I

DPS - MIS, DOHA-

SECOND PREBOARD QATAR
EXAMINA TroN (2020 _ 21)
CLASS: X
Subj ecr MATHEMATICS
\
(STANDARD-041)
-!
Time Allowed: 3 Hours Date: 04.09.2021,
Maximum Marks: B0
General instructions

1. This question paper

contains fwo parts
2. Both part A and part A and B.
n nur" irternal choices.
check that this q""rnor,
paper contains 10 printed pases.
:J:::
1,. It consists of two
sections_
2. Section I has 16 questioru I and II.
of tr't<urs
mark eacn'
each. htemal
questions Inte choice is provided
in 5
3' section II has 4 questions
on case sfudy. Each
sub-parts' An examinee question has 5 case-based
is to attemptany 4 0utof 5 sub_parts.
Part-B:
1. It consists of three sections-
2. Question No 2'r' to 26 arc III, ry and V.
Very short Answer Typu
each. questions of 2 marks
3. Question No 22 to 33 are Sho
4 euesrion rvo s+ to 36*"
;|f,T"TffJftT.?Tj ;j;ffff
5' There is no overat :;"7
"hri"". ;;;ever,
questions of 2 marks, ir;;; is provided in 2
2 questions of 3 marks "nor""
You have to attempt and 1 question of 5
onl; one of the arternuti;;, marks.
in alr such questions.

Page 1 of 10
 P

SECTION - I
choice is provided
in
of l mark each.Internal
section I has 16 questions
5 questions'

and 404
1. Find the HCF of'96 OR
gCf?a LCM of two numbers?
between
What is the relation

2.Findthevalue(s)ofkforwhichthequadraticequationx2+zokx*18=
0 has equal roots'
e x' is as follows:
polynomial in variabl
r), where p(r) is a
3. The graPh Y = P(
nurnber of ,.to"t
of P(x)'
Find the

1-60 t-t2q
A'P:
the common difference of the given fi ' 3q'3q
4. Find
- Ar'7'find its nth
.f is given by s" = 3n2
n terms
If S^ the sum of first "toiu
term.
tree)'
of x and y in the grven figure (factor
5. Find the values
1tlu1

l,l3

dl I

that AE =
Find the length of side AD' given
BC.
6. In the figure' DE ll
1.8 cm, BD = 7.2
cmand CE = 5'4 cm

Page 2 of 10
 l.Il rrl

7.2
cn

7. If sind * cos 0 = tD, cos0 , (0 +

C

90\ then find the value

of hn 0.

rf k + 1 = sec2 0 (1'
+sin d)(1 - ,,:;, then find the varue
of k.

'i;{;:i:"::l*?y}'ffi ;:,:f
},ff \o,},,ffi :f
"T'*fiil"J'""'

9. In the given flg"-r"

RS is
If zNML = 3ot dJ;;" the.oir.
tanjent to the circle
at L and MN is a
diameter.
lt

.1'
n'

10.A number is chosen

at random from the
-5, -4, _9, _2, _1,0, i,;,-;:;,';. numbers
the probabiliry that
number is less than..d;;i;; it"O square of this

11. Consider the

following frequency
of a class' Find th" distribution heights or 60students
"p;".;ffi
of the median d:jl"
No. of cturlrntr
lS0rtFs
t5
I55-160
t3
. _ l_fr0_ffl5
165-170
170-'175
I
't75-180 I
F

]2.Evaluate: sin2 600 _ 2tan4ilo _cosz

300
Page 3 of 10
 against is twice the distance by
the radder placed "y-"1] ** ri"a tr't" angle made
13.If the rength of ladder-La"r''" wall',
of the
between tl.r" foot
the horizontal'
,f'r" f"aa"r with

7'Jfim'then tinf tne value of BC'

14.In the given figure'
if AD =

?v/3 trr

t, ('
tI

then find PR'

andBC = L5cm'
15'If A ABc - a Qw'ffiB =2
OR

InAABC,AC=BC.IfAB2=2Ac2,,h."findthemeasureofangleC.
equations . - :'Tl{':':tfl3$"i":tJtL vour
16.state whether
th: *""'l:l::: perl
ii''""coincident lines or
represent parallel
answer' oR
y+B=0and6x-ky--t6
Forwhatofvalueofk,dotheequations3x-
,"ft"t"tt coincident lines'

SECTION - II sub-parts
Attempt any four
questions are to*p*t'ory'
Case studY based mark'
question' iu.tt sub-Part carries 1
of each

17.Case study based

- 1": ' h't rlecided to maintain social distancrng
" '"fl:";Tl"l'ffi p"i'o P' Q and R
*n':#***"Smffi
Page 4 of 10
 rtt t,
-.,"-fl
lrt. t-

If the coordinates of p, rt
answer the following: e and a
R re (4' -3),(7,3)
and (& S; respectively,
then
(a) I4rhar is the distaice
befween p and
(i) 16units e?
(iit 3rE units Gi) 4\E units
(iv) None of these
(b) rf a free is located
at the point X,lyingon the straight line joining
*"';'once Q and
ffit":T::r*ff' berw""., thu- in the
rario of 1: 2. rhe

0 (?,+) (tt) (?,f)

(iii) (6,1)
(iv) (9,1)
(c) Themid-point of the line
segment joining
e and R is
{n (},+)
(ii) (6,1)
Qii) (+,2) (iv) None of rhese
(d) The rafio in which
Q divides the Iine segment
joining p and R is
(i) z:1.
(ii) 3:1
(iij) r:2
(iv) None of these
(e) The points p,
e and R are the vertices of
(i) an isosceles triangle
(iii) a scalene triangl"e (ii) an equilateral triangle
(iv) none of these

Page 5 of 10
 - 2: t 5' The game consists
lg.case study based r rvher€ the entry fee
is
":fi3i;1*:**ir?q:Iffi :f;""n!io*op,swe'fage'isherenff
or else she
fees
v

r"."i""' aoout"
throws 3 t","uar,lr'r. "t'tty
fee back. If ,h"
will lose the game

Toesing of Coln

for this eame?

of possible outcomes
(a) What is the
total number
(iii) 6 li"; ttott" of these
(t8 (i')2
fee is:
that she loses the entry (iv) none of these
(b) Probability
(ii) 1/8 0r) 3/4
1
(r)

entry fee is:

that she gets {1uble (iv) none of these
iii; r7s
(c) Probability (r') 3/4
(,) 1
entry fee back is
:

that she iust gets her (iv) none of these

(d) ProbabilitY
(') 1 liil r7e (u') 3/4

cases ls:
probabtii:,"'lr* tI'" ub'l:iiH;e (iv) none of these
(") s'(,T)T the

- 3:
19.Case studY based

rohnandJily^t:pr"{113x*,*:,T:ffi
of them lost 5 n
ffI;",'iffi lin^':"mberof
*urUt"t' Both
have rs124'
marbles theY now
Page 6 of 10
 o€tnl

(a) If John had x

marbres, then number
of marbles Vrjay had:
(i) x _ asgi) as _ x
(b) rf Iohnlf"il:rbles
Qii) abx (iv) x _ 5
then the number of
marbres reft with vijay,when

(i)x-aS Q\ao_x gii)a|_x (iv)x_40

(c) The quadratic
equation rerated to
the given probrem
is:
(1) xt - 45x*324 =
0 $i)xz *4sx*324 =
(iii)x2 -4sx_324 g Q
= $v) _262 _4sx*324
(d) Number of marbles = Q
/ohn had
(i)10 Gi) s (iii) 35 (iv) 30
(e) If John had 36
marbles, then the number
of marbles Vijay had
(i)10 G\ s (iii) 36 (iv) 35
20.Case study based _
4

Page 7 of 1,0
(u)IfthegivenproblemisbasedonA.P,thenwhatisthefirsttermand
t"'
;";";;; tfrerenceffit#*t 100,100 (iv) 1000,1000
,,,n

months the loan

will be cleared?
(b) In how many
(ii) 30 (iii) 40 (rv) 5o
(i) 20 is:
(in <) paid by him in 30ft installment
(c) The amount
(ii) 3500 (iO 3000 (iv) 3600
(i) 3e00 is:
<) paid by him in 30 installments
(d) The amount
(in
(ii) 73500 (iii) 75300 (iv) 53700
(i) 37000
(e)Whatamount(i,,<)doeshestillhavetopayafter30thinstallment?
(iv) aa500
(ii) 44000
(iii) 54500
(i) 45500
PART - B
choices, attempt any one.
In case of internal
All questions afe compulsory. (6x2=1:2\
sECTroN - rlr
number'
number, if G is an irrational
that 3 + Z^lE is an irrational
21.Prove
and
possible that HCF
. uSrnon*"tic' Is it
answer'
", Justify your
be24
LcM of rwo r,rr*u"r, ""ail;.rp"tti"ury'
the points
the line segment joining
divides
,'.E'fid,the ratio in which P(4,9
"-'-[c,u) and Hence find m'
B(6'-3)'

t'iuoid so formed'
23.Twocubesof.side5cmeacharekepttog:q:,joiningldg.toedgetoform
surfac" *";;f *t''"
a cuboid' Fin'c the
division
expansion of ffi without actual
24. Write the decimal
Page 8 of 10
 25' rn the given figYe,a circre inr.ri6ed
is in AABcsuch that it touches
iii:: f3; 35 ;tt8f :ifffll*H*j ili".n"ery r*r,"1"ig,l,s the
o*he
lengths of AD - -*r7 cm orld
o 1u cm respectively,
find the
BE and cr.

26.rfsin (A + B) =
f ura sin (A _ B) =7
7
0s A + B < 90" and A >
and B. D then find A

fi, sin e - -L,find the value of sec2 g _olr"",

g.

sECTroN_rV
VXg=21)
" ::;:ifi:'l:;::;,?" porvnom iatp(x) 6x2
= - sx* k such that

28.State and prove

the Basic proportionality
theorem.
OR
If in AABC' AD is median
2AD2 +
and AE I BC, then prove that AB2
+ AC. _
|{nq,
29.Prove that (cot.A l-cos/
- cosec A),
= 1+cos/

OR
If sec 0 = x +
*,prove thatsec 0 * tan| = 2x or*.
eorrovethffi
j?lf:::n$?:"1#,::,pararrertangentstoa
"1[T."r#:
31'Draw a circre of radius
ot u points p and on one
each ut a"111"
extended diameter ry3 Q of its
tangents to the circle aistunce of z.*rro* its centre.
fro*tnJre two points p Draw
and e.
Page 9 of 10
 in a lake,lh" angle of
the level of water
32.From a point A,70*"T1q*e rj d".pr:tit" tt the reflection of the
of u'.iora is 30.. Th";;"- of the cloud from
A?
erevation
e i, oo;. ri,.d the ai,tu,.."
cloud in the
'J"'f,o* 3 cm, a- circle of diameter 4t5 cm
and a
each of diameter Find the area of
33.Three semicircres
4.5 .*r J;";;;;r,
i" tr'" s;;t""te'
semicircle of radius
the shaded region'

+- +-,

(3X5=t5)
SECTION-V
area of the
mlet the x-axi'. ri"a
the
Also, ma tne?o,I,,--*t.,"," !1.
il;
34.Drawthegraphsofthepairgflinearequations:x+2Y=5.Td2x-3y=-4,
il' ;;e
equations and the
; Ii"Jir"'""ti"s
iliffi
re gion uoo'ala
x-axis.

if 8 cm of
*iu it i,,igu1" i"io minutes
35.Waterinacanal6mwideandl.5mdeepisflowing]Mithu.'|""dof10
km/h. How;;.h;"" qJ.J;;"
standing water is neededt Ora

of the base'is 4 cm'

Asolidtoyisintheformofahemisphere-surmountedbyarightcircular
; t.* u'"'d tt"'" Jiu*eter
cone. The height
of the
":T
Determine":^""'"r":l t
jltTi*;: j:ffffi ;IHHffiT:ii',"'
the toY, find the difference
(Take n = 3'14)

distribution is given below: 70-E0

36. An incomPlete 30-40 40-50 50-60 60-70
r0-20 20-30 25 l8
Ct$c hErvd 4 65
then find the
t2 30
rs 230,
is 46 andthe total number of items
If the median value
missing frequencres

Page 10 of 10
 Drs-M ODERN INDIAN SCHOOL, DOHA- QATAR
rIRS T PRE BOARD EXAMTNATTON p}?r._Wl
CLASS X
$ubjcch IVIATI{EMATICS (STANDARD-
0{U Date: f.S,IZZALI
Time r\llorsccl: 3 l,lours
Maximum lvlarks: gfl
Gcneral Irrrtnrcl iorrs
1. Tlris Quc'stion Pa;rey
has F Sections A, B, C, D
L Section r\ lras Z0 h{Ces .nrr1,ing and E.
3.Scctiun B lrns E questionr.niryirrg
I mark eaeh
4" Section C has 6 qucstiorl,
0l nrarks eaclr,
.orrl,iog 03 marks cach.
5' sectior D lras4 rirrestiorrs..rriir',g
05 r'rarks eaelr.
6' scction E has * ti:::
lrascc{ intcgr*tnel units of
assessment (04 rnarks eaeh)
r. illti:*;ffi}'or
the valrtes oil, I anrt z nrarks eacrr respectivery.
s**r..,to;"1;;:Tru:X?J':H.,-'rT'Jil[TlSf j:,i1.::','"nsor
An internalchoicc ltas lrecn pro"iJ"J fj
$jfirllfi' in ilrc 2 nrarks quesrio's of
- figures whcre'er requircd.Tnke
* +rvhercver requirerl if not
:_:il:eat

seerion A csnsisr':Fil?[fl"= of r mark each

1' The HcF.f ttre sntallest nunrbcr of
tr*o cligits ancr rargest nrrrrtipre
is less than 40 is of 5 rvrricrr

(A) 3 F),t (q 5 (D) 10

?' Fcr what *alue of 'a'rviil rhe rines rcpresentect by equaticns
3x + 4y = 12 ancl
9x + a1' = ?0 Lre parallel?
(A) 10 G) 12 (c) ls (D) 24
Ilage I of 11

L,.

Scanned wfrth frarnscanner

 a O.

l(J

quaclratic o
the csrresponding
3. ln the given Eraph' 1
polYnornrnlhas: a
t
{B} 1 zero
(A) no r-ero
(D) 3 zrros
(c) 2 zeros trl rti 6

,1. For rvlrat I'altte o[ "rc'', tle toots o[ tfie equation

xx+4x+k=0arecqrrnl?
(A) 3 (B) 4 (c) 5 (D) 7

5" Tlre llth tcrm of an AP -5, f,, 0,: .*. is:

(A) -20 (B) 20

(q -30 (D) 30

A(?, 5) and B(-3' 1)' then

6. If (:, S) is the midpoint of the line segment ioinirrg
the value of k is:
(B) 4
(A) 3

(q5 (D) 6

(3, a) and (4' 1) is r/T0' then the positive

7. lf the clistance between tfre points
valuc of 'a' is:
(B) 3
(A) 2

(c)4 (D) 5

8. ln the given figure, LA = 9V, tB = 9S, OB = 4'5 cm'

OA = 6 cm, AP = 4 cm, then the value of QB is: n
n

(A) 12 cm P) 9 cm

(C) 3 cm (D) I cm

Fage 2oft1

-rr----

Ssanilled \rdfrth Canrn$f,fl nneffi

 'n the given figure, AB A
is a chorcl of the circle wittr
enhe O, AC is a tangent
at poinr A making an
tngle of 800 rvith AB,
then dlOB is

(A) 800 (Blstr

{C) 1000 {D} 160.}

' The lengtlr of a tangent from a point A at a clistance of F cm Fom

the cenlrc of
;r circle is 4 crn. Therr the radius of the circle is

(A)2 p)3
(c) 4
fD) s
." If sinx * cos y = 1; x = rF ancl y is an acute angle, then the value ofy is
(A) 300 p) 6oo
(c) +so (t}) eoo

.Z.If sec0 * tan9 = 7, then the value af secl

- tanl, if d is acute
(A)
+ p);
(c) I (D) 7

13. In the figure, a torver stands vertically on the

c
ground.
From a point on thc ground, tvhich is g0 m alvay
from
the foot of the tower, tlre angte of elevation
of the torver
is found to be 80.. Then the height
of the torver is ,t
I Xll ttt

(A) EorE rn p) 75rl5 m

(c) 80€ nr rs€

3 tD) m
3

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 whone afea is equal to the sur
1{.lt is 1.r1sp65ed t* Lruitrt n silgle circulnr park
nr *'el 24 m in a locality'
o{ rrre irrcas a[ trva cireurnr Farks af r[nmeters'10
Thc di*nreter crt the uerv pnrk woulel trc

m
(A) ?6 (B) ?7 m

(q 28 m (D) 30 nt

15. The lrrea of the square that can be inscribecl in n circle of radius 8 crn is

(A) 256 cm: (11) 1?8 cm?

(Q 6ar,E cmt (D) 6{ cm:

16- Thc probability that a non-leap year scleeted aL ranclom rvill contain 53
Sundays

(A)i p);
(c)i tD)*
12. One carcl is clrawn fronr a well-shuffled deck of 52 playing cards. The
probability of getting a king of reel colour is:

(A)* $)*
(q 3l2 (D) :5t

protrntrility of getting
18. If threc coins are tosscd sirnrrltaneously, tlrcn thc
at least two lreads is:

(B) I
(A) i 4

(c) 12 (D) I
4

Page 4of11

I
i
scanmed udfrth, fi auin$cff nner
 t)lltticrloN fnr t;ut'trli.tts l9 nrtrl z0:
l. t1u*srians rg and ?0, n stntement of
'lss('rtirlll (A) is fttll1r11'p.1 lrl'n slntcrrrerrl oJ ltcaseirr (lt). cprose tlre correct,
cption.

l9'Asscr!ion{A}: If trvc sollcl henrispheres

of snnre trase racli us, / are joined
tng*ther:rlong lltr-'ir [ras*s, llren curvecl surf;rce
ar€a of this new solid is 4trz

fteason(R|; Tttr-.surfnce arca sf a s5rl1c.1c, is Znrzs(luare units,

(A) Foth nssertion (A) and reason (R) are
true arrer reason (R) is the co'ect
c.xptirnation of assertion (A)
(B) llotlr irsserrion (A) ancr reason
(R) are true ancr reascn (R) is not
the
corrcct explanation of assertion (A)
(C) .,\ssertion (A) is rrue but rcilson
(R) is fatse.
(D) Assertion (A) is false bur rcason
{R} is rzue.

?'0. Assertion{A}: Trre firct negative rerm .f

an Ap 20, 17, 14, is -8
Reason(R): The nrh term of an Ap is on a * (n _
= l)tl
{A) Both assertion {A} ancl reason (R) are truc ancl reason (R) is thc correct
explanEllian of assertion (A)
(C) Bexlr asscrtion (A) ancl rcason (R) are true ancl reason (R) is not tlre
corrcct explanation of assertion (A)
(C) Assertion (A) is rrue but reason (R) is false.
(D) Assertion (rl) is false lrut rcason (R] is true.

Section ts
Sectinn B consists of 5 questions of 2 marks each

21" Irrove that 3 + 2fr is an irrational nunrbcr.

,
22 ln the given figurc, AABC ancl Ad#P are tn'o
c
righl anglect triangles rvith riglrt anglcs at fl ancl
M respcetivell'. Pror.e that CA x lrtP =IlA x llC. ,
ta t

Page 5of11

J
Scanfied with Camscanner
 point P

?3,n nrc 1y.',9l,:;:,lii,ff;: lllf.;lfJlft:'J:;1f:ff:'ffff''

o
strctr tlrnt PA
4 t,,^ t"

)^
s-'
A(Lt
'Tnh " i^

?4.If ? tnn S = 4, find

the value of ffi
OR
+ 0 . Find the value of sin26 'cospg
J5 tang = 3 sin6, \{here
sina

of ilre area af that circle' then find the

eenhal
tt
?5'lf area of a gector of a circle #
anglc of the sector.
OR
of raclius 49 m and eovers a distance
An athlete nrns on a circular tfack
of 3080 m along its boundary. Flow
many rouncls has he taken to cover this
distance?

Section C
each
Section C consists of 6 gueations of 3 marks

m€asure
26. ln a morning walk, three Persons step off together and their steps
80 cm,85 cm and 90 crn respectivety. what is the
rninimum distance each
should walk so tlrat each can cbver the same distance in
complete steps?

W,lta and p are the zeroes o[ a quadratic polynomial xz - 3x + 1, finc{ the va}ue
of
(i) d$+a,pz otF
F) trd
a,fl
(ii)
$'u

Fage 6uf11

Sc,anmsd \ndfrth f, ann$fffl nner

 3$,
'l'hr:rt'arrr $onlfr riuclt'rrls lrr lrvp t'xamllltlpn lralls A nnel B. To make thc
trullll)r:r of slrrrlt,nts rtlttnt irr lrrrllr llrc lrnllt, '10 *lrrclcnls nrc scnt from A to B'
llrrt irt r'rlsr,2[) slutlt,rrls flft. $r,lll fnrrrt l] kr A, tlte nUmller af gttrclentg in A
lqlrtrres tloubk, tht' trutrtlrt,r tlf stttrlrttrls irl ll. I:ind thr: numtrer Of stUdent$ in
caclt hall. 4
*\D i P''t tn
ott h -Ao . P'0i)
Sptl's; ini = 5 atrrli-* = 1

?9. lf a, [r, c arc the sielcs of a right-anglecl triangle, where c is hypotenuse, then
provl. that tlre rnditts of the circle rvhielr touches the sidcs of the triangle is
givetr Lrt' ,' = "nl-t
on
Prave that opposite sides of a qrradrilateral circumscribing a circle subtend
supplementary angles at the centrc of the circle.

30. Prove tt'at *#* - ffi= :ffi.

Fl. The follou'ing tal:le gives the number of participants in a yoga camP:
(in years) : 20-30 g0-40 40-50 50-60 60-?0
No. of Participants : I 40 58 90 83

Find the modal age of the participants.

Section D
section D conaists of { questions of 5 marks each

g?. Aperson on tsur has t 360 for t'ris expenses. If he extends his tour for four
da1,s, he has to cut dorvn his claily expeRses by t 3. Find the
original e{uration
of the tour.
OR

Page 7of11

scenned tftrfrth fi arnscanner

 a* * * 0 nnd r * -(n + b)
solve: *** =i+i** b 0,x

33.{a) Stnlc nrtrl provc basie proportiorrnlity tlteorsm.

fii) lrr tlrt'given figurc, if DU ll nC, find thc value of CE.

A
1.3 cru t crn
$
t cr*

r c
3{. A solir{ to3, is irr tlre fornr of a henrisplrere
sumrorrnteel b1, a right circular.o,r*. 2 cnt
The height
of the conlr is 2 cm ind the diameter
of the base is
4cnr. Dctcmrine the 2cm
(i) I'olumc of the toy.
(ii) surface area of the toy
(usen=3.14arff=r.i1
35. Find the vah.resof the frequencies x ancl y in the following
distribution table, if N = 100 and median frequency
is 32.
Marks : 0-10 10-20 20-30 30-40 40-s0 50 60 Total
o- nf Students: 10 x 25 30 v 10 100

OR

orthe rollorving frequency ctisrrirrution

$:,nT: usins step
EHI,:T
G.a.^.-

Class: l0-30 $0-50 50-70 ?0-$0 - 110

$0 110
-- - l$0
lhequency a

5 I 12 20 $ 2

I n\ Page 8oft1

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Snfr nmed wilrth fi annScanner

 Section E
C*se rIu cly-trnrr.d q uarl ion$ fl re eornprr I eory
36' A contest *ffers.'i5
t 250lcss than tlr,, Priz's,
thLr'lrt prizeis I sfixl, nnrl enclr successiv c prir.e is
1r,oo.tiffi.i"u.

\ \t, 1. /r'
.i
-:e"f--
't

{i) lVhat is rhe value of

thc l'rh prize?
(1}
(ii) lthich prize gets an amounr
of t 3000?
(iii) Find the totar anrount 0)
of money aistributed as prizes.
(2)
(iii) Find the total
"*t?X, of monel'erisrrilrutect fcrr rhe last four prizes.
37' Morning assembll' ;'
irn integral part of trre scr.:oors
schools conduct nrorning schedule. Almost a, trre
asscmtrlies rvhich incrude
latest happenings, inspiring prayers? i*forrnation
thoughts, speeclr, national of
scltool is alrt'ays particular antlem, etc. A goocl
about their morning assembly
assemblf is impartant tor a schedtrlc. I'loming
clrilcl's cle*elopment. It is essentiar
that nrorning assembly is to rrncrerstancr
not just about stancting in tong
prayers or nationar antrrem, qupucs arrcl singing
but it's sometrring bel,oncr just
actiuities carried out in morning pql1'ers. Ail the
asseml:ly by the scrroor staff ancr
Itave a great influe'ce in strrcrents
every poinr of lifc. Thc positi"*
scltool assemblies caR be felt oii*l,s of attendirrg
tiuoughnut lifc. Hopc
assernbly you alrvays stand 1,'u noticcrl t6at in scSool
in rorvs ancl colurnns and this rnake a
system. coorerinate

page g of1l

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nner
 tn 1$
thcy all assenrble for Prayer
ilnd
lff] strrtlents
SuPPose
a school have
lrclorr' ll
rot\'s As 6ivCn
,i
fi{.'
"t
SbSi
'-r1
.tl -!il

:r
tl
_f_ of four friends Anrar'
Bharat'
the positions
Herc ,\, B, C, anc{ D rePrcsent (1)
Colin arrd Dravid' AB'
(i) Find the distance betrveen
(1)
ans\\'et'
trianglc? Justify )'our
(iills AABC arr isosceles
the line segnrent iohring
ratio in rvhich lLo )'-n*i' cligicles (z)
t""";;;";i
ftiilFind the nnd D'
illirl
on tlre same srouncl

OR of p + q'
is (p, q), find the t'alue
(iii) If the ccntroia ofljgc
b)'heating the air inside the
It is
is a type of aircraft"
3s. A .ot air balloon ,u"igt..s
'iftedtlran tlre sanle volttnre of cold
less
firc. Hot ni,
balloon, usually rr,itlr or float'r'rrcrl tl'rere
rt,rricrr mcans trraihot air rv*l rise up
air (it is rcss crense), a pot.f rvntcr' Tlre
greater the
likc a lrrrlrblc of air in
is cold air arou'cl it, iust trre greater trre clifferetlce
in density'
arrcr tr.re c.lcr,
difference betrvce. the
'ot rvill pull up'
and tlre stronger the balloon
Aftcr reac*ing at'eight t nretres' at
Laksrrman is ricring
on a rrot air ba*oon. of depression
parkecl,at B on tl''* g'ounti ot"un arrglc hc spots the
point p, Irc spots a-lorry mctcrs * 1-int Q ancl now
furt'er by 50
of 30n. T'c lralloun ,ir"i o,'a o.."ul iJt"a at C
at an angle of
samc lorrl,at an angle of tlepres,t,";;;,
ctepression of 300'
Page 10of11

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 7
:rr a

i0t It

(?
t,
(i) Find the *
elistanroe drr r-_.-
lrehveen the
points A and B in rernrs
(ii)when thu
b"lt x.
(u
""h"*;
tfJl:rffi;,T,tfi,,lmr" then what is
thereration
(iii)lifhat is the naw
height of the (1)
the point
(iii)what is the e?
distance AB on ,r:;Tf,; (2)

tl*rt+**l}***++|i**
****t+*+r;+*rr15+**+

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 ,

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Class -X Brilliant Group of Institutions

Mathem atics -Revision
Worksheet

Real Numbers
1' IfLCM of (p,q)=6andHCF(p,
Q)=2thenfind (pq)r.
2' Find the LcM of the smatest composite 044)
number and smarest prime
number .
3' Anumberwhendividedby6l gives2Tasquotientand32asremainder.Findthenumber. g)
4' Find the HCF and LCM of 6, 72 andl20, (1679)
using the prime factonzationmethod.
5' HCF of 510 andg2and Q60,6)
verifv that LcM x HcF : produ* of the
ili'j.*t"*and two numbers.
6' that divides 12s1, s377
and. 15628reaving remainders
H:j|Hl:X;u'oo 1,2 and3,
7' Find the greatest number
that divides 49 and.3g
leaving the remainder
8' what is the smallest number
that' when divided by
4 in each case. (5)
35,56and 91 reaves remainders
9" tt"i-llrrttallest number which of 7 in each c,oe?(3647)
when increased bv 17
is exactry divisibre by borh 520 and, 468.
10. Prove that:..fZ,fi,rB and. O is inational number
I L Prove that;! is an irrational number

12. Prove thatlE - rDis an irrational number.

13. Explain why 7 x 11 x 13 + 13 are composite
numbers.
14' show that 12' cannot
end with the digit 0 0r
5 for any natural number
15' In a moming walk, n.
three persons step off
together and their steps
measure 40 cm,42 cm and45 cm,
distance each should
walk so that each can cover
::illJ:",'|r}Irj.l'minimum the same distance in
16' Two tankers contain
850 Iitres and 680litres
of kerosene oil respectivery.
container which can measure Find the maximum capacity
the kerosene oil of both of a
the t*ruwrr when
- --'v tankers wrlsrr used
uses
(170) an exact
exact number
nUt oftimes.
tt
ffil'"TTffi:;:t-.ff::,:' t" cm'
;:: 675cm and 450 cm respectivery.
Find the rongest
18' rhe traffic right at three
dirrerent,ouo
".oi;:;:n;:ffi;
change simultaneous
tv at 7 '00 am
or llll"o,, 72 seconds and r08 seconds
at whattime wlr
;;T;r; Jthev they change simurtaneousry
again?
 Polvnomials
Find the number of zeroes in each
ofthe following:
Find the quadratic polynomial whose
zeroes arc 7 +2 t/Z nd 7 _ 2r/Z (x2 - t4x+ +11

(t (i0
(iir)

(iv) (v) (vi)

3. Find the quadratic polynomial whose zeroes are r/i +2

e andrE _zr/2. g2 _ 2{ix _3)
4' If the zeroes of the polynomial x2 + px + q are double
in the value to the zeroes of 2x2 -5x - 3, find the
values ofp and q. ( _5, _6)
5' Ifzeroes aandp ofapolynomiarx2-7x+karesuchthat q-F:r,thenfindthevarueofk.(r2)
6' Find the zeroes of the following quadratic polynomial
and verify the relationship between
the zeroes and
the coefficients.
(i) x2 - 2x -I 1s (ii)
eii) 4uz + Bu 12 _
(iv)6x2-3-7x e)x2+Tx*L0 (vi)6x2*Tx*2
(i)4,-2 (ii) +V15 gii)0,_2 (ir);,+
@)-2,_5 fui)*,*
Find the quadratic polynomial, sum
of whose zeroes is 9 and product is lg.
Hence, find the zeroes of the
polynomial. (x, *
9x 78, - x:3,6)
8 If one of the solution of the equatio
n 3x2 - Bx * 2k* 1 is seven times the other. Find the
solutions and the
value ofk. (f )
9. rf m and n are the zeroes of the poryno
miar 3x2 * 71x- 4, find the value of 3 +
10. Find the quadratic polynomial, sum
#.e?r>
of whose zeroes is
-3 and product is 2. 1xz + 3x + 2)
Find the quadratic polynomial, sum
of whose zeroes is
f ana product is 2. (4x2 _x + B)
12' Find the quadratic porynomiar,
sum of whose zeroes is 0 and product
is vB. @2 + ,/i)
13. Find a quadratic polynomial
whose zeroes are 4 and _ 3. @z _ x _ 12)
14. rf a, p are zeroes of quadratic polynomial
kxz + 4x * 4,find the value of k such that
(a+D2-2ap=24. (-1,3)
a and p are the zeroes of the polyno mial
x2 t *
4x 3, find the value of
r'>j+f Gi)#+; (iiDfr+! (iv) a3 + B3
(v)(a-ilz (vi)a-p
 (i)-, L
(ii)+ (iii)
+ (iv) -28 (v) 4 (vi) +2

16.If aandp arethezeroes of thepolynomialxz *4x*3, fromthepolynomialwhosezeroes aret*Land

t ++.
tt
(3x' - L6x + L6)
17. Ifoneofthezeroesofthequadraticpolynomial (k-1)x2 * kx * 1is-3,thenthevalueoffr.
.- 10.
(l,T)
18.If2and-3arethezeroesofthepolynomialx2 + (a * 7)x + b,thenfindthevalueofaandb.
(0, -6)

19. If I isazeroofthepolynomialp(x) = axz -3(a-t)x-t,thenfindthevalueof 'a'. (l)

20. If the product of zeroes of the polyno mial ax2 - 6x - 6 is 4, find the value of 'a'. (-; )

21. Ifthe sum andproduct ofthe zeroes ofthe polynomial axz - 5r * c is equal to l0 each, findthe values of
a ana c. s1
Q,
22. lf a and B are the zeroes ofthe polynomial xz - 6x * p,findp if p = -2. (8)

23.The sum and the product of the zeroes of the polynomial p(x) = 4xz -27x*3k2 ate equal' Find the

value(s) of k. (+3 )

24.lfa,Barethezeroesofthepolynomialp(x)=x2-5x*ksuchthata-F=l,findthevalueofk.(6)

Pair of Linear E ouations In Variables

Solve the following pair of equations by substitution method:

(i) x*y=L4, x-Y=4

(ii) s-f,=3,'1+!=O
(iiD 3x-y=3,9x-3Y=9
(iv) 0.2x * 0,3y - 1.1 = 0, A.7x - 0.5y * 0.8 = 0
2 Use elimination method to find all possible solutions of the fotllowing pair of linear equations:

(i)x+/ = 5and 2x-3y - 4 (ii)2x*3y =Band4x*6y = 7 (iii)I*?- -L, *-"i=t

3 For each of the following systems of equations determine the value of k for which the given system of
equations has a unique solution

rtl rf i yr:'_, <i,>,nl;?i=+

each of the following system of equations determine the values of k for which the given system has no
solution.

^3x-4yl7 =0 ,..,2x-kY*3=0
(tt)3r
k*+ay-s = o
\') +zy-]-=o
5 For each of the following systems of equations determine the value of k for which the given system of
equations has infinitely many solutions.
 o f{***'lr==kr rro(u ;*?*u;Yr;u
6. Solve 2x + 3y - 11 and 2x - 4y = -24and hence find the value of 'm' forwhich y = mx + 3'
7. For which value of k will the following pair of linear equations have no solution?
3x t y = L and (2k - L)x + (k - L)Y = 2k + t.
8. For which values of a and b does the following pair of linear equations have an infinite number of
solutions?
2x I 3y = 7 and (a - b)x + (a + b)y = 3a I b - 2

9. Findthevalueofmforwhichthepairoflinearequation,2x*3y-7=0and(m-I)x+(m*I)y=
(3m - 1) has infinitely many solutions.
152x - 378Y = -74
lo. solve for x andy;
-378x+L52y=-604
ll.Solvethefollowingpairoflinearequationsforxandy:Lx*f,1=a'*b2,x*y=zab.
20. Solve for x andy: (a-b)x + (a*b)y = az -Zab - b2,(a+b)(xly) = az + b2.
12. Afraction becomes is added to both the numerator and the denominator. If 3 is added to both the
fi.rc2
numerator and the denominator becomes
f. fina the fraction.
13. Five years hence, the age of Jacob will be three times that of his son. Five years ago, Jacob's age was seven

times that of his son. What are their present ages?

14. Five years ago, Nuri was thrice as old as Sonu. Ten years later, Nuri will be twice as old as Sonu. How old
are Nuri and Sonu?

15. Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time"
If the cars travel in the same direction at different speeds, they meet in 5 hours. If they travel towards each
other, they meet in I hour. What are the speeds of the two cars?

taxi charges in a city consist ofa fixed charge together with the charge ofthe distance covered. For a

distance of l0 km, the charge paid is 1105 and for a journey of l5 km, the charge paid is {155. What are the

fixed charges and the charge per km? How much does a person have to pay for travelling a distance of 25
km?
17.34. A lending library has a fixed charge for the first three days and an additional charge for each day
thereafter. Saritha paid\27 for a book kept for seven days, while Susy paid t2l for the book, she kept for

five days. Find the fixed charge and the charge for each extra day.
18. The sum of a two-digit number and the number obtained by reversing the digit is 66. If the digits of the

number differ by 2, findthe number. How many such numbers are there?

19. The sum of the digits of a two-digit number is 9. Also, nine times this number is twice the number obtained
by reversing the order of the digits. Find the number
ratio of incomes of two persons is 9:7 and the ratio of their expenditure is 4:3. If each of them manages
to save 12000 per month, find their monthly incomes
2l . 38. Meena went to a bank to withdraw RS 2000. She asked the cashier to give her RS 50 and RS 100 notes

only. Meena got 25 notes in all. Find how many notes of {50 and t100 she received.
22.Yash scored 40 marks in a test, getting 3 marks for each right answer and losing I mark for each wrong

answer. Had 4 marks been awarded for each correct answer and 2 marks been deducted for each incorrect
answer, then Yash would have scored 50 marks. How many questions were there in the best?

23.The area of a rectangle gets reduced by 9 square units, if its length is reduced by 5 units and breadth is
increased by 3 units. If we increase the length by 3 units and the breadth by 2 units, the area increases by 67
square units. Find the dimensions of the rectangle.

24. Draw the graphs of the equations , - ! * L = 0 and 3z * 2y - L2 = 0. Determine the coordinates of the
vertices of the triangle formed by these lines and the line of y - 0 (i.e., x -axis), and shade the triangular
region. Also, find the area of triangle.

25. Draw the graphs of the pair of linear equations x - y+ 2:0 and4x-y - 4:0. Calculate the area of the

triangle formed by the lines so drawn and the x-axis.

26. Draw the graph of the pair of equations 2x + y: 4 and 2x - y: 4. Write the vertices of the triangle formed

by these lines and the y-axis. Also find the area of this triangle.

27 . Thereare some students in the two examination halls A and B. To make the number of students equal in
each hall, l0 students are sent from A to B. But if 20 students are sent from B to A, the number of students
in A becomes double the number of students in B. Find the number of students in the two halls.
the graphs of the lines x: -2 and y:3. Write the vertices of the figure formed by these lines, the x-

axis and the y-axis. Also, find the area of the figure.

29. A shopkeeper gives books on rent for reading. She takes a fixed charge for the first two days, and an
additional charge for each day thereafter. Latika paid Rs 22 for abook kept for six days, while Anand paid
Rs 16 for the book kept for four days. Find the fixed charges and the charge for each extra day.

30. In a competitive examination, one mark is awarded for each correct answer while ll2 mark is deducted for
every wrong answer. Jayanti answered 120 questions and got 90 marks. How many questions did she
answer correctly?

31. Two numbers are inthe ratio 5 : 6.If 8 is subtracted from each of the numbers, the ratio becomes 4 : 5. Find
the numbers.

32. .The age of the father is twice the sum of the ages of his two children. After 20 years, his age will be equal
to the sum of the ages of his children. Find the age of the father.

33. Two years ago, Salim was thrice as old as his daughter and six years later, he will be four years older than
twice her age. How old are theY now?
34. A two-digit number is obtained by either multiplying the sum of the digits by 8 and then subtracting 5 or by
multiplying the difference of the digits by 16 and then adding 3. Find the number.
35. Determine, algebraically, the vertices of the triangle formed by the lines
3x-Y-3, 2x-3Y=1, xI2Y-8
 OUADRATIC EOUATIONS

1. Check whether the following are quadratic equations:

(i)(x- 2)z +l= x2 - 3 (ii)(x+ 2)3 =x3 -4
2. Does (x - t)t + 2(x + L) = 0 have a real root? Justiff your answer'
3. Find the roots of the quadratic equation 2x2 - x **=O
4. Find the roots of the quadratic equations by factorization method:

(i)3x2 -z^loxt2 = 0 (iD\f3x2 -2^/1x -2tE = 0

(iii)x2 -z{zx-6= 0 (iv)x2 +2tl2x-6=0

5. Find the values of k for which the roots are real and equal in each of the following equations:

(a)kx2 t 4x* L= o (b)kx2 -ztlSx *4=0

(c)3x2 -Sx-2k = 0 (d)kx(x-2) * 6=0

6. Find the values of k for which the roots are real and distinct in each of the following equations:
kx2+2x*l=o
7. Find the values of k for which the QE kxz * 2x l1 = 0 has no real roots.
8. Find the values of k for which the QE kxz * 2x * 1 = 0 has real roots.
g. If(-s)isarootofthequadraticequation2x2+px-15=0andthequadraticequationp(x2+x)+k=
0 has equal roots, then find the value ofp and k.

10. Ifthe equation (L + m2)x2 * 2mcx + (c2 - a2) = 0,has equal roots, prove the c2 = a2 7l + m21'
I l. Find the roots of the following equations:

(a)x-I=2,* +0 G)* -+=ft,x+ -4,7

12. Solve for x:4 *# =+
13. Solve for x: 4 * #*= 3!; x + 2,4.
14. Solve for x:

@{ffi7 =x*2 (i\z,lNT,-Zx=!

15. Solve forr: 1 * * * i = ;fi;, a * o,b + o,x + o

16. Solve for ril#{= *+i* *

17. Solve - 4ac)x - bc = 0
forx:abxz + (b2
18. Solve for x:4x2 * 4bx - (a' - b2) = o.

le. sorve forr: 3 (#=) - r(#) = IL;x *i,+

20. Solve for x:Y| - n(#) = ,

21. Solve rbrxt6;|,.-.4-@=;
22. Solvethe following equation for x:9x2 - 9(a + b)x + (2a2 * Sab * 2b21 = g.
23.Thesum of two numbers is 16. The sum of their reciprocals is f. finO the numbers

24. Thedifference of two numbers is 4. If the difference of their reciprocals is f,, nna the numbers.

25. Find two consecutive odd positive integers, sum of whose square is290.

26.Thesum of reciprocals of Rehman's ages, (in years) 3 years ago and 5 years from now is J. ninO his present

age.

27 .ln a class test, the sum of Shefali's marks in Mathematics and English is 30. Had she got 2 marks more in

Mathematics and 3 marks less in English, the product of their marks would have been 210. Find her marks in

.-.[he two subjects.

'..I produces a certain number of pottery articles in a day. It was observed on a particular day
(g/ncofiage industry
that the cost of production of each article (in rupees) was 3 more than twice the number of articles produced
on that day. If the total cost of production on that day was RS 90, find the number of articles produced and
the cost ofeach article.

29. The altitude of a right triangle is 7cm less than its base. If the hypotenuse is l3cm, find the other two sides.
30. The diagonal of a rectangular field is 60 metres more than the shorter side. If the longer side is 30 metres

more than the shorter side, find the sides of the field.

3 1. The denominator of a fraction exceeds its numerator by 3. If one is added to both numerator and denominator,

the difference between the new and the original fraction is finO ttre original fraction.
f.
32. Atraintravels 360 km at a uniform speed. If the speed had been 5 km./h more, it would have taken I hour less
for the same journey. Find the speed of the train.
33. In a flight of 2,800 km, an aircraft was slowed down due to bad weather. Its average speed for the trip was
reduced by 100 km/tr and time increased by 30 minutes. Find the original duration of the flight.

34. A motor boat whose speed is 18 km/h in still water takes I hour more to go 24 km upstream than to return

downstream to the same spot. Find the speed of the stream.

35. Some students planned a picnic. The budget for food was t500. But 5 of them failed to go and thus the total
cost of food for each member increased by 15. How many students attended the picnic?

36. 16500 were divided equally among a certain number of persons. Had there been 15 more persons, each would

have got {30 less. Find the original number of a person.

37. The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger
number. Find the two numbers.

38. Sum of the areas of two squares is 468 m2. If the difference of their perimeters is24 m, find the sides of the
two squares.

39. Two water taps together can fill a tank in e I hours. The tap of larger diameter takes 10 hours less than the

smaller one to fill the tank separately. Find the time in which each tap can separately frll the tank'
 COORDINATE GEOMETRY

I . Find the distance between the following pair of points: (a, b), (-a' -b)
2. The distance of the point (-3, 2) from origin is
3 . By distance formula determine if the points (L,5), (2,3) and (-2, -LI) are
collinear.

4. prove that the points (7,L0),(-2,5) and (3, -4) arcthe vertices of an isosceles right triangle.
Showthatthepoints (L,7),(4,2),(-1,-1) and(-4,4) aretheverticesofasquare.
Find the value of y for which the distance between the points P (2, -3) and Q(I},y) is 10 units'

7. Find the point on the x - axis which is equidistant from (2, -5) and (-2,9).

8. Find a relation (equation) between x and y such that the point (x, y) is equidistant from the points (7,1) and

(3,5).
g. Find the areaof arhombus if its vertices are (3,0), (4,5),(-L,4) and (-2,-1) taken in order'

10. If ,4(4,3) , B (-L,y), C (3,4) are the vertices of a right triangle ABC, right angled at A, then find y.

1 1. Find the coordinates of the points which divide the line segment join ng A(-2,2) and B (2,8) into four equal

parts.

{t}rc (t,Z)is the midpoint of the line segment joining the points (2,0) and (0,3), find p.
lJ \ '3./
13. Find the coordinates of a point A, where AI| is the diameter of a circle whose centre is (2,-3) and B is (1,4).

14.If (L,Z), (4,y), (x,6) and(3,5) are the vertices of a parallelogram taken in order. Find x and y.

15. If the three vertices of a parallelogram, taken in order, ate (2,0), (-6, -Z) and(4, -2) find the fourth vertex'

16. Find the coordinates of the vertices of the triangle, the midpoints of whose sides are (I,2),(0,-1) and

(2,-L).
lT.FindthelengthofthemedianCFofthetriangleA.4BCwhosevertices ateA(L,-L),8(0,4) andC(-5,3).
18. Find the coordinates of the point which divides the join of (-1,7) and (4, -3) in the ratio 2:3.

19. If the point C(- 1,2) divides internally the line segment joining the points .A(2,5) and B (x, y) in the ratio 3:4,

find the val:ue of x2 + Y2.

20. Find the coordinates of the points of trisection (i.e., points dividing in three equal parts) of the line segment
joining the points A(2,-2) and B(-7 '4)
A and B arc (-2,-2) and (2,-4)respectively, frnd the coordinates of P such that AP = ; AB and P lies

on the line segment AB

22. point M(1lry) lies on the line segment joining the points P(15,5), Q(9,20). Find the ratio in which point M
divides the line segment P0 and also find the value of 'y'.

23. Find the ratio in which the line segment joining A(1,-5) and B(-4,5) is divided by the x - axts- Also find

the coordinates of the point of division.

24. Determine the ratio in which the line 2x t y- 4 = 0 divides the line segment joining the points A(2, -2)
and (3,7). Also find the Points.

25.Thevertices of atriangle are (2,L),(5,2) and (3,4) Find the coordinates ofthe centroid.
 26.Thetwoverticesof atriangle are(2,1)and(5,2).If itscentroidis (5,3.5) findthethirdvertex.
27 . Find the centre of a circle passing through the points (6, -6) , (3, -7) and (3,3).

CASE STIIDY
" rl\'
i Zg. y'o conduct Sports Day activities, in your rectangular shaped school ground ABCD, lines have been drawn
l-/*nnchalk powder at a distance of lm each. 100 flower pots have been placed at a distance of lm from each
other along AD, as shown in figure. Niharika runs th the distance AD on the 2ndlineand posts a green flag.
f
Preet runs tn tnr distance AD on the eighth line and posts a red flag. What is the distance between both the
I
flags? If Rashmi has to post a blue flag exactly halfivay between the line segment joining the two flags, where

should she post her flag?

I ,t

29.The Class X students ofa secondary school in Krishinagar have been allotted a rectangular plot ofland for
their gardening activity. Sapling of Gulmohar are planted on the boundary at a distance of lm from each
other. There is a triangular grassy lawn in the plot as shown in the figure. The students are to sow seeds of
flowering plants on the remaining area of the plot.
(i) Taking A as origin, find the.coordinates of the vertices of the triangle.

{ii} What will be the coordinates of the vertices of A PQR if C is the origin?

.{I 567
 TRIANGLES
l" State and prove Thale's theorem (BPT)

2. In figure PQ ll nC. Find 0C

1.3cm
1.5cm

3. Inthefigure, DE llBC.If AD --x,Dfi =x-2,A8 =x+2andEC =x- l,findthevalueofx

I

4. In triangle ABC,DE ll BC and #=i ff AC = 5,6. Find AE.

5. M and N are points on the sides PQ and PR respectively of a APQR. For each of the following cases, state
whether MN ll QR.
(i) PM = 4 cm,QM = 4.5 cm,PN = 4 cm,NR = 4.5 cm
(ii) PQ = 7.28 cm,PR = 2.56 cm,PM = 0.16 cm,PN = 0.32 cm
6. In the given hgure if AC :2m,OC: 3 m and OD:7 m, then find BD.

7. In figure, if AB ll CD find the value of r

A B

8. Inthefigure,LM ll AB.If AL = x-3,AC = 2x,BM = x-2andBC =2x13, findthevalueofx

A
9. Inthe figure,LM ll AB.If AL = 3 cm, CL = 2 cm,LM = 3 cmandAB = r, findx.

A B

10. In the figure, the points D and E divide the side AB and AC in the ratio 1:3. If DE : 2.4 cm, then find the
length of BC

1 l. In figure, MBC is right angled at C and DE L AB .Findthe lengths ofAE and DE.

E
cm

cm

Cm-

12.Averticalpoleoflength6mcastsashadow4mlongonthegroundandatthesametimeatowercastsa
shadow 28 m long. Find the height of the tower.

13. In the figure, DE ll AC and DC ll AP. Prove thatBj

EC= *.
CP

14. S and T are points on sides PR and QR of APQR such that LP = zRTS. Show that ARPQ - ART.S.

15. In figure. A, B and C are points on OP,OQ and OR respectively such that AB ll P0 and BC ll 0R. Show

that AC ll PR.

16. E is a point on the side AD produced of a parallelogram ABCD and BE intersects CD at F. Show that A^ABE -
ACFB
17. In figure,0A.OB = OC.OD. Prove thal zA = zC and LB = LD
A

18. DisapointonthesideBCofatriangleABCsuch thatLADC = LBAC. Showthat CAz = CB.CD

19. In figure , if BD I AC and CE I AB ,prove +=
that: AB DB
=.

20. In the figure, if{ = # *O z! = zL.Prove that APQS - ATQR

21. In figure,if LABE

=
AACD, prove that MDE - LABC
22. lJsing BPT, prove that a line drawn through the mid-point of one side of a triangle parallel to another side
bisects the third side.

23. Using BPT, prove that a line drawn through the mid-point of any two sides of a triangle is parallel to the third
side.

24. ABCD is a trapezium in which AB ll DC and its diagonals intersect each other at the point O. Show that
AO CO
-=-
BO DO.
25. Thediagonals of a quadrilateral ABCD intersect each other at the point O such thatffi= 9. Show that

ABCD is afapezium.
26. lf AD and PM are medians of triangles ABC and PQR, respectively where LABC - LPQR, prove that:
AB AD
-=-
PQ PM.

27. Sides AB, BC and median AD of a triangle ABC are respectively where proportional to sides PQ, QR and
median PM of APQR. Show that LABC - APQR.

28. Sides AB, AC and median AD of a triangle ABC are respectively proportional to sides PQ, PR and median
PM of another triangle PQR. Show that A,ABC - APQR.

29. A girl of height 90 cm is walking away from the base of a lamp-post at a speed of 1.2 m/s. If the lamp is 3.6
m above the ground, find the length ofher shadow after 4 seconds.
30. CD and GH are respectively medians of AABC and AEFG.If AABC - AFEG. Prove that

(i) LADC - AFHG (iDn=# Qii) ACDB - LGHE

31. If CD and GH (D and H lie on AB and FE) are respectively bisectors of zACB and zEGF and LABC -
AFEG, prove that

(i) LDCA - AHGF (iD#=# Qii) ADCB - LHGE

32. In figure , LACB = 900 and CD I AB.prou" 111at4 =

y.

c
33. In figure, express x interms of 4 b and c, ML : a, NP :x, MN : b, NK : c.

I 1 I
34. In figure, PA, QB and RC are each perpendicular to AC. Prove that * = .
xzy
PROGRESSION
1. Write the next term of the rf&y'T6, tfT2, ....
2. Find the 1Oft term from the end of the A.P 8,10,12, ... .. ,L26
3. The third term ofan AP is 12 and the seventh term is 24, then the lOth term is
4. How many terms of an AP must be taken for their sum to be equal Io 120 if its third term is 9 and the
difference between the seventh and second term is 20 ?
5. Find the sum of the firstZ2 terms of the AP: 8,3, -2,.. ...
6. In a certain AP, 5 times the 5th term is equal to 8 times the 8th term, then its l3th term is equal to
7. A man receives Rs. 60 for the first week and Rs. 3 more each week than the preceeding week. How much
does he earns bythe 20th week ?

8. Determine the AP whose 3td ter- is 5 and the 7th term is 9.

g. If the 10rt term of an AP is 52 and the lTth term is 20 more than the 13th ternu find the AP.
10. An AP consists of 50 terms of which 3rd term is l2 and the last term is 106. Find the 29thterm.
Il. If 2x, x * 10, 3x+ 2 are in A.P., find the value ofx.
12. Find the sum of all two-digit odd positive numbers.
13. If the sum of the first p terms of an AP is the same as the sum of first q terms (where p !q) then show that
the sum of its fust (p + q) terms is 0.
14. Find the sum of first 24 terms of the list of numbers whose nth term is given by an = 3 * 2n.

15. The sum of n terms of an AP is (5n2 - 3n). Find the AP and hence find its 10th term.
16. The first and last terms of an AP are 4 and 81 respectively. If the common difference is 7, howmany
terms are there in the AP and what is their sum?
17. Which term of the AP \21,Lt7,LL3, ..., is its first negative term.
18. Which terms of the AP:2L3,207,20I,..., is last positive term.
19. If the 3'd and the 9ft terms of an AP are 4 and- 8 respectively, which term of this AP is zero.
20. How many multiples of 4 lie between l0 and250?
21. Find the number of 3 digit-numbers, which leaves a remainder 1, when divided by 3.
22. The first term of an AP is 5, the last term is 45 and the sum is 400. Find the number of terms and the
common difference.
23. If the sum of first 7 terms of an AP is 49 and that of 17 terms is289, find the sum of the first n terms.
24. How many terms of the AP: 9,17,25,... must be taken to give a sum of 636?
25. The sum of three numbers in AP is 27 andtheir product us 405. Find the numbers.
26. The sum of the first 7 terms of an AP is 63 and the sum of its next 7 terms is 16l. Find the 28ft term of this
AP.
27. Findthe value of the middle most term(s) of the AP -LL,-7 ,-3, "' ' ',49.
28. A sum of Rs. 280 is to be used to give four cash prizes to students of a school for their overall academic
performance. each prize is Rs. 20 less than its preceding prize, find the value of each ofthe prizes.
If
29. In a school, students thought of planting trees in and around the school to reduce air pollution. It was decided
that the number of trees, that each section of each class will plant, will be the same as the class, in which
they are studying, e.g., a section of Class I will plant I
tree, a section of Class II will plant2 trees and
so on till Class XII. There are three sections of each class. How many trees will be planted by the students?
30. A spiral is made up of successive semicircles, with centres alternately at A and B, starting with centre at
A, of radii 0.5 cm, 1.0 cm, 1.5 cm, 2.0 cm,. . .. What is the total length of such a spiral made up of thirteen
consecutive semicircles? (Take tt ='+ )

CTRCLES

1. In Fig. XY and X'Y' are two parallel tangents to a circle with centre O and another tangent AB with point
of contact C intersecting XY at A and X'Y'at B. Prove that z AOB :90".

2. The radius of the in-circle of a triangle is 4cm and the segments into which one side is
divided by the point of contact are 6cm and \cm. Determine the other two sides of the
triangle. c

3. Prove that the parallelogram circumscribing a circle is a rhombus.

4. If AB, AC, PQ are tangents in below figure and AB : 5 cm, find the perimeter ofAAPQ.

5. Two tangents TP and TQ are drawn to a circle with centre O from an extemal point T. Prove that ZPTQ:
2AOPQ.
6. A quadrilateral ABCD is drawn to circumscribe a circle. Prove that AB + CD : AD + BC.
7 . Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.

8. Two tangents PA and PB are drawn to the circle with center O, such that zAPB : n00 .Prove that OP :
zAP,
9. PA and PB are the two tangents to a circle with centre O in which OP is equal to the diameter of the circle.
Prove that APB is an equilateral triangle.
I 0. Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the
centre ofthe circle.
1 1. The tangent at a noint C of a circle and a diameter AB when extended intersect at P
If ZPCA: I lo0, find L]BA
3cm

12. In the figure, quadrilateral ABCD is circumscribed, ftrd the value ofx.
0
13. In the given figure TAS is a tangent to the circle, with centre O, at the point A. If LOBA:32 , find the
value ofx and y.

14. In the adjoining figure, ABCD is a cyclic quadrilateral. AC is a diameter of the circle. MN is tangent to
the circle atD, LCAD:400,zACB :550

15. Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to
the angle subtended by the line-segmentjoining the points ofcontact at the centre.
: :
16. In the given figure, PT is a tangent to the circle at T. If PA 4cm and AB 5cm, find PT.

17. A circle touches the side of BC of LABC atP and touches AB and AC produced at Q and R respectively.
Prove that, g = j {eerimeter of MBC) or (Perimeter of MBC = 2AQ)
18. In the figure, two circle touches externally at a point P. From a point T on the tangent at P, tangents TQ and
TR are drawn to the circle with point of contact Q and R resprectively. Prove that TQ: TR.

R
19. In the figure, AB is a diameter of circle with centre O and QC is a tangent to the circle atC.If LCAB = 300,
find (i) LCQA,(ii) LCBA.

20.In the given figure, TBP and TCQ are tangents to the circle whose centre is O. Also LPBA = 600 and
LACQ = 700. Determine zBAC and zBTC.

AREAS RELATED TO CIRCLES

l. Find the area of the sector of a circle with radius 4 cm and of angle 300. Also find the area of the
corresponding major sector. (Use n = 3.L4)
2. Find the area of a quadrant of a circle whose circumferenc e is 22 cm.
3. The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand in 5 minutes.
4. A chord ofa circle ofradius 10 cm subtends a right angle at the centre. Find the area ofthe corresponding:
(i) minor segment and (ii) major segment. (Use n = 3.1,4)
5. In a circle of radius 2l cm, an arc subtends an angle of 600 at the centre. Find:
(i) the length of the arc
(ii) area of the sector formed by the arc
(iii) area of the segment formed by the corresponding chord.
6. A chord of a circle of radius 12 cm subtends an angle of 1.200 at the centre. Find the area ofthe corresponding
segment of thecircle. (tlse n = 3.1.4,8 = IJ3)
7. A horse is tied to a peg at one corner ofa square shaped grass field ofside 15m by means ofa 5m long rope.
Find:
(i) the area of that part of the field in which the horse can graze.
(ii) the increase in the grazing area if the rope were 10m long instead of 5m. (Use n = 3.14).
8. A brooch is made with silver wire in the form of a circle with diameter 35 mm. The wire is also used in
making 5 diameters which divide the circle into 10 equal sectors as shown in figure: Find:
(i) the total length ofthe silver wire required (ii) the area of each sector of the brooch

g. A round table cover has six equal designs as shown in figure. If the radius of the cover is 28cm, find the cost
of making the designs at the rate of RS 0.35 per cmz. (Use rtr = IJ)

10. A car has two wipers which do not overlap. Each wiper has a blade of length 25 cm sweeping through an
angle of 1150. Find the total area cleaned at each sweep ofthe blades.
 INTRODUCTION TO TRIGONOMETRY
X
If cos d : find the other five trigonometric ratios.
t7',
2. If tan: A = Ji -l,show that sin AcosA = 4 I .

5sina -3cosa 1
3. If 5tana = 4, show that
Ssinrz +2cosa 6
4. In LPQR, right angled at Q, PR + QR: 25 cm and PQ : 5 cm. Determine the values of sin P, cos P and
tan P

5. If /.A and /.P are acute angles such that tan A: tan P, then show that ZA: lP
6. IftanA: I andtanB: "6, evaluatecosAcosB-sinAsinB.
7. Find acute angles A and B, sin(A+2il)= f uoa cos (A + 48) : 0, A > B
2

8 In a right triangle ABC, right angled at B, if /ACB = 300 and AB

: 5 cm. Find the remaining angles and
sides
tanz 60 +4 cosz 45+3cosec2 50 +2 cosz 90
9. Find
zcosec3o + 3 0
3

10. Find the value of x in the following: (i) 2sin3x = f

1 1. Write the value of cos 10 cos 20 cos 30...............cos 1790 cos I 800.

l2.ln APQR, right angled at Q, PQ:3cm and PR:6cm. Determine ZQPRand /.PRQ .

13. Express the ratios cos A, tan A and sec A in terms of sin A.
14. Prove that sec A (l - sin A)(sec A + tan A) l. :
A *l
cos + sin A
15. Prove that: - 2 sec A
l+ SinA Cos A

16. Prove that:

Cot CosecA-l
A-Cos A
CotA+CosA CosecA+l
17. Provethat:
Sin9-Cos9+l 1

Sin? + Cos 0 -l Sec9 -Tan0

tan?+sec9-l 1+sind
18. Provethat:
tanQ -sec0 +l cosd
19. Find the value of: (l + tan 0 + sec 0) (1 + cot 0 - cosec 0).

20. Provethat:
A;st"n = Sec A+TanA.
- SinA
1+SecA Sin2 A
21. Prove that: =
Sec A l-Cos A
r-sino-:2
22. Prove that:-lr+tlno= + seco
{ l-sina -.
I+sing

23. Prove
ane * hno
that:
sec?-l sec9+l =2coseco
.

24. Prove that: (cosec 0 -cot0\2 = l-totl .

' l+cos9
25. Prove that: (sind+ cosec1)' +(cosd+se c0)2 =7 +tanz 0 +cotz 0
 (t+ron'A\ (l-TanA\' a
^ l=l _
.
26. Prove that:l I ='lan'A.
l1+Cot'A) \l-CotA)
27. Prove that:(Co secA - Sin A)(Sec A - Cos A)= -
' =-:-
TanA+Cot-A
28. If tan7 + cot? =2, thenfind the value of tanz 0 + cot2 0 .

tol9- :4;0<
29. Findthe value of 0,if
' -1-sind+-!9sP- 900.
1+sind

30. Provethat: (secd

' -cosec?--!
-tznl)'z cosec9 +l
.

31. lf 7 stnz 0 +3cos2 0 = 4,then find the value of tan?

32- Prove
tanQ
that: """ " + '"^' " :
tan9
2 cos ec9 .
sec0-1 secd+1
33. lf 0is an acute angle andsin 0 =cos?, find the value of 2tanz 0 + snz 0 -1.
34. If sin? +cosd = p and secg+cosec 0 = q, show that q(p'-l):2p
35. If cos ec g - sin 0 = m and sec d - cos I = n, prove that (m2n)2t3 + (mfif/3
:I
36. If tan 0 : !U, prove that
asin? -bcos? a' -b'
asin0 +bcos9 a2 +b'
37. lf x : asin0 andy = btanl,then show tnut {x' - \y" = t

SOME APPLICATIONS OF TRIGO NOMETRY

1 . The angle of elevation of the top of a tower from a point on the ground, which is 3 0 m away from the foot
of the tower, is 30o. Find the height of the tower.
2. A tree breaks due to storm and the broken part bends so that the top ofthe tree touches the ground making
an angle 30' with it. The distance between the foot of the tree to the point where the top touches the
ground is 8 m. Find the height of the tree.

3. FromapointPonthegroundtheangleofelevationofthetopofal0mtallbuildingis30o.Aflagis
hoisted at the top ofthe building and the angle ofelevation ofthe top of the flagstaff from P is 45o.

Find the length of the flagstaff and the distance of the building from the point P.(Take Jl =1.732)
4. Theshadowofatowerstandingonalevelgroundisfoundtobe40mlongerwhentheSun'saltitudeis30o
than when it is 60'. Find the height of the tower.

5. The angles of depression of the top and the bottom of an 8 m tall building from the top of a multi-storeyed
building are 30o and 45o, respectively. Find the height of the multi-storeyed building and the distance
between the two buildings.

6 . From a point on a bridge across a river, the angles of depression of the banks on opposite sides of the river
are 30o and 45", respectively. If the bridge is at a height of 3 m from the banks, find the width ofthe river.
7. From the top of a 7 m high building, the angle of elevation of the top of a cable tower is 60o and the angle
of depression of its foot is 45o. Determine the height of the tower.
t

8. A straight highway leads to the foot of a tower. A man standing at the top of the tower observes acar alan
angle of depression of 30o, which is approaching the foot of the tower with a uniform speed. Six seconds
later, the angle ofdepression ofthe car is found to be 60o. Find the time taken by the car to reach the foot of
the tower from this point.

9. As observed from the top of a 75 m high lighthouse from the sea-level, the angles of depression of two
ships are 30o and 45o. If one ship is exactly behind the other on the same side of the lighthouse, find the
distance between the two ships.
10. A person walking 45m towards a tower in a horizontal line thLrough its base observes that angle of elevation
of the top of the tower changes from 450 to 600. Find the height of the tower.
11. Theangleofelevationof ajetfighterfromapointAonthegroundis600.Afteraflightof l0secondsthe
angle ofelevation changes to 300. Ifthe jet is flying at a speed of648kmlhr, find the constant height at
which the plane is flying.
12. Anaeroplane, flying horizontally I km above the ground, is observed at an angle ofelevation of600 from a
point on the ground. After a flight of l0 seconds, the angle ofelevation at the point ofobservations changes
to 300. Find the speed of the plane inmf s.
I 3. Two ships are approaching from a light house from opposite directions. The angles of depressions ofthe
two ships from the top of the light-house are 300 and 450. If the distance between the two ships is 100m,
find the height of the light-house. [use\6 =1.7321.
14. Theangleofelevationofacloudfromapoint 60mabovealakeis300andtheangleofdepressionofthe
reflection of cloud in the lake is 600 . Find the height of the cloud from the lake.
I 5. The angle of elevation of the top of a tower as observed from a point on the ground is 'a' and on moving 'a'

metres towards the tower, the angle of elevation is 'B'. Prove that the height of the tower it #ffi.
16. If the angle of elevation of a cloud from apointhmeters above a lake isa and the angle of depression of its
reflection in the lake be B ,provethatthe distance ofthe cloud from the point ofobservation is
2hseca
tanB -tana

,* * d( !*,* d(,1.'1. * * i( * * !* 1. *,f * * :lt :t 1. 1. ** tl. * t * !*,t * * * * *,* d. **

O J', b v €_ *ha,ts ( - (4s'g) .fiec"cl = {a'-= e-
()p
€) ,/.t- ta< ,!- +- c'*L4 = I,

4,^ru 4 -t J,'n44 = I
Lr_
5ln P-tus P='(,
@ -g g +aa,=t ^">
E rf;C P
L
1a-s ;) : +
2-
e= ? /
-
n
3/5|nP On 9-
z7

9+ ll-f'^Q P: /
*t4a/- La-rPo= | @
Vo*
-)
-z-

&inun 4A"* .zS'n'9 1 ,Ecqy =

+rr-" T f .z,'- & CYj l! r E,

leca* /* S,'n6.
[y19 /D^f lf *
C-),g
/f 4e-cD + {a-g

*t c,o/-,F
8) P
T n2^
3ec B 4- usrc 'e- = #arg

/,
C rnk+ lu x--9, Ac = a?L
U Arb . Z{ 4 L=
ta
l- r3 C = ?x+3 1f'"1 ?L
s,Y = =c -Z c
H

pE ll e)c, *-J cTu EF

Cnfnb4 A 6

lzr{ L tlnql 1D' = Au x AF

A
F
E
P

*"/ veo,"a a{ x
( s) 9n 6^3; eo ll r,c,^
c
2

A
F ovr fr^, S,t)e BC
(.+> P,9L i4 +t"{ fi-r"
#a't-
^# € ArEe 73wgn
,J
A >-io
+F,*
F'*
I
t
I .+DC = b6+c.

Ct-, LB X c-D I v c
 .D E(l bc
o T,,, A lroc. A P= 6c IA ,&

I +-r*-^ #'^'4 4f.,

6^ "!n laoF As= -L +
C

n
P
rb
4cn /" BC:3cP
en A knc. Ace: 1oo Ac =
CD cpxftB )-5 --
llrv6 +1*, Vzlue I A

4>
+

? ''a
b) 9' A urq ^t, l! = 6e
o
In q oo et .dLU N Nd rye
fi*4 /_A i*
h. 4aqD , rt a lr' r 1r v+* in ,sf., ,'c)/\ ArzllDc
?

r&q 6'J\-Q.- nb F'' AP, & b C_ zt-9

( (ts
I..'f"'4
-9 ^"h *fiol- PL rr Dc 2f p= t * crn-
r}q = Sfcn> 8, 4c = trc,n, fr n ,) 4p

5ri,'sa*e ( .4D
t1 n +- 6tt tW 9,'J
(noJ"c.) + h all e lt64 Ltr) rtrzcp L

bE i-l{r'9ecb cD at F" tho>o #^l-

A 4BEotacf B'
t

G2@
?EgqAu,

76w
AZ&Clt 6,
AmrrtMahotsar

ADDMONAL PRACTICE
CBSE
QUESTIONS
MATrmMATrcs STANDARD (041)
Class ){l2023_24

Time allowed: 3 Hours

Maximum marks: g0

General Instructions:

l. Itr.a*sdon paper contains _ five sections A, B, C, D and E.

2. section A has 1g Mces and, }2Assertion-Reason
based questions of 1 mark each
3' section B has 5 very sihort A*r;-cvsA)-type
quesriors of 2 n:arks eac'.
4' section c has 6 short Answet
rsaj-bee questions of 3 marks eacb.
5' Section D has 4 r'ong. fuygt
cr-aj-qe. questions of 5 marks eacrr-
6' section E has 3 case-based i"t g"i"jird
yalues of 1, 1 and 2 marks each r"Jpectively. "i;;;;;l+'*rt. each) wirh s'b parrs orrhe
7' AII questions are
"oT*oty: rro*.u.i, an internal choice in 2 questions of
t'as been provided. ao int",nut
5 marks, 2 es of 3
fffil3t1trtrt:?r.%:ffi1-s "i;i;;;u."" il;oed
in the

SECTION A
(This section of Multi ple-choi ce questions M of l mark each.
Serial
No.
7 Which of the Marks
following could be tlrc gaph of the polyromial? 7
(x I )2(x + 2)?
 G2@
q34r\ar^

7
Aeadt xt
AmritMahotsa!

a)
il

,'t-'

@)

r
 G2@
7W
-AZaOl Ka
ffirrtMahotsal

(c)

(d)

2 The lines kt kz and kt represent

three different equations as
gaph beiow The solution of the shown n the 1
equations represented by the
a
x J and v 0 while the solution lines kt and ks
of the equations represented by the lines $
and kt $ x 4 and kz
v
 G2@'
iqgEaur

76w.
A26dl 62
AmntMahotsa!

Which of these is the equation of the line frs?

(a) x-y:3
(b) x -y:-3
(c) I +y: 3
(d) x+y: 1

3 What is/are the roots of :6x?. 1

(a) only 2
(b) onb 3
(c) 0 and 6
(d) 0 and 2

4 The coordinates of the cente of the circle, O, and a point on the circle, N, are I
shown in the figure below.
 G2@q3/4Borr

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Amrit MahOtSa\

0(-4,3)
a

N(-2.4,1.8)

What is the radius of the circle?

1a; {0.+ urits

(b) 2 udts
(c) a,units
(d) \142.4

5 APQR is shown below. ST is drawn such that zPRQ: zSTQ I

20

Q{ote: Thefigure is not to scale,)

If ST divides QR in a ratio of 2:3, then what is the length of ST?

-.10
{a)
.,3 cm
-
(b) 8 cm
(c) 12 cm
(d) cm
f,
6 Two scalene tiangles are given below. 1
 -
G2@Tqg/ar$h

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AmritMahqtsa\,

3
3cm

B
c
R
(Note: The figures qre not to scale.)

Anas and Rishi observed them and said the following:

Anas: APQR is similar to ACBA

Rishi: APQR is congruent to ACBA

Which ofthem is/are conect?

(a) Only Anas

(b) only Rishi
(c) Both Anas and Rishi
(d) Neither of thern, as two scalene tiangles can never be similar or
congruent.

7 Harsha n:ade a wind chinre r:sing a frane and nretal rods. She purched 8 holes 1

in the frame, each 2 cmapart, and then hung 6 rnetal rods from the fame, as
shoum in the figure below. The ends of the netal rods are aligned over a line,
shown by the dotted line in the figure.

ot.^
fit
* & alcsJt &
Lrr

A = 3+ ,,.'!+ *r \*/: ,{

qd .1 6
?j
cm 6rtot' f 's
t_ d ri
b"" :p

29 0.m
Bo-dF
^)?qtryi 't\i{
' *'
1

rr{
-t" nr
 G2@-aqgaNur

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(Note: Thefigureis notto scale.)

If all ofthe rods are staigfnt and not swaying then what is the length ofRod
P?

.- 69
cm
\a) 7
53
(b) ; cm

76
(c) ;cm
171
(dl
-cm
7

8 Two circles with centes O and N touch each other at point P as shown O, P 1

andNarecollinear. The radius of thecircle with cenfe Oishx'ice ftatofthe

:
circle with cente N. OX is a tangent to the circle with cenfe N, and OX 18
cm

(Note: Thefigure is itot to scale.)

What is the radius ofthe circle with centre N?

,.18
\a) ,cm
@)9cm
(c)
f cm
(Q
fficm
9 Shown below is a circle with cenhe O having tangents at points P, T and S. 1
 G2@'
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76w.
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AmrtMahotsa\i

(Note: Thefigure is not to scale.)

If QR: l2cmand the radius of the circle is 7 crq what is the perineter ofthe
po$gon PQTRSO?

(a) 26 cm
(b) 31cm
(c) 38 cm
(d) (cannot saywith the given inforrnation)

10 shown below is atable with vahres of cosecant and secant of different angles. 1

0 350 650
cosec 0 P 1.1
sec (90' - 0) T,7 o
What are the values of P and Q respectively?

(a) aand 1
7.7 7.1
(b) 1.1 and 1.7
(c) 1.7 and 1.1
(d) (canrot be fourd with the given ffinnation)

tl In the figure below, PQRS is a square. 1

 G2@-lqgqNqr

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A.dtMlfrotsar,

'17
crtr a
Q{ote : Thefigure is not to scale.)

What is the vafue ofsin

zSpT?

@)*

ft)g
"15

(c)l#
(d) (cannot be for.nd with
the given inforrnation)
12 Shown below is a sohed problem
7
g*=:#+.-+a#
cosec d+cotd- \
cosec (step 1) \'
cotd+ cosec d _ cotd O"
cosec - t
cot cosec 0
(step 2)
0+ (
1- cotd + cosec
cosec + (step 3) :A.

= cotd + cosecd (step 4)

In which step is there an
enor in sohing?
(a) Step I
(b) Step 2
(c) Step 3
(d) There is no error
 G2@
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area of regpn ln
A 6cm is shown
1
the circle.
circle is of the area of

(Note: The figure is not

to scale')

circle's minor arc?

Whatis the length of the

(alf cm
tb) '#r*
(c) 16n cm

[d) 20n cm 1
5 crnL, as
n a circle with cente O ,of radius
14 A regular pentagon 1S

shown below

shaded part of the circle?

What is the area of the

2ncn*
 G2@-qg€Nh

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(c) 5z cnf
(d) l0zcnf
I5 A cuboid ofbase atea P sq rndfs
A sphere of vohune R cu rxlits ls wirtl water rpto a height
of a rxlits.
ts dropped nto the 1
conpletely subnrerged. cuboid such tlnt it xs
A representation of the submerged
below. sphere ls shovrn

which ofthese represents the increase in t.e height ofwater?

(a) 0 udts
Ol f u'lm
(c) Rurirs
(d)0+ R
; uruts
16 Sweety, Niteslq and
Ashraf visited a hospital
which incfuded a blood presslre for their annual body checkup,
7
evaluation The resuits
pressure readings are
as follows
of their systolic blood

Sweety: t2l nrnHs

Nitesh: 147 mnIIs'
Ashraf 160 nxnFl;

The table below depicts

the systolic blood pressure
who visited the ranges of all the patients
on the same
Blood
Number of
LIs - 125
10
tzs - 13s
9
135 - 145
12
145 - 1s5
19
155 - 165
10

the three friends tnve a blood pressure reading

that frlls in the
 G2@'
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(a) Sweety
@) Nitesh
(c) Astraf
Both and Asliraf
17 The table below depicts the weiglrt ofthe students of class 6 of Red Bricks 7
Public School There are 18 students in the class that weig! above the median
weiglrt.

lt-Vejgh!]n kg
llu.mbqr of Students

lil;28 b

28-31 I
3J _M v

34 =37 '10

37-40 .|

If there are no shrdents with the sanre weigirt as median weigtrt, how many
students weigfr between the range of 37 - 40 kgs?

(a) s
(b) 7
(c) 18
d 31
18 Ginny flipped a fiir coin tlree times and tails came up each time. Ginny wants 1
to flip the coin again.

What is the probability of getting heads in the next coin flip?

(a) o
(b) 0.2s
(c) 0.s
1

I9 A number q is prime frctorised as 32 x 72 x b, where b is a prinre rurnber 1

other than 3 and 7.

.Based ou tlrc abovo ffirmatinn, two staternents are given below - one
labelled Assertion (A) attd the other labelled Reason (R). Read the statenrcnts
caref.rlly and choose the option that conectly descrbes statements (A) and
(R).
 G2@qj4hur

7M
AzECll 66
AmritMahotsa\

Assertion (4: q rs defrritely an odd nrrnber.

Reason E), 3'x 72 ts an odd nrunber'
(a) Both (A) and (R) are hue and @) is ttre correct explanation for (A).
tti gorh iAi anO ii) *"true but (R) is not the correct explanation for (A)'
(c) (A) is rue brt' R) is filse.
ts filse but s true.
P (-2,5 (2, are two on the plane. 1
1 )

Two staternents are given below - one labelled Assertion (A) *d the other
labelled Reason (R). Read the statenrents carefirlly and choose the option that
correctly describes staternents (A) and (R).
Assertion (A):T]he midpoint (0,2) is the only point equidistant from P and Q.

Reason (R): There are many poinrc (;r,y) where (x + 2)2 + (y - 5)2 = (x - 2)2
+
U+ t)2 are equidistant from P and Q.
(a) Both (A) and R) ate true and (R) is tlre conect explanation for (A).
lUj notn (A) *d (R) are true and @) is not the correct e>rplanation for (A).
(c) (A) is tue but (R) is frlse.
is frlse bd is true.

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

Serial
No. tion Marks
Check whether the staternent below is true or filse. 2
2I

'The square root of every conposite nurrjber is rational"

Justify your answer by proving rationality or inationality as applicable.

22 Kimaya and Heena started walking from the point P at the same moment n 2
opposite directions on a 800 m long circular path AS shown below. Kimaya
walked to the chb house at an average speed of 1 00 n/min and Heena
walked to the badminton court at an average speed of 80 fi/min The length
of the circular tack between the chrbhouse and the badminton cowt is 180
 G2@-rqg4 \1r

7W.
AZQCII 6s
AmntMahOtSat!

Kimaya, Heena

Club house

180 m Badminton court

(Note: Thefigure is not to scale.)

If Heena took 1 mirute more tban Kimaya to reach her destinatiorq frrd the
time taken by Heena to reach the badminton court. Show yor:r work.

23 Shown below is a circle wittr cenke. O and three tangents drawn at points A, 2
Eandc.AEis adiameter ofthe circle. The tangents intersect atpoints B
and D.

c
E D

Based on the above ffirmation, evafuate whether the following statement is

tue or filse. Jrstify your answer.

Atleast nne pair of opposite sides ofAEDB is parallel

24 Shown below is a riglt circular cone of volume 13,600 cnf . 2

 G2@q3tsrq^

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AmritMahotsa!

(Note: Thefigure is not to scale')

your
Find the angle which the slant heiglrt nrakes with the base radius' Show
work.

Q,{ote: Taken as 3,42 as 1'4 and43 as 1.7.)

OR
2
Shown below are two rigfot tiangles.

(Note: Thefigure is not to scale')

Find the length ofthe unknown side marked'?'. Show yow work'

25 ABCD is a rhombus with side 3 cm Two arcs are drawn from points A and 2
C respectively such that the radius equals the side of tlre rhombus. The figure
is shown below.
 G2@-qgr4L$h

7M
AZAOI 6a
AmfltMahotsa!

i3

(Note: Thefigure is not to scale.)

If BD is aline of syrnrntry for tlre figure, then find the area ofthe shaded
part of the figure in terms of z. Show your work.

OR

Wasim rnade amodel of Pac-Man, after playing the fimous video game ofthe 2
same rumrc. The area of the nndel is 120n cm2. Pac-Man--s mor-ilh forns an
angfe of 60" at the cente ofthe circle.

A pichre of the model is shown below

Qt[ote: Thefigure is not to scale.)

wasim wants to decorate the model by attaching a coloured nhhon to ttre

entire boundary of the shape. what is the minimum lengttr of the nbbon
in terms of ? Show work.
 G2@ rqg4riah

7qw
AZadl 63
AmritMAhotsa\

sEcTloN c
(This section comprises of short answertype questions (SAlof 3 marks each)

Serial
No. Question Marks
26 Prime frctorisation of tlnee numbers A, B and C is given below: J

A: (2, x fr x Ja)
B=@x3"x5p)
c=QqxJ4 x 9)such that, p<q<randp, q, & r arenatural numbers..

t The largest nurnber that divides A, B and C witlrorfr leaving a renrainder is

30.

0 The smallest mnnber tlnt leaves a rernainder of 2 when divided by each of

A, B and C is 5402,
Find A, B and C. Show your work.

27 Riddhi throws a stone in the air such that it follows a parabolic path before it J
lands at P on the grourd as depicted by the graph below

Hqisht (in units)

Distance

0 P

Q{ote: Thefigure is not to scale.)

 G2@.iQSEhur

76w
AZECII gg
AmritMahOtSar!

D The above gaph is represented by a potynomial where the surn of its

zeroes is 1 and the sum of the squares of its zeroes is 25. Find the
coordinates ofP and Q.
i| If one urit on the graph represents 25 metes, how fir tom Riddhi does
the stone land?

Show your work.

28 Given below is a pair of linear 3

2x - my:9
4x-ny:9
Find at least one pair of the possfule values of m and n, if exists, for which
*re above pair of linear equations has:

I a unique sohdion
i| infinitely many sohrtions
iii) no solution

Show your work.

OR
(6,0) and (0,2) are two ofthe points of intersections of trvo lines represented 3
by apair of linear equations.

I How many points of intersections does the pair of linear equations bave in
total? Justifi yow answer.

0 Find the equation that represents one of the lines of the above pair. show
your work.

29 In the given flgure, PQ is the diareter circle wittr cente O. Rls apoint 3
onthe bor.ndary of the circle, atwhich atangent is drawn Aline segnrent is
drawn parallel to PR ttnough o, such that it intersects the tangent at s.
 G2@-tr4gf terr

7W
Azadl g"
AmritMahotsa!

Show that SQ is a tangent to the circle.

OR
Shown below is a circle wift cenffe O. Tangents are drawn at points A and
3

C, such tbat they intersect at point B

If OA J. OC, then show that quadrilateral OABC is a square.

3
30 Shown below is a semicircle ofradius 1 rndt.

unit
Q'{ote: Thefigure is not to scale.)

Make necessary constructions and show that:

+^^ 0
Lctrr Z --
Slnd
T+Eo57
 G2@.8qg€(sr

75w
Azldl p
Amnt Mahotsa\

31 Nairm is playing a game and has two identical 6-sided dice. The frces of the 3
dice have 3 even numbers and 3 odd nwnbers.

She has to roll tlre two dice sinrultaneously and has two options to choose
from before rolling the dice. She wins aprun tf.

option 1: the sum of the two nunibers appearing onthe top of the two dice is
odd.
option 2: the product of the two numbers appearing on top of tlre two dice is
odd.

which option should Naima choose so that her chances ofwinning aprize is
higfrer? Show yow work.

SECTION D
(This section comprises of long answer-type questions (LA) of 5 marks each)

Serial
No. Question Marks
32 Manu and Atza are conpeting in a 60 km cycling race. Aiza's average speed 5
is 10km/hr greater than Manu's speed and she finished the race in
hours less tlnn Manu ^verage

Find the time taken by Manu to finish the race. Show your work.

OR
Shown below is a cuboid with water in two diffbrent orientatiors. The length, 5
breadth and heiglrt ofthe cuboid are distinct. The cuboid has 480 cm3 of water.

cm
l-

oriehJation r orientation ll
(Note; TheJigurgs are not to scale.)
 G2@-8O{54t$r

7&w.
AZidl g6
AmritMahotsa\

the water in tsi of L

the heigfrts ofwater in both orientations Show your work.

-tJ In the following figure, AABC is a rigfot-angled friangle, such tbat: 5

I AC:25 cm
i PT ll AB and SR ll BC

BTC
(Note: Thefigure is not to scale.)

Find the area of APQR. Show your work.

34 Two rectangular sheets of dinrensions 45 cm X i 55 cm are folded to make 5

hollow right circular cylindrical PFCS, such that there ls exactly cm of

i
overlap when sticking the ends of the sheet. Sheet 15 folded along its
.|
lengtll while Sheet ls folded along its width That ls, the top edge of the
sheet joined with iK bottom edge n both the sheets, AS depicted by the
arow m the below. Both plpes are closed on both ends to forrn
fuure
cylinders.

( 155 cm
45 cm

45 cm
,SheFt'l Slleet 2
Qr{ote: Thefigures are not to scale.)
 c2@Tqgqr\trr

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.ft?dlxa
AmntMahotsa!

em sufi.ce areas o two

D Find the ratio of the vohrmes oftlre two cylinders formed.

Show your work.

(Note: [Jse n as ']. Attr*" that the sheets have negligible thickness.)

OR

Shown below is a cylindrical canplaced in a cubical container 5

E
(J
a-

p/2on

i) How many of these cans can be packed in the container such that no nnre
cars are fitted?

i| If the capacity of one can is 539 mL find ttre internal volume of the
cr:bical container.

Show your work.

Taker
Q,{ote:
"t }.)
35 A car assemb$ r.urit assembles a limited nunber of cars daily, depending on 5
the prevailing demand. The following table presents an anal5nis of the
nurnber of csro asscriblcd by tlrc urit over tlree consecutive rnonths:
I
l

G2@ 'Iqg9.$r^

7M
AZAdI Ka
AmritMahotsatu

i) If tlre dernand ofthe cars is doubled, estimate how many cars on an

average should be assembled per day such tlat the increased demand is rnet?

if At least on how rnany days, less than average nurnber of caIS were
assenibled?

Show your work.

SECTION E

(This section comprises of 3 case-study/passage-based questions of 4 marks

each with two sub-questions. Firsttwo case study questions have three sub
questions of marks !,L,2 respectively. The third case study question has two
sub questions of 2 marks each,)

Serial
No. Marks

36 Answer the questions based on the given infonnation.

An interior desipper, Sana, hired two painters, Manan and Bhinrra to make
paintings for her buildings. Both painters were asked to nrake 50 diftrent
paintings each.

The prices quoted by both the painters are given below:

i Manan asked for Rs 6000 for the frst painting and an increment of Rs 200
for each following painting.
0 Bhima asked fol Rs 4000 for the first painting and an increrrent of Rs 400
for each following painting.

How much did Manan for his 25th Show your work. 1

How rnrch did Bhima in all? Show work. 1

 G2@:qsE.$ri

7W
AZEdI 63
Amnt Mahotsa!

(iif Ifboth Manan and Bhima make paintings at the same pace, frid the frst 2
painting for which Bhirna will get more money than Manan. show your
stePs'
oR
(o sana's friend, Aarti hired Manan and Bhima to make paintinp for her at 2
the same rates as for Sana. Aarti bad both painters rnake the same nurrjber of
paint'ngs, and paid them the exact same arnount in totaL

How rnany paintings did Aarti get each painter to make? Show your work.

37 Answer the questions based on the given information.

In the garne of archery a bow is used to shoot arrows at atarget board. The
player stands frr away from the board and airs the arrow so that it hits ttre
board.
one such board, which is divided into 4 concentric circular sectiors, is dravm
on a coordinate grid as shown- Each section carries different points as shown
in the figure. If an arrow lands on the boundary, the inner section points are
awarded.

: S Boints

I 10 p-qints

ffi 20 poinis
tr,30poinls

x
 G2@-llqgtsNr^

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Az?dl sg
Amrit.MahOtSar!

shooting two arrows, Rohan scored 25 points. 1

write one set of coordinates for each affow that landed on the
(0 one S AITOW onQ,2.5), nnny I
to the player? Show your work.

(ri!One ofRohan's arrow landed on(1.2, 1.6). He wants his second arrow to 2
Iand on the line joining fte origin and frst anow such that he gets l0 Points
for it.
Find one possible pair of coordinates of the second arrow's landing rnark'
Show your work.
OR
(xD An arrow landed on the boundary and is worth 20 points. The 2
coordinates of the landing mark were of the form (m, -m)

Find all such coordinates. Show your steps'

38 Answer the questions basbd on the given infonnation.

A drone, js an aircraft witlrout any hunran pilot and is controlled by a rernte-

confol device. Its various applications inchrde policing swveillance,
photography, precision agriculture, forest fire rrnnitoring river nronitoring
and so on

David rlsed an advanced drone with high resohrtion camera furing arl
expedition in a forest region which could ffy rpto 100 m heiglrt above the
ground level David rode on an open jeep to go deeper into the forest. The
initial position of drone with respect to the open jeep on which David was
riding is shown below.

?fl

s6
 G2@.aEJBNqr

76w,
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AmritMahotsa!

David's jeep rnoving to enter at an average speed of 10 n/s

He Sinrultaneousiy started fuing the drone in the same direction as that of the
jeep.

(D David reached near one of the tallest fuees n the forest. He stopped the
1
drone at a horizontal distance of sr/g m tom th" top of the tree and at
vertical distance of 65 m below its finxlmum vertical range.

Q{ote: Thefigure is not to scale.)

If the angle ofelevation of tlre drone from ftre top of the fee a
was J 00, find
the heightof the tee. Show your work.
(D The drone was fiying at a heieht of 3 rnehes at a constant speed n the 1
horizontal direction when it sp otted a zebra near a pond, right below the
drone.
The drone travelled for 3Ometes from there and it could see the znbra, attrte
same place, atanangfe ofdepression of dfrom it.

Draw a diagram to represent this situation and find d. Show you work.

0'D After 2 minutes of starting the expeditio n both the drone and the J eep 2
stopped at the same nnrnent so that the drone can caphne some mages The
position of the drone and the jeep when they stopped ls as shovm b elow.
I

G2@
7M
AZiCll 66
Amrit Mahotsa!

7{
i

E
€o
!r)

Q,{ote: Thefigure is not to scale.)

Find the average speedof the drone in rn/s rounded offrpto Zdecl.rlwl
places. Show your work.

OR

(D At some point during the e4pedition, David kept the drone stationary for 2
some time to capture the inrages of atiger . The angle of depression from the
drone to the tiger changed from 30o to 45 " in 3 seconds as shown below.

E
aa
-7
o

sH
Qtlote: Thefigure is not to scale.)

What was the average speed of the tiger dwing that time? Show your work.

(Note: Take',13 as L73.)

G2@qgqBur

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 Blueprint r Maths Standard, X (for March 20.24 board exams)

Section A Section B Section C Section D Section E

Total Marks
(l mark) (2marks) (3 morls) (5 morl<s) (4mdrls)

MCQ IBQ l8 marks

Assertion/ 2Q 2 marks
Reason

sQ
Very Short Answer (2 internalchoices)
lO marks

6Q
Short Answer 18 marks
(2 internal choices)

4Q
Long Answer 20 marks
(2 internalchoices)

3Q
case/ Data-based (l internal choice) 12 marks
per case-study
FN
o_ TOTAL 20Q sQ 6Q 4Q 3Q 80 marks
c
's
-t
Note:- The poper contains competency-bosed questions in the form of Stond-olone, Assertion/ Reoson, Toble-bosed ond Cctse-bosed queslions in oll
Seclions.

-r
 Series WYXZ11S SET-1
vq-{-T{ *ts
Q.P. Code 30t1t1
1t-n i.
Roll No.
qftm$ y*T-wr +ts q1 efl-qfuT *
ge-W q{ srq{q ftrri r

Candidates must write the Q.P. Code

on the title page of the answer-book-

+c NOTE
(I) FWT qiq s{ Et ts Fs n*r-qr ii 0 Please checle that this question
gRdyE12tr paper contains 12 printed pages.

(II) vw-qr it qTfri E-rq 61 3fr{ frq AD Q.P. Code giuen on the right hand
y*r-q:{ sts s} qfratr$ wr-5ft-c*t side of the question paper should
be written on the title page of the
* gq-yu qr fui r
answer-boolz by the candidate.
(III) EwT qiq s{ ti fu Fq sw-q{ i[ ss QII) Please checlz that this question
s{qtt poper contains 38 questions.

(I\D Fqqr yl+ sr wr frqqT {6 e-G t (N) Please write down the serial
q-C, enr-gfta-enr t[ y*r q-r ruumber of the question in. the
answer-book before attempting it.
3icRrr ful t

I
(vl {q s*r-wr s1 q-.dt * fr\ 15 ft M 15 minute time has been allotted
g
Te sT vrtir ftrn qqr t t yq{-q{ sT to read this question paper.
a
it ro.1b {S frqi qrq.n The question paper will be
Q
fr-dtq Wt6 r
distributed at 10.15 a.m. From
"<

E
10.1b t
qq 1o.Bo c$ a* g'm *-+o 10.15 a.rn. to 10.30 e.ffi., the
yw-q:{ q.b} 3i{ gu en+fu + *{q students will read the question
e

4
d

I iwr-gRr+r qr *t$ gtr qfr ftfi I

paper only and will not write
i any answer on the answer-book,
E
f during this period.

.rFrd (qrro)
MATHEMATICS (STAND A R,D)
ftqtRn{rnr r BqQ 3TfYf,dq sis': Bo
Times allowed: 3 hours Maximum Marh,s : B0

.301111 1 P.T.O.
Gendral Instructions:
(i) This Question paper contains - fiue sections A, B, C, D and E.
(ii) SectionAhas 18 MCQs and 02 Assertion-Reasonbased questions of 1 marlz
each.

(iii) Section, B has 5 Very Short Answer USA)-type questions of 2 marks each.

(iu) Section C has 6 Short Answer (SA)-type questions of 3 marks each.

(u) Section D has 4 Long Answer (LA)-type questions of 5 marks each.

(ui) Section E has 3 case based, intngrated, units of assessmerfi (4 marks each)
with sub parts of the ualues of 1, 1 and 2 marks each respectiuely.
(uii) AII questions are conxpulsory. Howeuer, an internal choice in, 2 questions of 5
marks, 2 Qs of 3 marles and 2 questions of 2 marks has been prouided. An
internal choice has been prouided in the 2 marks questions of Section E.

SECTION A

Question numbers I to 20 carry L mark each.

1. If two positive integers p and q ca:n be expressed as p = ab2 and q = a?b; a,
b being prime numbers, then LCM (p, q) is:
(a) ab (b) a2b2
a3b2 (d) asbg I
2. In graph, a polynomial P(r) is shown. The number of zeroes of P(r) is:
v
P(x)
A
x

v'

(a) 1 0o) 2
,-@r'e (d) 4 I
3 The value of k for which the system of linear equations r* 2y - 3, 5x + ky
* 7 = 0 is inconsistent, is:
*3
\-
(a)
t4
(b)
? J ld +
3 5
k= lD
(c) 5 10 I

.3O1111 2
 4. The roots of the equation x2 - 3x - m(m + 3) = O,where

d
nL rs a constant,
are:
rrl- .;pr-rl,r""nBm(a)
_, m * B m,m*3
(c) m, -(m + 3) (d)
-m, -(m + 3) I
o Two A.P.s have the same common difference. The first term of one of these
is -1 and that of the other is -8. Then the difference between their 4th
terms is: trd & I C'Td
-p "-l
6y+
-*--
(a) -1 (b) -8 | - r*7.,;f . --8"r
3rr" z (d) _g -t-t",x. ry I
,3
6. The perimeter of a triangle with verticeq and (3, 1S

(a) 5 units *M unl

(c) 1l- units (d) g+G)units I
7 Therelationbetweenxandysothatthepoint(x,y)isequidistantfromthepoints
(-4,-4) and (-2,4) is:
(a)2e)5
(c)3(d)8 1

8 ABCD is a trapezium with AD I I BC and AD = 4cm.If the diagonals AC

and BD intersect each other at O such that 49 = DO
oB
1
=;'thenBC=
oC

(a).
'"/ 6 cm &) 7 cm
JO 8cm (d) 9cm 1

9. At one end A of diameter AB of a circle of radius 5 cm, tangent XAY is

drawn to the circle. The length of the chord CD parallel to XY and at a
distance B cm from A is:
(a) 4 cm (b) 5 cm
(c) 6 cm (d) 8 cm. 1

10. The minute hand of a clock is 84 cm 1ong, The cl.istance covered by thc tip of
minute hand from 10:10 am to 10:25 am is :
(A) 44 cm 0o) 88 cm
,.W l-32 cm (d) L76 cm I
.3}l1l1 3 P.T.O.
 ?

cotvo
11. In the given figure, D is the mid-point of BC, then the value of a lSl
cotxo
rc A.c
Ett
e4\ \))
(\., cg) 79 {tz 78
t-./z: .>
c. cD) ri'i-1 "1,,Q
t \_ +r' /
-

(a) 2
1 & a\ (i,
-)
2 5-, \tQ

(c) T (d) T -;)

'2
r*\
3 4 I
L2. While eating sandwich, Chetna jokingly remarked that she can find out
the value of any trigonometric ratio if just one ratio is known to her, as
the sandwich is a right-angled triangle.

-Jsinh f, r X,'' r'

c-a$ +1 .'o)tt
% Sl'u"r |i .',\-. ll
3 sin A + 2 cos A
If 3 tan A= 4, then the value of rs: f et, -/l cDst
(a) 4
Ssin A-zcosA
(b) fi ,1

q
+{ r't;r ':
t?
\,a
15 a
(c) rt 3 I
--{
13. A 1.6 rn tall girl stands at a distance of 3.2 m from a lamppost and casts a
shadow of 4.8 m on the ground. Find the height of the lamppost. 1

L4. If the radii of two circles are in the ratio of4 : 3, then their areas are in the
ratio of:
(") 4:3 (b) 8 : 3
16:9
J9//perimeter .- (d) 9:16 I
15. The of a sector of a circle having radius r and angle 60o is:

(") '[;.') (b) 2r

;-')

(c) (d) 2y I+r 1

'[;.,) 6

.301111 4
 16. Ram Sewak is a wholesale dealer in eggs. He procures eggs directiy from
the poultry farms and sells them to the nearby stores

The probobiLitg of getting o bod egg in o lot of 400 is 0.035.The number of bod eggs in the lot is:
(a) 7 (b) 14
(c) 2L (d) 58 1

L7, From a pack of 52 playing cards jacks, queens, kings and aces of red
colour are removed. From the remaining, a card is drawn at random. Find
the probability that the card drawn is a red card.

(a) I 4
(b) T
t1
I

(c) (d)
I
7
22 I
18. For the fotl.owing distributions, the sum of the lower limits of the median and
modal class is: -!
r ti.

Closs FrequencA Cirt'l'. ,, .,, ,1,, :

I !,,'
0!5 10 (>"'-'
5-10 15 'lr lr
'/'v'; ' ,\ f ;..
i\

10-15 1.2
i...., I.;l ,",
1.),,, ,;1)i,*',,,,,,

1\,,1'j, "
15-20- 20 i':l :,
\ ,.. i
20-25 9
,

(a) 15 _(":ai ,

(c) 30 ;" (d) 35 I

DIRECTION: In the question number 19 and 20, a statement of assertion
(A) is followed by a statement of reason (R).
Choose the correct option.
(o) Roth assertion (A) and reasnn (R) are trrre and reason (R) ir the correct
explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the
correct explanation of assertion (A).

.301111 5 P.TO.
 (c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
L9. Assertion (A): If two identical solid cube of side 7 cm are joined end to end. Then
the total surface area of the resulting cuboid is 490 cm2'
Reason (R): Total surface area of cuboid = Ib + bh + hI I
t
20. v' v
Assertion ({): Let the positive numbers a,' b, r*r'' then
c be inA.P., "^^"^' +, , L
ac' ab'
are also in A.P. bc'
Reason (R): Ifeach term of the given A.P. is multiplied by abc, then the
resulting sequence is also in A.P. 1

SECTION B

Question numbers 21 to 25 caruy 2 marks each.

2L. Write the smallest number which is divisible by both 306 and 657. 2

22. In the figure, Dtr I I AC and DF I I AE. Prove that ## = #

A 2
D

B a
E

23. From a point P, two tangents PA and PB are drawn to a circle C(O, r). If
OP = 2r, then find IAPB. What type of triangle is APB?
2

24. In the given figure, D is the mid-point of BC, then the value of 99; is
cotxo

1 1 1
(a) 2 e)t (.)
E
(d)
;
.3o1111 6
 OR
If K + 1 = sec20(1 + sin0x1 - sinO), find the value of K. I

25. A 3.5 cm chord subtends an angle of 60" at the centre of a circle. What is
the arc length of the minor sector? Draw a rough figure and show your'
,.r.
stePs. (Note: Take TE as -)
t'
OR
In the radius 7cm each with vertices
gr.ven figure, arcs have been drawn of
A, B, C and D of quadrilateral ABCD as centres. Find the area of the shaded
region. t
.,/i
'r':1,1,/' D-
9t'
<
zl
c;l'- L
hr't::.' lt-i(t'l
)

C 1*-* 2

SECTION C

Question numbers 26 to 37 carcy 3 marh,s each.

26. During a sale, colour pencils were being sold in the pack of 24 each and
crayons in the pack of 32 each. If you want fu1l packs of both and the same
number of pencils and crayons, how many packets of each would you need to
buy? 3
27. Find a quadratic polynomial whose zeroes are reciprocals of the zeroes of
the polynomial f(x) = axz+ Sv * c, e. * 0, c + 0. 3
28. A train covered. a certain distance at a uniform speed. If the train
would have been 6 km/h faster, it would have taken 4 hours less than
the scheduled time. And, if the train were slower by 6 km/hr ; it would
have taken 6 hours more than the scheduled time. Find the length of the
journey.
OR
If we add 1 to the numerator and subtract 1 from the denominator, a
fraction red.uces to 1. It bect-rnr", l, if we only ad"d 1 to the denominator.
What is tSe fraction?
, y;.i.
2 3
29, Prove that a parallelogram circumscribing a circle is a rhombus,

.3o1111 7 P.T.O.
 OR
'In the figure XY and X'Y' are two parallel tangentsto a circle with centre O
and anofher tangent AB with poinl of contact C interesting XY at A and X'Y'
at B, what is the measure of /.AOB?
XPY

Prove the following that:

tans 0* cots 0 = seco cosec0 - 2 sino coso
l+tan20'1+cot20 3

31. The below table shows the ages of persons who visited a museum on a
certain day.
Find the median age of the person visiting the museum.
Age (Years) No. of persons
o
Less than 10 o

Less than 20 10
qq
L"ss than 30
Less ihart 40 40
Less than 50 54
Less than 60 7L
3

SECTION D

Question ruumbers 32 to 35 carcy 5 marks each.

32. To fill a swimming pool two pipes are used. If the pipe of larger diameter
used for 4 hours and the pipe of smaller diameter for t hours, only half of
the pool can be filled. Find, how long it would take for each pipe to filI the
pool separately, if the pipe of smaller diameter takes 10 hours more than
the pipe of larger diameter to fiIl the pool?
OR
A train covers a distance of 360 km at a uniform speed. Had the speed
been 5 kmflrour more, it would have taken 48 minutes less for the journey.
Find the original speed of the train. 5

.301111 I
33. In the given figure, I I I and line segments AB, CD and EF are
n1,
AC CE
concurrent at point P. Prove that S =
"' BF BD= FD'
Im

A
D
P
E F

5
34. There are two identical solid cubical boxes of side 7cm. From the top face
of the first cube a hemisphere of diameter equal to the side of the cube is
scooped out. This hemisphere is inverted and placed on the top of the second
cube's surface to form a dome. Find :
(A)the ratio of the total surface atea of the two new solid formed.
(B)volume of each new solid formed.
OR
Ramesh made a bird-bath for his garden in the shape of a cylinder with a
hemispherical depression at one end. The height of the cylinder is 1.45 m
and its radius is 30 cm. Find the total surface area of the bird-bath.
30 cm

1.45 m

35 The median of the following data is 16. Find the missing frequencies a and
b, if the total of the frequencies is 70.
Class 0-5 5-10 10-15 L5-20 20-25 25-30 30-35 35-40
Frequency T2 a T2 15 b 6 6 4
5

.301111 I P.T.O.
 SECTION E

Case Study-l
36. The school auditorium was to be constructed to accommodate at least 1500
people. The chairs are to be placed in concentric circular arrangement in
such a way that each succeeding circular row has l-0 seats more than the
previous one.

(A) If the first circular row has 30 seats, how many seats will be there
in the 10ih row?
(B) For 1500 seats in the auditorium, how many rows need to be there?
OR

If1500 seats are to be arranged in the auditorium, how many seats

are still left to be put after 10th row?
(C) If there were 17 rows in the auditorium, how many seats will be there
in the middle row? 4

Case Study-2
37. A tiling or tessellation of a flat surface is the covering of a plane using
one or more geometric shapes, called tiles, with no overlaps and no gaps.
Historically, tessellations were used in ancient Rome and in Islamic art.
You may find tessellation patterns on floors, walls, paintings etc. Shown
below is a tiled. floor in the archaeological Museum of Seville, made using
squares, triangles and hexagons.

.3o1111 10
 A craftsman thought of making a floor pattern after being inspired by the
above design. To ensure accuracy in his work, he made the pattern on the
Cartesian plane. He used regular octagons, squares and triangles for his
floor tessellation pattern
li:l

-!
-1'

Use the above figure to answer the questions that follow:

(A) What is the length of the line segment joining points B and tr'?
(B) The centre'Z' of the figure will be the point of intersection of the
diagonals of quadrilateral WXOP. Then what are the coordinates of
Z?
(C) What are the coordinates of the point on y axis equidistant from A
and G?
OR
What is the area of trapezLum AFGH? 4

.301111 11 P.T.O
 Case StudY-3
38. Lakshaman Jhula is located 5 kilometers north'east of the city of
tlRishikesh in the Indian state of Uttarakhand. The bridge connects the
villages of Tapovan to Jonk. Tapovan is in Tehri Garhwal district, on the
*u.t bunk of the river, while Jonk is in Pauri Garhwal district, on the east
bank. Lakshman Jhula is a pedestrian bridge also used by motorbikes. It
is a landmark of Rishikesh. A group of Class X students visited Rishikesh
in Uttarakhand on a trip. They observed from a point (P) on a river bridge
that the angles of depression of opposite banks of the river are 60o and 30o
respectivety. tfre height of the bridge is about 18 meters from the river.

Based on the above information answer the following questions'

(A)Find the distance PA.
(B)Find the distance PB.

(C)Find the width AB of the river.

OR
Find the height BQ if the angle of the elevation from P to Q be 30"' 4

.3o1111 12
 (a t au'\4 .j

-_U"

(Nl6--lJ.gfJId-rg-r uu3_o 6
KtNoBLl rNrrnNxnoNAL scHool
MODEL EXAMT NATTON (2022_20231

GRADE: X
MATHEMATICS
TIME ALLOTTED: 3 HOURS
MAX. MARKS: 80

General lnstructions:
1. This Question paper has 5 Sections
A, B, C, D, and E.
2. section A has 20 Murtipre choice euestions
(MCes) carrying 1 mark each.
3' section B has b short Answer-r (sA-r) type questions
carrying 2 marks each.
4' section c has 6 short Answer-il (sA-il) type questions
carrying 3 marks each.
5' section D has 4 Long Answer (LA) type questions
carrying 5 marks each.
6' section E has 3 case Based integrated units
of assessment (4 marks each) with sub-
parts of the varues of 1, 1 and 2 marks
each respectivery.
7 ' AII Questions are compulsory. However,
an internar choice in 2 es of 2 marks, 2
es of
3 marks and 2 Questions of 5 marks has
been provided. An internal choice has been
provided in the 2marks questions of Section
E.
=--: =-L =Di'aw-=ileatiigu-ie--s v-v:he.-Levs:- re=q-uireq'-Taite
a:22iv vvj:ei.q-qei- i-equireci if noJsi..€t _ -
Section A a1
$l no. Section A consists of 20 questions of 1 mark I ?
I
each. :l
The HCF of g6 and 404 is ... ... ";1 l

(a) 1
1

@) 2 (c) 3 ,,(d) 4
{ what is the greatest possible speed at which a
man can walk 52 kmand g1 km in an
1
exact number of hours?
(a) 17 km/hours (b) 7 km/hours
.,d
Jrf,3 tm)nours (d) 26 km/hours
lf one zero of the quadratic porynomiar xz
+ 3x* k is 2 then the varue of k is
(a) 10 ,.t6 -,0 (c) 5
(d) _5
d lf the system of equations 3x+y
=1 and 6x+ky=g is inconsistent, then k=
(a)-1 (b)o (c)1 gfz
1

Page I of8
 !d a
t ?,tn)
IE,t)

./ (n.'!l . (q,e)
/r.
ord er are P(3,4), Q(5, 4) and R(4,2);
then 1

/ lf the verti""J oti P"'"r lelogram PQRS taken in

the coordinates of its fourth vertex S are
\4 Q,r) (c) (2'-1) (d) none of these
1
lf AM and PN are altitudes of AABC
and APQR respectively
,{
^ABC-APQR.
and ABz , PQ' = 4 9, then AM:
PN =

(a) 3:2
"
,(b) 16:81
(c) 4:e g$ z'a
1

lf x tan60" cos 60o = sin60' cot 60o then x =

{{ '
(d) cot30'
(a) cos3o " #)tan30"
(c) sin30"

' lr\' :
o l/r. '1./ l
1
+
lf sinO + cos0 = {2, then tanO cot 0 =
.{
(a) 1 (c)
'&) 2
(d) none of these 3
BC = y units 1

ln the given figure, DE ll BC, AE = a

units, EC =b units, DE =x units and
\g
Which of the following is true?
s,

)r.'' \

rl\
{4 *=#
! dlC'! ;r.r

ta) *=;fa*b ,Ji{;=t 't-

A
,rl , Lt
t
a

4cm. lf the diagonals AC and B D intersect

10 D is a traPezium with AD tl BC and AD = Cg 'c.
DO/O B 1/2, then BC =
each other at O su ch that AO/OC =
--(tt6-in-- -@
7cm (c) Bcm (d) none of these
the
60o are drawn to a circle of radius
3cm' then 1

,y( lf two tangents inclined at an angle of

J.
length of each tangent is equalto
arfe (b) 3crn (c) 6cm ,{a6.lr**
ta) 2 crn
1

J,,{ TheareaofthecirclethatcanbeinscribedinaSquareofside6cmis
(c) 12 n t*t gn *t#
(a) 36n crf (b) l8n cm! Jai
of a square, then the ratio of thei r areas
ls 1

lf the perimeter of a circle is equal to that

/ :22
Jd) ZZ :t (b) 14 : (c)7 11 (d) none of these

of 4 3, then their areas are in the

ratio of : 1

lf the radii of two circles are in the ratio

"
(b).8:3 (d)e:16
(a)4:3 ,pfiato
'r .t

l\'i? " {t l-n.{ t

hF
Page 2 of 8
'i-l
ti
!r
1( ' r'-;. I
"" d .., ri

-t
e'
1,5
The total surface area of a solid hemisphere of radius 7 cm is 1

(a) 447n cm2 (b) 239n cm2 (c) 174n cmz gldi t+tn cmz

r For the following distribution:

Class 0-5 5-10 10-15 15 - 20 20 -25

1

Frequency 10 15 12 2s) I
the lower limit of the modal class is
a) 10 ,rsfis (c) 20 (d) 25

lf the mean of the following distribution is 2.6, then the value of y is 1

Variable (x) 1 2 3 4 5

Frequency 4 5 v 1 2

(a) 3 (b) B 13
*.(d) none of these
(c)
A card is selected at random from a well shuffled deck of 52 cards. The probability of its 1

being a red face card is

{ar* 1n)* (c)i (d);

Direction for questions 19 & 20: ln question numbers 19 and 20, a statement of
Assertion (A) is followed by a statement of Reason (R). Ghoose the correct option.
n
Assertion: The number 4 cannot end rvith the digit zero:, where n is a natural number- 1

o
Reason: The prime factorization of number 4 have only the prirne 2.

. \Zfro1f1'3ssgfio1r (A) a!]d tegson ([) are trge aq{, reason (R) is.the correct explalatlo4 9t

assertion (A)
(b) Both assertion (A) and reason (R) are true and reason (R) is not the correct
explanation of assertion (A) . r-.
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
e0 Assertion: C is themid-pointofPQ, ifP is (4,x),C is (y,- l) and Q is (- 2,4),thenxandy
1

respectively are -6 and l.

Reason , 'ih* mid-point of the line segment joining the points P(xr , 1l. ) and Q(xz , y:) is
lx1+x2 /r+Ie\
\--, 2 )
.(l eotnAssertion (A) and Reason (R) are true and Reason (R) is the correct explanation
of Assertion (A)
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct
Page 3 of 8
 .i

L'
:.b,'L
":
explanation of Assertion (A). ,t"
'i {
(c) Assertion (A) is true but Reason (R) is false. [*. r il,
\\ ""
(d) Assertion (A) is false but Reason (R) is true. ot:,'A'J '' r'l
c)

Section B fi
Section B consists of S questions of 2 marks each. "&
/ lf 49x+51y= 499, 51 x+49 y= 501, then find the value of x and y 2

tORI

I Find the 20th term from the last term of the AP 3,8, 1g,... .,252

ln Fig., if LM ll CB and LN ll CD, prove that # =# 2

B
lvl
A C

/ The length of a tangent from a point A at distance 5 cm from the centre of the circle is 4 2
cm. Find the radius of the circle.

/ The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand 2
in 5 minutes

TORI
trinr,l
r rrrv {hs
tttv
areq
qrgq a
vr u_ qqqdra.ltgla
nd ni.ol^
9I crrhAcA r.ir,*r:nr{,rran.-n(tl
I VIE, YY I I\JOIJ vlMl I ll9l l\r[
i.* ')4 a*r
lD l--a l)ltt

lf sin q= LlZ and cos B=1/2 , then the value of ( o + B) is 2

Section G
Section C consists of 6 questions of 3 marks each.
6
v Prove that
^h is an irrational number. 3
fr Find the zeroes of the quadratic polynomial 6x 2
- 7x - 3 and verify the relationship 3
between the zeroes and the coefficients.
,/ Five years hence, the age of Jacob will be three times that of his son. Five years ago, 3
Jacobis age was seven times that of his son. what are their present ages?

IORI
A fraction becomes 9111 , if 2 is added to both the numerator and the denominator. lf, 3 is
added to both the numerator and the denominator it becbmbs 5/6 . Find the fraction.

Page 4 of 8
I

3
A quadrilateralABCD is drawn to circumscribe a circle (see Fig'). il
Prove that AB + CD = AD + BC
T
A'l

d
t\
* tA,
rv
nq\
a/
i"[ -

\!w
I /, v,f,
r*4 q '/
?
fL
{'i

Prove that
co$ A +
l+sinA 2secA
l+sinA cos A

Prove that
1-cos0
(cosec 0 - cot 0)= :
I +cos 0
Two dice are thrown at the same time
numbers appearing on the top of the d
(i) 6?
(ii) 12?
U
(iii) less than 12?

Section D consists
'behfle'eh -:: - 5
eifiilGTialn-tilne-ed-t"-hourless'thaF a passenger train tii traVel 1'32 km
at
Mysore and Bangalore (without taking into consideration the time they stop
more than that
intermediate stations). lf the average speed of the express train is 1 1km/h
of the passenger train, find the average speed of the two trains. a'

IOR]
A motor boat whose speed is 1B km/h in still water takes t hour more to go 24 km
stream.
upstream than to return downstream to the same spot. Find the speed of the
'/ ..-^- -^ n^-:^ rr-^^^,+i^^^li*.r fhanram 5
V{g State and prove Basic Proportionality theorem'
/I4 Rachel, an engineering student, was asked to make a model shaped like a
cylinder with 5

of the
two cones attached at its two ends by using a thin aluminium sheet. The diameter
, model is 3 cm and its length is 12 cm. lf each cone has a height of 2 cm,
find the volume

Page 5 of 8
 of air contained in the model that Rachel made. (Assume the outer and inner dimensions
of the model to be nearly the same.)

tORI
A juice seller was serving his customers using glasses as shown in Fig. 13.13. The inner
diameter of the cylindrical glass was 5 cm, but the bottom of the glass had a
hemispherical raised portion which reduced the capacity of the glass. lf the height of a
glass was 10 cm, find the apparent capacity of the glass and its actual capacity.
(Use n = 3.14.)

3,5 A life insurance agent found the following data for distribution of ages of 100 policy 5
-.d
holders' Calculate the median age, if policies are given only to persons having age 1B
years onwards but less than 60 years.

Age (in years) Number of holders

Below 20 2
20-25 4
25-30
$u-3b
35'{0 33
4045 1'l
45-50 3
50-55 6
55-60 2
-- [co '

Page 6 of 8
 Section E
Case study based questions are compulsory'
Case StudY - 1

v36 The production of sets in a factory increases uniformly by a fixed

number every year' lt

produced 16000 sets in 6th year and 22600 in 9th year

2.76t;a
fr+ tC ' t6As9, - t(oilo
!; ( tia
{. q 1 .4,d,,', Jlt $n',u,
'
)
-t''.t'-6{go
a,l: , e.?oa "

questions
Based on the above information answer the following

-tl. Find the production during first year

1
Jt. Find the production during Bth year.

W'., Find the total Producton in B Years

1
loRl
2
Find-tHotaFpro'du'etioftin +eYealr
Gase Study -2
.bg9n,alloted a tu-Gta-xqlA!l$4-
Jhs=qnss4gqd_=95l_ _.s." [a schogl_-iq,--F"ajiqder $3se1hane, --. .

planted on the boundary at the

land for their gardening activity. sapling of Mango are
in the plot as shown in
distance of 1m from each other. There is a triangular grassy lawn
the figure.

s I rG o
.o
o
l} o
o

t' Ll

> t
oo
) otlo .\ D 0
0
0
o
t) x'
1 56 B et o ?4-
\
h

questions
Based on the above information answer the following
9' Page 7 of 8
to^\u

W
 \l{ What are the coordinates of p if A is origin?
tt/ Taking A as origin the coordinates of vertices of the triangle are
1
,il(. What are the coordinates of p if D is origin?
1

IORI
€eripneterof ApeR
2
3fi Case Study - 3
v
A truss is a structure that consists of members organized into connected
triangles so that
the overall assembly behaves as a single object. Trusses are most
commonly used in
bridges, roofs and towers. .

\l
HH H,i'i ,l , '
Ei?FitF,!i{'rrt,qr- r /4
';: lt

Consider the line diagram of truss shown below and find the following -.t
length I

vll
(Use r/r = 1.732)

'v't'

$;& '_t
n **+F- * *-*l** l0 --*l
o[1' What is the length a?
d.1". What is the length b ? 1

ilt *-l&hat=is the length c?^ , 31 1

F-! 'L"
IORI v 2

,.What is the length d? 6q

3h

! I .', i'r

I
i

I
Page 8 of 8
 ,f a",r

GRADE: X
IUIATHEMAfl
TIII'IE ALLOTTED
cs ts TAN DAR DI
3 HOURS
rnsrrucrions:
General
n{Axrtt'un{
'r';^/.rr'rUrv MARKs:g0,
j. 5 Se*ions A-E.
#i$Tffli:r'.*s carrying
3. s..ri;;;#J;:.y9Qs 1 mark each.
cattvinsoz
4. section
ffi ; ::::.'i""t carrving .urt, "u.r,.
;
s' s"tioni;;; :::::ons o: marks Ja'JlL
u. r"rri", fi* ,"llltlions.carryine
os nt*t.
e based intesrated
"Jri.
val ues
;i;;
I orlrlorr"n uni*
respectively. ----'^rw't (04 marks each) wrrh subparrs
^cas
"f each
7. AII
a;"rfi"#;i1*s
compulsory' How.uian of rhe
yarks ;o t a;;: intemal choi
.12 mar;#;Jlli*ffi;:;::1fril:"#"#,T::in 2 or5 marks,2 e, es or3

SECTION
:ffi::ff*.'
(Section A consists A
"rf . Ifp und q are the of 20 questions
zeroes of the quadratic of I mark each.)
3 polynomial
, (a) 2x2 - 5x + 3, then the
,'/2. The area of a quadrant
5
value ofl+ 1
"6) 3 (c) 3
(d) 7
p q
1s

of
':,!'r f-le , a circle, whose circumferen
ce is 22 cm,
"'(4S "m' is
flit t
'"
(b)
79
cm2
/"
1B
*!ei! L *o t )r{
lt
/t3 . The empirical
relationship between "m,
(d)
27
B
cm2
n
the three measures
(a)mode=median_2mean of central tendency .1rrY z -i
1S

G) mode=2median_3mean
,t<61 3 median =
mode *2 mean 'a
f '; Z
$ \{T L
/^. (d) median=mode*mean
,-W 77
For the foltowing
distribution
?w
*E u-f j -1i'
,a-l

Below
1o Below
-4 'z

o.ro 2o lb-y_n 30 Below

No. of Bo -:r,L.l 40 _.
;.1 j r a.t,;1 50
11
?0*E&
the model class
IS rf
,i q, L\
(a) 10 20
- (qto-_30 \r--
u6 qo
- s,o
(d) 50
-)60

1
 cos A sin C =
cot A ='J3'thenthen
/5.^AABC righ't angled at B' if
,::'^ (c)-1 (ofi/2
(a)1 Jol:l+ angle' then the
value of A is
is an acute
/U.rrsin 3A =l'#45o where A
to) 4s' (c) 3o'
(d) 1s'

6sro" 2x+ 4l+ L0 = 0 and -3x -

6y '21:0 have t,l t]
,/l.fn"pair of equations solutions >(:)
solution Yh'eo; r"
P,

(a) aunique rhr "or

"EA":3i# r i L a rr"+

**"'i"?
; :::J::::" "'^-6'''''u
"ffi .l;r
probabitity'f
"'*"
it" ticket has
lliH'LTHffit.1"'T
0
" "' a nrrnber wh
(d) 1
';t;cted (c) a/35
p and q are
@)7135 and b = x3v
t'yfle
*'otl::'":-:"^
j *"'' * :
positive integers
,.n;T';;ffit' t
,/g.wa and b be two then p
rt = ;,i;'
w*=Lrrnbers' "'i,*J' - rrt ,h;:* value of m is
in the form 65m
#;;;rtrt
/ro. .rrrE of
HcF nf 65.. anllll is e'-'
-::l' expressible
(c) 1 (d) 3
+*ionole.S €rro
"
j]vftlz tl:",the two triangles
z
a DE.
DEF,= zE,tF = zc andAB
(a) = 3
- and 1* ,u
,h.tntriangtes ABc 1|; rtt"tt*
rT:1,:ri"tJ:l#t*
(a) congruent but
not t*t11^,'^ (d; as we

congruent nor
simrlar "ottg*"nt
point P such
i"1 o"ith"r her at the

two line segments

Ac-Ti::r'li1':"H*l%;; = 3b"' rhen'
,/t .rn the below riBurl, cm' PD
: ) cl
that PA = ;"?);l-'' "^'Pc=2'5
is equal to
zPBA
p
A
I
q c21
c
B
.16 too"
(c) 60'
(b) 30"
(a) 50'
and
of AXYZ and AABC respectively
A]'{ are medians
rt If AXYZ -
XM
&\BC and XM anrl
tt
xYz L b
AB2 4'
then
1;
.f'i (c) 4:9
(d) 2'.3
(b) 1 6:8 1 2
Tt,z
 ( 14. The points A(0, 6), B(-5,
3) andc(3, 1) are the vertices of
a triangle which is
isoscg;es equilateral
(b) (c) scarene
"la) ciu"n'thatsin
rts.
(d)
' right
e angled,
0: |,thentan 0: .,'
nd " @# *)rrTF g1@
"tie' The base radii ofa cone and a cylinder are equai. Iftheir
e5trnal' then the ratio of curved surface areas are arso
the rr"t r,"igti;rri" ,i#lJ'in"
t7(a)2:r @)t:2 (c)1:3
height of the cylinder is:
(d)3:1
/i7.rn" area of acircre that canbe inscribed
in a square of side 10 cm is
(a) 40n cm2 @) lOn cml fc) 10bn cm, $2sn cm,
x2 + bx 'r 72:0 and the equation x2 + bx + q = 0 has equal
{;rli?#"
(a) 12
-;::1f"#uation
.16 rc @) 4 @) 20
r'
ttt' The value of v is 6, for which
*'Affilt i,ff"fion): the distance berween the points pe,1)
Statement R(Reason Distance
) between two grven points
A (xt,y) andB (x, ,y) isgiven
by AB (*. - *r)" *(v, -
v,)'
(a) Both assertion (A)
and reason (R) ar" true and
reason (R) is the correct expranation
assertion (A) of
(b) Both assertion (A) and
reason (R) are true and reason
(R) is not the correct explanation
assertion (A) of
(c) Assertion (A) is true
but reason (R) is false.
r{d) Assertion (A) is false but reason (R)
is true.

,&,0. St^t"ment A (Assertion) : If HCF ( g0,144): 1g, then LCM (90,144):720 i

Statement R(Reason) : HCF
,/
(a, b) x LCM (a, b) : axb :

Both Assertion (A) and Reason

ika) (R) ar" true and Reason (R)
is the correct expranation
Assertion (A). of
(b) Both Assertion (A) and
Reason (R) are true but
Reason (R) is not the correct
expranation of
Assertion (A).
(c) Assertion (A) is true
but Reason (R) is false.
(d) Assertion (A) is false
but Reason (R) is true"-.

3
 SECTIOI\ B
questions'of 2 marks each')
(Section B consists of 5
Find the length of Bc'
AABC is circumscribing a circle'
./zt. In the below figure,

t
4cm o t1 cm
t
'l h^ It
3 cm
t B
7
/zr. Solve for x and Y: lCI
ax+bY:a-b
bx-aY:3*b
and zPST = zPRQ' Prove
that P:* * an isosceles triangle'
In figure, H=H
0 ll
t
,i .'
,l', ', .lt'
Ir
i:.
1A
f' ,il
g,r \\Qf" t'
\.l

T *tr

R
a
v>

4+. frr
"otA
=
+, prove tnxffi: i *
OR r
F
o'frIfr , rcr ;,'a1,,3*",?* "
F,1

, ,, ti.;."ffi1'd'g';?i'Y:;il"d :,,I
4, y..#'"1"'ffi of*di"' *"uf;"t.$"H"#i" 14 cm
s
cente. Find the area of the
' r/ I
'a.::
'fi;q
xl
conesponding, qL Y jri: i.-tYr '. t.6rt
*
(i)minorsegment -Tb/.q-{}
""

(ii) major segment

OR
from ,t"t"
of a clockis 8 cm Findthe:;i'ffi:r"#T:,il{"-s 5?
x/-t.'
The lengthoftheminutehand
"9b*h,;*";#t,
8:30P'm. to 9i05 P'm'
sECrroN c '-
4
Al ct
(a,t r{} j t

L4 t')
.t
1'
,{6h
,- rWF"*'"
-*S*9, ,,.:"
;,) zzryL
.*
I/ C
./
1) I r.;.
IFJr
/ze.Findthe,","i,TlT#;,""H,fl .,i,t"m;:t:l1ffi ffi;thererationshipberween
the zeroes and the coefficients of the polynomial

,/Zl. Two different dice are tossed together. Find the probability of getting

(i) numbers appearing on two dice are odd.

(ii) the sum of numbers appearing on two dice is 12.
(iii) the product of numbers appearing on two dice is less than 9.

4s. Prove that rE is irrational. Hence, prove thatlE - 2 is also irrational.

{r. A fraction is such that if the numerator is multipliedby 2 and denominator is increased by 2,
we get But if numerator is increased by 1 and the denominator is doubled, we get j. find
I
the fraction.
vL-"& ) ru+.t
qx?-
, IL
OR y'n"'' t

Xtr o numbers are in the ratio 5:6. If 8 is subtracted from each of the numbers, they become
in the ratio of 4:5. Find the numbers.

A. As shown in the figure given below, two tangents TP and TQ are drawn to a circle with
centre O from an external point T. Prove that zPTQ:2 z OPQ. o
Arq+Arq= ?o"
P
/jiPQ
i* ATPQ,
rfu /asi
T /vT6, + A?Ql ziPq=
41q+ R ('J-i:'YO -- 8:{
a lfqa.+lgso - 7,0Pq' ms-
' /:'I Q'. -- z/StQ
OR -+=#*
pto,r" that opposite sides of a quadrilateral circumscribing a circle subtend supplementary
angles at the centre of the circle.

/31. Prove that

ttil-rl.,a"il- 2 cosec A
-secA-.1"iecA*1 =
SECTION D
Section D consists of 4 questions of 5 marks each.
The median of the following data is 868. If the total frequency is 100, find the values of x and y,
'/r.

5
 Clnss Frequency
800 * s20 7 (;.
s20 *t40 t4
s40 * 860 x
860 * 8S0 ?{
8S0 * 900 V
900 * 9?0 IO
920 * 9{0 5
,/
v/33. State and prove Basic Proportionality theorem.

By using the above theorem, find EC in the following figure if DE ll BC

T)E )lP,C 2cm 3cm

lb -flt
"r:fiVa-g 6cm
" [.tt A-1 "*i'-!.]*-

l,( '.,i:'' I
60. two water taps together can fill a tank in 6 hours. The tap of larger diameter takes t hours less
than the smaller one to filIthe tank separately. Find the time in which each tap can separately
frll the tank.
OR
A motorboat whose speed in still water is 18kmihr, takes t hour more to go 24km upstream
than to return downstream to the same spot. Find the speed of the stream.
{ SS.lfhe length, breadth and height of a cuboid are in the ratio 5: 4:2.If the total surface area of
cuboid is 1216 cmz, find the length, breadth and height of the cuboid. Also, find the voiume of
the cuboid.

OR
".\L
,- .Ip.: In toys manufacturing company, one specific wooden toy is in the shape of a cone mounted
a
on a cylinder with both their radii being equal to 8 cm. The total height of the toy is 26 cm and
the height of its conical part is 6 cm. If the cost of painting is 3 Rs per sq cm, then find the cost
of painting the toy. Also, find the volume of the wood used in making this toy. (Use n :3.I4)

/'
SECTION E
Section E cuusists of 3 case study-bascd qucstions of 4 marks each.
36. As per a report, in the month of April to June 2022, the exports of passenger cars from India
increased by 26% in the corresponding quarter of 2021-22. A car manufacturing company
'#,
planned to produce 2000 cars in 5th ftar and 3000 cars in 1Oth year, assuming that the
production increases uniformly by a fixed number gvery year. *t 11. o
-,t0- e. ,
A, A-+ 7C -
a-+ qd 3C,O b '" ,tooO
_.)
,,> : aol,;, ?|j,,n,y',,,,
6

''/ .l --. fi,o '')

t.\
 i

I
I
I

i
l

Based on the above information

answer the following questions.
(i) Find the production in the lst vear. --)128ic
(ii) Find the ;;;;;i;;;;; ;;ii;:; .--) \\00,^ _ D c !11
(iii) Find the total producrion in nrrt io y"*r..2 7.w"' (1)
(2\
(iii) h howmany
oR'
years w'l the totar productionreach 43200
carc? 1(, yeaax, Q)
37' Radio towers are used for transmitting a range
a
of communication services including
television' The tower will either act radio and
as an antenna itself or support
skucture' including microwave dishes. one or more antennas on its
They *" *ong the tallest human-made
There arc 2 maintypes: guyed and self-supporting skucfures.
,oo-.ror"r. on a similar concept, a radio
station tower was built in two sections
A and B. Tower is supported by wires from
Distance between the base of the tower apoint o.
and point o is 36 m. From point
elevation of the top of section B is 30o o, the angle of
and tie angle oielevation of the
top of section A is 45".

86 rn
On the basis of the above information,
answer the following questions:
(iI What is the height of the section B?
li,
(ii) What is the length of the wire from the point
lh'{z (1)
O to the top of section B?
(1)
t01't
t"
wl
7
 total len$h of the
(ii1) As shown in the figure, what is the
T?:11iil*tffifil1wetz(z)
Q)"
What is the height of the section A? \6, 7,1(g{v
sit in a row
Chetan are best friends childhood. TheY always want to
38. AbhaY, Bala and everY daY
But teacher doesn' t allow them and rotate the seats row-wise
in the classroom. day. He considers the
verY goo d inmaths andhe does calculation ofdistance every
Chetan is sYstem' One daY
their Position on a Paper in a co-ordinate
centre of class as origin and marks B and
of their seating Position marked AbhaY as A, Baia as
Chetan make the following diagram
Chetan as C.

AI t o

I IC N

t I t 0 t t l t
t
t

B
I it n

coordinate geometry'
the above information answer the folrowing questions using the
Based on (1)
and B?
itl What is the distance between A (1)
: 4 :3 ' what
points A and B such that AD:DB
' - n:*
ill, $:,:i1ff;"t"#',#.iliKeen
ut" the coordinates of Point D? Q)
OR

IfthepointP(k,0)dividesthe}inesegmentjoiningthepointsAQ,_2)andB(_7,"'aj*
rafio l:2, then find the value of
k'

If\/t!****: *{.***
*
*** * * {.* **tl.*****c**{<{'
* d< *. *c {. {. * * * x *+* * x * * d' * * * * * {' * {c tc *( d'
'!
(
I
..{:}.iJ"
.,r ii 'l p . ,,|,
v d 'I
i'] 8
(\
/ -:, rf) I! i
7 t, .r..--r/ 7 /
-t d
/vl,tl
I
1
'f
j !4

'1-'
 t COIMBATORE SAHODAYA A
SCHOOLS COMPLEX

+VFtf-na nAtrriarte
pRE - BOARD EXAMINATION (DEC - 2022)
GRADE: X MARKS:80

DATE : MATITEMATTCS(041) TIME :3 HRS

General Instructions:

1. This Question Paper has 5 Sections A-E.

2. Section A has 20 MCQs carrying 1 mark each
3. Section B has 5 questions carrying 02 marks each.
4. Section C has 6 questions carrying 03 marks each.
5. Section D has 4 questions carrying 05 marks each.
6. Section E has 3 case based integrated units of assessment (04 marks each) with subparts of.
the values of L, L and2 marks each respectively.
T. All Questions are compulsory. However, an internal choice in 2 Qs of 5 marks, 2 Qs of 3
-"rkr and 2 Questions of 2 marks has been provided. An internal choice has been provided
in the 2marks questions of Section E
8. Draw neat figures wherever required. Take n =2217 wherever required if not stated.

SECTION A

Section A consists of 20 questions of 1 mark each'

Al 1. ff two positive integers m and n are expressible in the form m = pq3 and n = p3q2
where p and q are prime numbers then HCF (m, n) =

(a) pq ,,6'iq' (.) p3q3 (d) p2q3

(tt Z.lf one of the zeroes of a quadratic polynomial of the form x2 * ax+ b is the negative
of the other then it

(a) has no linear term and the constant term is negative.

(b) has no linear term and the constant term is positive.

(c) can have a linear term but the constant term is negative'

term is positive
*(d) can have a linear term but the constant
i-;t
L-') S.The value of k for which the system of equations 2x + 3y = 5 and 4x + kV = 10 has
infinite number of solutions.

(a) 1 (b) 3 (d) 0

(.1 4. The largest number which divides 70 & 125 leaving remainders 5 & B respectively is

3 (b) 65 (c) 875 (d) 1750

( \ 5. lf a & b are the roots of the equation x2 * ax* b = 0 then a * b =
t-

(b) 2 (c)-2 (d)-1

6' The coordinates of the fourth vertex of the rectangle formed by the points O (0, 0),
A(2,0)&B(0,3)are
(a) (3, 0) (b) (0, 2) (c) (2, 3) (d) (3,2)

7. The perpendicular bisector of the line segment joining the points A(2, 3) & B(S, 6) cuts
the y axis at

(a) (8, 0) (b) (0, B) (c) (0, -B) (d) (0, 7)

8. lf AABC and ADEF are similar such that zA= 47" and lE = g3. then Zc =
(a) 50' (b)60' (c) 70' (d) 80.

9. ln the given figure ABCD is a trapezium such that BC ll AD & AD = 4cm. lf the
AO DO
diagonals AC and BD intersect at o such that - :1rn"n BC = -----
OCOB2

(a) 7cm (b) Scm (c) 9cm (d) 6cm

10. lf tan (A+ B) = tF Atan (A- B) =*O)B then the value of Ais

(a)30' (b) 45' (c) 60'

\r (d) 90"

11.lf tan245" - cos230" = x sin45"cos45. then x =

(b)-2 1 I
(a\ 2 (c) -, (d),
12. When the sun's elevation is 30" the shadow of a tower is 30m long, if the sun's
elevation is 60' then the length of the shadow is

(a) 35m (b) 20rn (u) 10nr (d) 1bm

13. The length of the major arc of the circle with radius 14 cm and chord length 14 cm is

(a) 14.7cm (b) 73.3cm (c) 146.Tcm (d) 216.3cm

 14' lf the radii of the bases of a cylinder
and a cone are in the ratio 3 : 4 andtheir
ratio 2: 3 then theiatio between ftrevotume heights
of the cytindei to that of the
SiiJ,jnt
(a)7:S (b)5:7 (c)s:e (d) 9:B
15' A right triangle with gides 3 cm,
4 cm & 5 cm is rotated qvvur
about the
trre ortrl
side of 3cm to form a
cone. The vorume of the cone so formed
is
(a) 12rr cm3 cm3
(b) 15n
16' The mean of n observation is . lf
(c) 16n cm3 (d) 20n cm3
-7 the first term is rrincreased
rvr vquvv vv
by 1I , D.'
and so on then the new mean is ' second term by 2 _, _
{ L]

(a)x*n @r++
\ut&,t : (.)r+!+7
(c/x+_r_ x*n*7
(d)_J_
17 'The empirical relationship between
mean, median and mode for a distribution is
(a) mode = 2 median - 3mean (b) mode = median -2 mean
(c) mode = 2 median - mean (d) mode = 3 media n -2mean
1B' A three digit number is chosen at random. The probability
that it is divisible by both
2and3is
(a) 1tB (b) 1/e (c) 1t6 @) 1t12
DIRECTTON: rn the question number
1g and 20, astatement of assertion (A)
followed by a statement of ReasonlR). ' - -'- is

Choose the correct option

" fi:i"#i:1tJn:ffrtion):
A tansent to a circle is perpendicutar
to the radius rhroush

statement R( Reason) : The length of tangents

drawn from an external point to a
circle are equal.

(a) Both assertion (A) and reason (R)

are true and reason (R) is the correct
explanation of assertion (A)
(b) Both assertio,n (A) and reason (R)
are true and reason (R) is not the correct
explanation of assertion (A)
(c) Assertion (A) is true but reason (R)
is false.
(d) Assertion (A) is false but reason (R)
is true.
20. statement A (Assertion): rf in aABC,
D & E are points on sides AB & AC

respectively such that DE BC then AD AE

ff
AB- AC
statement R( Reason) : rf a rine is drawn parailer
to one side of a triangre
intersecting the other two sides then it divides the two sides
in the same ratio
 reason (R) is the correct
(a) Both assertion (A) and reason (R) are true ind
exPlanation of assertion (A)
(R) is not the correct
(b) Both assertion (A) and reason (R) are true and reason
explanation of assertion (A)
(c) Assertion (A) is true but reason
(R) is false'

(d)Assertion (A) is false but reason (R) is true'

SECTION B

2 marks each'
Section B consists of 5 questions of
L'\ ++ 99v -= 5u1 the value of x and Y
find rrr
501 Tlno
21.|f 99x + 101y = 499 and 101x 99y
the value of x'
22.lnthe given figure if AB ll DC then find

AC at P, Q and
figure a circle is inscribed in a AABC touching sides AB, BC and
--
23. ln the
R r".p"1tiu"ly. lf AB = 10 cm' AR =
7 cm' and CR = 5 cm find the length
of the side

BC.
A

B a C

OR

perpendicular to the radius

prove that the tangents drawn at any point of a circle is
through the Point of contact'
required
35 cm How manY revolutions are
24.Thewheel of a motorcycle is of radius
to travel a distance of 11 m ?
OR
and the
area of the corresponding major sector of a circle of radius 28 cm
Find the
central angle being 45'

+
o+ o

25. Evaluate:
c 5
 SECTION C

Section C consists of 6 questions

of 3 marks each.
26. Prove that1fi is an irrational numbe r (- |

27 ' Find the zeros

of the quadratic polynomial sx2 +
Bx - 4.and verify the relationship
between the zeros and the co-efficl:ent
oiir,u'poryiromiar. (-L
2B. Solve forxand
" U
y' +Y,- - q
7 x-7 v+1
2 ' 3 -7' 3 -J--o
.

OR Gb
Determine graphicaily whether
the foilowing pair of rinear equations
2x + 5y -1 has a unique solution, 3x - y = 7 and
infinitely
=
29' ln the given figure two tangents
many solutions or no solution. - C-3
TP and rQ are drawn to -v s
a v"v'!e
circle with
an external point r. prove that vv,,r centre o from
zpTe = 2 lope
P

30. lf cos0 + sinO.=

\8, cosl,show that cosO _ sinO
= 1f) sinL
OR

Prove that :
sin d - cos d -, tid2 + cos d
'
sind*cosd sinpls;A =rmq
31' Three different coins are tossed
together. Find the probabirity of getting
i) at least one head ii) exactly two tails iii) atmost
two heads

SECTION D

Section D consists of 4 questions

of S marks each.
32' Atrain covers a distance of 480 km at a uniform
speed. lf the speed had been
,ffJy'ji5:;*Tilil:1i1,?:"" t;[;;; ffi;;;''"
to cover *re-same distance Find

(oR) ('-1
 then he
/c"
,'/ t4200 for his expenses' lf he extend his tour for 3 days'
( A person on tour has
Find the original duration of
the tour'
has to cut down his daily ;t?o
"*pJntu.
prove that a
prove the Basic Proportionality theorem' using this theorem
33. State and
of tre oiag;n"r. parallel to the base of the
line through the point of intersSciion "ti
trapeziumdividesthenon-parallelsidesintheSameratio.

34.Arighttrianglewhosesidesare20"Tunl]5:mismadetorevolveaboutthe
hypotenuse.rinothevolumeandsurfaceu,."ofthedoubleconesoformed.
(oR)
24cm is
pole consists of a cylinder of heigh l220cm and base diameter of the
A solid iron ano raoiur gcm. Find the mass
surmounted by another cyrinder or r,"iJr,iooim
Bgm mass ( use n = 3'14)
pipe given that 1.r! of iron rras uppro*i*utely

of 280 persons
35. The table shows the salaries

Salary (t in thousands) No of Persons

49
5-10
133
10-15
63
15-20
15
20-25
b
25-30
7
30-35
4
35-40
2
40-45
45-50 1

the data'
Calculate the median salary of
SECTION E

are Gompulsory'
Case study based questions

/' .q r..r-.-^-a job vou find that firm Awill start you
annnr.rr rnifias you
,^1. opportrrnities,
u 36. Satary : ln lnvestigating diffcront
you ,3i." oi Rt 1,200 each
year whereas
25,000 per year and guarahte" 3
at Rs
at n, p"tV"ut nui*iff guut"ni"" you a raise of only
firm B will start you Z6.OOO
Rs 800 each Year
 (i) Over a period of
1S years, h ow much
would you receive from
firm A?
OR
(ii) over a period of
15 years, how much
wourd you receive from
firm B?
(iii) what wourd be your
annuar sarary at firm
A for the tenth year?
(iv) what wourd be your
annuar sarary at firm
B for the tenth year?

37' The diagrams tl9,*

the plans for a sun room.
The four wats of the rt wilr be buirt onto
the wa, of a house.
";;;; *or"r" crear grass paners. The roof
"? is made using
'Four clear glass panels,
trapezium in shape,
all the same size
. One tinted glass
panel, half a regular
octagon in shape
V

*fr*fl6-

-'**-
lcm

ilot to $cala

Scale lcm=im x
Refer to top view

(i) Find the midpoint of

the segment joining the
points J(6, 1 7) andr(g, 16)
(ii) The distance of
the point p from y axis
is

OR
 the point A and S
(iii) Find the distance between B)
from the points Q(9, B) and S(17,
point (x, y) is equidistant
(iv) Find the relation if a
pretty hard
at any major construction site. They're
a common fixture out just as far.
38. ToWer Cranes are into the air, and can reach
hundreds of feet
to miss -' they often rise
other buirding materials'
Theconstructioncrewusestrretowercranetoliftsteel,concrete,largetoolslike
generutorr, uno a wide variety of
acetyrene torches and 24 m and
represent"o nv a tower AB' of height
rever grouno.'tt is
A crane stands on a plane about B' Avertical
't9.; and can rotate in a vertical
a iib BR. The jib is of
tengt! position of the iib' cable
and
S The diagram shows tu*"nt
cable, RS , carries a load '
load R

L6
8m

24m

BS ?
(i) What is the distance
?
angle that the jib' BR, makes with the horizontal
(ii) What is the
of the ZBRS ?
(iii) What is the measure
OR

increased?
by which RS has been
(iv) What is the length