TN 11th Question Paper 2022 Maths

No. of Printed Pages : 15
6012
A £vÄ Gs
Register Number

!6012IstYearMathematics!
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

6012 2

3 2x − 3
1. x= &À f (x ) = Gߣx :
2 2x − 3

(A) ÁøP°hzuUPx (B) öuõhºa]¯õÚx

(C) §ä⯩ØÓx (D) öuõhºa]¯ØÓx

3 2x − 3
At x = the function f (x ) = is :
2 2x − 3

(a) differentiable (b) continuous
(c) non-zero (d) discontinuous

2. A, B Gß£Ú n Á›ø\²ÒÍ \©a^º AoPÒ. C[S (A ≠ B) GÛÀ :

(A) A+B Gߣx J¸ ‰ø»Âmh Ao
(B) A+B BÚx Kº Gvº \©a^º Ao

(C) A+B Gߣx J¸ §ä⯠Ao
(D) A+B Gߣx J¸ \©a^º Ao
If A and B are symmetric matrices of order n, where (A ≠ B), then :
(a) A+B is a diagonal matrix
(b) A+B is skew-symmetric
(c) A+B is a zero matrix
(d) A+B is symmetric

x
 x 2 + 5x + 3 
3. lim   :
x →∞  x 2 + x + 3 

(A) e 3 (B) e 4 (C) 1 (D) e 2

x
 x 2 + 5x + 3 
lim   is :
x →∞  x 2 + x + 3 

(a) e3 (b) e4 (c) 1 (d) e2

A

3 6012

4. log 2 512 &ß ©v¨¦ :

(A) 9 (B) 16 (C) 12 (D) 18
The value of log 2 512 is :

(a) 9 (b) 16 (c) 12 (d) 18

dy
5. y=f (x2+2) ©ØÖ® f 9(3)=5 GÛÀ, x=1&À Gߣx :
dx

(A) 15 (B) 5 (C) 10 (D) 25

dy
If y=f (x2+2) and f 9(3)=5 then, at x=1 is :
dx
(a) 15 (b) 5 (c) 10 (d) 25

6. C¯À GsPÎß AøÚzxUPn® N &US A ©ØÖ® B EmPn[PÒ GÛÀ
A9∪[(A∩B)∪B9] Gߣx :

(A) B (B) A (C) N (D) A9
Let A and B be subsets of the universal set N, the set of natural numbers. Then
A9∪[(A∩B)∪B9] is :
(a) B (b) A (c) N (d) A9

 x +2, −1< x <3

7. f (x ) =  5, x =3 , x=3 &À f 9(x) Gߣx :
8−x , x >3


(A) 0 (B) 1
(C) QøhUP¨ ö£Óõx (D) −1

 x +2, −1< x <3

If f (x ) =  5, x =3 , then at x=3, f 9(x) is :
8−x , x >3


(a) 0 (b) 1
(c) does not exist (d) −1

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6012 4

8. tan908 &ß ©v¨¦ :

3
(A) (B) 0 (C) 1 (D) ∞
2

The value of tan908 is :

3
(a) (b) 0 (c) 1 (d) ∞
2

x2 2
9. ∫ f
(x )e dx = (x −1)ex + c GÛÀ, f (x) Gߣx :

x2
(A) x3+4x2+6x+c (B) 2x 3 − + x +c
2

2x 3 x3
(C) − x2 +x + c (D) + 3x 2 + 4x + c
3 2

2 2
If ∫ f
(x )ex dx = (x −1)ex + c , then f (x) is :

x2
(a) x3+4x2+6x+c (b) 2x 3 − + x +c
2

2x 3 x3
(c) − x2 +x + c (d) + 3x 2 + 4x + c
3 2

10. A ©ØÖ® B GßÓ C¸ {PÌa]PÐUS P(A)=0.4, P(B)=0.8 ©ØÖ® P(B/A)=0.6 GÛÀ,
P ( A ∩ B ) &ß ©v¨¦ :

(A) 0.56 (B) 0.96 (C) 0.66 (D) 0.24
If A and B are two events such that P(A)=0.4, P(B)=0.8 and P(B/A)=0.6, then
P ( A ∩ B ) is :

(a) 0.56 (b) 0.96 (c) 0.66 (d) 0.24

A

5 6012

11. sec(−θ) Gߣx :
(A) secθ (B) cosθ (C) sec(−θ) (D) cos(−θ)
sec(−θ) is :
(a) secθ (b) cosθ (c) sec(−θ) (d) cos(−θ)

∧ ∧ ∧
12. J¸ •U÷Põnzvß Cµsk •øÚ¨¦ÒÎPÎß {ø» öÁUhºPÒ 3 i + 4 j − 4k
∧ ∧ ∧ ∧ ∧ ∧
©ØÖ® 2 i + 3 j + 4 k . ø©¯U÷Põmk \¢v°ß {ø»öÁUhº i + 2 j + 3 k GÛÀ,
‰ßÓõÁx •øÚ¨ ¦Ò롧 {ø» öÁUhº :
∧ ∧ ∧ ∧ ∧ ∧
(A) 2 i − j + 6 k (B) −2 i − j + 9 k
∧ ∧ ∧ ∧ ∧ ∧
(C) −2 i + j + 6 k (D) −2 i − j − 6 k
∧ ∧ ∧ ∧ ∧ ∧
Two vertices of a triangle have position vectors 3 i + 4 j − 4 k and 2 i + 3 j + 4 k .
∧ ∧ ∧
If the position vector of the centroid is i + 2 j + 3 k , then the position vector of
the third vertex is :
∧ ∧ ∧ ∧ ∧ ∧
(a) 2 i − j + 6k (b) −2 i − j + 9 k
∧ ∧ ∧ ∧ ∧ ∧
(c) −2 i + j + 6 k (d) −2 i − j − 6 k

x
2
13. ∫x 2
e dx =

x x x
x x x
2
(A) 2x 2 e 2 − 8xe 2 + 16e 2 + c (B) x 2 2 2
e − 4xe − 8e + c

x x x x x x
(C) x 2 e2 xe 2 e2 (D) 2x 2
e − 8xe − 16e 2 + c
2 2
− + +c
2 4 8

x
2
∫x 2
e dx is :

x x x
x x x
(a) 2x 2
e − 8xe + 16e 2 + c
2 2 (b) x 2 e 2 − 4xe 2 − 8e 2 + c

x x x x x x
(c) 2 e2 xe 2 e2 (d) 2x 2
e − 8xe − 16e 2 + c
2 2
x − + +c
2 4 8

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14. AøÚzøu²® JØøÓ GsPÍõPU öPõsh 5 C»UP GsPÎß GsoUøP :

(A) 56 (B) 25 (C) 625 (D) 55

The number of 5 digit numbers all digits of which are odd is :

(a) 56 (b) 25 (c) 625 (d) 55

1 1 1
15. X ©ØÖ® Y GßÓ C¸ {PÌa]PÐUS P(X/Y)= , P(Y/X)= , P(X∩Y)= GÛÀ
2 3 6
P(X∪Y) &ß ©v¨¦ :

1 1 2 2
(A) (B) (C) (D)
6 3 3 5

1 1 1
If X and Y be two events such that P(X/Y)= , P(Y/X)= and P(X∩Y)= , then
2 3 6
P(X∪Y) is :

1 1 2 2
(a) (b) (c) (d)
6 3 3 5

16. 3x−y=−5 GßÓ ÷Põmkhß 458 ÷Põn® HØ£kzx® ÷Põmiß \õ´ÄPÒ :

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

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

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

2

A

7 6012

→ →
17. a ©ØÖ® b &I Akzukzu £UP[PÍõPU öPõsh CønPµ® ABCD &ß J¸
→ →
‰ø»Âmh® a + b GÛÀ ©ØöÓõ¸ ‰ø»Âmh® BÚx :

→ →
→ → → → → →
(A) a + b (B) a − b (C) a + b (D) b − a
2

→ →
One of the diagonals of parallelogram ABCD with a and b as adjacent sides is
→ →
a + b . The other diagonal is :

→ →
→ → → → → →
(a) (b) (c) a + b (d) b − a
a + b a − b
2

ex −2 e7+x 
18. A=   Gߣx J¸ §äâ¯U ÷PõøÁ Ao GÛÀ, x&ß ©v¨¦ :
e2+x e2x +3 

(A) 7 (B) 9 (C) 6 (D) 8

ex −2 e7+x 
The value of x, for which the matrix A =   is singular :
e2+x e2x +3 

(a) 7 (b) 9 (c) 6 (d) 8

∧ ∧ ∧ ∧ ∧ ∧
©ØÖ® a i + 11 j BQ¯ {ø» öÁUhºPÎß ¦ÒÎPÒ J÷µ
19. 10 i + 3 j , 12 i − 5 j
÷PõmiÀ Aø©¢uõÀ ‘a’ &ß ©v¨¦ :
(A) 5 (B) 6 (C) 8 (D) 3

∧ ∧ ∧ ∧ ∧ ∧
If the points whose position vectors are 10 i + 3 j , 12 i − 5 j and a i + 11 j are
collinear then ‘a’ is equal to :

(a) 5 (b) 6 (c) 8 (d) 3

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20.
1 3 7 15 , ..... GßÓ öuõhº •øÓ°ß n&BÁx EÖ¨¦ :
, , ,
2 4 8 16

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

The nth term of the sequence
1 3 7 15 , ..... is :
, , ,
2 4 8 16

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

£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. ?2x−17?=3 &À x &US wºÄ PõsP.

Solve ?2x−17?=3 for x.

22. sin508+sin208 &ø¯ ö£¸UP»õP GÊxP.

Express sin508+sin208 as a product.

23. cos1358 &ß ©v¨¦ PõsP.

Find the value of cos1358.

24. 5 |õn¯[PøÍ J¸ •øÓ _sk®÷£õx HØ£k® ÂøÍÄPÎß ö©õzu
GsoUøPø¯U PõsP.
Find the total number of outcomes when 5 coins are tossed once.

A

9 6012

 n ; n Gߣx 1, 2 AÀ»
- x3
25. n &Áx EÖ¨¦ a n = 
a n−1 + a n−2 + a n−3 ; n > 3


&IU öPõsh öuõhº•øÓPÎÀ •uÀ 4 EÖ¨¦PøÍU PõsP.
Write the first 4 terms of the sequences whose nth term an is given as.

n ; if n is 1, 2 or 3
an = 
 a n−1 + a n−2 + a n−3 ; if n > 3


26. (1, 1) ©ØÖ® (−2, 3) GßÓ ¦ÒÎPÒ ÁÈ÷¯ ö\À»UTi¯ ÷|ºU÷Põmiß
\©ß£õmøhU PõsP.
Find the equation of the lines passing through the points (1, 1) and (−2, 3).

 0 sinα cosα 

A =  sinα 0 sinβ  GÛÀ, ?A?&I PõsP.
27.
cosα −sinβ 0 

 0 sinα cosα 
 sinβ  .
Find ?A? if A =  sinα 0
cosα −sinβ 0 

∧ ∧ ∧
28. 5 i − 3 j + 4 k &ß vø\°À EÒÍ Kº Kµ»S öÁUhøµU PõsP.

∧ ∧ ∧
Find a unit vector along the direction of the vector 5 i − 3 j + 4 k .

29. y=x3+5x2+3x+7 &I x &I ö£õ¸zx ÁøP°kP.

Differentiate y=x3+5x2+3x+7 with respect to x.

30. (x−11)7&I, x &I ö£õ¸zx öuõøP°kP.

Integrate (x−11)7 with respect to x.

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6012 10

£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. Pn® A BÚx A={x : x=4n+1, 2≤n≤5, n∈N} GÛÀ, A &ß EmPn[PÎß
GsoUøPø¯U PõsP.
Find the number of subsets of A if A={x : x=4n+1, 2≤n≤5, n∈N}.

x
32. £Sv ¤ßÚ[PÍõP¨ ¤›UPÄ® : .
(x + 3) (x − 4)

x
Resolve into partial fractions :
(x + 3) (x − 4)

33. ACCESSIBILITY GßÓ Áõºzøu°À EÒÍ GÊzxPøͨ £¯ß£kzv GzuøÚ
öÁÆ÷ÁÓõÚ Á›ø\ ©õØÓ[PøÍ E¸ÁõUP»õ® ?
Find the distinct permutations of the letters of the word ACCESSIBILITY.

34. (x+2)−2/3 &I x &ß AkUSPÍõP ›ÁõUP® ö\´P.

Expand (x+2)−2/3 in powers of x.

35. 5x+12y−3=0 GßÓ ÷PõmiØS® (1, 2) GßÓ ¦ÒÎUS® Cøh÷¯ EÒÍ yµ®
PõsP.
Find the distance from a point (1, 2) to the line 5x+12y−3=0.

A

11 6012

1 1 1
x y z
36. =(x−y)(y−z)(z−x) GÚ {ÖÄP.
x2 y2 z2

1 1 1
x y z
Prove that =(x−y)(y−z)(z−x).
x 2 y2 z2

∧ ∧ ∧ ∧ ∧ ∧
37. 5 i + 3 j + 4 k ©ØÖ® 6 i − 8 j − k BQ¯ öÁUhºPÐUS Cøh¨£mh ÷PõnzøuU
PõsP.

∧ ∧ ∧ ∧ ∧ ∧
Find the angle between the vectors 5 i + 3 j + 4 k and 6 i − 8 j − k .

dy
38. x2+y2=1 GÛÀ, PõsP.
dx

dy
Find if x2+y2=1.
dx

39. f 9(x)=4x−5 ©ØÖ® f (2)=1 GÛÀ, f (x) PõsP.

If f 9(x)=4x−5 and f (2)=1, find f (x).

40. J¸ £Pøhø¯ J¸ •øÓ E¸mk®÷£õx J¸ Cµmøh¨£øh Gs QøhUS®
GÛÀ 6 Qøh¨£uØPõÚ {PÌuPÄ GßÚ ?

A die is rolled. If it shows an even number, then find the probability of getting 6.

A [ v¸¨¦P / Turn over

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£Sv & IV/PART - IV

SÔ¨¦ : AøÚzx ÂÚõUPÐUS® Âøh¯ÎUPÄ®. 7x5=35

Note : Answer all the questions.

41. (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

2x + 4
(B) ©v¨¤kP : ∫ 2
dx
x + 4x + 6

(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 knows only language A.
OR

2x + 4
(b) Evaluate : ∫ 2 dx
x + 4x + 6

cot(180
+θ) sin(90
−θ) cos(−θ) 2
42. (A) sin(270
+θ) tan(−θ) cosec(360
+θ) = cos θ cotθ

AÀ»x

(B) ©v¨¤kP : ∫ x cosx dx

cot(180
+θ) sin(90
−θ) cos(−θ)
(a) Prove that : = cos2θ cotθ
sin(270
+θ) tan(−θ) cosec(360
+θ)

OR

(b) Evaluate : ∫ x cosx dx

A

13 6012

43. (A) Pouz öuõSzuÔuÀ •øÓ°À n/1 &US

2
 n(n+1) 
13+23+33+....+n3=   GÚ {¹¤UP.
 2 

AÀ»x

dy
(B) x=a(t−sin t), y=a(1−cos t) GÛÀ, PõsP.
dx

(a) By the principle of mathematical induction, prove that, for n/1

2
3 3 3 3  n(n+1) 
1 +2 +3 +....+n =  
 2 

OR

dy
(b) Find , if x=a(t−sin t), y=a(1−cos t).
dx

44. (A) 12x2+2kxy+2y2+11x−5y+2=0 GßÓ \©ß£õk Cµmøh ÷|ºU÷Põmiß
\©ß£õmøhU SÔzuõÀ k &ß ©v¨ø£U PõsP.

AÀ»x

tan2x
(B) lim &ß ©v¨¦ PõsP.
x →0 sin5x

(a) For what value of k does the equation 12x 2 +2kxy+2y 2 +11x−5y+2=0
represent two straight lines.

OR

tan2x
(b) Evaluate : lim
x → 0 sin5x

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2bc−a 2 c2 b2 a b c
2

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

AÀ»x

(B) log  75 −2log  5 +log  32 =log2 GÚ {ÖÄP.
 16  9  243 

2bc−a 2 c2 b2 a b c
2

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

OR

 75  5  32 
(b) Prove that log  −2log  +log  =log2 .
 16  9  243 

∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧
46. (A) 2 i + 4 j + 3 k , 4 i + j + 9 k , 10 i − j + 6 k GßÓ öÁUhºPøÍ {ø»
öÁUhºPÍõPU öPõsh ¦ÒÎPÒ J¸ ö\[÷Põn •U÷Põnzøu Aø©US®
GÚ {ÖÄP.

AÀ»x

(B) ©v¨¦ PõsP (i) cos158 (ii) tan1658.

∧ ∧ ∧ ∧ ∧ ∧
(a) Prove that the points whose position vectors 2 i + 4 j + 3 k , 4 i + j + 9 k and
∧ ∧ ∧
10 i − j + 6 k form a right angled triangle.

OR

(b) Find the values of (i) cos158 and (ii) tan1658.

A

15 6012

47. (A) 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.
(i) A¨£¢x P¸¨£õP C¸¨£uØPõÚ {PÌuPøÁU PõsP.

(ii) GkUP¨£mh £¢x P¸¨¦ GÛÀ •uÀ áõi°¼¸¢x
GkUP¨£mhuØPõÚ {PÌuPÄ ¯õx ?

AÀ»x

(B) 3x − y + 4 = 0 GßÓ ÷Põmøha ö\[Szx ÁiÁzvØS ©õØÖP.

(a) 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. (i) find the probability that the ball is black (ii) if the ball is black, what is
the probability that it is from the first urn ?
OR

(b) Rewrite 3x − y + 4 = 0 into normal form.

-o0o-

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