TN 12th Question Paper 2022 Maths

No. of Printed Pages : 15
5912
£vÄ Gs
A Register Number

!5912Mathematics!
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

5912 2

1. J¸ \©- Á õ´¨¦ ©õ- Ô - ° ß {PÌ- u - P Ä Ahºzv \õº¦
 2x 0
x
a
f (x ) =  GÛÀ, a &ß ©v¨¦ :
 0 ¤Ó ©v¨¦PÐUS
(A) 3 (B) 1 (C) 4 (D) 2

If f (x ) = 
2x 0
x
a
is a probability density function of a random variable,
 0 otherwise
then the value of a is :
(a) 3 (b) 1 (c) 4 (d) 2

2. X Gß-£x uÛ-{ø» \©-Áõ´¨¦ ©õ-Ô-¯õP C¸U-S® ÷£õx RÌU-Põ-q® TØ-Ö-P-ÎÀ
Gx uÁ-Óõ-Úx ?
(A) xlim
→∞
F(x ) = F(∞) = 1

(B) 0 £ F(x) £ 1, AøÚz-x x e R
(C) F(x) Gß-£x ö©´-©-v¨-¦-øh¯ SøÓ-²® \õº-¦.
(D) lim F(x ) = F(−∞) = 0
x →−∞

Which one of the following is not true in the case of discrete random variable X ?

(a) lim F(x ) = F(∞) = 1
x→ ∞

(b) 0 £ F(x) £ 1 for all x e R
(c) F(x) is real valued decreasing function.

(d) lim F(x ) = F(−∞) = 0
x →−∞

3.
x GÛÀ, Auß ÁøP-±-k :
f (x ) =
x +1

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

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

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

A

3 5912

1
99
4. ∫ x (1 − x ) dx &Cß ©v¨¦ :
0

1 1 1 1
(A) (B) (C) (D)
10010 11000 10001 10100

1
The value of ∫ x (1 − x )99 dx is :
0

1 1 1 1
(a) (b) (c) (d)
10010 11000 10001 10100

 3
5. cos−1   &ß •ußø© ©v¨-¦ :
 2 

π π 5π π
(A) (B) (C) (D)
2 3 6 6

 3
The principal value of cos−1   is :
 2 

π π 5π π
(a) (b) (c) (d)
2 3 6 6

2 3
 ©Ø-Ö® λA =A GÛÀ, λ &ß ©v¨¦ :
6. A=  −1
5 −2 

(A) 19 (B) 17 (C) 21 (D) 14

2 3 −1
If A =   be such that λA =A, then λ is :
5 −2 

(a) 19 (b) 17 (c) 21 (d) 14

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1
7. x 3 +px 2 +qx+r &U- S α, β ©Ø- Ö ® γ Gß- £ øÁ §a- ] - ¯ - © õU- Q - P Ò GÛÀ ∑ α &ß
©v¨-¦ :

q q q p
(A) (B) − (C) − p (D) −
r r r

1
If α, β and γ are the zeros of x3+px2+qx+r, then ∑ is :
α

q q q p
(a) (b) − (c) − (d) −
r r p r

8. (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 the value 2⋅5⋅10 ... (1+n2) is :

(a) x2+y2 (b) 1 (c) 1+n2 (d) i

9. ?3−x ?+9 GßÓ \õº-¤ß SøÓ¢u ©v-¨-¦ :

(A) 6 (B) 0 (C) 9 (D) 3
The minimum value of the function ?3−x ?+9 is :

(a) 6 (b) 0 (c) 9 (d) 3

12
10. ∑ i n &ß ©v¨-¦ :
n=1

(A) 0 (B) 1 (C) −1 (D) i

12
The value of ∑ i n is :
n=1

(a) 0 (b) 1 (c) −1 (d) i

A

5 5912

∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧
11. 2 i − j + 3k , 3 i + 2 j + k , i + m j + 4k GßÓ öÁU-hº-PÒ J¸ uÍ öÁU-hº-P-Ò
GÛÀ, m &ß ©v¨¦ :
(A) 2 (B) 3 (C) −2 (D) −3
∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧
If the vectors 2 i − j + 3 k , 3 i + 2 j + k , i + m j + 4 k are coplanar, then the
value of m is :
(a) 2 (b) 3 (c) −2 (d) −3

12. ø©¯® (−3, −4) ©Ø-Ö® Bµ® 3 A»-S-PÒ öPõsh Ám-hz-vß ö£õx Ái-Á-a
\©ß-£õk :
(A) x2+y2−6x+8y−16=0 (B) x2+y2−6x−8y+16=0
(C) x2+y2+6x−8y+16=0 (D) x2+y2+6x+8y+16=0
The general equation of a circle with centre (−3, −4) and radius 3 units is :
(a) x2+y2−6x+8y−16=0 (b) x2+y2−6x−8y+16=0
(c) x2+y2+6x−8y+16=0 (d) x2+y2+6x+8y+16=0

dy
13. + p(x )y = 0 &Cß wº-Ä :
dx
(A) x=ce−∫pdy (B) y=ce∫pdx (C) x=ce∫pdy (D) y=ce−∫pdx
dy
The solution of + p(x )y = 0 is :
dx
(a) x=ce−∫pdy (b) y=ce∫pdx (c) x=ce∫pdy (d) y=ce−∫pdx

∞
14.
∫e
−3x
x 2dx &ß ©v¨-¦ :
0

4 7 2 5
(A) (B) (C) (D)
27 27 27 27

∞
−3x
The value of
∫e x 2dx is :
0

4 7 2 5
(a) (b) (c) (d)
27 27 27 27

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15. y=(x−1)3 GßÓ ÁøÍ-Á-øµ-°ß ÁøÍ-Ä ©õØ-Ó¨-¦-ÒÎ :

(A) (1, 0) (B) (0, 0) (C) (1, 1) (D) (0, 1)

The point of inflection of the curve y=(x−1)3 is :

(a) (1, 0) (b) (0, 0) (c) (1, 1) (d) (0, 1)

16.
x−4 y z + 1 ©Ø-Ö® x − 1 y +1 z − 2 GßÓ C¸ ÷|ºU ÷Põ-k-P-ÐUS
= = = =
2 1 −2 4 −4 2
Cøh¨-£mh ÷Põ-n® :

π π 2π π
(A) (B) (C) (D)
2 4 3 3

The angle between the lines

x−4 y z + 1 and x − 1 y +1 z − 2 is :
= = = =
2 1 −2 4 −4 2

π π 2π π
(a) (b) (c) (d)
2 4 3 3

17. ¤ß-Á-¸-£-øÁ-P-ÎÀ Gx N &ß «x Kº D¸-Ö¨-¦a ö\¯-¼ BS® ?

(A) ö£¸U-PÀ (B) ÁSz-uÀ (C) PÈz-uÀ (D) AøÚz-x®

Which one of the following is a binary operation on N ?

(a) Multiplication (b) Division (c) Subtraction (d) All the above

A

7 5912

18. RÌ-U-Põ-q® TØ-Ö-P-ÎÀ Gx Es-ø©-¯À» ?

(A) A Gß-£-x n Á›-ø\-²-øh¯ J¸ \xµ Ao ©Ø-Ö® λ Gß-£x J¸ vø\-°¼
GÛÀ Adj (λA)=λn (Adj A).

(B) J¸ \©a-^º Ao°ß ÷\º¨¦ Ao \©a-^-µõP C¸U-S®.

(C) A(Adj A)=(Adj A)A=?A?I.

(D) J¸ ‰ø»-Âmh Ao-°ß ÷\º¨¦ Ao ‰ø»-Âmh Ao-¯õP C¸U-S®.-
Which one of the following is incorrect ?

(a) If A is a square matrix of order n, and λ is a scalar, then Adj (λA)=λn (Adj A).

(b) Adjoint of a symmetric matrix is also a symmetric matrix.

(c) A(Adj A)=(Adj A)A=?A?I.

(d) Adjoint of a diagonal matrix is also a diagonal matrix.

dy
19. + Py = Q GÝ® ÷|›-¯À ÁøPU-öP-Êa \©ß-£õm-iß öuõ-øP-±m-kU Põ-µo sinx
dx
GÛÀ, P Gß-£-x :

(A) tan x (B) log sin x (C) cot x (D) cos x

dy
If sinx is the integrating factor of the linear differential equation + Py = Q ,
dx
then P is :

(a) tan x (b) log sin x (c) cot x (d) cos x

20. x2=24y GßÓ £µ-Á-øÍ-¯z-vß ö\Æ-Á-P-»-}Í® :

(A) 8 (B) 24 (C) 6 (D) 12
The length of the latus rectum of the parabola x2=24y is :

(a) 8 (b) 24 (c) 6 (d) 12

A [ v¸¨¦P / Turn over

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£S-v - 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. RÌ-UPõ-q® £s-¦-PøÍ {Ö-Ä-P :

z+z z−z
Re(z ) = ©ØÖ® Im(z ) =
2 2i

Prove the following properties :

z+z z−z
Re(z ) = and Im(z ) =
2 2i

22. 2− 3 &I ‰»-©õ-PU öPõsh SøÓ¢-u-£m\ £i-²-hß ÂQ-u-•Ö öPÊU-P-Ð-øh¯
£À-¾-Ö¨-¦U-÷Põ-øÁa \©ß-£õm-øhU Põs-P.
Find a polynomial equation of minimum degree with rational coefficients, having
2 − 3 as a root.

23. tan−1 ( 3 ) &ß •ußø© ©v¨¦ Põs-P.

Find the principal value of tan−1 ( 3 ) .

24. y=x3−3x2+x−2 GßÓ ÁøÍ-Á-øµUS, G¢-öu¢u ¦Ò-Î-P-ÎÀ Áøµ-¯¨-£-k® öuõ-k-
÷Põ-k y=x GßÓ ÷Põm-iØS Cøn-¯õP C¸U-S® ?
Find the points on the curve y=x3−3x2+x−2 at which the tangent is parallel to
the line y=x.

25. f (x)=x2+3x GßÓ \õº-¤Ø-S, x=2 ©Ø-Ö® dx=0.1 GÝ® ÷£õx df & I Põs-P.

Find df for f (x)=x2+3x and evaluate it for x=2 and dx=0.1.

A

9 5912

26. y=Aex+Be−x, (A ©Ø-Ö® B BQ-¯øÁ H÷uaø\ ©õ-Ô-¼-PÒ) GßÓ ÁøÍÁøµPÎß
2
öuõ-S¨-¤ß ÁøPU-öPÊa \©-ß-£õk d y − y = 0 GÚ {Ö-Ä-P.
dx 2
Show that the differential equation of the family of curves y=Aex+Be−x, where
2
A and B are arbitrary constants, is d y − y = 0 .
dx 2

27. wºU-P : dy = 1 − y2
dx 1 − x2

2
Solve : dy = 1 − y
dx 1 − x2

28. J¸ uÛ-{ø» \õº¦ X &ß {PÌ-u-PÄ {øÓ \õº-£õ-Úx :
x 1 2 3 4 5 6

f (x ) k 2k 6k 5k 6k 10k

GÛÀ, k &ß ©v¨-ø£U Põs-P.
A random variable X has the following probability mass function.

x 1 2 3 4 5 6

f (x ) k 2k 6k 5k 6k 10k

Find k.

29. X Gß-£x ‰ßÖ ^µõÚ |õ-n-¯[-PøÍ J÷µ \©-¯z-vÀ J¸ •øÓ _s-k® ÷£õx
ÂÊ® §U-P-Îß Gs-oUøP GßP. \©-Áõ´¨¦ ©õÔ¯õÚ X &Cß ©v¨-¦-P-øÍ-²®
Auß ÷|º-©õÖ ¤®-£[-P-ÎÀ EÒÍ ¦Ò-Î-P-Î-ß Gs-oU-øP-ø¯-²® Põs-P.
X is the number of tails occurred when three fair coins are tossed simultaneously.
Find the values of the random variable X and number of points in its reverse
images.

30. Bv¨-¦Ò-Î-°-¼-¸¢x 3x+6y+2z+7=0 GßÓ uÍ-z-vØS EÒÍ öuõ-ø»-Ä 1 GÚ
{Ö-Ä-P.

Show that the distance from the origin to the plane 3x+6y+2z+7=0 is 1.

A [ v¸¨¦P / Turn over

5912 10

£S-v - 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 2 −1
3 −1 2
31. ¤ß-Á-¸® Ao-°ß Aoz-u-µ® 3 GÚU Põm-kP.  
 1 −2 3 
 
 1 −1 1

1 2 −1
3 −1 2
Show that the rank of the matrix   is 3.
 1 −2 3 
 
 1 −1 1

32. ¤ß- Á - ¸ ® ÷|›- ¯ a \©ß- £ õm- k z öuõ- S ¨ø£ ÷|º- © õÖ Ao Põ- n À •øÓ- ø ¯
£¯ß-£-kzv wºUP :
5x+2y=3, 3x+2y=5.

Solve the following system of linear equations, using matrix inversion method :

5x+2y=3, 3x+2y=5.

33. 10−8i, 11+6i BQ¯ ¦Ò-ÎP
- Î
- À G¨-¦Ò-Î 1+i &US ªP A¸-Põ-ø©-°À C¸U-S® ?
Which one of the points 10−8i, 11+6i is closest to 1+i.

34. 2x3−9x2+10x=3 GßÓ \©ß-£õm-iØS, 1 J¸ ‰»-ö©-ÛÀ ¤Ó ‰»[-PøÍU Põs-P.

Solve the equation 2x3−9x2+10x=3, if 1 is a root, find the other roots.

A

11 5912

∧ ∧ ∧
35. 2i + j − k Gß- Ý ® Âø\ Bv¨- ¦ ÒÎ ÁÈ- ¯ õ- P a ö\¯À- £ - k - Q - Ó x GÛÀ,
(2, 0, −1) GßÓ ¦Ò-Î-ø¯¨ ö£õ-Özx AÆ-Â-ø\-°ß v¸¨-¦-Â-ø\-°ß Gs-nÍÄ
©Ø-Ö® vø\U öPõ-ø\ß-PøÍU Põs-P.

Find the magnitude and the direction cosines of the torque about the point
∧ ∧ ∧
(2, 0, −1) of a force 2 i + j − k , whose line of action passes through the origin.

2x 2 − 3
36. ©v¨-¤-kP : lim
x → ∞ x 2 − 5x + 3

2x 2 − 3
Evaluate : lim
x → ∞ x 2 − 5x + 3

37. ©Û-u-Ûß Cµz-uU SÇõ-°ß (u-©-Û-°ß) SÖUS öÁm-hõ-Úx Ámh Ái-Á® GÚU
öPõÒP. J¸ ÷|õ-¯õ-ÎUS Cµz-uU SÇõ´ ›-Á-øh-Á-uØ-PõÚ ©¸¢x öPõ-kU-P¨&-
£m-kÒÍx. Cµz-uU SÇõ-°ß Bµ® 2 ª.« C¼-¸¢x 2.1 ª.« BP Av-P-›U-S®
÷£õx Auß SÖUS öÁm-iß £µ¨¦ ÷uõ-µõ-¯-©õP G¢u AÍÄ Av-P-›U-S® ?

Assume that the cross section of the artery of human is circular. A drug is given
to a patient to dilate his arteries. If the radius of an artery is increased from
2 mm to 2.1 mm, how much is cross-sectional area increased approximately.

π
3
sec x tan x −1 π
38. ∫ 1 + sec2x dx = tan (2) − 4 GÚU Põm-k-P.
0

π
3
sec x tan x −1 π
Show that ∫ 1 + sec2x dx = tan (2) − 4 .
0

A [ v¸¨¦P / Turn over

5912 12

39. R &ß «x * BÚ-x (a*b)=a+b+ab−7 GÚ Áøµ-¯-ÖU-P¨-£m-hõÀ, *, R &ß «x

AøhÄ ö£Ø-ÖÒÍ-uõ ? AÆ-Áõ-öÓ-ÛÀ, 3* −7  &I Põs-P.
 15 

Let * be defined on R by (a*b)=a+b+ab−7. Is * binary on R ? If so, find 3* 
−7 
.
 15 

40. (−4, −2) ©Ø-Ö® (−1, −1) GßÓ ¦Ò-Î-PøÍ Âm-hz-vß •øÚ-PÍõ-PU öPõsh
Ám-hz-vß ö£õ-xa \©ß-£õ-k x2+y2+5x+3y+6=0 GÚ {Ö-Ä-P.

Prove that the general equation of the circle whose diameter is the line segment
joining the points (−4, −2) and (−1, −1), is x2+y2+5x+3y+6=0.

£S-v - IV / PART - IV

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

Note : Answer all the questions.

41. (A) 3x+y+z=2, x−3y+2z=1, 7x−y+4z=5 GßÓ \©ß-£õm-kz öuõ-S¨-¤øÚ
wºÄ Põn Qµõ-©-›ß Âv-ø¯ £¯ß-£-kzu C¯-»õx. Hß ?

AÀ»-x

(B) f (x)=4x6−6x4 GßÓ \õº-¤ØS Chg-\õº¢u ]Ö-© ©v¨-¦P
- Ò x=−1, x=1 BQ-¯
¦Ò-Î-P-ÎÀ QøhU-S® GÚ {¹-¤U-P.

(a) Cramer’s rule is not applicable to solve the system 3x+y+z=2, x−3y+2z=1,
7x−y+4z=5. Why ?

OR

(b) Prove that the local minimum values for the function f (x)=4x6−6x4 attain at
−1 and 1.

A

13 5912

42. (A) ?z+i ?=?z−1? GßÓ {£¢-uø
- ÚU-S z=x+iy &ß {¯-©¨-£õøu x+y=0 GÚ {Ö-ÄP
- .

AÀ»-x

a
(B) ∫ f (x ) a GÚU Põm-k-P.
dx =
f (x ) + f (a − x ) 2
0

(a) Show that the locus of z=x+iy if ?z+i ?=?z−1?, is x+y=0.

OR

a
f (x ) a
(b) Show that
∫ f (x ) + f (a − x ) dx = 2 .
0

43. (A) SÂ- ¯ ® (− 2, 0 ) ©Ø- Ö ® C¯U- S - Á - ø µ x = 2 Eøh¯ £µ- Á - ø Í- ¯ z- v ß
\©ß-£õk y 2 = −4 2x GÚU Põm-k-P.

AÀ»-x

 3
(B) cot−1(1) + sin−1 −  − sec (− 2 ) &ß ©v¨¦ Põs-P.
−1
 2 

(a) Show that the equation of the parabola with focus (− 2, 0 ) and directrix

x = 2 is y 2 = −4 2x .

OR

 3
Find the value of cot−1(1) + sin−1 −  − sec (− 2 ) .
−1
(b)
 2 

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5912 14

44. (A) `›-¯-Û-¼-¸¢x §ª-°ß Av-P-£m-\ ©Ø-Ö® SøÓ¢u£m\ yµ[-PÒ •øÓ-÷¯
152×106 Q.«. ©Ø-Ö® 94.5×106 Q.«. }Ò-Ám-h¨ £õ-øu-°ß J¸ SÂ-¯z-vÀ
`›- ¯ ß EÒÍx. `›- ¯ - Ý U- S ® ©Ø- ö Óõ¸ SÂ- ¯ z- v Ø- S - © õÚ yµ®
575×105 Q.«. GÚU Põm-k-P.

AÀ»-x

(B) öÁU-hº •øÓ-°À {Ö-Ä-P.
sin(A+B)=sinA.cosB+cosA.sinB.

(a) 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. Show that the distance from the Sun to the other focus is
575×105 km.

OR

(b) Prove by vector method

sin(A+B)=sinA.cosB+cosA.sinB.

45. (A) (2, 2, 1) ©Ø-Ö® (9, 3, 6) GßÓ ¦Ò-Î-PÒ ÁÈ-¯õ-Pa ö\À-Á-x®, 2x+6y+6z=9
GßÓ uÍz- v ØS ö\[- S z- u õ- P - Ä ® EÒÍ uÍz- v ß öÁU- h º \- © ß- £ õk
(H-÷u-Ý® J¸ Ái-Á®) AÀ-»x Põº-j-]-¯ß \©ß-£õk Põs-P.

AÀ»-x

(B) y=x2 ©Ø-Ö® x=y2 GßÓ ÁøÍ-Áø
- µ-PÐ
- US Cøh¨-£mh ÷Põ-n® (1, 1) Gß-Ý®
¦Ò-Î-°À tan−1  3  GÚU Põm-k-P.
4

(a) Find the vector equation (any form) or Cartesian equation of a plane passing
through the points (2, 2, 1), (9, 3, 6) and perpendicular to the plane
2x+6y+6z=9.

OR

3
(b) Show that the angle between the curves y=x2 and x=y2 at (1, 1) is tan−1   .
4

A

15 5912

46. (A) J¸ öuõ-hº \©-Áõ´¨¦ ©õ-Ô X &Cß £µ-ÁÀ \õº-¦ :
 0 , x <1
x − 1

F(x ) =  , 1x 5
 4

 1 , x >5

GÛÀ (i) P(X < 3) (ii) P(2 < X < 4) (iii) P(3 £ X) Põs-P.

AÀ»-x
(B) 3x−2y+6=0, x=−3, x=1 ©Ø- Ö ® x - Aa- _ BQ- ¯ - Á Ø- Ó õÀ Aøh- £ k®
Aµ[-Pz-vß £µ¨¦ 15 G-ÚU Põm-k-P.
2
(a) The distribution function of a continuous random variable X is :

 0 , x <1
x − 1

F(x ) =  , 1x 5
 4

 1 , x >5

Find (i) P(X < 3) (ii) P(2 < X < 4) (iii) P(3 £ X)

OR
(b) Show that the area of the region bounded by 3x−2y+6=0, x=−3, x=1 and
15
x-axis, is .
2

47. (A) (1 + x 2 ) dy = 1 + y 2 GßÓ ÁøPU- ö PÊ \©ß- £ õm- i ß wº- Ä
dx
tan y=tan−1x+C AÀ-»-x tan−1x=tan−1y+C GÚ {Ö-Ä-P.
−1

AÀ»-x
(B) ö©´ø© Am-h-Á-øn-ø¯¨ £¯ß-£-kz-v p→(q→r) ≡ (p ∧ q)→r GÚ {Ö-Ä-P.

Show that the solution of the differential equation (1 + x 2 )
dy
(a) = 1 + y 2 is
dx
tan−1y=tan−1x+C (or) tan−1x=tan−1y+C.
OR
(b) Prove p→(q→r) ≡ (p ∧ q)→r using truth table.

-o0o-
A [ v¸¨¦P / Turn over