Series EF1GH/4 SET~3

Q.P. Code 65/4/3

Roll No. narjmWu àíZ-nÌ H$moS> >H$mo CÎma-nwpñVH$m Ho$
_wI-n¥ð >na Adí` {bIo§ &
Candidates must write the Q.P. Code on
the title page of the answer-book.

J{UV
MATHEMATICS
*
:3 : 80
Time allowed : 3 hours Maximum Marks : 80

NOTE :
(i) - 23
Please check that this question paper contains 23 printed pages.
(ii) - - -
-
Q.P. Code given on the right hand side of the question paper should be written on the title
page of the answer-book by the candidate.
(iii) - 38
Please check that this question paper contains 38 questions.
(iv) -

Please write down the serial number of the question in the answer-book before
attempting it.
(v) - 15 -
10.15 10.15 10.30 -
-
15 minute time has been allotted to read this question paper. The question paper will be
distributed at 10.15 a.m. From 10.15 a.m. to 10.30 a.m., the students will read the
question paper only and will not write any answer on the answer-book during this period.

65/4/3 JJJJ Page 1 P.T.O.

 :
:
(i) 38
(ii)
(iii) 1 18 19 20

(iv) 21 25 (VSA)

(v) 26 31 (SA)

(vi) 32 35 (LA)
(vii) 36 38

(viii) 2 3
2 2

(ix)

IÊS> H$
1

1. `{X A EH$ 3 4 Amì`yh h¡ VWm B EH$ Eogm Amì`yh h¡ {H AB VWm AB XmoZm| n[a^m{fV
h¢, Vmo Amì`yh B H$s H$mo{Q> h¡ :
(a) 3 4 (b) 3 3

(c) 4 4 (d) 4 3

2. `{X erfmªo (2, 6), (5, 4) VWm (k, 4) dmbo {Ì^wO H$m joÌ\$b 35 dJ© BH$mB© h¡, Vmo
k ~am~a h¡ :
(a) 12 (b) 2

(c) 12, 2 (d) 12, 2

65/4/3 JJJJ Page 2

General Instructions :
Read the following instructions very carefully and strictly follow them :
(i) This question paper contains 38 questions. All questions are compulsory.
(ii) This question paper is divided into five Sections A, B, C, D and E.
(iii) In Section A, Questions no. 1 to 18 are multiple choice questions (MCQs) and
questions number 19 and 20 are Assertion-Reason based questions of 1 mark
each.
(iv) In Section B, Questions no. 21 to 25 are very short answer (VSA) type
questions, carrying 2 marks each.
(v) In Section C, Questions no. 26 to 31 are short answer (SA) type questions,
carrying 3 marks each.
(vi) In Section D, Questions no. 32 to 35 are long answer (LA) type questions
carrying 5 marks each.
(vii) In Section E, Questions no. 36 to 38 are case study based questions carrying
4 marks each.
(viii) There is no overall choice. However, an internal choice has been provided in
2 questions in Section B, 3 questions in Section C, 2 questions in Section D and
2 questions in Section E.
(ix) Use of calculators is not allowed.

SECTION A
This section comprises multiple choice questions (MCQs) of 1 mark each.

1. If A is a 3 4 matrix and B is a matrix such that A B and AB are both

defined, then the order of the matrix B is :

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

2. If the area of a triangle with vertices (2, 6), (5, 4) and (k, 4) is
35 sq units, then k is
(a) 12 (b) 2
(c) 12, 2 (d) 12, 2

65/4/3 JJJJ Page 3 P.T.O.

3. `{X f(x) = 2 x + 3 sin x + 6 h¡, Vmo f(x) H$m x = 0 na XmE± nj H$m AdH$bO h¡ :
(a) 6 (b) 5

(c) 3 (d) 2

1 2 4
4. `{X x y h¡, Vmo :
2 5 9

(a) x = 1, y = 2 (b) x = 2, y = 1
(c) x = 1, y = 1 (d) x = 3, y = 2

5. `{X Amì`yh A = [1 2 3] h¡, Vmo Amì`yh AA h¡ (Ohm± A Amì`yh A H$m n[adV© h¡) :
1 0 0
(a) 14 (b) 0 2 0
0 0 3

1 2 3
(c) 2 3 1 (d) 14
3 1 2

a b a b
6. JwUZ\$b ~am~a h¡ :
b a b a

a2 b2 0 a b2 0
(a) (b)
0 a2 b2 a b2 0

a2 b2 0 a 0
(c) (d)
a2 b2 0 0 b

7. q~Xþ (p, q, r) H$s y-Aj go Xÿar h¡ :

(a) q (b) q

(c) q + r (d) p2 r2

65/4/3 JJJJ Page 4

3. If f(x) = 2 x + 3 sin x + 6, then the right hand derivative of f(x) at x = 0
is :
(a) 6 (b) 5
(c) 3 (d) 2

1 2 4
4. If x y , then :
2 5 9

(a) x = 1, y = 2 (b) x = 2, y = 1
(c) x = 1, y = 1 (d) x = 3, y = 2

5. If a matrix A = [1 2 3], then the matrix AA (where A is the transpose

of A) is :
1 0 0
(a) 14 (b) 0 2 0
0 0 3
1 2 3
(c) 2 3 1 (d) 14
3 1 2

a b a b
6. The product is equal to :
b a b a

a2 b2 0 a b2 0
(a) (b)
0 a2 b2 a b2 0

a2 b2 0 a 0
(c) (d)
a2 b2 0 0 b

7. Distance of the point (p, q, r) from y-axis is :

(a) q (b) q

(c) q + r (d) p2 r2

65/4/3 JJJJ Page 5 P.T.O.

8. Ag{_H$m 3x + 5y 7

(a) aoIm 3x + 5y = 7 xy-Vb

(b) aoIm 3x + 5y = 7 na pñWV q~XþAm| Ho$ gmW nyam xy-Vb

(c) aoIm 3x + 5y = 7 na
_yb-q~Xþ ^r h¡ &
(d) dh Iwbm AmYm Vb {Og_| _yb-q~Xþ Zht h¡ &
a

9. `{X 3x 2 dx = 8 h¡, Vmo H$m _mZ h¡ :

0

(a) 2 (b) 4
(c) 8 (d) 10

^ ^ ^ ^ ^ ^
10. g{Xem| a = 3i + j + 2k VWm b = i + j + 2k Ho$ ~rM Ho$ H$moU H$m Á`m (gmBZ) h¡ :

5 5
(a) (b)
21 21

3 4
(c) (d)
21 21

2 3
d 2y dy dy
11. AdH$b g_rH$aU 2
x sin Ho$ H$mo{Q> d KmV (`{X n[a^m{fV h¢)
dx dx dx
H«$_e: h¢ :
(a) 2, 2 (b) 1, 3
(c) 2, 3 (d) 2, KmV n[a^m{fV Zht

12. e 5 log x dx ~am~a h¡ :

x5 x6
(a) +C (b) +C
5 6
(c) 5x4 + C (d) 6x5 + C
65/4/3 JJJJ Page 6
8. The solution set of the inequation 3x + 5y 7 is :

(a) whole xy-plane except the points lying on the line 3x + 5y = 7.

(b) whole xy-plane along with the points lying on the line 3x + 5y = 7.

(c) open half plane containing the origin except the points of line
3x + 5y = 7.

(d) open half plane not containing the origin.

a

9. If 3x 2 dx = 8,
0

(a) 2 (b) 4
(c) 8 (d) 10

^ ^ ^
10. The sine of the angle between the vectors a = 3 i + j + 2 k and
^ ^ ^
b = i + j + 2 k is :

5 5
(a) (b)
21 21
3 4
(c) (d)
21 21
11. The order and degree (if defined) of the differential equation,
2 3
d 2y dy dy
2
x sin respectively are :
dx dx dx

(a) 2, 2 (b) 1, 3
(c) 2, 3 (d) 2, degree not defined

12. e5 log x dx is equal to :

x5 x6
(a) +C (b) +C
5 6
(c) 5x4 + C (d) 6x5 + C

65/4/3 JJJJ Page 7 P.T.O.

13. g{Xe 4 ^i ^
3k H$s {Xem _| EH$ _mÌH$ g{Xe h¡ :
1 ^ ^
(a) (4 i 3k )
7

1 ^ ^
(b) (4 i 3k )
5

1 ^ ^
(c) (4 i 3k )
7

1 ^ ^
(d) (4 i 3k )
5

14. {ZåZ{b{IV _| go H$m¡Z-gm q~Xþ {ZåZ XmoZm| Ag{_H$mAm| H$mo g§VwîQ> H$aVm h¡ ?
2x + y 10 VWm x + 2y 8

(a) ( 2, 4) (b) (3, 2)

(c) ( 5, 6) (d) (4, 2)

dy
15. `{X y = sin2 (x3) h¡, Vmo ~am~a h¡ :
dx

(a) 2 sin x3 cos x3 (b) 3x3 sin x3 cos x3

(c) 6x2 sin x3 cos x3 (d) 2x2 sin2 (x3)

16. xy-Vb H$m H$moB© q~Xþ (x, y, 0), q~XþAm| (1, 2, 3) VWm (3, 2, 1) H$mo {_bmZo dmbo aoImIÊS>
H$mo {Og AZwnmV _| ~m±Q>Vm h¡, dh h¡ :
(a) 1:2 AÝV: (b) 2 : 1 AÝV:

(c) 3:1 AÝV: (d) 3 : 1 ~mø

65/4/3 JJJJ Page 8

 ^ ^
13. A unit vector along the vector 4 i 3 k is :

1 ^ ^
(a) (4 i 3k )
7

1 ^ ^
(b) (4 i 3k )
5

1 ^ ^
(c) (4 i 3k )
7

1 ^ ^
(d) (4 i 3k )
5

14. Which of the following points satisfies both the inequations 2x + y 10

and x + 2y 8?

(a) ( 2, 4) (b) (3, 2)

(c) ( 5, 6) (d) (4, 2)

dy
15. If y = sin2 (x3), then is equal to :
dx

(a) 2 sin x3 cos x3 (b) 3x3 sin x3 cos x3

(c) 6x2 sin x3 cos x3 (d) 2x2 sin2 (x3)

16. The point (x, y, 0) on the xy-plane divides the line segment joining the
points (1, 2, 3) and (3, 2, 1) in the ratio :

(a) 1 : 2 internally (b) 2 : 1 internally

(c) 3 : 1 internally (d) 3 : 1 externally

65/4/3 JJJJ Page 9 P.T.O.

17. ñdV§Ì KQ>ZmAm| E VWm F Ho$ {bE, `{X P(E) = 0·3 VWm P(E F) = 0·5 h¡, Vmo
P(E/F) P(F/E) ~am~a h¡ :
1 2
(a) (b)
7 7
3 1
(c) (d)
35 70
dy
18. AdH$b g_rH$aU x y 2x 2 H$mo hb H$aZo Ho$ {bE g_mH$bZ JwUH$ h¡ :
dx
(a) e y (b) e x
1
(c) x (d)
x

19 20 1
(A) (R)
(a), (b), (c) (d)
(a) A{^H$WZ (A) Am¡a VH©$ (R) XmoZm| ghr h¢ Am¡a VH©$ (R), A{^H$WZ (A) H$s ghr
ì¶m»¶m H$aVm h¡ &
(b) A{^H$WZ (A) Am¡a VH©$ (R) XmoZm| ghr h¢, naÝVw VH©$ (R), A{^H$WZ (A) H$s ghr
ì¶m»¶m H$aVm h¡ &
(c) A{^H$WZ (A) ghr h¡ VWm VH©$ (R) µJbV h¡ &
(d) A{^H$WZ (A) µJbV h¡ VWm VH©$ (R) ghr h¡ &

19. (A) : aoImE± r = a1 + b1 VWm r = a2 + b2 nañna b§~dV² h¢, O~

b1 . b 2 = 0 h¡ &

(R) : aoImAm| r = a1 + b 1 VWm r = a2 + b2 Ho$ ~rM H$m H$moU ,

b1 . b 2
cos Ûmam àXÎm h¡ &
| b1 || b |
2

20. (A) : g^r {ÌH$moU{_Vr` \$bZm| Ho$ AnZo àmÝV _| ì`wËH«$_ hmoVo h¢ &
(R) : tan 1 x Ho$ {H$gr x Ho$ {bE ì`wËH«$_ H$m ApñVËd h¡ &
65/4/3 JJJJ Page 10
17. The events E and F are independent. If P(E) = 0·3 and P(E F) = 0·5,
then P(E/F) P(F/E) equals :
1 2
(a) (b)
7 7
3 1
(c) (d)
35 70
18. The integrating factor for solving the differential equation
dy
x y 2x 2 is :
dx
(a) e y (b) e x
1
(c) x (d)
x

Questions number 19 and 20 are Assertion and Reason based questions carrying
1 mark each. Two statements are given, one labelled Assertion (A) and the other
labelled Reason (R). Select the correct answer from the codes (a), (b), (c) and (d)
as given below.

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the
correct explanation of the Assertion (A).
(b) Both Assertion (A) and Reason (R) are true, but Reason (R) is not
the correct explanation of the Assertion (A).
(c) Assertion (A) is true and Reason (R) is false.
(d) Assertion (A) is false and Reason (R) is true.

19. Assertion (A) : The lines r = a1 + b1 and r = a2 + b 2 are

perpendicular, when b1 . b 2 = 0.

Reason (R) : The angle between the lines r = a1 + b1 and

b1 . b 2
r = a 2 + b 2 is given by cos
| b1 || b |
2

20. Assertion (A) : All trigonometric functions have their inverses over their
respective domains.

Reason (R) : The inverse of tan 1 x exists for some x .

65/4/3 JJJJ Page 11 P.T.O.

 IÊS> I

(VSA) 2

dy y 1
21. `{X xy = ex y h¡, Vmo Xem©BE {H$ .
dx xy 1

22. (H$) y = sin 1 (x2 4) H$m àmÝV kmV H$s{OE &

AWdm
(I) _mZ kmV H$s{OE :
7
cos 1 cos
3

^ ^ ^ ^ ^ ^ 1
23. `{X g{Xe i + j + k H$m g{Xe pi + j 2k na àjon h¡, Vmo p H$m/Ho$ _mZ
3
kmV H$s{OE &

24. dH«$ y2 = 8x na dh q~Xþ kmV H$s{OE Ohm± x-{ZX}em§H$ VWm y-{ZX}em§H$ g_mZ Xa go
~XbVo h¢ &

x 1 y 2 z 3
25. (H$) {~ÝXþ (2, 1, 3) go hmoH$a OmZo dmbr VWm aoImAm| ;
1 2 3
x y z
XmoZm| Ho$ b§~dV² EH$ aoIm H$m g{Xe g_rH$aU kmV H$s{OE &$
3 2 5

AWdm

(I) EH$ aoIm Ho$ g_rH$aU 5x 3 = 15y + 7 = 3 10z h¢ & Bg aoIm Ho$
{XH²$-H$mogmBZ {b{IE VWm Bg na pñWV EH$ q~Xþ Ho$ {ZX}em§H$ kmV H$s{OE &

65/4/3 JJJJ Page 12

 SECTION B

This section comprises very short answer (VSA) type questions of 2 marks each.

dy y 1
21. If xy = ex y, then show that .
dx xy 1

22. (a) Find the domain of y = sin 1 (x2 4).

OR
(b) Evaluate :

7
cos 1 cos
3

^ ^ ^ ^ ^ ^ 1
23. If the projection of the vector i + j + k on the vector p i + j 2 k is ,
3
then find the value(s) of p.

24. Find the point on the curve y2 = 8x for which the abscissa and ordinate
change at the same rate.

25. (a) Find the vector equation of the line passing through the point
(2, 1, 3) and perpendicular to both the lines

x 1 y 2 z 3 x y z
; .
1 2 3 3 2 5
OR
(b) The equations of a line are 5x 3 = 15y + 7 = 3 10z. Write the
direction cosines of the line and find the coordinates of a point
through which it passes.

65/4/3 JJJJ Page 13 P.T.O.

 IÊS> J
(SA) 3

26. kmV H$s{OE :

2
dx
(1 x) (1 x2 )

27. (H$) _mZ kmV H$s{OE :

1
(x x 3 )1 / 3
dx
x4
1/3

AWdm
(I) _mZ kmV H$s{OE :
3

1 (x 2) dx
1

28. {ZåZ{b{IV a¡{IH$ àmoJ«m_Z g_ñ`m H$mo AmboI Ûmam hb H$s{OE :

ì`damoYm| 3x + 5y 15,

5x + 2y 10,

x, y 0
Ho$ A§VJ©V z = 5x + 3y H$m A{YH$V_ _mZ kmV H$s{OE &

29. 30 ~ë~m| H$s EH$ T>oar _| go, {Og_| 6 ~ë~ Iam~ h¢, 2 ~ë~m| H$m EH$ Z_yZm `mÑÀN>`m
EH$-EH$ H$aHo$ à{VñWmnZ g{hV {ZH$mbm J`m & Iam~ ~ë~m| H$s g§»`m H$m àm{`H$Vm ~§Q>Z
kmV H$s{OE, AV: Iam~ ~ë~m| H$s g§»`m H$m _mÜ` kmV H$s{OE &

65/4/3 JJJJ Page 14

 SECTION C
This section comprises short answer (SA) type questions of 3 marks each.

26. Find :

2
dx
(1 x) (1 x2 )

27. (a) Evaluate :

1
(x x 3 )1 / 3
dx
x4
1/3

OR
(b) Evaluate :
3

1 (x 2) dx
1

28. Solve the following linear programming problem graphically :

Maximise z = 5x + 3y

subject to the constraints

3x + 5y 15,
5x + 2y 10,
x, y 0.

29. From a lot of 30 bulbs which include 6 defective bulbs, a sample of

2 bulbs is drawn at random one by one with replacement. Find the
probability distribution of the number of defective bulbs and hence find
the mean number of defective bulbs.

65/4/3 JJJJ Page 15 P.T.O.

 dy x y
30. (H$) AdH$b g_rH$aU , y(1) = 0 H$m {d{eîQ> hb kmV H$s{OE &$
dx x

AWdm
(I) AdH$b g_rH$aU ex tan y dx + (1 ex) sec2 y dy = 0 H$m ì`mnH$> hb kmV
H$s{OE &$
31. (H$) _mZ kmV H$s{OE :
/2
1 sin 2x
e 2x dx
1 cos 2x
/4

AWdm
(I) _mZ kmV H$s{OE :
2
x2
dx
1 5x
2

IÊS> K
(LA) 5

11 y 2 z 8
32. (H$) q~Xþ (2, 1, 5) H$m aoIm _| à{V{~å~ kmV H$s{OE &
10 4 11

AWdm
x 2 y 1 z
(I) EH$ ABC Ho$ erf© B VWm C aoIm na pñWV h¢ &
2 1 4
ABC H$m joÌ\$b kmV H$s{OE O~{H$ {X`m J`m h¡ {H$ q~Xþ A Ho$ {ZX}em§H$
(1, 1, 2) h¢ VWm aoImIÊS> BC H$s b§~mB© 5 BH$mB© h¡ &
1 1 2
33. Amì`yh A 0 2 3 H$m ì`wËH«$_ kmV H$s{OE & ì`wËH«$_ A 1 Ho$ à`moJ go,
3 2 4
a¡{IH$ g_rH$aU {ZH$m` x y + 2z = 1; 2y 3z = 1; 3x 2y + 4z = 3 H$mo hb
H$s{OE &

65/4/3 JJJJ Page 16

30. (a) Find the particular solution of the differential equation
dy x y
, y(1) = 0.
dx x
OR
(b) Find the general solution of the differential equation
ex tan y dx + (1 ex) sec2 y dy = 0.

31. (a) Evaluate :

/2
1 sin 2x
e 2x dx
1 cos 2x
/4

OR
(b) Evaluate :
2
x2
dx
1 5x
2
SECTION D

This section comprises long answer (LA) type questions of 5 marks each.
32. (a) Find the image of the point (2, 1, 5) in the line
11 y 2 z 8
.
10 4 11
OR
x 2 y 1 z
(b) . Find
2 1 4
point A has coordinates (1, 1, 2) and
the line segment BC has length of 5 units.
1 1 2
33. Find the inverse of the matrix A 0 2 3 . Using the inverse,
3 2 4
1
A , solve the system of linear equations

x y + 2z = 1; 2y 3z = 1; 3x 2y + 4z = 3.

65/4/3 JJJJ Page 17 P.T.O.

34. g_mH$bZ Ho$ à`moJ go, nadb` y2 = 4ax VWm BgHo$ Zm{^b§~ go {Kao joÌ H$m joÌ\$b
kmV H$s{OE &

35. (H$) `{X N, g^r N N _| EH$

g§~§Y R, Bg àH$ma n[a^m{fV h¡ {H$ (a, b) R (c, d), `{X ad(b + c) = bc(a + d).
Xem©BE {H$ R EH$ Vwë`Vm g§~§Y h¡ &$
AWdm
4 4x
(I) _mZm f: f (x ) Ûmam n[a^m{fV EH$ \$bZ h¡ & Xem©BE
3 3x 4
{H$ f EH$ EH¡$H$s \$bZ h¡ & `h ^r Om±M H$s{OE {H$ f EH$ AmÀN>mXH$ \$bZ h¡ `m
Zht &
IÊS> L>

3 4

àH$aU AÜ``Z 1

36. EH$ ^dZ ~ZmZo dmbm R>oHo$Xma, EH$ ßbm°Q na 4 âb¡Q> VWm nm{Hª$J joÌ H$m H$m_ boVm h¡ &

0·65 h¡ & ~hþV go

l{_H$m| Ho$ Z hmoZo na ^r H$m`© Ho$ g_` na nyam hmo OmZo H$s àm{`H$Vm >0·35 h¡ & g^r
l{_H$m| Ho$ H$m_ na AmZo na H$m`© g_` na nyam hmoZo H$s àm{`H$Vm 0·80 h¡ &

_mZm : E1 : {Zê${nV H$aVm h¡ Cg KQ>Zm H$mo O~ ~hþV go l{_H$ H$m_ na Zht AmE;

E2 : {Zê${nV H$aVm h¡ dh KQ>Zm O~ g^r l{_H$ H$m_ na AmE; Am¡a

E : {Zê${nV H$aVm h¡ {H$ H$m`© g_` na nyam hmo OmVm h¡ &

65/4/3 JJJJ Page 18

34. Using integration, find the area of the region bounded by the parabola
y2 = 4ax and its latus rectum.

35. (a) If N denotes the set of all natural numbers and R is the relation on
N N defined by (a, b) R (c, d), if ad(b + c) = bc(a + d). Show that R
is an equivalence relation.
OR
4 4x
(b) Let f : be a function defined as f (x ) . Show
3 3x 4
that f is a one-one function. Also, check whether f is an onto
function or not.

SECTION E

This section comprises 3 case study based questions of 4 marks each.

Case Study 1

36. A building contractor undertakes a job to construct 4 flats on a plot along

with parking area. Due to strike the probability of many construction

workers not being present for the job is 0·65. The probability that many

are not present and still the work gets completed on time is 0·35. The

probability that work will be completed on time when all workers are

present is 0·80.

Let : E1 : represent the event when many workers were not present for
the job;

E2 : represent the event when all workers were present; and

E : represent completing the construction work on time.

65/4/3 JJJJ Page 19 P.T.O.

 Cn`w©º$ gyMZm Ho$ AmYma na, {ZåZ{b{IV àíZm| Ho$ CÎma Xr{OE :

(i) ? 1

(ii) H$m`© g_` na nyam hmo OmZo ? 1

(iii) (H$) {X`m J`m h¡ {H$ H$m`© g_` na nyam hmo J`m, Vmo ~hþV go l{_H$m| Ho$ H$m_ na

Z AmZo ? 2

AWdm

(iii) (I) {X`m J`m h¡ {H$ H$m`© g_` na nyam hmo J`m, Vmo g^r l{_H$m| Ho$ H$m_ na

CnpñWV hmoZo H$s m h¡ ? 2

àH$aU AÜ``Z 2

37. _mZm f(x) EH$ dmñV{dH$ _mZ dmbm \$bZ h¡ & Vmo BgH$m
f (a h) f (a)
~mE± nj H$m AdH$bO (L.H.D.) : Lf (a) = lim
h 0 h
f (a h) f (a)
XmE± nj H$m AdH$bO (R.H.D.) : Rf (a) = lim
h 0 h

gmW hr, EH$ \$bZ f(x), x = a na AdH$bZr` H$hbmVm h¡ `{X x=a na BgHo$ L.H.D.
Am¡a R.H.D. H$m ApñVËd h¡ VWm XmoZm| g_mZ h¢ &
x 3, x 1
\$bZ f( x ) 2
x 3x 13
,x 1
4 2 4
Ho$ {bE {ZåZ{b{IV àíZm| Ho$ CÎma Xr{OE :
(i) f(x) H$m x = 1 na XmE± nj H$m AdH$bO (R.H.D.) ? 1

(ii) f(x) H$m x = 1 na ~mE± nj H$m AdH$bO (L.H.D.) ? 1

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 Based on the above information, answer the following questions :

(i) What is the probability that all the workers are present for the job ? 1

(ii) What is the probability that construction will be completed on time ? 1

(iii) (a) What is the probability that many workers are not present
given that the construction work is completed on time ? 2

OR

(iii) (b) What is the probability that all workers were present given
that the construction job was completed on time ? 2

Case Study 2

37. Let f(x) be a real valued function. Then its

f (a h) f (a)
Left Hand Derivative (L.H.D.) : Lf (a) = lim
h 0 h
f (a h) f (a)
Right Hand Derivative (R.H.D.) : Rf (a) = lim
h 0 h

Also, a function f(x) is said to be differentiable at x = a if its L.H.D. and

R.H.D. at x = a exist and both are equal.
x 3, x 1
For the function f(x) x2 3x 13
,x 1
4 2 4

answer the following questions :

(i) What is R.H.D. of f(x) at x = 1 ? 1

(ii) What is L.H.D. of f(x) at x = 1 ? 1
65/4/3 JJJJ Page 21 P.T.O.
 (iii) (H$) x = 1 na \$bZ f(x) AdH$bZr` h¡ & 2

AWdm
(iii) (I) f (2) VWm f ( 1) kmV H$s{OE & 2

àH$aU AÜ``Z 3

38. gyaO Ho$ {nVm EH$ BªQ>m| H$s Xrdma H$mo EH$ gmBS> boH$a, EH$ Am`VmH$ma ~mJ ~ZmZm MmhVo h¢
bJmZm MmhVo h¢ (O¡gm {MÌ _| {XIm`m h¡) &
Ho$ {bE 200 _rQ>a H$s Vma h¡ &

Cn`w©º$ gyMZm Ho$ AmYma na, {ZåZ{b{IV àíZm| Ho$ CÎma Xr{OE :
(i) _mZm ~mJ _| BªQ> H$s Xrdma Ho$ b§~dV² gmBS> H$s b§~mB© _rQ>a h¡ VWm BªQ> H$s
Xrdma Ho$ g_m§Va gmBS> H$s b§~mB© _rQ>a h¡ & Hw$b bJZo dmbr Vma H$s b§~mB© H$m
gyÌ (g§~§Y) kmV H$s{OE VWm ~mJ H$m joÌ\$b A(x) ^r {b{IE & 2

(ii) A(x) H$m A{YH$V_ _mZ kmV H$s{OE & 2

65/4/3 JJJJ Page 22

 (iii) (a) Check if the function f(x) is differentiable at x = 1. 2

OR

(iii) (b) Find the f (2) and f ( 1). 2

Case Study 3

38.
on one side of the garden and wire fencing for the other three sides as
shown in the figure. He has 200 metres of fencing wire.

Based on the above information, answer the following questions :

(i)

the side parallel to the brick wall. Determine the relation

representing the total length of fencing wire and also write A(x),
the area of the garden. 2

(ii) Determine the maximum value of A(x). 2

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