2014 question...comprises of 12 questions of four marks each and section c comprises of 7 questions...
TRANSCRIPT
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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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