Tamil Nadu 12th Std Model Question Paper Maths

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No. of Printed Pages : 16
3262 (NS)
!3262NSMathematics! £vÄ Gs
Register Number

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

3262 (NS) 2

 12 −17 
©Ø-Ö® A−1=
1 −1
1. (AB)−1 =   GÛÀ, B =
−1
− 19 27  − 2 3 

 8 − 5  2 −5  8 5  3 1
(1) − 3 2  (2) −3 8  (3) 3 2  (4)  2 1
       

 12 −17   1 −1
If (AB)−1 =  and A−1= −1
 , then B =
− 19 27  − 2 3 

 8 − 5  2 −5  8 5  3 1
(1) − 3 2  (2) −3 8  (3) 3 2  (4)  2 1
       

2. ρ(A)=ρ([A|B]) GÛÀ, AX=B GßÓ ÷|›-¯a \©ß-£õ-k-P-Îß öuõ-S¨-£õ-Úx :
(1) J¸[-P-ø©-ÁØ-Ó-x
(2) J¸[-P-ø©-Ä-øh-¯x ©Ø-Ö® J÷µ J¸ wºÄ ö£Ø-Ô-¸U-S®
(3) J¸[-P-ø©-Ä-øh-¯-x
(4) J¸[-P-ø©-Ä-øh-¯x ©Ø-Ö® Gs-nØÓ wº-Ä-PÒ ö£Ø-Ô-¸U-S®
If ρ(A)=ρ([A|B]), then the system AX=B of linear equations is :
(1) inconsistent
(2) consistent and has a unique solution
(3) consistent
(4) consistent and has infinitely many solutions

13
3. ∑ ( i n +i n−1 ) &ß ©v¨¦ :
i=1

(1) 0 (2) 1+i (3) i (4) 1
13
( )
The value of ∑ i n +i n−1 is :
i=1

(1) 0 (2) 1+i (3) i (4) 1

4. arg(0) &ß ©v¨¦ :
(1) ∞ (2) 0
(3) π (4) Áøµ-¯-ÖU-P¨-£-h-ÂÀ-ø»
arg(0) is :
(1) ∞ (2) 0
(3) π (4) undefined

3 3262 (NS)

5. n £i-²ÒÍ J¸ £À-¾-Ö¨-¦U-÷Põ-øÁa \©ß-£õk ö£Ø-ÖÒÍ ‰»[-PÒ :
(1) \›-¯õP n ‰»[-PÒ (2) n öÁÆ-÷ÁÖ ‰»[-PÒ
(3) n ö©´-ö¯s ‰»[-PÒ (4) n P»¨-ö£s ‰»[-PÒ
A polynomial equation of degree n always has :
(1) exactly n roots (2) n distinct roots
(3) n real roots (4) n imaginary roots

2π
6. sin−1 x+sin−1 y = ; GÛÀ cos−1x+cos−1y Gß-£-uß ©v¨¦ :
3
2π π π
(1) π (2) (3) (4)
3 3 6

−1 2π
If sin x+sin−1 y = ; then cos−1x+cos−1y is equal to :
3
2π π π
(1) π (2) (3) (4)
3 3 6

1 2
7. tan−1  +tan−1   =
4 9

1 1 −1  3 
(1) tan−1   (2) cos  
2 2  5

1 −1  3  1 3
(3) sin   (4) tan−1  
2 5 2 5

1 2
tan−1  +tan−1   is :
4 9

1 1 −1  3 
(1) tan−1   (2) cos  
2 2  5

1 −1  3  1 3
(3) sin   (4) tan−1  
2 5 2 5

8. 3x2+by2+4bx−6by+b2=0 GßÓ Ám-hz-vß Bµ® :
(1) 11 (2) 1 (3) 3 (4) 10
The radius of the circle 3x +by +4bx−6by+b2=0 is :
2 2

(1) 11 (2) 1 (3) 3 (4) 10

[ v¸¨¦P / Turn over

3262 (NS) 4

9. x2=8y−1 GßÓ £µ-Á-øÍ-¯z-vß •øÚ :
 1  1  1   1
(1)  0, −  (2) − , 0  (3)  , 0 (4)  0, 
 8  8  8   8
2
The vertex of the parabola x =8y−1 is :
 1  1  1   1
(1)  0, −  (2) − , 0  (3)  , 0 (4)  0, 
 8  8  8   8

→ ∧ ∧
10. r = s i + t j (C[S s, t Gß-£øÁ xøn-¯-»S--PÒ) GßÓ \©ß-£õk :
(1) zox uÍ®
∧ ∧
(2) i, j BQ-¯-ÁØøÓ CønU-S® ÷|º-÷Põ-k
(3) xoy uÍ®
(4) yoz uÍ®
→ ∧ ∧
r = s i + t j is the equation of (s, t are parameters) :
(1) zox plane
∧ ∧
(2) a straight line joining the points i and j
(3) xoy plane
(4) yoz plane

11. x+2y+3z+7=0 ©Ø- Ö ® 2x+4y+6z+7=0 BQ- ¯ uÍ[- P - Ð US Cøh¨- £ mh
öuõ-ø»Ä :
7 7 7 7
(1) (2) (3) (4)
2 2 2 2 2 2
The distance between the planes x+2y+3z+7=0 and 2x+4y+6z+7=0 is :
7 7 7 7
(1) (2) (3) (4)
2 2 2 2 2 2

12. t GßÓ Põ-»z-vÀ Qøh-©m-h-©õP |P-¸® xP-Îß {ø» s(t)=3t2−2t−8 GÚU
öPõ-kU-P¨-£m-kÒÍx. xPÒ K´Ä {ø»US Á¸® ÷|µ® :
1
(1) t=3 (2) t=0 (3) t= (4) t=1
3
The position of a particle moving along a horizontal line of any time t is given by
s(t)=3t2−2t−8. The time at which the particle is at rest, is :
1
(1) t=3 (2) t=0 (3) t= (4) t=1
3

5 3262 (NS)

13. 100 m2 £µ¨-£ÍÄ öPõsh ö\Æ-Á-Pz-vß «a-]Ö _Ø-ÓÍÄ («m-h-›À) :
(1) 50 (2) 10 (3) 20 (4) 40
The least possible perimeter (in meter) of a rectangle of area 100 m2 is :
(1) 50 (2) 10 (3) 20 (4) 40

2 2 ∂u
14. u(x, y)=ex +y GÛÀ ∂x &ß ©v¨¦ :
2 2
(1) y2u (2) ex +y (3) 2xu (4) x2u

2 2 ∂u
If u(x, y)=ex +y , then is equal to :
∂x
2 2
(1) y2u (2) ex +y (3) 2xu (4) x2u

2
3
dx
15. ∫ &Cß ©v¨¦ :
0 4−9x 2

π π π
(1) π (2) (3) (4)
6 2 4
2
3
dx
The value of ∫ is :
0 4−9x 2

π π π
(1) π (2) (3) (4)
6 2 4

π
4
16. ∫ sin x dx &Cß ©v¨¦ :
0

3π 3π 3π 3π
(1) (2) (3) (4)
2 10 8 4
π
4
The value of ∫ sin x dx is :
0

3π 3π 3π 3π
(1) (2) (3) (4)
2 10 8 4

[ v¸¨¦P / Turn over

3262 (NS) 6

17. ø©¯® (h, k) ©Ø-Ö® Bµ® ‘a’ öPõsh GÀ-»õ Ám-h[-P-Îß ÁøPU-öP-Êa
\©ß-£õm-iß Á›-ø\ (C[S h, k, a BQ-¯øÁ ©õ-Óz-uUP ©õ-Ô-¼-PÒ AÀ-»x
H÷uaø\¯õÚ ©õ-Ô-¼-PÒ).
(1) 1 (2) 2 (3) 3 (4) 4
The order of the differential equation of all circles with centre at (h, k) and radius ‘a’,
where h, k and a are arbitrary constants, is :

(1) 1 (2) 2 (3) 3 (4) 4

dx dy
18. + = 0 GßÓ ÁøPU-öPÊ \©ß-£õm-iß Á›-ø\ ©Ø-Ö® £i •øÓ-÷¯ :
dy dx

(1) 2, £i Áøµ-¯-ÖUP C-¯»õ-x (2) 1, 2

(3) 2, 1 (4) 2, 2

The order and degree of the differential equation dx +
dy
= 0 are :
dy dx

(1) 2, degree not defined (2) 1, 2
(3) 2, 1 (4) 2, 2

19. n=25 ©Ø- Ö ® p=0.8 GÚ EÒÍ D¸- Ö ¨¦ £µ- Á À öPõsh \©- Á õ´¨¦
©õÔ X&ß vmh »U-Pz-vß ©v¨¦ :
(1) 2 (2) 6 (3) 4 (4) 3
A random variable X has binomial distribution with n=25 and p=0.8, then the standard
deviation of X is :
(1) 2 (2) 6 (3) 4 (4) 3

20. PÈz-u-¼ß Aøh-Ĩ-£s¦ ö£Óõu Pn® :
(1) ℚ (2) ℝ (3) ℤ (4) ℕ
Subtraction is not a binary operation in :

(1) ℚ (2) ℝ (3) ℤ (4) ℕ

7 3262 (NS)

£Sv & II / PART - II

SÔ¨¦ : (i) GøÁ÷¯Ý® HÊ ÂÚõUPÐUS Âøh¯ÎUPÄ®. 7x2=14

(ii) ÂÚõ Gs 30 &US Psi¨£õP Âøh¯ÎUPÄ®.
Note : (i) Answer any seven questions.

(ii) Question number 30 is compulsory.

3 3
 1+i   1−i 
21.  1−i  −  1+i  = − 2i GÚ {¹-¤U-P.
   

3 3
 1+i   1−i 
Prove that   −  1+i  = − 2i .
 1−i   

22. (1+i) (1+2i) .......... (1+ni)=x+iy GÛÀ 2⋅5⋅10⋅ ............ (1+n2)=x2+y2 GÚ {Ö-Ä-P.

If (1+i) (1+2i) .......... (1+ni)=x+iy, then prove that 2⋅5⋅10⋅ ............ (1+n2)=x2+y2.

  5π  
23. sin−1  sin    &ß ©v¨¦ Põs-P.
  4 

  5π  
Find the value of sin−1  sin   .
  4 

∧ ∧ ∧
24. 2 i + 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- Ö zx AÆ-  - ø \- ° ß •ÖU- S z vÓ- Û 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.

[ v¸¨¦P / Turn over

3262 (NS) 8

1 1  1 
25. f ( x )=x+ , x ∈ , 2  GßÓ \õº-¤ØS  , 2  GßÓ Cøh-öÁ-Î-°À ÷µõ-¼ß
x 2  2 
÷uØ-Ózøu {øÓ-Äa ö\´-²® ©v¨-ø£U Põs-P.
1 
Find the value in the interval  , 2  satisfied by the Rolle’s theorem for the function
2 

1 1 
f ( x )=x+ , x ∈ , 2 
x 2 

26. f (x)=x2+3x GßÓ \õº-¤ØS x=2, dx=0.1 GÝ® ÷£õx df &I ©v¨-¤-k-P.
For the function f (x)=x2+3x, calculate the differential df when x=2 and dx=0.1.

π
2
f (sin x ) π
27. ∫ f (sin x)+ f (cos x) dx = 4 GÚ {Ö-Ä-P.
0

π
2
f (sin x ) π
Prove that
∫ f (sin x)+ f (cos x) dx = 4 .
0

28. y2=4ax GÝ® £µ-Á-øÍ-¯z öuõ-S-v-°ß ÁøPU-öP-Êa \©ß-£õm-øhU PõsP.
C[S ‘a’ Gß-£x ©õ-Óz-uUP ©õ-Ô¼ AÀ»x H÷uaø\ ©õ-Ô¼ BS®.
Find the differential equation of the family of parabolas y2=4ax, where ‘a’ is an arbitrary
constant.

29. Kº C¯Ø-P-ou Aø©¨-¤À \©-Û EÖ¨¦ C¸U-S® GÛÀ Ax J¸-ø©z-ußø©
Áõ´¢-ux & GÚ {Ö-Ä-P.
Prove that the identity element is unique if it exists.

30. •øÚ (2, 1) ©Ø-Ö® (1, 3) GßÓ ¦ÒÎ ÁÈ-¯õP ö\À-Á-x®, Ch-¨£U-P® vÓ¨¦
Eøh-¯-x-©õÚ £µ-Á-øÍ-¯z-vß \©ß-£õk Põs-P.
Find the equation of the parabola if the curve is open leftward, vertex is (2, 1) and
passing through the point (1, 3).

9 3262 (NS)

£Sv & III / PART - III

SÔ¨¦ : (i) GøÁ÷¯Ý® HÊ ÂÚõUPÐUS Âøh¯ÎUPÄ®. 7x3=21

(ii) ÂÚõ Gs 40 &US Psi¨£õP Âøh¯ÎUPÄ®.

Note : (i) Answer any seven questions.

(ii) Question number 40 is compulsory.

2 9
 GÛÀ (A ) =(A ) GÚ {Ö-Ä-P .
31. A= T −1 −1 T
1 7 

2 9 T −1 −1 T
If A=  then prove that (A ) =(A ) .
1 7 

32. p Gß- £ x J¸ ö©´- ö ¯s GÛÀ, 4x 2+4px+p+2=0 GÝ® \©ß- £ õm- i ß
‰»[-P-Îß uß-ø©ø¯ p &ß Ai¨-£-øh-°À Bµõ´-P.

If p is real, discuss the nature of the roots of the equation 4x2+4px+p+2=0, in terms
of p.

33. J¸ PõßQŸm £õ- » ® £µ- Á - ø ͯ Ái-  À EÒÍx. \õ- ø »- ° ß- ÷ ©À EÒÍ
£õ- » z- v ß }Í® 40 « ©Ø- Ö ® Auß Av- P - £ m\ E¯- µ ® 15 « GÛÀ A¢u
£µ- Á - ø ͯ ÁøÍ- Â ß \©ß- £ õk PõsP. •øÚ- ° øÚ (0, 0) GÚ Gkz- x U
öPõÒ-P.

A concrete bridge is designed as a parabolic arch. The road over bridge is 40 m long
and the maximum height of the arch is 15 m. Write the equation of the parabolic arch.
Take (0, 0) as the vertex.

[ v¸¨¦P / Turn over

3262 (NS) 10

34. (−5, 7, −4) ©Ø- Ö ® (13, −5, 2) GßÓ ¦Ò- Î - P Ò ÁÈ- ¯ õ- P a ö\À- ¾ ®
÷|ºU-÷Põm-iß öÁU-hº ©Ø-Ö® Põº-j-]-¯ß \©ß-£õ-k-PøÍU PõsP. ÷©¾®,
C¢u ÷|ºU-÷Põk xy &uÍzøu öÁm-k® ¦Ò-Î-ø¯U PõsP.
Find the Vector and Cartesian equations of a straight line passing through the points
(−5, 7, −4) and (13, −5, 2). Find the point where the straight line crosses the
xy - plane.

4
35. f ( x )=x 5 ( x−4)2 GßÓ \õº-¤ß {ø»¨-¦ÒÎ Gs-PøÍU (x &ß ©v¨-¦-PÒ)
PõsP.
4
Find the critical numbers (only x values) of the function f ( x )=x 5 ( x−4)2 .

∂U ∂U ∂U
36. U=log(x3+y3+z3), GÛÀ + + &I Põs-P.
∂x ∂y ∂z

∂U ∂U ∂U
If U=log(x3+y3+z3) then find + + .
∂x ∂y ∂z

37. J¸ uÛ-{ø» \õº¦ X &ß {PÌ-u-PÄ {øÓ \õº-£õ-Úx :

X 1 2 3 4 5 6
P(X=x ) k 2k 6k 5k 6k 10k

GÛÀ P(2 < X < 6) &ß ©v¨-¦U Põs-P.
A random variable X has the following probability mass function :

X 1 2 3 4 5 6
P(X=x ) k 2k 6k 5k 6k 10k

then find P(2 < X < 6).

11 3262 (NS)

38. X GßÓ öuõ-hº \©-Áõ´¨¦ ©õÔ

kx (1−x )10 , 0 < x < 1
f ( x )=

0 , ¤Ó

GÚ Áøµ-¯-ÖU-P¨-£-iß, k &ß ©v¨-¤-øÚU Põs-P.

Let X be a continuous random variable and f (x) is defined as :

kx (1−x )10 , 0 < x < 1
f ( x )=
0 , otherwise

find the value of k.

39. p → q ≡ ¬p ∨ q GÚ {Ö-Ä-P.

Prove that p → q ≡ ¬p ∨ q.

x−x1 y−y1 z−z1
40. öPõ- k U- P ¨- £ mh C¸ ÷Põ- k - P Ò = = ©Ø- Ö ®
l1 m1 n1

x−x2 y−y2 z−z2
= = J¸ uÍz- v ß «x Aø©- ² - © õ- Ú õÀ Az- u Íz- v ß
l2 m2 n2

Põº-j]
- ¯
- ß \©ß-£õm-iøÚ Gz-uøÚ ÁÈ-PÎ
- À Põ-n»
- õ® ? ÁÈ-Pø
- Í TÓ-Ä®.

x−x1 y−y1 z−z1 x−x2 y−y2 z−z2
If the lines = = and = = lie on the same
l1 m1 n1 l2 m2 n2
plane, then write the number of ways to find the Cartesian equation of the above plane
and explain in detail.

[ v¸¨¦P / Turn over

3262 (NS) 12

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

41. (A) ¤ß-Á-¸® ÷|›-¯a \©ß-£õm-kz öuõ-S¨-£õ-Úx J¸[-P-ø©Ä Eøh-¯-uõ
Gß-£øu uµ •øÓ-°À Bµõ-´P.
x−y+z=−9
2x−y+z=4
3x−y+z=6
4x−y+2z=7
AÀ»x
(B) 2 cos α = x + 1 ©Ø-Ö® 2 cos β = y + 1 GÛÀ
x y

xm yn
(i) − = 2 i sin ( m α−nβ )
yn xm

1
(ii) x m y n + m n = 2cos ( m α+nβ ) GÚ {Ö-Ä-P.
x y

(a) Test the consistency of the following system of linear equations by rank method.
x−y+z=−9
2x−y+z=4
3x−y+z=6
4x−y+2z=7
OR
1 1
(b) If 2 cos α = x + and 2 cos β = y + , show that :
x y

xm yn
(i) − = 2 i sin ( m α−nβ )
yn xm

1
(ii) x m y n + m n = 2 cos ( m α+nβ )
x y

13 3262 (NS)

42. (A) cosx &ß Áøµ-£-hzøu [0, π] GßÓ Cøh-öÁ-Î-°-¾® ÷©¾® cos−1 x &ß
Áøµ-£-hzøu [−1, 1] GßÓ Cøh-öÁ-Î-°-¾® Áøµ-P.

AÀ»x

(B) (1, 1), (2, −1) ©Ø- Ö ® (3, 2) GßÓ ‰ßÖ ¦Ò- Î - P Ò ÁÈa- ö \À- ¾ ®
Ám-hz-vß \©ß-£õk Põs-P.

(a) Draw the graph of cosx in [0, π] and cos−1 x in [−1, 1].

OR

(b) Find the equation of the circle passing through the points (1, 1), (2, −1) and
(3, 2).

43. (A) uøµ-©m-hz-v¼
- ¸
- ¢x 7.5 « E¯-µz-vÀ uøµUS Cøn-¯õ-P¨ ö£õ-¸z-u¨£mh
J¸ SÇõ- ° - ¼ - ¸ ¢x öÁÎ- ÷ ¯- Ö ® }º uøµ- ø ¯z öuõ- k ® £õøu J¸
£µ-Á-øÍ-¯zøu HØ-£-kz-x-Q-Óx. ÷©¾® C¢-u¨ £µ-Á-øÍ-¯¨ £õ-øu-°ß
•øÚ SÇõ-°ß Áõ-°À Aø©-Q-Ó-x. SÇõ´ ©m-hz-vØS 2.5 « R÷Ç }›ß
£õ´-Áõ-Úx SÇõ-°ß •øÚ ÁÈ-¯õPa ö\À-¾® {ø» Sz-xU ÷Põm-iØS
3 « yµz-vÀ EÒÍx GÛÀ Sz-xU ÷Põm-i-¼-¸¢x GÆ-ÁÍÄ yµz-vØS
A¨-£õÀ }µõ-Úx uøµ-°À ÂÊ® Gß-£-øuU Põs-P.

AÀ»x

(B) öÁU-hº •øÓ-°À cos(α+β)=cosα cosβ−sinα sinβ GÚ {Ö-Ä-P.

(a) Assume that water issuing from the end of a 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

(b) By vector method, prove that, cos(α+β)=cosα cosβ−sinα sinβ.

[ v¸¨¦P / Turn over

3262 (NS) 14

44. (A) (0, 1, −5) GßÓ ¦ÒÎ ÁÈa ö\À-¾® r = i +2 j−4 k + s 2 i + 3 j + 6 k
→
( ∧ ∧ ∧
) ( ∧ ∧ ∧
)
→
( ∧ ∧ ∧
) ( ∧
©Ø-Ö® r = i − 3 j + 5 k + t i + j−k GßÓ ÷Põ-k-P-ÐUS Cøn-¯õP
∧ ∧
)
EÒÍ- x - © õÚ uÍz- v ß öÁU- h º ©Ø- Ö ® Põº- j - ] - ¯ ß \©ß- £ õ- k - P øÍU
Põs-P.
AÀ»x
cos 2 x
π
(B) ©v¨-¤-kP : ∫ dx
1 + ax −π
(a) Find the vector and Cartesian equation of the plane passing through the point
(0, 1, −5) and parallel to the straight lines
→
( ∧ ∧ ∧
) ( ∧ ∧ ∧
)
r = i +2 j−4 k + s 2 i + 3 j + 6 k and r = i − 3 j + 5 k + t i + j−k
→
( ∧ ∧ ∧
) ( ∧ ∧ ∧
)
OR

cos 2 x
π
(b) Evaluate : ∫ 1 + ax dx
−π

45. (A) Áh vø\-°¼ - ¸- ¢x J¸ ö\[-÷Põn \¢-v¨ø£ Aq-S® J¸ Põ-ÁÀ-xøÓ
Áõ-PÚ
- ® ÷ÁP-©õ-Pa ö\ßÖ v¸®¤ QÇUS ÷|õU-Qa ö\À-¾® J¸ ©Q-Ê¢øu
xµz- x - Q - Ó x. \õø» \¢- v ¨- ¤ ß ÁhU÷P 0.6 Q.«. öuõ- ø »- Â À Põ- Á À
xøÓ-°ß Áõ-P-Ú-•® QÇU÷P 0.8 Q.«. öuõ-ø»-ÂÀ ©Q-Ê¢-x® EÒÍ
ö£õ-Êx, ªß- P õ¢u Aø»U P¸-  - ° - ß xøn öPõsk Põ- Á À- x øÓ
u[-PÍx Áõ-PÚ - z-vØ-S® ©Q-Ê¢-xU-S® Cøh¨-£mh yµ® ©oUS 20 Q.«.
Ãuz-vÀ Av-P-›U-Q-Óx GÚz wº-©õ-ÛU-Qß-Ó-Úº. Põ-ÁÀ-xøÓ Áõ-P-Ú®
©oUS 60 Q.«. ÷ÁPz-vÀ |Pº-QÓ - x GÛÀ ©Q-Ê¢-vß- ÷Á-P® GßÚ ?
AÀ»x
(B) y=|cosx| GßÓ ÁøÍ- Á øµ x &Aa_, ÷Põ- k - P Ò x=0 ©Ø- Ö ® x=π
BQ-¯-ÁØ-ÓõÀ Aøh-£-k® Aµ[-Pz-vß £µ¨-ø£U Põs-P.
(a) A police jeep, approaching an orthogonal intersection from the northern direction,
is chasing a speeding car that has turned and moving straight east. When the
jeep is 0.6 km north of the intersection and the car is 0.8 km to the east, the police
determine with a radar that the distance between the jeep and the car is increasing
at 20 km/hr. If the jeep is moving at 60 km/hr at the instant of measurement,
what is the speed of the car ?
OR
(b) Find the area of the region bounded by x-axis, the curve y=|cosx|, the lines
x=0 and x=π.

15 3262 (NS)

46. (A) £µ¨- £ ÍÄ 196 \x- µ A»- S - P Ò öPõsh J¸ \xµ uPm- i øÚ Auß
JÆ- ö Áõ¸ ‰ø»- ° - ¾ ® \©- © õÚ ]Ö \x- µ [- P øÍ }UQ, ©izx J¸
ö£m-i¯õP ©õØ-Ó¨-£-k-Q-Óx. ö£m-i-°ß PÚ AÍÄ Ea-\-©õP C¸UP
7
÷Ás-k-©õ-°ß öÁmi }U-P¨-£mh \x-µz-vß £U-Pz-vß AÍÄ GÚ
3
{¹-¤UP.

AÀ»x

(B) {øÓ M Eøh-¯ J¸ uõ-Û¯
- [Q C¯¢-vµ- z-vß C¯U-Q¯
- õÀ E¸-ÁõU-P¨-£k
- ®
dV
©õ-Óõu Âø\ F GÛÀ Au-Ý-øh¯ vø\-÷Á-P® V Gß-£x M = F − kV
dt
GÝ® \©ß-£õm-hõÀ SÔU-P¨-£-k-Q-Óx. k Gß-£x ©õ-Ô-¼-¯õ-S®. t=0 GÝ®
−kt
÷£õx V=0 GÛÀ V = F
k
( 1− e Μ ) GÚ {¹-¤U-P.
(a) A square shaped thin material with area 196 sq. units to make into an open box
by cutting small equal squares from the four corners and folding the sides upward.
7
Prove that the length of the side of a removed square is when the volume of the
3
box is maximum.

OR

(b) If F is the constant force generated by the motor of an automobile of mass M, its
dV
velocity V is given by M = F − kV , where k is a constant. Prove that
dt

−kt
V=
F
k
( 1− e Μ ) when t=0 and V=0.

[ v¸¨¦P / Turn over

3262 (NS) 16

47. (A) J¸ x¨- £ - Ô - Á õͺ ¦»ß Â\õ- µ - ø n- ° ß ÷£õx, J¸- Á - › ß E°- µ ØÓ
Eh-ø» \›-¯õP ¤Ø-£-PÀ 8 ©oUS Põs-Q-Óõº. •ß-öÚa-\-›U-øP-¯õP
x¨-£-Ô-Áõͺ AÆ-Ä-h-¼ß öÁ¨-£-{ø»ø¯ AÍ¢x 708F GÚU SÔz-xU
öPõÒ-Q-Óõº. 2 ©o ÷|µ® PÈzx A¢u Eh-¼ß öÁ¨-£-{-ø» 608F BP
C¸¨-£ø - uU Põs-QÓ- õº. EhÀ C¸¢u AøÓ-°ß öÁ¨-£{ - ø» 508F BS®,
©Ø-Ö® CÓ¨-£-uØS •ß¦ A¢-|-£-›ß EhÀ öÁ¨-£-{ø» 98.68F GÛÀ,
A¢- | - £ º CÓ¢u ÷|µ® ¤Ø- £ - P À 5 ©o 26 {ª- h ® GÚ {¹- ¤ UP
(÷uõ-µõ-¯-©õP).
 log(2.43) 
 log(2) ≃ 1.28 
 
AÀ»x
(B) ‰ßÖ ^µõ-Ú |õ-n-¯[-PÒ J¸ •øÓ _s-h¨-£-k-Qß-ÓÚ. uø»-P-Îß
Gs-oUøP {PÌ-ÂØS, {PÌ-u-PÄ {øÓ \õº¦, \µõ-\› ©Ø-Ö® £µ-ÁØ-£i
PõsP. ÷©¾® D¸-Ö¨¦ £µ-ÁÀ ‰»® CÁØ-ÔøÚ ÷\õ-vU-P.
(a) In an investigation, a corpse was found by a detective at exactly 8 p.m. Being
alert, the detective also measured the body temperature and found it to be 708F.
Two hours later, the detective measured the body temperature again and found
it to be 608F. If the room temperature is 508F, and assuming that the body
temperature of the person before death was 98.68F, prove that the time of death

 log(2.43) 
is 5.26 p.m. (5 hrs 26 minutes) (app.).  ≃ 1.28 
 log(2) 

OR
(b) Three fair coins are tossed once. Find the probability mass function, mean and
variance for number of heads occurred. Verify the results by binomial distribution.

-oOo-