65/1 1 P.T.O.

narjmWu H$moS >H$mo CÎma-nwpñVH$m Ho$ _wI-n¥ð >na Adí` {bIo§ & Candidates must write the Code on the

title page of the answer-book.

Series OSR/C H$moS> Z§. 65/1

Code No.

amob Z§. Roll No.

J{UV MATHEMATICS

{ZYm©[aV g_` : 3 KÊQ>o A{YH$V_ A§H$ : 100

Time allowed : 3 hours Maximum Marks : 100

H¥$n`m Om±M H$a b| {H$ Bg àíZ-nÌ _o§ _w{ÐV n¥ð> 12 h¢ &

àíZ-nÌ _| Xm{hZo hmW H$s Amoa {XE JE H$moS >Zå~a H$mo N>mÌ CÎma-nwpñVH$m Ho$ _wI-n¥ð> na {bI| &

H¥$n`m Om±M H$a b| {H$ Bg àíZ-nÌ _| >29 àíZ h¢ &

H¥$n`m àíZ H$m CÎma {bIZm ewê$ H$aZo go nhbo, àíZ H$m H«$_m§H$ Adí` {bI| & Bg àíZ-nÌ H$mo n‹T>Zo Ho$ {bE 15 {_ZQ >H$m g_` {X`m J`m h¡ & àíZ-nÌ H$m {dVaU nydm©•

_| 10.15 ~Oo {H$`m OmEJm & 10.15 ~Oo go 10.30 ~Oo VH$ N>mÌ Ho$db àíZ-nÌ H$mo n‹T>|Jo Am¡a Bg Ad{Y Ho$ Xm¡amZ do CÎma-nwpñVH$m na H$moB© CÎma Zht {bI|Jo &

Please check that this question paper contains 12 printed pages.

Code number given on the right hand side of the question paper should be

written on the title page of the answer-book by the candidate.

Please check that this question paper contains 29 questions.

Please write down the Serial Number of the question before

attempting it.

15 minutes 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.

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65/1 2

gm_mÝ` {ZX}e :

(i) g^r àíZ A{Zdm`© h¢ &

(ii) Bg àíZ nÌ _| 29 àíZ h¢ Omo VrZ IÊS>m| _| {d^m{OV h¢ : A, ~ VWm g & IÊS> A _|

10 àíZ h¢ {OZ_| go àË`oH$ EH$ A§H$ H$m h¡ & IÊS> ~ _| 12 àíZ h¢ {OZ_| go àË`oH$ Mma

A§H$ H$m h¡ & IÊS> g _| 7 àíZ h¢ {OZ_| go àË`oH$ N>… A§H$ H$m h¡ &

(iii) IÊS> A _| g^r àíZm| Ho$ CÎma EH$ eãX, EH$ dmŠ` AWdm àíZ H$s Amdí`H$Vm AZwgma

{XE Om gH$Vo h¢ &

(iv) nyU© àíZ nÌ _| {dH$ën Zht h¢ & {\$a ^r Mma A§H$m| dmbo 4 àíZm| _| VWm N>… A§H$m| dmbo

2 àíZm| _| AmÝV[aH$ {dH$ën h¡ & Eogo g^r àíZm| _| go AmnH$mo EH$ hr {dH$ën hb H$aZm

h¡ &

(v) H¡$bHw$boQ>a Ho$ à`moJ H$s AZw_{V Zht h¡ & `{X Amdí`H$ hmo Vmo Amn bKwJUH$s` gma{U`m±

_m±J gH$Vo h¢ &

General Instructions :

(i) All questions are compulsory.

(ii) The question paper consists of 29 questions divided into three sections A,

B and C. Section A comprises of 10 questions of one mark each, Section B

comprises of 12 questions of four marks each and Section C comprises

of 7 questions of six marks each.

(iii) All questions in Section A are to be answered in one word, one sentence or

as per the exact requirement of the question.

(iv) There is no overall choice. However, internal choice has been provided in

4 questions of four marks each and 2 questions of six marks each. You

have to attempt only one of the alternatives in all such questions.

(v) Use of calculators is not permitted. You may ask for logarithmic tables, if

required.

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65/1 3 P.T.O.

IÊS> A

SECTION A

àíZ g§»`m 1 go 10 VH$ àË`oH$ àíZ 1 A§H$ H$m h¡ &

Question numbers 1 to 10 carry 1 mark each.

1. _mZm * : R R R, (a, b) a + 4b2 Ûmam àXÎm EH$ {ÛAmYmar g§{H«$`m h¡ & (– 5) * (2 * 0) H$m n[aH$bZ H$s{OE &

Let * : R R R given by (a, b) a + 4b2 is a binary operation.

Compute (– 5) * (2 * 0).

2. EH$ 3 3 Amì`yh Ho$ Ad`d aij = 2

1|– 3i + j| Ûmam àXÎm h¢ & Ad`d a32 H$m _mZ

{b{IE &

The elements aij of a 3 3 matrix are given by aij = 2

1|– 3i + j|. Write

the value of element a32.

3. tan–1

2–sin H$m _w»` _mZ {b{IE &

Write the principal value of tan–1

2–sin .

4. `{X (2x 4)

8–

x = 0 h¡, Vmo x H$m YZmË_H$ _mZ kmV H$s{OE &

If (2x 4)

8–

x = 0, find the positive value of x.

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65/1 4

5. gma{UH$

8695

7583

6572

H$m _mZ {b{IE &

Write the value of

8695

7583

6572

.

6. dxxcos

xsin

8

6

kmV H$s{OE &

Find .dxxcos

xsin

8

6

7. _mZ kmV H$s{OE :

1

0

2

1–

dxx1

xtan

Evaluate :

1

0

2

1–

dxx1

xtan

8. `{X |a | = 8, |

b | = 3 VWm |

a

b | = 12 h¡, Vmo

a VWm

b Ho$ ~rM H$m

H$moU kmV H$s{OE &

If |a | = 8, |

b | = 3 and |

a

b | = 12, find the angle between

a

and b .

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65/1 5 P.T.O.

9. g{Xe ^i +

^j +

^k VWm x-Aj Ho$ ~rM H$m H$moU kmV H$s{OE &

Find the angle between x-axis and the vector ^i +

^j +

^k .

10. {~ÝXþ (, , ) go JwµOaZo dmbr Cg gab aoIm H$m g_rH$aU {b{IE Omo z-Aj Ho$ g_mÝVa

h¡ &

Write the equation of the straight line through the point (, , ) and

parallel to z-axis.

IÊS> ~

SECTION B

àíZ g§»`m 11 go 22 VH$ àË`oH$ àíZ 4 A§H$ H$m h¡ & Question numbers 11 to 22 carry 4 marks each.

11. _mZ br{OE Xmo \$bZ f , g : R R {ZåZ ê$n go n[a^m{fV h¢ : f (x) = |x| + x VWm

g(x) = |x| – x, g^r x R Ho$ {bE & Vmo fog VWm gof kmV H$s{OE &

Let f, g : R R be two functions defined as f (x) = |x| + x and

g(x) = |x| – x, for all x R. Then find fog and gof.

12. {gÕ H$s{OE {H$ :

cos–1 (x) + cos–1 32

x3–3

2

x2

AWdm

x Ho$ {bE hb H$s{OE :

tan–1 x + 2 cot–1 x = 3

2

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65/1 6

Prove that :

cos–1 (x) + cos–1 32

x3–3

2

x2

OR

Solve for x :

tan–1 x + 2 cot–1 x = 3

2

13. gma{UH$m| Ho$ JwUY_m] Ho$ à`moJ go {ZåZ H$mo {gÕ H$s{OE :

.abc4

bacc

bacb

aacb

Using properties of determinants, prove the following :

.abc4

bacc

bacb

aacb

14. AMa am{e k H$m _mZ kmV H$s{OE {OgHo$ {bE {ZåZ Ûmam n[a^m{fV \$bZ f, {~ÝXþ x = 0

na gVV hmo :

f(x) =

0x,k

0x,x8

x4cos–1

2

`{X

`{X

Find the value of the constant k so that the function f, defined below, is

continuous at x = 0, where

f(x) =

0xif,k

0xif,x8

x4cos–1

2

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65/1 7 P.T.O.

15. `{X y = tan– 1

x

a + log

ax

a–x

h¡, Vmo {gÕ H$s{OE {H$

44

3

a–x

a2

dx

dy h¡ &

If y = tan– 1

x

a + log

ax

a–x

, prove that

44

3

a–x

a2

dx

dy .

16. CZ A§Vambm| H$mo kmV H$s{OE {OZ_| \$bZ f(x) = 11x5

36x3–x

5

4–x

10

3 234

(a) {Za§Va dY©_mZ h¡, (b) {Za§Va õmg_mZ h¡ &

AWdm

EH$ g_~mhþ {Ì^wO H$s ^wOmE± 2 go_r/goH$ÊS> H$s Xa go ~‹T> ahr h¢ & Bg {Ì^wO H$m joÌ\$b

{H$g Xa go ~‹T> ahm h¡, O~ {Ì^wO H$s wOm 10 go_r h¡ &

Find the intervals in which the function given by

f(x) = 11x5

36x3–x

5

4–x

10

3 234 is (a) strictly increasing (b) strictly

decreasing.

OR

The sides of an equilateral triangle are increasing at the rate of 2 cm/sec.

Find the rate at which the area increases, when the side is 10 cm.

17. {XImBE {H$ :

2/

0

2

12log2

1dx

xcosxsin

xsin

AWdm

kmV H$s{OE :

2x3x

x24

3

dx

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65/1 8

Show that :

2/

0

2

12log2

1dx

xcosxsin

xsin

OR

Find :

2x3x

x24

3

dx

18. AdH$b g_rH$aU ;0x

yeccosxy–

dx

dyx

H$m {d{eï> hb kmV H$s{OE; {X`m h¡

{H$ O~ x = 1 h¡, Vmo y = 0 h¡ &

Find the particular solution of the differential equation

;0x

yeccosxy–

dx

dyx

given that y = 0 when x = 1.

19. AdH$b g_rH$aU ,xsinxcosxydx

dyx {X`m h¡ {H$ y

2 = 1, H$mo hb

H$s{OE &

Solve the differential equation ,xsinxcosxydx

dyx given y

2 = 1.

20. aoIm r = 2

^i –

^j + 2

^k + (3

^i + 4

^j + 2

^k ) VWm g_Vb

r . (

^i –

^j +

^k ) = 5

Ho$ à{VÀN>oXZ {~ÝXþ H$s {~ÝXþ (– 1, – 5, – 10) go Xÿar kmV H$s{OE &

Find the distance of the point (– 1, –5, –10) from the point of

intersection of the line r = 2

^i –

^j + 2

^k + (3

^i + 4

^j + 2

^k ) and the

plane r . (

^i –

^j +

^k ) = 5.

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65/1 9 P.T.O.

21. g{Xe p kmV H$s{OE Omo g{Xe

= 4

^i + 5

^j –

^k VWm

=

^i – 4

^j + 5

^k

XmoZm| Ho$ bå~dV² hmo, VWm p .

q = 21 hmo, Ohm±

q = 3

^i +

^j –

^k h¡ &

AWdm

EH$ _mÌH$ g{Xe, Omo g_Vb ABC Ho$ bå~dV² hmo, kmV H$s{OE, O~{H A, B Am¡a C Ho$

pñW{V g{Xe H«$_e… 2^i –

^j +

^k ,

^i +

^j + 2

^k VWm 2^

i + 3^k h¢ &

Find the vector p which is perpendicular to both

= 4

^i + 5

^j –

^k

and =

^i – 4

^j + 5

^k and

p .

q = 21, where

q = 3

^i +

^j –

^k .

OR

Find the unit vector perpendicular to the plane ABC where the position

vectors of A, B and C are 2^i –

^j +

^k ,

^i +

^j + 2

^k and 2

^i + 3

^k

respectively.

22. EH$ H$jm _| 15 N>mÌ h¢ {OZH$s Am`w 14, 17, 15, 14, 21, 17, 19, 20, 16, 18, 20, 17,

16, 19 Am¡a 20 df© h¡ & EH$ N>mÌ H$mo Bg àH$ma MwZm J`m h¡ {H$ àË`oH$ N>mÌ Ho$ MwZo OmZo

H$s g§^mdZm g_mZ h¡ Am¡a MwZo JE N>mÌ H$s Am`w X H$mo {bIm J`m h¡ & `mÑpÀN>H$ Ma X

H$m àm{`H$Vm ~§Q>Z kmV H$s{OE & X H$m _mÜ` ^r kmV H$s{OE &

A class has 15 students whose ages are 14, 17, 15, 14, 21, 17, 19, 20, 16,

18, 20, 17, 16, 19 and 20 years. One student is selected in such a manner

that each has the same chance of being chosen and the age X of the

selected student is recorded. What is the probability distribution of the

random variable X ? Find the mean of X.

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65/1 10

IÊS> g

SECTION C

àíZ g§»`m 23 go 29 VH$ àË`oH$ àíZ Ho$ 6 A§H$ h¢ & Question numbers 23 to 29 carry 6 marks each.

23. AB EH$ d¥Îm H$m ì`mg h¡ Am¡a {~ÝXþ C d¥Îm H$m EH$ {~ÝXþ h¡ & {XImBE {H$ {Ì^wO ABC H$m joÌ\$b A{YH$V_ hmoJm, O~ {Ì^wO EH$ g_{Û~mhþ {Ì^wO hmoJm &

AB is a diameter of a circle and C is any point on the circle. Show that

the area of ABC is maximum, when it is isosceles.

24. g_mH$bZ H$m à`moJ H$aHo$ EH$ Eogo {Ì^wO PQR H$m joÌ\$b kmV H$s{OE {OgHo$ erf© P(2, 0), Q(4, 5) VWm R(6, 3) h¢ &

Using integration, find the area of the triangle PQR, coordinates of whose

vertices are P(2, 0), Q(4, 5) and R(6, 3).

25. kmV H$s{OE :

dxx

)xlog2–)1x((log1x

4

22

AWdm

kmV H$s{OE :

]1,0[x,dxxcosxsin

xcos–xsin

1–1–

1–1–

Find :

dxx

)xlog2–)1x((log1x

4

22

OR

Find :

]1,0[x,dxxcosxsin

xcos–xsin

1–1–

1–1–

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65/1 11 P.T.O.

26. Cg g_Vb H$m g_rH$aU kmV H$s{OE Omo g_Vbm| r . (

^i +

^j +

^k ) = 1 VWm

r . (2

^i + 3

^j –

^k ) + 4 = 0 H$s à{VÀN>oXZ aoIm go hmoH$a JwµOaVm hmo VWm x-Aj Ho$

g_mÝVa hmo &

Find the equation of the plane passing through the line of intersection of

the planes r . (

^i +

^j +

^k ) = 1 and

r . (2

^i + 3

^j –

^k ) + 4 = 0 and

parallel to x-axis.

27. EH$ W¡bo _| 4 J|X| h¢ & W¡bo _| go `mÑpÀN>H$ ê$n go (à{VñWmnZm a{hV) {H$Ýht Xmo J|Xm| H$mo {ZH$mbZo na kmV hmoVm h¡ {H$ `o XmoZm| J|X| g\o$X a§J H$s h¢ & W¡bo H$s 4 J|Xm| Ho$ g\o$X a§J Ho$ hmoZo H$s àm{`H$Vm kmV H$s{OE &

AWdm

EH$ Iob _| {H$gr ì`{º$ H$mo EH$ Ý`mæ` nmgo H$mo CN>mbZo Ho$ ~mX N>… AmZo na nm±M énE {_bVo h¢ Am¡a AÝ` H$moB© g§»`m AmZo na dh EH$ én`m hma OmVm h¡ & `h ì`{º$ `h {ZU©` boVm h¡, {H$ dh nmgo H$mo VrZ ~ma \|$Ho$Jm bo{H$Z O~ ^r N>… àmá hmoJm, dh IobZm N>mo‹S> XoJm & CgHo$ Ûmam OrVr/hmar JB© am{e H$s àË`mem kmV H$s{OE &

An urn contains 4 balls. Two balls are drawn at random from the urn

(without replacement) and are found to be white. What is the probability

that all the four balls in the urn are white ?

OR

In a game, a man wins rupees five for a six and loses rupee one for any

other number, when a fair die is thrown. The man decided to throw a die

thrice but to quit as and when he gets a six. Find the expected value of

the amount he wins/loses.

28. Xmo {dÚmb`, P VWm Q, AnZo MwZo JE Hw$N> {dÚm{W©`m| H$mo {ZîH$nQ>Vm, gË`dm{XVm VWm n[al_r hmoZo Ho$ _yë`m| Ho$ {bE à{V {dÚmWu H«$_e… < x, < y VWm < z XoZm MmhVo h¢ & {dÚmb` P AnZo H«$_e… 2, 3 VWm 4 {dÚm{W©`m| H$mo Cnamoº$ _yë`m| Ho$ {bE Hw$b < 4,600

nwañH$ma ñdê$n XoZm MmhVm h¡ O~{H$ {dÚmb` Q AnZo H«$_e… 3, 2, 3 {dÚm{W©`m| H$mo Cnamoº$ _yë`m| Ho$ {bE Hw$b < 4,100 nwañH$ma ñdê$n XoZm MmhVm h¡ & `{X BZ _yë`m| Ho$ EH$-EH$ nwañH$ma H$s Hw$b am{e < 1,500 h¡, Vmo Amì`yhm| Ho$ à`moJ go àË`oH$ _yë` H$s nwañH$ma am{e kmV H$s{OE & {dÚmb`m| H$mo nwañH$ma XoZo Ho$ {bE Amn EH$ AÝ` _yë` gwPmBE &

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65/1 12

Two schools, P and Q, want to award their selected students for the

values of sincerity, truthfulness and hard work at the rate of < x, < y

and < z for each respective value per student. School P awards its 2, 3

and 4 students on the above respective values with a total prize money of

< 4,600. School Q wants to award its 3, 2 and 3 students on the

respective values with a total award money of < 4,100. If the total

amount of award money for one prize on each value is < 1,500, using

matrices find the award money for each value. Suggest one other value

which the school can consider for awarding the students.

29. EH$ àH$ma Ho$ Ho$H$ Ho$ {bE 200 J«m_ AmQ>m VWm 25 J«m_ dgm (fat) H$s Amdí`H$Vm hmoVr

h¡ VWm Xÿgar àH$ma Ho$ Ho$H$ Ho$ {bE 100 J«m_ AmQ>m VWm 50 J«m_ dgm H$s Amdí`H$Vm

hmoVr h¡ & Ho$H$m| H$s A{YH$V_ g§»`m kmV H$s{OE Omo 5 {H$bmoJ«m_ AmQ>m d 1 {H$bmoJ«m_

dgm go ~Z gH$Vo h¢ & `h _mZ {b`m J`m h¡ {H$ Ho$H$m| H$mo ~ZmZo Ho$ {bE AÝ` nXmWm] H$s

H$moB© H$_r Zht ahoJr & Cnamoº$ H$mo EH$ a¡{IH$ àmoJ«m_Z g_ñ`m ~Zm H$a J«m\$ Ûmam hb

H$s{OE &

One kind of cake requires 200 g of flour and 25 g of fat, another kind of

cake requires 100 g of flour and 50 g of fat. Find the maximum number of

cakes which can be made from 5 kg of flour and 1 kg of fat, assuming that

there is no shortage of the other ingredients used in making the cakes.

Make it an LPP and solve it graphically.

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strictly conficrentiar --- (For Internal and Restricted use

senior school certificate Examination

t_ L l-March - 2014 CenP@Mar ng Scheme -,-- Mathematics (Outside) 6511, 65/2, 6513

onU

General uctions :

ing Scheme provides general guidelines to reduce subjectivity in the marking. The

given inthe Marking Scheme are suggested answers. The content is thus indicative.

ent has glven any other answer which is different from the one given in the Markingbut conveys the meaning, such answers should be grven firll weightage.

ion is to be done as per instructions provided inthe marking scheme. It should not

according to one's own interpretation or any other consideration -

Markinsshould be strictly adhered to and religiously followed.

ive methods are accepted. proportionar marks are to be awarded.

ion(s) on differential equations, constant ofintegrationhas to be written.

idate has attempted an exta question" marks obtained in the-question attempted

ould be retained and the other answer should be scored out.

scale of marks - 0 to 100 has to be used. Please do not hesitate to award fullifthe answer deserves it.

Marking Scheme for all the three sets has been given.

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@

es\ r

SEcf tO N-A

3. -+ 4, z-= 4" 5. o (zn")i,l- ?,

**o

77

v, * 8' + ?'-dht to. ry=ry=+8 E=c.x;tpinli) +lLl)

l,l l Xto

=lO.

fic*r=l :,jrT:"

*=\:;fff:"

{L-LZ

LtL.I- 2

Llte =

J-+12 -z-

us + cclo, p?ol: A\=:) =oo + ?** -_D #xeR tla1a€.-izs) EI"fcx) =fu(o) :

rl

C-d x.:d -', ealgcf :2C

c( + cd l*'o,hy3+ $/*do( + er;= L*t% c{rs d+ A ',,Yo s""{c( + eu3 (e"t L\;xt) e- c(+f%-or:r7. '--Rl+s

+ 2.#* j5*+ 2t[--ta;*) =oVz

i

hfrl".= AY=-fr= - Ws "r },# *=V,?L = h.nYg=,{T

(us^gv-Ra R-cgn&t L

=

rlt(Jn?,-r.f+

| -l-I-:/-I ui ^\t|LL. ) '^-

€

!,

!,+ !,

L

lurc a. q. 1

f u e+q bl =I e c e+ulh-+ Cf, en4 Rs--t'b Rt

= z*l o ^c u\bc lbc t+aL bglvv

I u. be- ab*f I

= c>'+ *:3"\ i -,

fo-2clb e+4lc e

12r"1 1

;,1:- 4&b c. I

Ye fa*r = { 2x'' *_*>o andt o) +7s-<o

-:llblosl

ffe+a e

bat

= -zbe(-za)

q

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e slr

[r; t- be{x [,; z srirry7-+O 6= x,+o -ffi;= 0-,lr pryf - -{-.>c*o G_/f rs c"' \' iows *:r=s :' brf*l= $rc1 -- k

-.. l:k of l'<"=l-

/ =Jbal ( 9) * +l W(tc-") - bE,=+ql

*= +'(-u*tlI -**lcJ)c I * _-

= #?*+& = */

S&r=- Ef--D-;-6x-+4= e (tt -t) (:c* z) Lzc-z)

l^-\A,3.{txl :o + ?c= -2, t'

.,, Irl-*-unJt d*e C- to.,, '4 G7t t), Lt'21' (7'{.)

.fc=-.r bf;-reasrngl rh t-2, t) U (3r4)

Cl^d., /gL' 'J *ttJ' s<il6 to (- rc7) U Ll '22

oR-[+ s be ft'. &-,.{, $ etl^, Is]Edtn r^6lt

lh"n # = ' o.'>l*u

Oad lvng,tA) =

4 .=2Es* dsdr-&= +,lo.z , LNhen s=toLrn)

L

!,

T

l+{

L2L'Lzl_/

I2Iz

J-2Ir5g

4

!,

L2-

rW-s,I*J"ffi*d*=I^W-af = l': | = dx

J 6 -9-x_1 euar

c)

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aslr

1

!2I2-

FI--|'.

--Lzfr-

t

htr"lB. JzLJ2-

Il-zI2

IztIzT

IzI7

*=L,y=l + l:lj. go Ufo;, i6

Lq)

r = * t["Al *". Lx-W) +z.nc:c- nlill|

Lqln+r I ualle-,1]+bl#\qll=*rl = # W\r=+r\

oR.

= + lJ€D - t.;r"] = * t = 9"A\t+=t - ry lt+rli nL r+ I

= hlR\ rc= h\#1*-*-vd- Dc o,tsec^ ( *)V+ >c d! = V - g) F,ec.V, l,^lbet - *, =V.d>c

r-Js-,.a' =I*+ cosv = 0"srxr*c =) e-us(*l : fulr\ r c')c,- lr U:o + C=l.

.' I.F= Jl'^ = Jw =- x-

so brfro^ t )cJ -- Je cos>c + So lr-) dx

+ )c:PI-', V = .$nz.-t C'&'+e+ +C=o

J =- &tn>c,

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2a,

22.

Fo';r eD the qrvbn l.rn-- ts 3w6n *s(ztBL *t+ 4A" z+tl)

tbr€ poinl- Lts s itv, I=t" D€, ftr"^Q+zy - (-r +4A) +(z-+2^) = s+

^- o

n'. Pur 3P io,h-s.tha l's (21 - rr z)

L

T

-Drbf6;,".. { Pt A- t,-) +r-, e(_rr_5 to) ;,{ tz+r1t (_t+sJr*€+tg, :m

=dun J-*t-odrcA* E b"n ? o,.ap = ;2-FI : ) (c(iF) =) (=, t - =, j- -rE14?==, ) A(eg-:1 +;<t)=2t +,\=5' T"_7t'-7j-74

OR

^ $ t- =__> _)Kq1*red Vec.lar

*9

A'B:

ffix.

lffixEl.-t+i+A and. tr&- ot+f+aQ

AAAvi +2J k

u2?il- veda = I- (= |=!14 \st

t7 tB tq3t2ls Gts

Meo.D**= )rpr"; =

= Ll=s+ts+=2+

F"

:13 , t'uni E

I2tL!.2

tLJ_2

Iz

I

1

!,

!

2"

IzI{2

.', hqyreel

X : 14 tS;

Pur, ?, '5

A ^+2j -k/

)-0 21.

-a.Jrs ts,

t62ls l

sl +lg + 38 +6o+2t

= rt-L 2GB] = l7.ss

CrJ

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esf r

)"t tt* s.*", f at.A ABc-22)-: >C +3 :4 L qzDcd

A= Ates*L -+,)c-yt"+ s- ffu:t*"'

tr F -+L+i;-*9-tls t*-4e#=e ) *=r* crr >c:Jfa-

and gL 4*--rl*=zn]=-:y a.ad db =

:) Y&4'ffii-- 6t*39 -F e

=7*-q* <o,T)ex,^r^rtxAeo 2\ is f ggec-ej e_a _

t CerrsrrtaQ- Dc

L2-

*" ds&

Le.

-4-=o tE't4,g q

?c2p;

sI

:

I\

J-zT

Lz1I

II'>

l!I.7

Pg : 9=E(---) )gR: Sce_x_. I

f_ Pa, v=€-e-g A )J,+(:c-z;dx -f*_;^; _l?.. _z; dx

b,-ill* * tt-*flj $."*r"f5 + B 6 =Y -I,tirl;

R.g./

Q"', + P8,8R q^a ffiRcr"re-.H-fo j-

qre:

ttR" aY€.e

=

25.dx =J/'E(k t'n#)J, *""^ 1

T

1

i.

J

I

30"dt.t3*+t,+c

l+#=Ft (rt'dr) rar = -{tgt .L,d + -*.a**BL-'-l-jd-:- !f_etEr g{l+

S d:

* rr**F (-uatr*b.) " 3) +c

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6sll

-ar-"jl x- - dgJ=OR

crx - +[Ml= -(%- sil.r)_] a^f lJ

-Jr dx.4,n +c-Eg,z ['^ ^-tfrJ 2strJ"[-'dx

)c= st&A + dx. = 2Srzrt9 dae dQ

=+ Lo*=,-l; _ z [ g1 .2u,,0 e,s dp1_ =_r c

i U*4n -ja*&s>s) deJ ->c-+ c

^ftd .f^ pla"- J-"*r;? th*,^71" th. !"u-f tnfers.*w?De E .c,i +J +t) :l aD d n .tr; +f,_ft) * 4 _*-o

,(d+i+A) -r +) (8.(z,i+{ +)

+

l-2-

LzI2-

L

Il.L

-9 l,n^'\U-r4Dt * (r+qrj _+ /, r.r \,-Pl,.'i.,J. .'- , . r.u-alx,l ='**;i,;*''3 i=l;i;h" S P^n. ra

E,(-ii+] ft) = 3

:-Lto ? o'6

#rLu

84.- |'6-:t.I

L'

LL

.{+

lJ.

T

l"t27.

ry 2,(-i+3ft)=eEl I trvo ha.s 2 r^6,b bqlls \F> t tlvn l'q6 3 r^rP*fz, bo;t* (E3: u-rrn hqs 4 L$dt bqll6 )A I 2 batl" cft.a-r^,,-' c:"y-e- t,.trhJa

Pc-u,;: pcrrl- pce.;-5Pc*lEr): -a''- I4: E ., PCAI p-)= R',

= + pf, ==*, f(A/4rrln) = @ r,'l Drr: r flr'/^ r. = 3P cu,) Prn/es+ prE ?t{r;@

tr!

* b*5 L+* t

(+\

J*1

1

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as/r

oR

ha"* ilTi:iz:ii,;(? ? r\fr[)iL:'-Rru\i;',)lz)=\fl#) 6rAx^B

x.Pa) = L' YY - t;6 (t so+ lao +?s - BZs) .- o.'. Fr.peclEd ka,lr* :&o

)c c-*kes f -ft="F* anc+ t/t-PP & Maxrn, fsy-, t Z: >c+ 1,

-8-f {oo>c *lctca2f?L +EDy

x>o

F,FS trtrFtu.=/- Rsc-g/

7n Vr't/e 76 7, V4

7s/zta -#

cokes f, ="c \fu

Ve.lig"o #ffi,-ffi;,"i:' Z ri

''?nex . aF L>nr lo1

8D L.e 2D e,aks S.fi"*" FhA4 @ c'{s i Zd bt^

s,Bs s/_

It/e

'r '/t

F,s.R'T,/r,y,

*/o

tL

tL

tL

rl48.

d'zI_,)-

Ilzl*t

+

=)

- 2(*') -3(q + 4Q/ = z *o,. x - ;f g@".n ^e. l:,1

-,, f,l=:.

,ll1=,,

4 "t

= | fts__ a A,"r=_ s

* (-+ r, J") (iiry)=/ T )\ I 1*/ \i-/)L = s, g=- 4ffit ):={'-

F, nrdffiq oba__raore. r,qlt^,

tL

1

-!"

\

se

Y4o

3t€ tooo IYzro j

2c-LT a)$(gro7

b)

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