TN 12th Question Paper 2024 Maths

TN PUBLIC EXAM

QUESTION
PAPER
2024

No. of Printed Pages : 15
7412
£vÄ Gs
A Register Number

!7412Mathematics!
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®,
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 |õßS ©õØÖ ÂøhPÎÀ ªPÄ® Hئøh¯
Âøhø¯z ÷uº¢öukzxU SÔ±mkhß Âøh°øÚ²® ÷\ºzx
GÊuÄ®.
Note : (i) All questions are compulsory.
(ii) Choose the most appropriate answer from the given four alternatives and write
the option code and the corresponding answer.

[ v¸¨¦P / Turn over

7412 2

1. y2=4x GßÓ £µÁøͯzvØS® Auß ö\ÆÁP»zvØS® Cøh÷¯ £µ¨£õÚx :

8 2 5 4
(A) 3 (B) 3 (C) 3 (D) 3

The area between y2=4x and its latus rectum is :

8 2 5 4
(a) (b) (c) (d)
3 3 3 3

a 3
∫ ( a − x ) dx &Cß ©v¨¦ :
2. 2 2
0

3πa 2 πa 3 3πa 4 3πa 4
(A) (B) (C) (D)
8 16 8 16

a 3
∫ ( a − x ) dx is :
The value of 2 2
0

3πa 2 πa 3 3πa 4 3πa 4
(a) (b) (c) (d)
8 16 8 16

3. P(x, y) GßÓ ¦ÒÎ S¯[PÒ F1(3, 0) ©ØÖ® F2(−3, 0) öPõsh T®¦ ÁøÍÄ
16x2+25y2=400 &ß «xÒÍ ¦ÒÎ GÛÀ PF1+PF2 &ß ©v¨¦ :

(A) 10 (B) 8 (C) 12 (D) 6
If P(x, y) be any point on 16x2+25y2=400 with foci F1(3, 0) and F2(−3, 0), then PF1+PF2
is :

(a) 10 (b) 8 (c) 12 (d) 6

4. ?z1?=1, ?z2?=2, ?z3?=3 ©ØÖ® ?9z1z2+4z1z3+z2z3?=12 GÛÀ, ?z1+z2+z3? &ß ©v¨¦ :

(A) 3 (B) 1 (C) 4 (D) 2
If ?z1?=1, ?z2?=2, ?z3?=3 and ?9z1z2+4z1z3+z2z3?=12 then the value of ?z1+z2+z3? is :

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

A

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5. (p∨q) ∧ (p∨r) GßÓ ö©´ AmhÁøn°À EÒÍ {øµPÎß GsoUøP :

(A) 6 (B) 9 (C) 3 (D) 8

The number of rows in the truth table of (p∨q) ∧ (p∨r) is :

(a) 6 (b) 9 (c) 3 (d) 8

→ → →
6. β ©ØÖ® γ BQ¯øÁ Aø©US® uÍzvÀ α Aø©¢xÒÍx GÛÀ,

(A)  α , β , γ  = 0 (B)  α , β , γ  = 1
→ → → → → →

(C)  α , β , γ  = 2 → → → 
→ → →
(D)  α , β , γ  =− 1

→ → →
If a vector α lies in the plane of β and γ , then :

(a) → → → (b) → → → 
α , β , γ  =0 α , β , γ  =1

(c) → → → (d) → → → 
α , β , γ  =2  α , β , γ  =− 1

7. y=Ae x+Be −x, C[S A, B Gß£Ú H÷uÝ® C¸ ©õÔ¼PÒ, GÝ® ÁøÍÁøµz
öuõSv°ß ÁøPUöPÊa \©ß£õk :

dy d2 y dy d2 y
(A) +y =0 (B) 2
+y =0 (C) −y =0 (D) −y =0
dx dx dx dx 2

The differential equation of the family of curves y=Aex+Be−x, where A and B are arbitrary
constants is :

dy d2 y dy d2 y
(a) +y =0 (b) 2
+y =0 (c) −y =0 (d) −y =0
dx dx dx dx 2

A [ v¸¨¦P / Turn over

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1
8. f (x) = &ß Qøh©mhz öuõø»z öuõk÷Põk :
x

(A) x=c (B) y=0 (C) y=c (D) x=0

1
The horizontal asymptote of f ( x ) = is :
x

(a) x=c (b) y=0 (c) y=c (d) x=0

x
9. f (x) = GÛÀ, Auß ÁøP±k :
x +1

1 −1 −1 1
(A) x + 1 dx (B) ( x + 1)2 dx (C) x + 1 dx (D) ( x + 1 )2
dx

x
If f ( x ) = , then its differential is given by :
x +1

1 −1 −1 1
(a) dx (b) dx (c) dx (d) dx
x +1 (x + 1 )2 x +1 ( x + 1 )2

10. (1+i) (1+2i) (1+3i) ... (1+ni)=x+iy GÛÀ, 2⋅5⋅10 ... (1+n2) &ß ©v¨¦ :

(A) x2+y2 (B) 1 (C) 1+n2 (D) i
If (1+i) (1+2i) (1+3i) ... (1+ni)=x+iy then 2⋅5⋅10 ... (1+n2) is :

(a) x 2+y 2 (b) 1 (c) 1+n 2 (d) i

11. x3−3x2, x∈[0, 3] GßÓ \õº¤ØS ÷µõ¼ß ÷uØÓzøu {øÓÄ ö\´²® Gs :

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

The number given by the Rolle’s theorem for the function x3−3x2, x∈[0, 3] is :

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

A

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12. x2−3=5x+3y GßÓ T®¦ ÁøÍÂß ÁøP :

(A) Av£µÁøͯ® (B) }ÒÁmh®

(C) Ámh® (D) £µÁøͯ®
The type of conic section for x2−3=5x+3y is :

(a) hyperbola (b) ellipse

(c) circle (d) parabola

 5 3 
13. A Gߣx §a]¯©ØÓU ÷PõøÁ Ao ©ØÖ® A−1 =   GÛÀ, (A ) =
T −1
−2 −1

−1 −3  −5 3   5 −2   5 3
(A)  (B)  (C)  (D)
2 5   2 1
 
 3 −1 
−2 −1
 

 5 3 
If A is a non-singular matrix such that A−1 =  T −1
 , then (A ) =
− 2 − 1 

−1 −3  −5 3   5 −2   5 3
(a) 2 (b) (c) (d)
 5   2 1
 
 3 −1 
 
−2 −1
 

14.
→
( → → →
r = i +2 j −3 k +t 2 i + j −2 k ) ( → → →
) GßÓ ÷PõmiØS® r ⋅ ( i + j ) + 4 = 0 GßÓ
→ → →

uÍzvØS® Cøh¨£mh ÷Põn® :

(A) 458 (B) 08 (C) 908 (D) 308

→
( →
The angle between the line r = i + 2 j − 3 k + t 2 i + j − 2 k
→ →
) ( → → →
) and the plane
→
( → →
r ⋅ i + j + 4 = 0 is : )
(a) 458 (b) 08 (c) 908 (d) 308

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7412 6

1 π
15. sin−1 x + cot−1   = GÛÀ x &ß ©v¨¦ :
2 2

2 1 3 1
(A) 5
(B) 2 (C) (D) 5
2

1 π
If sin−1 x + cot−1   = , then x is equal to :
2 2

2 1 3 1
(a) (b) (c) (d)
5 2 2 5

1
16. x3+px2+qx+r &US α, β ©ØÖ® γ GߣøÁ §a]¯©õUQPÒ GÛÀ, ∑ α &ß ©v¨¦ :
q q q p
(A) (B) − (C) − p (D) −
r r r
1
If α, β and γ are zeros of x3+px2+qx+r then ∑ α is :
q q q p
(a) (b) − (c) −
p (d) −
r r r

17. Var(3) &ß ©v¨¦ :

(A) 0 (B) 3 (C) Var(3) (D) 9
The value of Var(3) is :
(a) 0 (b) 3 (c) Var(3) (d) 9

18. n=25 ©ØÖ® p=0.8 GßÖ EÒÍ D¸Ö¨¦ £µÁÀ öPõsh \©Áõ´¨¦ ©õÔ X GÛÀ,
X &ß vmh»UPzvß ©v¨¦ :

(A) 3 (B) 6 (C) 2 (D) 4
The random variable X has binomial distribution with n=25 and p=0.8 then standard
deviation of X is :

(a) 3 (b) 6 (c) 2 (d) 4

A

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19. A, B ©ØÖ® C Gß£Ú ÷|º©õÖ PõnzuUPÁõÖ H÷uÝö©õ¸ Á›ø\°À C¸¨¤ß
¤ßÁ¸ÁÚÁØÔÀ Gx Esø©¯À» ?
(A) det A−1=(det A)−1 (B) adj A=?A?A−1
(C) (ABC)−1=C−1B−1A−1 (D) adj (AB)=(adj A) (adj B)
If A, B and C are invertible matrices of some order, then which one of the following is not
true ?
(a) det A−1=(det A)−1 (b) adj A=?A?A−1
(c) (ABC)−1=C−1B−1A−1 (d) adj (AB)=(adj A) (adj B)

20. x3+64 &ß J¸ §a]¯©õUQ :

(A) 4i (B) 0 (C) −4 (D) 4
A zero of x3+64 is :
(a) 4i (b) 0 (c) −4 (d) 4

£S-v & II / PART - II

SÔ¨¦ : GøÁ÷¯Ý® HÊ ÂÚõU- P - Ð US Âøh- ¯ - Î U- P - Ä ®. ÂÚõ Gs 30 &US
Pm-hõ-¯-©õP Âøh-¯-ÎU-P-Ä®. 7x2=14
Note : Answer any seven questions. Question No. 30 is compulsory.

12
n
21. _¸USP : ∑ i
n=1

12
Simplify : ∑ in
n=1

22. 2x2−7x+13=0 GÝ® C¸£ia \©ß£õmiß ‰»[PÒ α ©ØÖ® β GÛÀ α2 ©ØÖ®
β2 BQ¯ÁØøÓ ‰»[PÍõPU öPõsh J¸ C¸£ia \©ß£õmøh E¸ÁõUPÄ®.

If α and β are the roots of the quadratic equation 2x2−7x+13=0, construct a quadratic
equation whose roots are α2 and β2.

A [ v¸¨¦P / Turn over

7412 8

23. f (x)=x 2+3x GßÓ \õº¤ØS df PõsP. ©ØÖ® x=3 ©ØÖ® dx=0.02 GÝ® ÷£õx
df &I ©v¨¤kP.

Find df for f (x)=x2+3x and evaluate it for x=3 and dx=0.02.

24. Bv ÁȯõPa ö\À¾® ÷|ºU÷PõkPÎß öuõSv°ß ÁøPUöPÊa \©ß£õmøhU
PõsP.

Find the differential equation for the family of all straight lines passing through the origin.

 2(x − 1) 1< x < 2
25. \©Áõ´¨¦ ©õÔ X &ß {PÌuPÄ Ahºzv \õº¦ f ( x ) = 
 0 ¤Ó ©v¨¦ PÐUS
\µõ\› PõsP.

For the random variable X with the given probability mass function.

 2(x − 1) 1 < x < 2
f (x) = 
 0 otherwise

Find the Mean.

26. ø©¯® (−3, −4) ©ØÖ® Bµ® 3 A»SPÒ öPõsh Ámhzvß ö£õxÁiÁa
\©ß£õk PõsP.

Find the general equation of a circle with centre (−3, −4) and radius 3 units.

A

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−1 3 
27.  4 −7  GßÓ Ao°ß Aozuµ® PõsP.
 
 3 −4 

−1 3 
Find the rank of the matrix  4 −7  .
 
 3 −4 

π
2
28. ©v¨¤kP : ∫ sin 10 x dx
0

π
2

∫ sin x dx
Evaluate : 10

0

x 2 − 3x + 2
29. PnUQkP : lim
x →1 x 2 − 4x + 3

x 2 − 3x + 2
Evaluate : lim
x →1 x 2 − 4x + 3

→ → → → → → → →
30. 2 i − j +3 k , i − j , 3 i − j +6 k BQ¯ öÁUhºPÒ J¸ uÍ öÁUhºPÍõS®
GÚ {¹¤UPÄ®.

→ → → → → → → →
Show that the vectors 2 i − j + 3 k , i − j and 3 i − j + 6 k are coplanar.

A [ v¸¨¦P / Turn over

7412 10

£S-v & III / PART - III

SÔ¨¦ : GøÁ÷¯Ý® HÊ ÂÚõU- P - Ð US Âøh- ¯ - Î U- P - Ä ®. ÂÚõ Gs 40 &US
Pm-hõ-¯-©õP Âøh-¯-ÎU-P-Ä®. 7x3=21

Note : Answer any seven questions. Question No. 40 is compulsory.

 1 
31. cot−1  −1
 = sec x , ?x? > 1 GÚU PõmkP.
2
 x −1 

Show that cot−1  
1 −1
2  = sec x , ?x? > 1.
 x −1 

32. x2+6x+4y+5=0 GßÓ £µÁøͯzvØS (1, −3) GßÓ ¦ÒΰÀ öuõk÷Põk ©ØÖ®
ö\[÷Põmka \©ß£õkPøÍU PõsP.
Find the equation of tangent and normal to the parabola x2+6x+4y+5=0 at (1, −3).

→ → → → → →
33.  a − b , b − c , c − a  = 0 GÚ {ÖÄP.

→ → → → → →
Prove that  a − b , b − c , c − a  = 0 .

x2 + y 2
34. u(x , y ) = GÛÀ, x ∂u + y ∂u = 3 u GÚ {ÖÄP.
x +y ∂x ∂y 2

x2 + y 2 ∂u ∂u 3
If u(x , y ) = , prove that x +y = u.
x +y ∂x ∂y 2

35. Cµsk ªøP GsPÎß Tmkz öuõøP 12, ÷©¾® Auß ö£¸USz öuõøP
ö£¸©® GÛÀ, A¢u GsPøÍU PõsP.
Find two positive numbers whose sum is 12 and their product is maximum.

A

11 7412

3π
1
36. ©v¨¦ PõsP : ∫ 8
π 1 + tanx
dx
8

3π
1
Evaluate : ∫ 8
π 1 + tanx
dx
8

3 3
 1+i   1−i 
37.   −   &I ö\ÆÁP ÁiÂÀ _¸USP.
1 − i 1+ i

3 3
 1+i   1−i 
Simplify   −   into rectangular form.
1 − i 1+ i

38. wºUP : (1 + x2 ) dy = 1 + y 2
dx

dy
Solve : (1 + x 2 ) = 1 + y2
dx

39. ‰ßÖ ^µõÚ |õn¯[PÒ J÷µ ÷|µzvÀ _sh¨£kQßÓÚ. QøhUS® uø»PÎß
GsoUøPUPõÚ {PÌuPÄ {øÓ \õº¤øÚU PõsP.
Three fair coins are tossed simultaneously. Find the probability mass function for number of
heads occurred.

 2 −1 3 
40. A = −5 3 1  GÛÀ ?adj (adj A)? &øÁ PõsP.
 
−3 2 3 

 2 −1 3 
If A = −5 3 1  , then find ?adj (adj A)?.
 
−3 2 3 

A [ v¸¨¦P / Turn over

7412 12

£S-v & IV / PART - IV
SÔ¨¦ : AøÚz-x ÂÚõU-P-ÐU-S® Âøh-¯-ÎU-P-Ä®. 7x5=35
Note : Answer all the questions.

41. (A) y=x2 ©ØÖ® y=(x−3)2 BQ¯ ÁøÍÁøµPÐUS Cøh¨£mh ÷Põn® PõsP.
AÀ-»-x

(B) wºUP : tan−1 
x −1  −1  x + 1  π
 + tan  =
 x −2   x +2  4
(a) Find the angle between the curves y=x2 and y=(x−3)2.
OR

 x −1  −1  x + 1  π
(b) Solve : tan−1   + tan  =
 x −2   x +2  4

42. (A) Kº AÖ£UP £Pøh°ß J¸ £UPzvÀ ‘1’ GÚÄ®, C¸ £UP[PÎÀ ‘3’ GÚÄ®
©ØÖ® HøÚ¯ ‰ßÖ £UP[PÎÀ ‘5’ GÚÄ® SÔUP¨£mkÒÍx. £Pøh
C¸ •øÓ Ã\¨£kQÓx. C¸ •øÓ Ã\¨£mhvß ö©õzu GsoUøPø¯
X SÔUQÓx GÛÀ,
(i) {PÌuPÄ {øÓ \õº¦
(ii) SÂÄ £µÁÀ \õº¦
(iii) P(4 ≤ X < 10) &ø¯ PõsP.
AÀ-»-x

(B) z=x+iy GßÓ H÷uÝ® J¸ P»¨ö£s Im 
2z + 1 
 = 0 GÝ©õÖ Aø©¢uõÀ
 iz + 1 
z &ß {¯©¨£õøu 2x2+2y2+x−2y=0 GÚU PõmkP.
(a) A six sided die is marked ‘1’ on one face, ‘3’ on two of its faces, and ‘5’ on remaining
three faces. The die is thrown twice. If X denotes the total score in two throws, find :
(i) the probability mass function
(ii) the cumulative distribution function
(iii) P(4 ≤ X < 10)
OR

 2z + 1 
(b) If z=x+iy is a complex number such that Im   = 0 , show that the locus of z is
 iz + 1 
2x 2+2y 2+x−2y=0.

A

13 7412

43. (A) uø»RÇõP øÁUP¨£mh J¸ ÷|ºÁmhU T®¤ß ÁiÂÀ EÒÍ J¸ }º{ø»z
öuõmi°ß BÇ® 12 «mhº ©ØÖ® ÷©¾ÒÍ Ámhzvß Bµ® 5 «mhº GßP.
{ªhzvØS 10 PÚ«mhº ÷ÁPzvÀ }º £õ´a\¨£kQÓx GÛÀ, 8 «mhº
BÇzvÀ }º C¸US®÷£õx }›ß BÇ® AvP›US® Ãu® GßÚ ?
AÀ-»-x
(B) öÁUhº •øÓ°À sin(α−β)=sinα cosβ−cosα sinβ GÚ {ÖÄP.
(a) A conical water tank with vertex down of 12 meters height has a radius of 5 meters at
the top. If water flows into the tank at a rate 10 cubic m/min, how fast is the depth of
the water increases when the water is 8 metres deep ?

OR

(b) Prove by vector method that sin(α−β)=sinα cosβ−cosα sinβ.

44. (A) uøµ©mhzv¼¸¢x 7.5 « E¯µzvÀ uøµUS Cøn¯õP¨ ö£õ¸zu¨£mh
J¸ SÇõ°¼¸¢x öÁÎ÷¯Ö® }º uøµø¯z öuõk® £õøu J¸
£µÁøͯzøu HØ£kzxQÓx. ÷©¾® C¢u¨ £µÁøͯ¨ £õøu°ß •øÚ
SÇõ°ß Áõ°À Aø©QÓx. SÇõ´ ©mhzvØS 2.5 « R÷Ç }›ß £õ´ÁõÚx
SÇõ°ß •øÚ ÁȯõPa ö\À¾® {ø» SzxU ÷PõmiØS 3 « yµzvÀ
EÒÍx GÛÀ, SzxU ÷Põmi¼¸¢x GÆÁÍÄ yµzvØS A¨£õÀ }µõÚx
uøµ°À ÂÊ® GߣøuU PõsP.
AÀ-»-x
(B) ¤ßÁ¸® ÷|›¯À ÁøPUöPÊa \©ß£õmiß wºÄ PõsP

dy y
+ = sinx .
dx x

(a) Assume that water issuing from the end of horizontal pipe, 7.5 m above the ground,
describes a parabolic path. The vertex of the parabolic path is at the end of the pipe.
At a position 2.5 m below the line of the pipe, the flow of water has curved outward
3 m beyond the vertical line through the end of the pipe. How far beyond this vertical
line will the water strike the ground ?

OR

dy y
(b) Solve the Linear differential equation + = sinx .
dx x

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7412 14

45. (A) ~sq°ºPÎß ö£¸UPzvÀ, £õUj›¯õUPÎß GsoUøP°ß ö£¸UP
Ãu©õÚx AvÀ Põn¨£k® £õUj›¯õUPÎß GsoUøP°ß ÂQu©õP
EÒÍx. C¨ö£¸UPzuõÀ £õUj›¯õÂß GsoUøP 5 ©o ÷|µzvÀ
•®©h[PõQÓx GÛÀ, 10 ©o ÷|µ •iÂÀ £õUj›¯õUPÎß GsoUøP
GßÚÁõP C¸US® ?

AÀ-»-x

x −2 y −2 z −1
(B) = = GßÓ ÷Põmøh EÒÍhUQ¯x® x + 1 = y − 1 = z + 1
2 3 3 3 2 1
GßÓ ÷PõmiØS Cøn¯õÚx©õÚ uÍzvß öÁUhº ©ØÖ® Põºj]¯ß
\©ß£õkPøÍU PõsP.
(a) The rate of increase in the number of bacteria in a certain bacteria culture is proportional
to the number present. Given that the number triples in 5 hours, find how many
bacteria will be present after 10 hours ?

OR

x −2 y −2 z −1
(b) Find the vector and Cartesian equations of the plane containing = =
2 3 3
y −1
and parallel to the line x + 1 = =
z +1 .
3 2 1

x2y2
46. (A) 2 + = 1 GßÓ }ÒÁmhzvÚõÀ Aøh£k® Aµ[Pzvß £µ¨ø£U PõsP.
a b2

AÀ-»-x

(B) y2−4y−8x+12=0 GßÓ £µÁøͯzvß •øÚ, S¯®, C¯USÁøµ°ß
\©ß£õk PõsP.
(a) Find the area of the region bounded by the ellipse

x2 y2
+ =1
a2 b2

OR

(b) Find the vertex, focus and equation of the directrix of the parabola y2−4y−8x+12=0.

A

15 7412

47. (A) p ↔ q ≡ ((~p)∨q) ∧ ((~q)∨p) GÚU PõmkP.
AÀ-»-x
(B) ¤ßÁ¸® ÷|›¯a \©ß£õkPÎß öuõS¨ø£ Qµõ©›ß Âv¨£i wºUP.
3 4 2
− − − 1 = 0,
x y z

1 2 1
+ + − 2 = 0,
x y z

2 5 4
− − +1=0
x y z

(a) Show that
p ↔ q ≡ ((~p)∨q) ∧ ((~q)∨p)
OR
(b) Solve the system of linear equations by Cramer’s Rule.

3 4 2
− − − 1 = 0,
x y z

1 2 1
+ + − 2 = 0,
x y z

2 5 4
− − +1=0
x y z

-oOo-

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