TN 11th Question Paper 2023 Maths

Tamil Nadu
State Board

2023
QUESTION
PAPER

No. of Printed Pages : 12
6712
A £vÄ Gs
Register Number

!6712IstYearMathematics!
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

6712 2

1. f (x)=x2 GßÓ \õº¦ C¸¦Óa \õº£õP Aø©¯ ÷Áskö©ÛÀ Auß \õº£P•®
xøna \õº£P•® •øÓ÷¯ :
(A) (0, ∞), R (B) R, R (C) [0, ∞), [0, ∞) (D) R, (0, ∞)
The rule f (x)=x2 is a bijection if the domain and the co-domain are given by :
(a) (0, ∞), R (b) R, R (c) [0, ∞), [0, ∞) (d) R, (0, ∞)

2. If f(x)=mx+c ©ØÖ® f (0)=f '(0)=1 GÛÀ f (3) Gߣx :
(A) 3 (B) 1 (C) 4 (D) 2
If f(x)=mx+c and f (0)=f '(0)=1 then f (3) is :
(a) 3 (b) 1 (c) 4 (d) 2

3. J¸ uÍzvÀ EÒÍ 8 ¦ÒÎPÎÀ 4 ¦ÒÎPÒ J¸ ÷Põhø©ÁÚ. H÷uÝ® C¸
¦ÒÎPøÍ Cønzx QøhUS® ÷PõkPÎß GsoUøP :
(A) 39 (B) 45 (C) 38 (D) 23
There are 8 points in a plane and 4 of them are collinear. The number of straight
lines joining any 2 points is :
(a) 39 (b) 45 (c) 38 (d) 23

4. 3x 2 +3y 2 −8x−12y+17=0 GßÓ {¯©¨ £õøu°ß «x Aø©¢v¸US®
¦ÒÎPÒ :
(A) (1, 2) (B) (0, 0) (C) (0, −1) (D) (−2, 3)
The points lie on the locus of 3x2+3y2−8x−12y+17=0.
(a) (1, 2) (b) (0, 0) (c) (0, −1) (d) (−2, 3)

1 0 0 
0 0 0 
5.   GßÓ AoUS ¤ßÁ¸ÁÚÁØÔÀ Gx Esø©¯À» ?
0 0 5 

(A) J¸ ÷©À •U÷Põn ÁiÁ Ao (B) J¸ vø\°¼ Ao
(C) J¸ RÌ •U÷Põn ÁiÁ Ao (D) J¸ ‰ø»Âmh Ao

1 0 0 
 
Which of the following is not true about the matrix 0 0 0  ?
0 0 5 

(a) an upper triangular matrix (b) a scalar matrix
(c) a lower triangular matrix (d) a diagonal matrix

A

3 6712

6. ¤ßÁ¸ÁÚÁØÔÀ Gx \›¯õÚuÀ» ?
3 1
(A) tanθ=25 (B) sinθ = − (C) secθ = (D) cosθ=−1
4 4
Which of the following is not true ?

3 1
(a) tanθ=25 (b) sinθ = − (c) secθ = (d) cosθ=−1
4 4

7. 44 ‰ø»Âmh[PÒ EÒÍ £»÷Põnzvß £UP[PÎß GsoUøP :
(A) 11 (B) 4 (C) 22 (D) 4!
Number of sides of a polygon having 44 diagonals is :
(a) 11 (b) 4 (c) 22 (d) 4!

8. 72n+33n−3⋅3n−1, n e N Gߣx G¢u GsnõÀ ÁS£k® ?
(A) 45 (B) 25 (C) 55 (D) 35
If n e N, then 72n+33n−3⋅3n−1 is always divisible by :
(a) 45 (b) 25 (c) 55 (d) 35

→ → → →
9. AB + BC + DA + CD Gߣuß ©v¨¦ :
→ → → →
(A) 0 (B) AD (C) − AD (D) CA
→ → → →
The value of AB + BC + DA + CD is :

→ → → →
(a) 0 (b) AD (c) − AD (d) CA

sin x
10. ∫ x
dx =

(A) − 2sin x + c (B) 2cos x + c (C) − 2cos x + c (D) 2sin x + c

sin x
∫ x
dx =

(a) − 2sin x + c (b) 2cos x + c (c) − 2cos x + c (d) 2sin x + c

A [ v¸¨¦P / Turn over

6712 4

1 − cos 2x
11. lim =
x →0 x

(A) 1 (B) 2
(C) 0 (D) CÁØÔÀ HxªÀø»
1 − cos 2x
lim =
x →0 x

(a) 1 (b) 2
(c) 0 (d) None of the above

3
12. ∫ sin x dx :
3 cos 3x 3 cos 3x
(A) − cos x + +c (B) − cos x − +c
4 12 4 12
3 sin3x 3 cos 3x
(C) − sin x − +c (D) cos x + +c
4 12 4 12

∫ sin x dx is :
3

3 cos 3x 3 cos 3x
(a) − cos x + +c (b) − cos x − +c
4 12 4 12
3 sin3x 3 cos 3x
(c) − sin x − +c (d) cos x + +c
4 12 4 12

1 1 1
13. , , , ... GßÓ öuõhº•øÓ :
3 3+ 2 3+2 2
(A) Cø\z öuõhº•øÓ
(B) Tmkz öuõhº•øÓ
(C) Tmk ö£¸USz öuõhº•øÓ
(D) ö£¸USz öuõhº•øÓ
1 1 1
The sequence , , , . . . form an :
3 3+ 2 3+2 2
(a) Harmonic Progression
(b) Arithmetic Progression

(c) Arithmetico-Geometric Progression
(d) Geometric Progression

A

5 6712

14. x2−3?x?+2=0 GßÓ \©ß£õmiß ö©´ö¯s wºÄPÎß GsoUøP :

(A) 4 (B) 2 (C) 1 (D) 3
The number of real solutions of the equation x2−3?x?+2=0 are :

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

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.
a
=13,
b
= 5 ©ØÖ® a ⋅ b = 60 GÛÀ
a × b
&ß ©v¨¦ :

(A) 45 (B) 15 (C) 25 (D) 35
→ → → → → →
If
a
=13,
b
= 5 and a ⋅ b = 60 then
a × b
is :

(a) 45 (b) 15 (c) 25 (d) 35

17. cos18+cos28+cos38+ . . .+cos1798=

(A) −1 (B) 0 (C) 89 (D) 1
cos18+cos28+cos38+ . . .+cos1798=

(a) −1 (b) 0 (c) 89 (d) 1

18. 6x2−xy+4cy2=0 GßÓ ÷PõkPÎÀ J¸ ÷PõhõÚx 3x+4y=0 GÛÀ c &ß ©v¨¦ :

(A) 3 (B) −3 (C) 1 (D) −1
If one of the lines given by 6x2−xy+4cy2=0 is 3x+4y=0, then c equals to :

(a) 3 (b) −3 (c) 1 (d) −1

A [ v¸¨¦P / Turn over

6712 6

19. n−1C3+n−1C4 > nC3 GÛÀ :
(A) n > 7 (B) n > 5 (C) n > 4 (D) n > 6
If n−1C3+n−1C4 > nC3 then :
(a) n>7 (b) n>5 (c) n>4 (d) n>6

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

£Sv - II / PART - II
SÔ¨¦ : GøÁ÷¯Ý® HÊ ÂÚõUPÐUS Âøh¯ÎUPÄ®. ÂÚõ Gs 30 &US
Pmhõ¯©õP Âøh¯ÎUPÄ®.
Note : Answer any seven questions. Question No. 30 is Compulsory. 7x2=14
21. A={1, 2, 3, 4}; B={3, 4, 5, 6} GÛÀ n((AcB)×(A1B)×(A∆B)) &IU PõsP.
If A={1, 2, 3, 4}; B={3, 4, 5, 6} find n((AcB)×(A1B)×(A∆B)).

22. (A) J¸ {PÌa] A {PÇ \õuP ÂQu® 5 &US 7 GÛÀ P(A) &I PõsP.
(B) P(B) =2 GÛÀ, {PÌa] B {PÇ \õuP ÂQuzøuU PõsP.
5
(a) The odds that the event A occurs is 5 to 7, then find P(A).
2
(b) Suppose P(B) = . Express the odds that the event B occurs.
5

n(n +1)
23. log a+log a2+log a3+. . . . +log an= log a GÚ {ÖÄP.
2
n(n +1)
Prove that log a+log a2+log a3+. . . . +log an= log a.
2

lim x 2 − 81
24. &ß GÀø» ©v¨ø£U PõsP.
x →3 x −3

x 2 − 81

Evaluate the limit lim .
x →3 x −3

A

7 6712

25. A+B=458 GÛÀ (1+tanA)(1+tanB)=2 GÚ {ÖÄP.
If A+B=458, show that (1+tanA)(1+tanB)=2.

26. nC4=495 GÛÀ n &ß ©v¨ø£U PõsP.
If nC4=495, find the value of n.

27. 3
1001 &ß ©v¨ø£z ÷uõµõ¯©õPU PõsP. (C¸ u\©v¸zu©õP)
Find 3 1001 approximately (two decimal places).

28. 3x 2 +2xy−y 2 =0 GßÓ Cµmøh ÷|º÷PõkPÎß uÛzuÛ ÷|º÷PõkPÎß
\©ß£õkPøÍU PõsP.
Find the separate equation of the pair of straight lines 3x2+2xy−y2=0.

 1 −2 3 
29. A =  1 2 1  J¸ §äâ¯U÷PõøÁ Ao GÛÀ, x &ß ©v¨ø£U PõsP.
 x 2 −3

 1 −2 3 
If A =  1 2 1  is singular, find the value of x.
 x 2 −3

1
30. ©v¨¦U PõsP : lim 6n + 5n  n
n →∞  
1
Evaluate : lim 6n + 5n  n
n →∞  

£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. A GßÓ {PÌa]°ß {PÌuPÄ 0.5, B GßÓ {PÌa]°ß {PÌuPÄ 0.3 ©ØÖ®
A &²®, B &²® JßøÓö¯õßÖ Â»US® {PÌa] GÛÀ RÌPõq® {PÌuPÄPøÍU
PõsP.
(A) P(Ac B) (B) P(A ∩ B) (C) P(A ∩ B)
The probability of an event A occurring is 0.5 and B occurring is 0.3. If A and B
are mutually exclusive events, then find the probability of :

(a) P(A c B) (b) P(A ∩ B) (c) P(A ∩ B)

A [ v¸¨¦P / Turn over

6712 8

1
32. x &I¨ ö£õÖzx öuõøP°kP :
x 2 − 4x + 5

1
Find the integral of :
x 2 − 4x + 5

33. 1 GßÓ \õº¤ß Ãa\PzøuU PõsP.
2cos x − 1

1
Find the range of the function .
2cos x − 1

cos x
34. x &I¨ ö£õÖzx ÁøPUöPÊøÁU PõsP. y =
x3

cos x
Differentiate with respect to x. y =
x3

 x 2− b 2 x <4
35. g( x ) =  GßÓ \õº¦ (−∞,∞) &À öuõhºa]¯õÚx GÛÀ, ©õÔ¼
 b x + 20 x 4
b &IU PõsP.

 x 2− b2 if x < 4
Find the constant b that makes g continuous on (−∞,∞) g(x ) =  .
bx + 20 if x  4

36. θ J¸ xøn¯»S GÛÀ, x=a cos3θ, y=a sin3θ BQ¯ B¯zöuõø»PøÍ Eøh¯
|P¸® ¦Ò롧 {¯©¨£õøu°ß \©ß£õmøhU PõsP.
If θ is a parameter, find the equation of the locus of a moving point, whose
coordinates are x=a cos3θ, y=a sin3θ.

→ ∧ ∧ ∧ → ∧ ∧ ∧
37. ©ØÖ® b = i − j + k BQ¯ÁØøÓ Akzukzu £UP[PÍõPU
a = 3 i + j + 4k
öPõsh CønPµzvß £µ¨£ÍøÁU PõsP.
→ ∧ ∧ ∧
Find the area of the parallelogram whose adjacent sides are a = 3 i + j + 4k
→ ∧ ∧ ∧

b = i − j +k

A

9 6712

38. {ÖÄP : sin 4x + sin2x = tan3x
cos 4x + cos 2x
sin 4x + sin2x
Prove that : = tan3x
cos 4x + cos 2x

39. 6 − 4x − x 2 = x +4 GßÓ \©ß£õmøhz wºUP.

Solve the equation 6 − 4x − x 2 = x +4 .

40. nCr−1=36, nCr=84 ©ØÖ® nCr+1=126 GÛÀ, r &Cß ©v¨¦ PõsP.
If nCr−1=36, nCr=84 and nCr+1=126 then find the value of r.

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

 −x + 4 ; −∞ < x ≤ − 3
 x+4; − 3 < x <−2

 2
41. (A) f ( x ) =  x −x ; −2≤x <1
 2
 x−x ; 1≤x <7
 0 ©ØÓ Ch[PÎÀ


GÚ Áøµ¯ÖUP¨£iß −4, 1,−2, 7, 0 BQ¯ÁØÔÀ f &ß ©v¨¦PøÍU PõsP.

AÀ»x
1 π θ
(B) θ J¸ SÖ[÷Põn® GÛÀ, sin θ = GÝ®÷£õx sin  −  &ß
25 4 2
©v¨ø£U PõsP.
(a) Write the values of f at −4, 1,−2, 7, 0 if

 −x + 4 if −∞ < x ≤ − 3
 x+4 if − 3 < x < − 2


f (x ) =  x 2 − x if − 2 ≤ x < 1
 2
 x−x if 1 ≤ x < 7
 0 otherwise

OR

π θ 1
(b) If θ is an acute angle, then find sin  −  when sin θ = .
4 2 25

A [ v¸¨¦P / Turn over

6712 10

42. (A) A+B+C=1808 GÛÀ

A B B C C A
tan tan + tan tan + tan tan = 1 GÚ {ÖÄP.
2 2 2 2 2 2

AÀ»x

sinθ
(B) lim θ
= 1 GÚ {ÖÄP.
θ→0

(a) If A+B+C=1808, prove that

A B B C C A
tan tan + tan tan + tan tan = 1.
2 2 2 2 2 2

OR

sinθ
(b) Prove that lim = 1.
θ→0 θ

43. (A) C¸ GsPÎß Tmka \µõ\›¯õÚx, ö£¸USa \µõ\›ø¯ Âh 10
AvP©õPÄ®, Cø\a \µõ\›ø¯ Âh 16 AvP©õPÄ® C¸US©õÚõÀ A¢u
C¸ GsPøÍU PõsP.

AÀ»x

(B) x sec θ+y cosec θ=2a ©ØÖ® x cos θ−y sin θ=a cos 2θ GßÓ ÷PõkPÐUS
Bv°¼¸¢x ö\[Szxz yµ[PÒ •øÓ÷¯ P1 ©ØÖ® P2 GÛÀ P12+P22=a2
GÚ {ÖÄP.

(a) The AM of two numbers exceeds their GM by 10 and HM by 16. Find the
numbers.

OR

(b) If P1 and P 2 are the lengths of the perpendiculars from the origin to the
straight lines x sec θ+y cosec θ=2a and x cos θ−y sin θ=a cos 2θ, then prove

that P12+P22=a2.

A

11 6712

44. (A) k(x−1)2=5x−7 Gߣuß J¸ ‰»® ©ØÓuß C¸©h[S GÛÀ, k=2 AÀ»x
−25 GÚU PõsP.

AÀ»x

 1 3 5
(B) A = − 6 8 3 GßÓ Aoø¯ \©a^º ©ØÖ® Gvº\©a^º AoPÎß
− 4 6 5 

Tku»õP GÊxP.
(a) If one root of k(x−1)2=5x−7 is double the other root, show that k=2 or −25.

OR

 1 3 5
 
(b) Express the matrix A = − 6 8 3 as the sum of a symmetric and a skew
− 4 6 5 
symmetric matrices.

∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧
45. (A) 4 i + 5 j + k , − j − k , 3 i + 9 j + 4 k ©ØÖ® − 4 i + 4 j + 4 k BQ¯&
ÁØøÓ {ø» öÁUhºPÍõPU öPõsh ¦ÒÎPÒ J¸uÍ Aø©ÁÚ GÚU
PõmkP.
AÀ»x
(B) 4 Pou¨ ¦zuP[PÒ, 3 C¯Ø¤¯À ¦zuP[PÒ, 2 ÷Áv°¯À ¦zuP[PÒ ©ØÖ®
1 E°›¯À ¦zuPzøu Kº A»©õ›°À J÷µ £õh ¦zuP[PÒ JßÓõP Á¸®
ÁøP°À GzuøÚ ÁÈPÎÀ AkUP»õ® ?

∧ ∧ ∧ ∧ ∧
(a) Show that the points whose position vectors 4 i + 5 j + k , − j − k ,
∧ ∧ ∧ ∧ ∧ ∧
3 i + 9 j + 4k and − 4 i + 4 j + 4 k are coplanar.

OR

(b) In how many ways 4 mathematics books, 3 physics books, 2 chemistry books

and 1 biology book can be arranged on a shelf so that all books of the same
subjects are together ?

A [ v¸¨¦P / Turn over

6712 12

6x + 5
46. (A) ©v¨¤kP : ∫ dx
1 − 4x − 4x 2
AÀ»x
1 dy −1
(B) y = sin−1 ( 1 + x + 1 − x ) GÛÀ dx = GÚU PõmkP.
2 2 1 − x2

6x + 5
(a) Evaluate : ∫ 1 − 4x − 4x 2 d x
OR

1 dy −1
(b) If y = sin
−1 ( 1 + x + 1 − x ) then show that dx = .
2 2 1 − x2

47. (A) J¸ öuõÈØ\õø»°À C¯¢vµ[PÒ I ©ØÖ® II GÚ C¸ÁøPPÒ EÒÍÚ.
C¯¢vµ® I öuõÈØ\õø»°ß EØ£zv°À 40% u¯õ›UQÓx ©ØÖ® C¯¢vµ®
II EØ£zv°À 60% u¯õ›UQÓx. ÷©¾® C¯¢vµ® I &ß ‰»® EØ£zv
ö\´¯¨£mh ö£õ¸mPÎÀ 4% SøÓ£õk EÒÍuõPÄ®, C¯¢vµ® II &ß ‰»®
EØ£zv ö\´¯¨£mh ö£õ¸mPÎÀ 5% SøÓ£õkÒÍuõPÄ® C¸UQßÓÚ.
EØ£zv ö\´¯¨£mh ö£õ¸mPμ¸¢x \©Áõ´¨¦ •øÓ°À ÷uº¢&
öukUP¨£mh J¸ ö£õ¸Ò SøÓ£õkÒÍuõP C¸¨¤ß A¨ö£õ¸Ò
C¯¢vµ® II &À EØ£zv ö\´uuØPõÚ {PÌuPÄ ¯õx ?
AÀ»x
d2y dy
(B) y=(cos−1x)2 GÛÀ, (1−x2) 2
−x
dx
−2=0 GÚ {¹¤UP. ÷©¾® x=0 &ß
dx
÷£õx y2 ©v¨ø£U PõsP.
(a) A factory has two machines I and II. Machine I produces 40% of items of the
output and Machine II produces 60% of the items. Further 4% of items
produced by Machine I are defective and 5% produced by Machine II are
defective. An item is drawn at random. If the drawn item is defective, find
the probability that it was produced by Machine II.
OR
(b) −1 2
If y=(cos x) prove that

d2y dy
(1−x2) 2
−x −2=0, hence find y2 when x=0.
dx dx

-o0o-

A