TN 11th Question Paper 2025 Mathematics

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8412
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!8412IstYearMathematics!
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 |õßS ©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

8412 2

1. |x−1| / |x−3| GßÓ A\©ß£õmiß wºÄU Pn® :
(A) (0, 2) (B) [0, 2] (C) (−∞, 2) (D) [2, ∞)
The solution set of the following inequality |x−1| / |x−3| is :
(a) (0, 2) (b) [0, 2] (c) (−∞, 2) (d) [2, ∞)

2. x=−3 À f(x)=x|x| &ß ÁøP°h¼ß ©v¨¦ :
(A) QøhUP¨ö£Óõx (B) 6
(C) 0 (D) −6
The derivative of f(x)=x|x|at x=−3 is :
(a) does not exist (b) 6
(c) 0 (d) −6

3. 2+4+6+ . . . +2n &ß ©v¨¦ :
2n(2n + 1) n(n − 1) n(n + 1)
(A) (B) (C) n(n+1) (D)
2 2 2
The value of 2+4+6+ . . . +2n is :
2n(2n + 1) n(n − 1) n(n + 1)
(a) (b) (c) n(n+1) (d)
2 2 2

4. y=logex GßÓ \õº¤ß ÷|º©õÖ :
(A) y=ex (B) y=logex (C) y=e−x (D) y=−logex
The inverse function of y=logex is :
(a) y=ex (b) y=logex (c) y=e−x (d) y=−logex

5. (x, −2), (5, 2), (8, 8) Gß£Ú J¸ ÷Põhø©¨ ¦ÒÎPÒ GÛÀ, x &ß ©v¨¦ :
1
(A) 1 (B) −3 (C) 3 (D) 3
If the points (x, −2), (5, 2), (8, 8) are collinear, then x is equal to :
1
(a) 1 (b) −3 (c) 3 (d)
3

6. ∆ ABC CÀ sin2A+sin2B+sin2C=2 GÛÀ A¢u •U÷Põn©õÚx :
(A) ö\[÷Põn •U÷Põn® (B) \©£UP •U÷Põn®
(C) A\©£UP •U÷Põn® (D) C¸\©£UP •U÷Põn®
In a triangle ABC, sin2A+sin2B+sin2C=2, then the triangle is :
(a) right triangle (b) equilateral triangle
(c) scalene triangle (d) isosceles triangle

A

3 8412

7. GÀ»õ® JØøÓ GsPÍõPU öPõsh 5 C»UP GsPÎß GsoUøP :
(A) 5 6 (B) 25 (C) 625 (D) 5 5
The number of 5 digit number, all digits of which are odd is :
(a) 56 (b) 25 (c) 625 (d) 55

→ → → → → →
8. → = 3, → = 4, → = 5 ©ØÖ® a + b + c = 0 GÛÀ a ©ØÖ® b &US Cøh÷¯
a b c
EÒÍ ÷Põn® :
(A) 608 (B) 0 (C) 458 (D) 908
→ → → → → → → →
If →
a
= 3,
b
= 4,
c
= 5 and a + b + c = 0 then the angle between a and b is :

(a) 608 (b) 0 (c) 458 (d) 908

→ → → →
9. a + 2 b ©ØÖ® 3 a + m b BQ¯øÁ Cøn GÛÀ, m &ß ©v¨¦ :
1 1
(A) 6 (B) 3 (C) 6 (D) 3

→ → → →
If a + 2 b and 3 a + m b are parallel, then the value of m is :

1 1
(a) 6 (b) 3 (c) (d)
6 3

1

∫ x 2 dx = k ( 3 x ) + c GÛÀ, k &ß ©v¨¦ :
10. 3x 1

1 1
(A) − (B) log 3 (C) log 3 (D) −log 3
log 3

1

∫ x 2 dx = k ( 3 x ) + c , then the value of k is :
If 3x 1

1 1
(a) − (b) log 3 (c) log 3 (d) −log 3
log 3

A [ v¸¨¦P / Turn over

8412 4

d 2 
11.  sin x
 :
dx  π 

π π 2 1
(A) 90 cos x8 (B) 180 cos x8 (C) π cos x8 (D) 90
cos x8

d 2 
 sin x
 is :
dx  π 

π π 2 1
(a) cos x8 (b) cos x8 (c) cos x8 (d) cos x8
90 180 π 90

3 x+5
12.
∫2 dx=

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

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

∫2
3 x+5 dx is :

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

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

13. (2, 3) ©ØÖ® (−1, 4) GßÓ ¦ÒÎPøÍ CønUS® ÷|ºU÷Põmiß «x (α, β) GßÓ
¦ÒÎ C¸¢uõÀ :
(A) α+3β=11 (B) α+2β=7 (C) 3α+β=11 (D) 3α+β=9
Straight line joining the points (2, 3) and (−1, 4) passes through the point (α, β) if :
(a) α+3β=11 (b) α+2β=7 (c) 3α+β=11 (d) 3α+β=9

A

5 8412

14. A ©ØÖ® B Gß£Ú C¸ {PÌa]PÒ GÛÀ, \›¯õP J¸ {PÌa] {PÌÁuØPõÚ
{PÌuPÁõÚx :
(A) P(A)+P(B)−P(A ∩ B) (B) P(A∪ B) + P(A∪B)

(C) P(A)+P(B)+2P(A ∩ B) (D) P(A ∩ B)+P (A ∩ B)

If A and B are any two events, then the probability that exactly one of them occurs is :

(a) P(A)+P(B)−P(A ∩ B) (b) P(A∪ B) + P(A∪B)

(c) P(A)+P(B)+2P(A ∩ B) (d) P(A ∩ B)+P (A ∩ B)

15. 3 EÖ¨¦PÒ öPõsh Pnzvß «uõÚ öuõhº¦PÎß GsoUøP :

(A) 512 (B) 9 (C) 1024 (D) 81
The number of relations on a set containing 3 elements is :
(a) 512 (b) 9 (c) 1024 (d) 81

16. lim x =
x →3

(A) ©v¨¦ CÀø» (B) 2
(C) 0 (D) 3
lim x =
x →3

(a) Value does not exist (b) 2
(c) 0 (d) 3

17. cos 288+sin 288=k3 GÛÀ cos 178 &ß ©v¨¦ :

k3 k3 k3 k3
(A) ± (B) (C) − (D) −
2 2 3 2

If cos 288+sin 288=k3, then cos 178 is equal to :

k3 k3 k3 k3
(a) ± (b) (c) − (d) −
2 2 3 2

A [ v¸¨¦P / Turn over

8412 6

18. (1, −1) GßÓ ¦ÒÎ ÁÈ÷¯ ö\ÀÁx® ©ØÖ® 3x+4y=6 &US ö\[SzuõÚx©õÚ
÷|ºU÷Põmiß \©ß£õk :
(A) 4x+3y+7=0 (B) 4x−3y−7=0 (C) 3x+4y+7=0 (D) 3x+4y−7=0
The equation of the line through the point (1, −1) and perpendicular to 3x+4y=6 is :
(a) 4x+3y+7=0 (b) 4x−3y−7=0 (c) 3x+4y+7=0 (d) 3x+4y−7=0

19. 343 &ß ©hUøP 3 GÛÀ, Auß Ai©õÚ® :
(A) 6 (B) 5 (C) 9 (D) 7
If 3 is the logarithm of 343, then the base is :
(a) 6 (b) 5 (c) 9 (d) 7

20. ‰ßÖ BsPÒ, C¸ ö£sPÒ ©ØÖ® |õßS SÇ¢øuPÒ EÒÍ J¸ Sʼ¸¢x
\©Áõ´¨¦ •øÓ°À |õßS |£ºPÒ ÷uº¢öukUP¨£kQßÓÚº. AÁºPÎÀ \›¯õP
C¸Áº ©mk® SÇ¢øuPÍõP C¸¨£uØPõÚ {PÌuPÄ :
1 3 10 10
(A) 2 (B) 4 (C) 21 (D) 23
Four persons are selected at random from a group of 3 men, 2 women and 4 children. The
probability that exactly two of them are children is :
1 3 10 10
(a) (b) (c) (d)
2 4 21 23
£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.

21. GÛÀ, P, Q, R BQ¯øÁ J÷µ ÷Põhø© ¦ÒÎPÒ GÚ {ÖÄP.
If , prove that the points P, Q, R are collinear.

22. ¤ßÁ¸® JßøÓö¯õßÖ Â»UQ¯ A, B, C ©ØÖ® D GßÓ |õßS {PÌa]PøÍ
©mk® öPõsh J¸ ÷\õuøÚ°ß {PÌa]PÎß {PÌuPÄPÒ \õzv¯©õÚøÁ¯õ
GÚz wº©õÛUPÄ®.
P(A)=0.15, P(B)=0.30, P(C)=0.43, P(D)=0.12
An experiment has the four possible mutually exclusive and exhaustive outcomes A, B, C
and D. Check if the following assignments of probability are permissible.
P(A)=0.15, P(B)=0.30, P(C)=0.43, P(D)=0.12

A

7 8412

23. ÁøP°kP : y=esinx
Differentiate : y=esinx

n n
24. lim x − 2 = 12 GÝ©õÖ EÒÍ ªøP •Ê Gs n &IU PõsP.
x→2 x −2

n n
Find the positive integer ‘n’ so that lim x − 2 = 12
x→2 x −2

25. aij =
( i − 2 j ) 2 , m=2, n=3 GÚ C¸US©õÖ EÖ¨¦PøÍU öPõsh m×n Á›ø\
2
Eøh¯ A=[aij] Ao°øÚ E¸ÁõUSP.
2
Construct an m×n matrix A=[aij], where aij is given by aij = (
i − 2 j)
with m=2, n=3.
2

2 3 4
26. ›ģkzuõ©À ©v¨ø£U PõsP : 5 6 8
6 x 9 x 12 x

2 3 4
Without expanding, evaluate 5 6 8
6 x 9 x 12 x

27. 23x < 100 &ß wºøÁ (i) xeN (ii) xeZ &US PõsP.
Solve 23x < 100 when (i) x is a natural number, (ii) x is an integer.

28. n &BÁx EÖ¨¦, an &IU öPõsh ¤ßÁ¸® öuõhº•øÓ°ß •uÀ
6 EÖ¨¦PøÍU PõsP.

n + 1 ; n JØøÓ¨£øh Gs GÛÀ
an =  n ; n Cµmøh¨£øh Gs GÛÀ



 n + 1 if n is odd
Write the first 6 terms of the sequences whose nth term, an =  n if n is even


A [ v¸¨¦P / Turn over

8412 8

29. (9 cosα, 9 sinα) GßÓ B¯z öuõø»PøÍ Eøh¯ |P¸® ¦ÒÎ P &ß {¯©¨£õøu°ß
\©ß£õmøhU PõsP. C[S α J¸ xøn¯»S BS®.
Find the equation of the locus of P, if for all values of α, the co-ordinates of a moving point
P is (9 cosα, 9 sinα), where α is a parameter.

30. 3!+4!+. . .+20! &ß JßÓõ® C»UP® GßÚ ?
What is the unit digit of the sum 3!+4!+. . .+20! ?

£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 − 3 cos x

1
Find the range of the function f (x ) =
1 − 3 cos x

32. x 2 − x − 2 = x + 1 GßÓ \©ß£õmøhz wºUP.

Solve : x2 − x − 2 = x + 1

33. {ÖÄP : sin(458+θ)−sin(458−θ)= 2 sinθ

Prove that sin(458+θ)−sin(458−θ)= 2 sinθ

34. (n+2)P4=42×nP2 GÛÀ, n &IU PõsP.
If (n+2)P4=42×nP2, find n.

35. ©v¨¦U PõsP : (102)4
Compute (102)4

A

9 8412

 3  1
36.  0, −  , (1, −1) ©ØÖ®  2, −  GßÓ ¦ÒÎPÒ J¸ ÷Põhø©¨ ¦ÒÎPÒ GÚU
 2  2
PõmkP.
 3  1
Show that the points  0, −  , (1, −1) and  2, −  are collinear.
 2  2

→ → → → → → → → →
37. a , b , c GßÓ A»S öÁUhºPÐUS a . b = a . c = 0 ©ØÖ® b &US® c &US®
π → 2
Cøh¨£mhU ÷Põn® 3 GÛÀ a = ± 3
( →b × →c ) GÚ {¹¤UP.
→ → → → → → → → →
Let a , b , c be unit vectors such that a . b = a . c = 0 and the angle between b and c
π → 2
is 3 , prove that a = ±
3
( →b × →c ) .

38. x &I¨ ö£õÖzx öuõøP°kP.

cos5x sin3x
Integrate with respect to x
cos5x sin3x

39. Gmk |õn¯[PÒ J¸ •øÓ _sh¨£kQßÓÚ.
(i) \›¯õP Cµsk §UPÒ.
(ii) AvP£m\©õP Cµsk §UPÒ
Qøh¨£uØPõÚ {PÌuPÄPøÍU PõsP.
Eight coins are tossed once, find the probability of getting :
(i) exactly two tails
(ii) atmost two tails

dy
40. y=cos−1(2cos2x−1) GÛÀ &IU PõsP.
dx

dy
Find , if y=cos−1(2cos2x−1)
dx

A [ v¸¨¦P / Turn over

8412 10

£Sv & IV / PART - IV

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

41. (A) £Sv ¤ßÚ[PÍõP ¤›zöuÊxP :
x2 + x + 1
x 2 − 5x + 6
AÀ»x
(B) 3 x − y + 4 = 0 GßÓ ÷Põmøh RÌUPõq® \©õÚ ÁiÁzvØS ©õØÖP.

(i) \õ´Ä ©ØÖ® öÁmkzxsk ÁiÁ®
(ii) öÁmkzxsk ÁiÁ®
(iii) ö\[Szx ÁiÁ®
x2 + x + 1
(a) Resolve into partial fractions :
x 2 − 5x + 6
OR

(b) Express the equation 3 x − y + 4 = 0 in the following equivalent form :
(i) Slope and intercept form
(ii) Intercept form
(iii) Normal form

(A) {¹¤UP : cot(180
+ θ) sin(90
− θ) cos(− θ)
42. = cos 2 θ cot θ
sin(270
+ θ) tan(− θ) cosec(360
+ θ)

AÀ»x
−1
(B) y = sin x
GÛÀ, (1−x2) y2−3xy1−y=0 GÚU PõmkP.
2
1−x

cot(180
+ θ) sin(90
− θ) cos(− θ)
(a) Prove that = cos 2 θ cot θ
sin(270
+ θ) tan(− θ) cosec(360
+ θ)
OR

sin−1 x
(b) If y = , show that (1−x2) y2−3xy1−y=0
2
1−x

A

11 8412

43. (A) lim sinθ = 1 GÚ {ÖÄP.
θ →0 θ
AÀ»x
(B) J¸ öuõÈØ\õø»°À C¯¢vµ[PÒ I ©ØÖ® II GÚ C¸ÁøP EÒÍÚ.
C¯¢vµ® -I öuõÈØ\õø»°ß EØ£zv°À 60% u¯õ›UQÓx ©ØÖ®
C¯¢vµ® -II EØ£zv°À 40% u¯õ›UQÓx. ÷©¾® C¯¢vµ® &I ß ‰»®
EØ£zv ö\´¯¨£mh ö£õ¸mPÎÀ 2% SøÓ£õkÒÍuõPÄ® C¯¢vµ® &II ß
‰»® EØ£zv ö\´¯¨£mh ö£õ¸mPÎÀ 4% SøÓ£õk EÒÍuõPÄ®
C¸UQßÓÚ. EØ£zv ö\´¯¨£mh ö£õ¸mPμ¸¢x, \©Áõ´¨¦ •øÓ°À
J¸ ö£õ¸Ò ÷uº¢öukUP¨£kQÓx. A¨ö£õ¸Ò SøÓ£õkhß C¸¨£uØPõÚ
{PÌuPÄ ¯õx ?
sinθ
(a) Prove that lim =1
θ →0 θ
OR
(b) A factory has two Machines - I and II. Machine - I produces 60% of items and
Machine - II produces 40% of the items of the total output. Further 2% of the items
produced by Machine - I are defective whereas 4% produced by Machine - II are
defective. If an item is drawn at random what is the probability that it is defective ?

3 3 1
44. (A) x J¸ ö£›¯ Gs GÛÀ, x3 + 7 − x 3 + 4 &ß ©v¨¦ ÷uõµõ¯©õP
x2
GÚ {ÖÄP.
AÀ»x
(B) k(x−1) 2= 5x−7 Gߣuß J¸ ‰»® ©ØÓuß C¸©h[S GÛÀ,
k=2 AÀ»x −25 GÚU PõsP.
1
(a) Prove that 3 x 3 + 7 − 3 x 3 + 4 is approximately equal to when x is large.
x2
OR
(b) If one root of k(x−1)2 = 5x−7 is double the other root, show that k=2 or −25.

45. (A) tan 208 tan 408 tan 608 tan 808=3 GÚ {ÖÄ.
AÀ»x
1
(B) öuõøP°kP :
x 2 + 5x + 4
(a) Show that tan 208 tan 408 tan 608 tan 808=3
OR
1
(b) Integrate :
2
x + 5x + 4

A [ v¸¨¦P / Turn over

8412 12

46. (A) ©UPÒöuõøP 5000 EÒÍ J¸ |PµzvÀ |hzu¨£mh J¸ PnUöPk¨¤À,
ö©õÈ A öu›¢uÁºPÒ 45%, ö©õÈ B öu›¢uÁºPÒ 25%, ö©õÈ C öu›¢uÁºPÒ
10%, A ©ØÖ® B ö©õÈPÒ öu›¢uÁºPÒ 5%, B ©ØÖ® C ö©õÈPÒ
öu›¢uÁºPÒ 4%, A ©ØÖ® C ö©õÈPÒ öu›¢uÁºPÒ 4% BS®. CvÀ ‰ßÖ
ö©õÈPøͲ® öu›¢uÁºPÒ 3% GÛÀ, ö©õÈ A ©mk® öu›¢uÁºPÒ
GzuøÚ ÷£º ?
AÀ»x
(B) J¸ •U÷Põnzvß |kU÷PõkPÒ J¸ ¦ÒΰÀ \¢vUS® GÚ {ÖÄP.
(a) In a survey of 5000 persons in a town, it was found that 45% of the persons know
Language A, 25% know Language B, 10% know Language C, 5% know Languages
A and B, 4% know Languages B and C and 4% know Languages A and C. If 3% of the
persons know all the three Languages, find the number of persons who know only
Language A.
OR
(b) Prove that the medians of a triangle are concurrent.

47. (A) Pouz öuõSzuÔuÀ •øÓ°À n/1 &US
2
13 + 2 3 + 33+ .... + n 3 =  n + 1 
n( )
GÚ {¹¤UP.
 2 
AÀ»x
1+a 1 1
= abc  1 + + +  GÚ {ÖÄP.
1 1 1
(B) 1 1+b 1
 a b c
1 1 1+c

(a) By the principle of mathematical induction, prove that, for n/1
2
13 + 2 3 + 33+ .... + n 3 =  n + 1 
n( )
 2 
OR

1+a 1 1
= abc  1 + + + 
1 1 1
(b) Prove that 1 1+b 1
 a b c
1 1 1+c

-oOo-

A

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