£mk Av¼¸¢x
J¸ £¢x GkUP¨£kQÓx GÛÀ, A¨£¢x P¸¨£õP C¸¨£uØPõÚ
{PÌuPøÁU PõsP.
5x − 2
(a) Integrate with respect to x.
2 + 2x + x2
OR
(b) There are two identical urns containing respectively 6 black and 4 red balls,
2 black and 2 red balls. An urn is chosen at random and a ball is drawn from it.
Find the probability that the ball is black.

-oOo-