TN 12th Question Paper 2023 Maths

Tamil Nadu
State Board

2023
QUESTION
PAPER

No. of Printed Pages : 12
6612
£vÄ Gs
A Register Number

!6612Mathematics!
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 ©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

6612 2

1. n Á›ø\²øh¯ J¸ \xµ AoUS ÷|º©õÖ Põnz ÷uøÁ¯õÚ ©ØÖ® ÷£õx©õÚ
{£¢uøÚ :
(A) ρ(A) > n (B) ρ(A)=n (C) ρ(A) ≠ n (D) ρ(A) < n
A square matrix A of order n has inverse if and only if :
(a) ρ(A) > n (b) ρ(A)=n (c) ρ(A) ≠ n (d) ρ(A) < n

2. Bv¨¦Òΰ¼¸¢x 3x−6y+2z+7=0 GßÓ uÍzvØS EÒÍ öuõø»Ä :
(A) 2 (B) 0 (C) 3 (D) 1
Distance from the origin to the plane 3x−6y+2z+7=0 is :
(a) 2 (b) 0 (c) 3 (d) 1

3. 3 cos−1x=cos−1 (4x3−3x) GÛÀ :

1  1  1 
(A) x
 2 , 1  (B) x
 2 , 1 (C) x e (−∞, 1] (D) x
 , ∞
2 
If 3 cos−1x=cos−1 (4x3−3x),

1  1  1 
(a) x
 , 1 (b) x
 , 1 (c) x e (−∞, 1] (d) x
 , ∞
2  2  2 

dy y
4. = GÝ® ÁøPUöPÊ \©ß£õmiß ö£õxz wºÄ :
dx x

(A) y=kx (B) xy=k (C) logy=kx (D) y=k logx
dy y
The general solution of the differential equation = is :
dx x
(a) y=kx (b) xy=k (c) logy=kx (d) y=k logx

5. öPõkUP¨£mh ¦Òΰ¼¸¢x y2=4ax GßÓ £µÁøͯzvØS Áøµ¯¨£k®
ö\[÷PõkPÎß GsoUøP :
(A) 3 (B) 2 (C) 0 (D) 1
The number of normals that can be drawn from a point to the parabola y2=4ax

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

A

3 6612

→ → → → → 
6. a ©ØÖ® b Gß£Ú Cøn öÁUhºPÒ GÛÀ  a , c , b  &ß ©v¨¦ :
(A) 1 (B) 2 (C) 0 (D) −1
→ → → → → 
If a and b are parallel vectors then  a , c , b  is equal to :
(a) 1 (b) 2 (c) 0 (d) −1

7. &À sin 4 x−2 sin 2 x+1 &ø¯ {øÓÄ ö\´²® ö©´ö¯sPÎß
[0, 2π]
GsoUøP :
(A) 1 (B) 2 (C) ∞ (D) 4
The number of real numbers in [0, 2π] satisfying sin4 x−2 sin2 x+1 is :
(a) 1 (b) 2 (c) ∞ (d) 4

8. 0, 1 ©ØÖ® 2 BQ¯ ©v¨¦PÎÀ JßøÓ X öPõÒQÓx GßP. H÷uõ J¸ ©õÔ¼
1
k &ÂØS, P(X=i) = kP(X=i−1), i=1, 2 ©ØÖ® P(X = 0) = GÛÀ k &ß ©v¨¦ :
7
(A) 3 (B) 1 (C) 4 (D) 2
Suppose that X takes on one of the values 0, 1, 2. If for some constant k,
1
P(X=i) = kP(X=i−1) for i=1, 2 and P(X = 0) = , then the value of k is :
7
(a) 3 (b) 1 (c) 4 (d) 2

9. x2 e−2x, x > 0 GßÓ \õº¤ß ö£¸© ©v¨¦ :
1 1 4 1
(A) 2 (B) e (C) 4 (D) 2e
e e
The maximum value of the function x2 e−2x, x > 0 is :
1 1 4 1
(a) 2 (b) (c) 4 (d)
e e e 2e

ab
10. * GßÓ D¸Ö¨¦a ö\¯¼ a * b = GÚ Áøµ¯ÖUP¨£kQÓx. * Guß «x D¸Ö¨¦a
7
ö\¯¼ BPõx ?
(A) R (B) Q+ (C) C (D) Z
ab
The operation * defined by a * b = is not a binary operation on :

7
(a) R (b) Q+ (c) C (d) Z

A [ v¸¨¦P / Turn over

6612 4

11. 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

12. Bv°À y2=x ©ØÖ® x2=y GßÓ ÁøÍÁøµPÐUS Cøh¨£mh ÷Põn® :
−1  3  4
tan−1  
π π
(A) 2 (B) tan   (C) (D)
4 4 3
Angle between the curves y2=x and x2=y at the origin is :

3 4
tan−1   tan−1  
π π
(a) (b) (c) (d)
2 4 4 3

13. ?adj (adjA)?=?A?16 GÛÀ, \xµ Ao A &ß Á›ø\¯õÚx :
(A) 2 (B) 3 (C) 5 (D) 4
?adj (adjA)?=?A?16, then the order of the square matrix A is :
(a) 2 (b) 3 (c) 5 (d) 4

8 8
 1+ i   1− i 
14.   +  &ß ©v¨¦ :
 2   2 
(A) 8 (B) 4 (C) 2 (D) 6
8 8
 1+ i   1− i 
The value of   +  is :
 2   2 
(a) 8 (b) 4 (c) 2 (d) 6

1+ z
15. ?z?=1 GÛÀ &ß ©v¨¦ :
1+ z
1
(A) z (B) z (C) 1 (D) z

1+ z
If ?z?=1, then the value of is :
1+ z

1
(a) (b) z (c) 1 (d) z
z

A

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16. GßÓ ÁøÍÁøµ°ß G¢u x &B¯zöuõø»ÂÀ Áøµ¯¨£mh
f ( x ) = 8 − 2x
öuõk÷Põmiß \õ´Ä −0.25 BP C¸US® ?
(A) −2 (B) −8 (C) 0 (D) −4
The abscissa of the point on the curve f ( x ) = 8 − 2x at which the slope of the
tangent is −0.25 ?
(a) −2 (b) −8 (c) 0 (d) −4

π/3
17. ∫ tan x dx &ß ©v¨¦ :
0

(A) −log 2 (B) log 2 (C) −log 3 (D) log 3
π/3
The value of ∫ tan x dx is :
0

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

n
18. ∑ n Cr (−1)r xr GÝ® £À¾Ö¨¦U ÷PõøÁ°ß ªøPö¯s §a]¯©õUQPÎß
r= 0
GsoUøP :
(A) < n (B) 0 (C) r (D) n
n
The number of positive zeros of the polynomial ∑ n Cr (−1)r xr is :
r= 0

(a) <n (b) 0 (c) r (d) n

 −1 
19. sin−1   &ß •ußø© ©v¨¦ :
 2 

−π −π π
(A) 6
(B) 0 (C) 2
(D) 2

 −1 
The Principal value of sin−1   is :
 2 

−π −π π
(a) (b) 0 (c) (d)
6 2 2

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6612 6

x2 y2
20. + = 1 GßÓ }ÒÁmhzvÝÒ Áøµ¯¨£k® ªP¨ ö£›¯ ö\ÆÁPzvß
a2 b2
£µ¨¦ :
a
(A) ab (B) 2ab (C) b (D) ab

x2 y2
Area of the greatest rectangle inscribed in the ellipse + = 1 is :
a2 b2

a
(a) ab (b) 2ab (c) (d) ab
b

£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. ?z?=2 GÛÀ, 3 ≤ ?z+3+4i? ≤ 7 GÚU PõmkP.
If ?z?=2, show that 3 ≤ ?z+3+4i? ≤ 7

p q n
22. lx2+nx+n=0 GÝ® \©ß£õmiß ‰»[PÒ p ©ØÖ® q GÛÀ, + + =0
q p l
GÚU PõmkP.
p q n
If p and q are the roots of the equation lx2+nx+n=0, show that + + =0
q p l

23. y=4x+c GßÓ ÷|ºU÷Põk x2+y2=9 GßÓ Ámhzvß öuõk÷Põk GÛÀ, c &ß
©v¨¦U PõsP.
If y=4x+c is a tangent to the circle x2+y2=9, find c.

24. 10 ö\.«. Bµ® EÒÍ ÷PõÍzvß Bµ® 0.1 ö\.«. SøÓQßÓx GÛÀ Auß PÚ
AÍÄ ÷uõµõ¯©õP GÆÁÍÄ SøÓ²® ?

If the radius of a sphere with radius 10 cm, has to decrease by 0.1 cm, approximately
how much will its volume decrease ?

A

7 6612

∞
25. ©v¨¤kP : ∫ 2 1 2 dx , a > 0, b e R.
a +xb

∞
1
Evaluate : ∫ d x , a > 0, b e R.
2 2
b a +x

26. Bv¨¦Òΰ¼¸¢x 7 A»SPÒ öuõø»ÂÀ EÒÍx®, ö\[Szvß vø\
ÂQu[PÒ 3, −4, 5 öPõshx©õÚ uÍzvß öÁUhº \©ß£õk PõsP.
Find the vector equation of a plane which is at a distance of 7 units from the
origin having 3, −4, 5 as direction ratios of a normal to it.

0 1  1 1
27. A=   , B=   BQ¯ Cµsk® J÷µ ÁøP¯õÚ §¼¯ß AoPÒ GÛÀ,
 1 1  0 1
A ∨ B ©ØÖ® A ∧ B BQ¯ÁØøÓU PõsP.

0 1  1 1
Let A =   , B=   be any two Boolean matrices of the same type. Find
 1 1  0 1
A ∨ B and A ∧ B.

 cosθ −sinθ 
28.  sinθ cosθ  Gߣx ö\[Szx Ao GÚ {ÖÄP.
 

 cosθ −sinθ 
Prove that   is orthogonal.
 sinθ cosθ 

29. y=x 2 +3x−2 GßÓ ÁøÍÁøµUS (1, 2) GßÓ ¦ÒΰÀ öuõk÷Põmiß
\©ß£õmøhU PõsP.
Find the equation of tangent to the curve y=x2+3x−2 at the point (1, 2).

30. ecosθ+i sinθ Gߣøu a+ib GßÓ ÁiÂÀ GÊxP.

Express ecosθ+i sinθ in a+ib form.

A [ v¸¨¦P / Turn over

6612 8

£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.

31. •øÚ (−1, −2), Aa_ y-Aa_US Cøn ©ØÖ® (3, 6) ÁÈaö\À¾® £µÁøͯzvß
\©ß£õk PõsP.
Find the equation of the parabola with vertex (−1, −2), axis parallel to y-axis and
passing through (3, 6).

32. `›¯Û¼¸¢x §ª°ß AvP£m\® ©ØÖ® SøÓ¢u£m\ yµ[PÒ •øÓ÷¯
152×106 Q.«. ©ØÖ® 94.5×106 Q.«. }ÒÁmh¨ £õøu°ß J¸ S¯zvÀ `›¯ß
EÒÍx. `›¯ÝUS® ©ØöÓõ¸ S¯zvØS©õÚ yµ® PõsP.
The maximum and minimum distances of the Earth from the Sun respectively
are 152×106 km and 94.5×106 km. The Sun is at one focus of the elliptical orbit.
Find the distance from the Sun to the other focus.

π
33. x &ß G¢u ©v¨¤ØS, \©Ûø» < cos−1 (3x−1) < π ö©´¯õS® ?
2

π
For what value of x, the inequality < cos−1 (3x−1) < π holds ?
2

x +3 y −1
34. = =− z GßÓ ÷|ºU÷Põk B¯ Aa_UPÐhß HØ£kzx®
2 2
÷Põn[PøÍU PõsP.
x +3 y −1
Find the angle made by the straight line = =− z with coordinate axes.
2 2
2
35. (123) 3 &ß ÷uõµõ¯ ©v¨¤øÚ ÷|›¯À ÷uõµõ¯ ©v¨¥mk •øÓ°À PõsP.

2
Use the linear approximation to find an approximate value of (123) 3 .

wºUP : x cosy dy=ex(x logx+1)dx

36.
Solve : x cosy dy=ex(x logx+1)dx

A

9 6612

 cosα 0 sin α 
37. F(α ) =  0 1 0  GÛÀ, [F(α)]−1=F(−α) GÚU PõmkP.
 
− sin α 0 cosα 

 cosα 0 sin α 
If F(α ) =  0 1 0  , show that [F(α)]−1=F(−α)
 
− sin α 0 cosα 

38. p → q ©ØÖ® q → p BQ¯øÁPÒ \©õÚ©ØÓøÁ GÚU PõmkP.
Show that p → q and q → p are not equivalent.

39. z=(2+3i) (1−i) GÛÀ z−1 &ø¯U PõsP.
If z=(2+3i) (1−i), then find z−1.

40. a+b+c=0 ©ØÖ® a, b, c BQ¯øÁ ÂQu•Ö GsPÒ GÛÀ
(b+c−a)x 2 +(c+a−b)x+(a+b−c)=0 GÝ® \©ß£õmiß ‰»[PÒ ÂQu•Ö
GsPÍõS® GÚU PõmkP.
If a+b+c=0 and a, b, c are rational numbers then, prove that the roots of the
equation (b+c−a)x2+(c+a−b)x+(a+b−c)=0 are rational numbers.

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

41. (A) z3+8i=0 GßÓ \©ß£õmøhz wºUP, C[S z e C.
AÀ»x

(B) wºUP : ( 1 + x + xy 2 ) dy + ( y + y 3 ) = 0 .
dx
(a) Solve the equation z3+8i=0, where z e C.
OR

(b) (
Solve : 1 + x + xy 2 ) ddyx + ( y + y 3 ) = 0 .
A [ v¸¨¦P / Turn over

6612 10

42. (A) öÁUhº •øÓ°À cos(α−β)=cosα cosβ+sinα sinβ GÚ {ÖÄP.
AÀ»x
(B) J¸ £õÀ ÂØ£øÚ¯PzvÀ ÂÛ÷¯õQUP¨£k® £õ¼ß AÍÄ \©Áõ´¨¦
©õÔ X GßP. SøÓ¢u£m\® 200 ¼mhºPÒ ©ØÖ® AvP£m\®
k 200 ≤ x ≤ 600
600 ¼mhºPÐhß {PÌuPÄ Ahºzv \õº¦ f ( x ) = 
0 ¤Ó ©v¨¦ PÐUS

(i) k ©v¨¦ PõsP.

(ii) £µÁÀ \õº¦ PõsP.
(iii) 300 ¼mhºPÒ ©ØÖ® 500 ¼mhºPÐUQøh÷¯ vÚ\› ÂØ£øÚ
C¸¨£uØPõÚ {PÌuPÄ PõsP.
(a) Using vector method, prove that cos(α−β)=cosα cosβ+sinα sinβ
OR
(b) Suppose the amount of milk sold daily at a milk booth is distributed with a
minimum of 200 litres and a maximum of 600 litres with probability density
k 200 ≤ x ≤ 600
function of random variable X is f ( x ) =  .
 0 otherwise

Find (i) the value of k
(ii) the distribution function
(iii) the probability that daily sales will fall between 300 litres and
500 litres.

43. (A) 18x2+12y2−144x+48y+120=0 GßÓ T®¦ ÁøÍÂß ÁøPø¯ PshÔ¢x,
AÁØÔß ø©¯®, S¯[PÒ ©ØÖ® •øÚPøÍU PõsP.
AÀ»x
(B) cos−1x+cos−1y+cos−1z=π ©ØÖ® 0 < x, y, z < 1 GÛÀ, x2+y2+z2+2xyz=1
GÚU PõmkP.
(a) Identify the type of conic and find centre, foci and vertices of
18x2+12y2−144x+48y+120=0
OR

(b) If cos−1x+cos−1y+cos−1z=π and 0 < x, y, z < 1, show that
x2+y2+z2+2xyz=1

A

11 6612

44. (A) J¸ ]ÖÁß y=ax2+bx+c GßÓ £õøu°À (−6, 8), (−2, −12) ©ØÖ® (3, 8)
GÝ® ¦ÒÎPÒ ÁȯõP ö\ÀQÓõß. P(7, 60) GßÓ ¦ÒΰÀ EÒÍ
AÁÝøh¯ |s£øÚ \¢vUP ¸®¦QÓõß. AÁß, AÁÝøh¯ |s£øÚ
\¢v¨£õÚõ ? (Põìê¯ß }UPÀ •øÓø¯ £¯ß£kzxP.)
AÀ»x
(B) x 2 +4y 2 =8 GßÓ }ÒÁmh•® x 2 −2y 2 =4 GßÓ Av£µÁøͯ•®
ö\[SzuõP öÁmiU öPõÒЮ GÚ {ÖÄP.
(a) A boy is walking along the path y=ax2+bx+c through the points (−6, 8),
(−2, −12) and (3, 8). He wants to meet his friend at P(7, 60). Will he meet
his friend ? (Use Gaussian Elimination method)

OR

(b) Prove that the ellipse x2 +4y2=8 and the hyperbola x2−2y2 =4 intersect
orthogonally.

45.
→
( ∧ ∧
(A) r = i − j + 3 k + t 2 i − j + 4 k
∧
) ( ∧ ∧ ∧
) GßÓ ÷Põmøh EÒÍhUQ¯x®
→ ∧
( ∧ ∧
)
r ⋅ i + 2 j + k = 8 GßÓ uÍzvØSa ö\[SzuõÚx©õÚ uÍzvß xøn¯»S
ÁiÁ öÁUhº ©ØÖ® Põºj]¯ß \©ß£õkPøÍU PõsP.
AÀ»x

1
(B) 6x 4 −5x 3 −38x 2 −5x+6=0 GÝ® \©ß£õmiß J¸ wºÄ GÛÀ,
3
\©ß£õmiß wºÄ PõsP.
(a) Find the parametric form of Vector equation and Cartesian equations of the

plane containing (
the line
→ ∧ ∧ ∧
) ( ∧
r = i − j +3k +t 2 i − j +4 k
∧ ∧
) and

perpendicular to the plane r ⋅ ( i + 2 j + k ) = 8 .
→ ∧ ∧ ∧

OR

1
(b) Solve the equation 6x 4 −5x 3 −38x 2 −5x+6=0 if it is known that is a
3

solution.

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6612 12

46. (A) p → (¬q ∨r) ≡ ¬p∨(¬q ∨r) Gߣøu ö©´ø© AmhÁønø¯¨ £¯ß£kzv
{ÖÄP.

AÀ»x

(B) Á¸hzvØS 5% öuõhº Tmk ÃuzvÀ J¸Áº ` 10,000 &zøu Á[Q PnUQÀ
•u½k ö\´QÓõº. 18 ©õu[PÐUS ¤ßÚº AÁº Á[Q PnUQÀ GÆÁÍÄ
öuõøP C¸US® ?
(a) Prove that p → (¬q∨r) ≡ ¬p∨(¬q∨r) using truth table.

OR

(b) Suppose a person deposits ` 10,000 in a bank account at the rate of 5% per
annum compounded continuously. How much money will be in his bank
account 18 months later ?

47. (A) log x GßÓ \õº¤ß «¨ö£¸ ©v¨¦ PõsP.
x

AÀ»x

x2 y2 x y
(B) + =1 GßÓ }ÒÁmhzvØS®, + =1 GßÓ ÷|ºU÷PõmiØS®
a2 b2 a b
ö£õxÁõÚ Aµ[Pzvß £µ¨ø£U PõsP.

log x
(a) Find the maximum value of
x

OR

x2 y2
(b) Find the area of the region common to the ellipse + = 1 and the
a2 b2
x y
straight line + =1 .
a b

-oOo-

A