Maths class x

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COURSE STRUCTURE
CLASS X
 As perCCE guidelines,the syllabusof MathematicsforclassX has beendividedtermwise.
 The unitsspecifiedforeachtermshall be assessedthroughbothformative andsummativeassessment.
 In eachterm,there shall be twoformative assessmentseachcarrying10% weightage.
 The summative assessmentinItermwill carry20% weightage andthe summative assessmentinthe IItermwill
carry 40% weightage.
 Listedlaboratoryactivitiesandprojectswill necessarilybe assessedthroughformativeassessments.
SUMMATIVE ASSESSMENT -1
FIRST TERM (SA I) MARKS: 80
UNITS MARKS
I NUMBER SYSTEM
Real Numbers
10
II ALGEBRA
Polynomials,pairof linearequationsintwovariables.
20
III GEOMETRY
Triangles
15
V TRIGONOMETRY
Introductiontotrigonometry,trigonometricidentity.
20
VII STATISTICS 15
TOTAL 80
SECONDTERM (SA II) MARKS: 80
UNITS MARKS
II ALGEBRA(contd)
Quadraticequations,arithmeticprogressions
20
III GEOMETRY(contd)
Circles,constructions
16
IV MENSURATION
AreasrelatedtoCircles,Surface Area&Volumes
20
V TRIGONOMETRY(Contd)
HeightsandDistances.
08
VI COORDINATEGEOMETRY 10
VIIPROBABILITY 06
TOTAL 80

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DETAILS OF THE CONCEPTS TO BE MASTERED BY EVERY CHILD OF CLASS X WITH
EXERCISE AND EXAMPLES OF NCERT TEXT BOOK
SA-I
SYMBOLS USED
*:-ImportantQuestions,**:- VeryimportantQuestions,***:- Veryvery importantQuestions
S.No TOPIC CONCEPTS DEGREE OF
IMPORTANCE
References(NCERT
BOOK)
01 Real Number
Euclid’sdivision
Lemma& Algorithm
*** Example -1,2,3,4
Ex:1.1 Q:1,2,4
Fundamental Theoremof
Arithmetic
*** Example -5,7,8
Ex:1.2 Q:4,5
RevisitingIrrational
Numbers
*** Example -9,10,11
Ex: 1.3 Q:1.2 Th:1.4
RevisitingRational
Numberandtheirdecimal
Expansion
** Ex -1.4
Q:1
02 Polynomials
Meaningof the zeroof
Polynomial
* Ex -2.1
Q:1
Relationship between
zeroesandcoefficientsof
a polynomial
** Example -2,3
Ex-2.2
Q:1
Forminga quadratic
polynomial
** Ex -2.2
Q:2
Divisionalgorithmfora
polynomial
* Ex -2.3
Q:1,2
Findingthe zeroesof a
polynomial
*** Example:9
Ex -2.3 Q:1,2,3,4,5
Ex-2.4,3,4,5
03 LinearEquation
intwo variables
Graphical and algebraic
representation
* Example:2,3
Ex -3.4 Q:1,3
Consistencyof pairof
linearequations
** Ex -3.2
Q:2,4
Graphical methodof
solution
*** Example:4,5
Ex -3.2 Q:7
Algebraicmethodsof
solution
a. Substitution
method
b. Elimination
method
c. Cross
multiplication
method
d. Equation
reducible topair
of linearequation
intwo variables
**
Ex -3.3 Q:1,3
Example-13Ex:3.4
Q:1,2
Example-15,16Ex:3.5
Q:1,2,4
Example-19Ex-3.6
Q :1(ii),(viii),2(ii),(iii)

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04 TRIANGLES
1) Similarityof
Triangles
*** Theo:6.1
Example:1,2,3
Ex:6.2 Q:2,4,6,9,10
2) Criteriafor
Similarityof
Triangles
** Example:6,7
Ex:6.3
Q:4,5,6,10,13,16
3) Areaof Similar
Triangles
*** Example:9The:6.6
Ex:6.4 Q:3,5,6,7
4) Pythagoras
Theorem
*** Theo:6.8 & 6.9
Example:10,12,14,
Ex:6.5
Q:4,5,6,7,13,14,15,16
05 Introductionto
Trigonometry
1) Trigonometric
Ratios
* Ex:8.1 Q:1,2,3,6,8,10
2) Trigonometric
ratiosof some
specificangles
** Example:10,11
Ex:8.2 Q:1,3
3) Trigonometric
ratiosof
complementary
angles
** Example:14,15
Ex:8.3 Q:2,3,4,6
4) Trigonometric
Identities
*** Ex:8.4 Q:5 (iii,v,viii)
06 STATISTICS
CONCEPT1Mean of
groupeddata
1. DirectMethod *** Example:2
Ex:14.1 Q:1&3
2. AssumedMean
Method
* Ex:14.1 Q:6
3. StepDeviation
Method
* Ex:14.1 Q:9
CONCEPT2
Mode of groupeddata *** Example:5
Ex:14.2 Q:1,5
CONCEPT3
Medianof groupeddata *** Example:7,8
Ex:14.3 Q1,3,5
CONCEPT4
Graphical representation
of c.f.(ogive)
** Example:9
Ex:14.4 Q:1,2,3

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REAL NUMBER
IMPORTANT CONCEPTS
TAKE A LOOK
1. Euclid’sDivisionlemma or
Euclid’sDivisionAlgorithm:-
For any twogivenPositiveintegersaandb there exitsunique wholenumberqandr such that.
a=bq +r, where 0 r<b
Here,we call a as dividend,basdivisorq,as quotientandr as remainder.
Dividend=(divisorx quotient)+Remainder.
2. Algorithm:- Algorithmisaseriesof well definedstepswhichgivesprocedureforsolvingatype of problem.
3. Lemma:- A lemmaisa provenstatementwhichisusedtoprove anotherstatement.
4. Euclid’sdivisionAlgorithm:- It isa technique tocompute (Calculate orfind) the Highestcommonfactor(HCF) of
twogivenpositive integers.
5. Fundamental theoremof Arithmetic:-
Everycomposite numbercanbe expressedasaproduct of primes,andtheirfactorizationisunique,
apart fromthe orderinwhichthe prime factor occur. For example 98280 can be factorizedasfollows:-
98280
2 49140
2 24570
2 12285
3 4095
3 1365
3 455
5 91
7 13
Therefore 98280=23
x33
x5x7x13as a productof powerof primes.

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6. Rational Numberand theirDecimal Expansion:-
i. If a=p/q, where pand q are co- prime and q=2n
x5m
(n and m whole numbers) then the rational number
has terminating decimal expansion.
ii. If a=p/q where p and q are co- prime and q cannot be written as 2n
x5m
(n and m whole number) then
the decimal expansion of a has non-terminating repeating decimal expansion.
LEVEL-I
1. Euclid’sDivisionlemmastatesthatforany twopositive integersaandb, there exist- unique integersqandr such that
a=bq+r, where r mustsatisfy. Ans-0≤r<b.
2. Express10010 and 140 as prime factors. Ans-10010=2x5x7x11x13
140=2x2x5x7
3. If p/qis a rational number(q≠0),whatis the conditionof qso that the decimal representationof p/qisterminating?
Ans-Qshouldbe inthe formof
2n
x5m
where nand m are +ve integer.
4. Write anyone rational numberbetween√2 and√3. Ans-1.5321
5. Findthe [HCFx LCM] for 105 and 120. Ans-12600
6. If twonumbersare 26 and91 andtheirH.C.F is13 thenLCM is. Ans-182
7. The decimal expansionof the rational number33/22
.5 will terminate. Ans-Twodecimal places
8. HCF of twoconsecutive integersx andx+1 is. Ans-1
LEVEL-II
Q1. Use Eucliddivisionalgorithmtofindthe HCFof:-
i) 196 and 38220 ii)867and 225 Ans:-i)196ii)51
Q2. Findthe greatestcommonfactor of 2730 and9350. Ans-10
Q3. IF HCF (90,144)=18 findLCM (90,144) Ans-720
Q4.Is 7x5x3x2+3 isa composite number?Justifyyouranswer. Ans-213
Q5. Write 98 as productof itsprime factors. Ans-2x7x7
Q6. Findthe largestnumberwhichdivides245and 1245 leavingremainder5in eachcase.Ans-40
Q7. There isa circular patharound a sportsfield.Geetatakes20 minutestodrive one roundof the field.While Ravi
takes14 minutesforthe same.Suppose theybothstartat the same pointandat the same time,andgo inthe same
direction.Afterhowmanyminuteswilltheymeetagain atstaringpoint?
Ans-42 minutes

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Level-III
1. UsingEuclid’sdivisionlemma,show thatthe cube of anypositive integerisof the form9q, 9q+1 or 9q+8 for
some integer‘q’.
2. Showthat one and onlyone outof n,n+2,n+4 isdivisibleby3.
3. Prove that 5+√3 is an irrational no.
4. UsingEuclid’salgorithmfindthe HCFof the following4052 and 12576
Ans-4
5. Showthat the square of anyodd integerisof the form4q+1 for some integerq.
6. Prove that √5+√3 isirrational.
Self evaluation questions
1. Draw the factor tree for 678.
2. The sum of twonumbersis1660 and HCF is20 findthe numbers.
3. Prove that √7 isirrational number.
4. Prove that is irrational number.
5. Write indecimal formand commentondecimal expansion.
6. If be an irrational numberprove that + isirrational.
7. Showthat any positive oddintegerwill be of the form4q+1 or 4q+3 where qis some integer.

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POLYNOMIALS
IMPORTANT CONCEPTS
TAKE A LOOK
1. POLYNOMIALS:Anexpressionof the form xn
+ xn-1
+ xn-2
+…….+ x+ where
………. are real numbersandn is whole number,iscalledaPolynomialinthe variable x.
2. Zeroes of a Polynomial,kissaidtobe zeroof a Polynomial p(x) if p(k) =0
3. Graph of Polynomial:
i. Graph of a linearpolynomial ax+bisstraightline.
ii. Graph of a quadraticpolynomial p(x)= isaparabolaopenupwardslike U if a>o.
iii. Graph of a quadraticpolynomial p(x)= isaparabolaopendownwardslike if a<0.
Discriminantof aquadratic polynomial.
For p(x)= , isknownas itsdiscriminant‘D’.
D=
i. If D>o graph of p(x) = will intersectthe x-axisattwodistinctpoints,x- coordinatesof
pointsof intersectionwithx-axisisknownas‘zeroes’of p(x).
ii. If D=o, graphof p(x) = will touchthe x-axisatone pointonly.
p(x) will have onlyone ‘zero’
iii. If D<o , graph of p(x) = will neithertouchnorintersectthe x-axis.
P(x) will nothave anyreal zeroes.
4. Relationshipbetweenthe zeroesandthe coefficientof apolynomial :
i. If are zeroesof p(x)= ,thensumof zeroes= = -b/a=
Productof zeroes= =c/a =
ii. If and are zeroesof p(x)= +cx +d
Then, sum of zeroes= + =-b/a =
+ =c/a = , Productof zeroes=-d/a= =
iii. If are zeroesof quadraticpolynomialp(x) ,thenp(x) =X2
–( )x +
iv. If and are zeroesof a cubicpolynomial p(x),then
P(x) = x3
- ( + )x2
+( x-( )
5. DivisionAlgorithmforpolynomials.
If f(x) andg(x) are anytwo polynomialswithg(x)≠0thenwe canfindpolynomialsq(x) and suchthat
f(x) = q( )x g( )+ where =0 or degree of < degree of g
If the remainderisoor degree of remainderislessthandivisor,thenwe can not continue the divisionany
further.

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LEVEL-I
1. The zeroesof the polynomial2x2
-3x-2are
a. 1,2
b. -1/2,1
c. ½,-2
d. -1/2,2 [Ans- (d)]
2. If are zeroesof the polynomial2x2
+7x-3,thenthe value of 2
+ 2
is
a. 49/4
b. 37/4
c. 61/4
d. 61/2 [Ans-( c) ]
3. If the polynomial6x3
+16x2
+px -5is exactlydivisible by3x+5, thenthe value of p is
a. -7
b. -5
c. 5
d. 7 [Ans- (d)]
4. If 2 is a zeroof boththe polynomials3x2
+ax-14and2x3
+bx2
+x-2,thenthe value of 2-2bis
a. -1
b. 5
c. 9
d. -9
[Ans-( c) ]
5. A quadraticpolynomial whose productandsumof zeroesare 1/3 and 2 respectivelyis
(a) 3x2
– x +32 (b) 3x2
+ x - 32 (c) 3x2
+ 32x +1 (d) 3x2
– 32x +1 Ans:(d)
LEVEL-II
1. If 1 is a zeroof the polynomial p(x) =ax2
-3(a-1)x -1,thenfindthe value of a.
Ans:a=1
2. For whatvalue of k, (-4) iszero of the polynomialx2
–x – (2k+2)?
Ans:k=9
3. Write a quadratic polynomial,the sumandproductof whose zeroesare 3 and -2.
Ans:x2
-3x-2
4. Findthe zeroesof the quadratic polynomial 2x2
-9-3x andverifythe relationshipbetweenthe zeroesandthe
coefficients.
Ans:3, -3/2
5. Write the polynomial whosezeroesare 2+3 and 2 - 3.
Ans:p(x)=x2
-4x+1
LEVEL – III
1. Findall the zeroesof the polynomial 2x3
+x2
-6x-3,if twoof its zeroesare -3and 3.
Ans:x=-1/2
2. If the polynomialx4
+2x3
+8x2
+12x+18is dividedbyanotherpolynomial x2
+5,the remaindercomesouttobe
px+q.Findthe value of p and q.
Ans:p=2,q=3
+8x3
+17x2
+21x+7 isdividedbyanotherpolynomial3x2
+4x+1,the remaindercomesoutto
be (ax+b),finda andb.
Ans:a=1, b=2
4. If two zeroesof the polynomial f(x)=x3
-4x2
-3x+12are 3 and-3,thenfinditsthirdzero.
Ans:4
5. If ,  are zeroesof the polynomial x2
-2x-15thenformaquadraticpolynomial whosezeroesare (2) and(2).
Ans:x2
-4x-60

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LEVEL – IV
1. Findotherzeroesof the polynomial p(x)=2x4
+7x3
-19x2
-14x +30 if twoof itszeroesare 2 and -2.
Ans: 3/2 and -5
2. Divide 30x4
+11x3
-82x2
-12x-48 by (3x2
+2x-4) and verifythe resultbydivisionalgorithm.
+8x3
-5x2
+ax+bisexactlydivisible bythe polynomial2x2
-5,thenfindthe value of aandb.
Ans:a=-20,b=-25
4. Obtainall otherzeroesof 3x4
-15x3
+13x2
+25x-30, if two of its zeroesare and - .
Ans: , , 2, 3
5. If ,  are zeroesof the quadraticpolynomial p(x)=kx2
+4x+4suchthat 2
+2
=24, findthe value of k.
Ans:k= or k= -1
SELF EVALUATION
1. If ,  are zeroesof the quadraticpolynomial ax2
+bx+cthenfind (a) (b) 2
+2
+bx+cthenfindthe value of 2
- 2
+bx+cthenfind the value of 3
+3
4. What mustbe addedto 6x5
+5x4
+11x3
-3x2
+x+5so that itmay be exactlydivisibleby3x2
-2x+4?
5. If the square of differenceof the zeroesof the quadraticpolynomial f(x)=x2
+px+45isequal to144, findthe value
of p.
---xxx---

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PAIR OF LINEAR EQUATIONS IN TWO VARIABLES
IMPORTANT CONCEPTS
TAKE A LOOK:
1. EQUATION:The statementof anequalityiscalledanequation.
2. Linearequationinone variable:Anequationof the formax+b=0,where a,bare real numbers,(a≠0) iscalleda
linearequationinone variable.
3. Linearequationintwovariables:Anequationof the formax+by+c=0, where a,b,care real
numbers(a≠0,b≠0)iscalledalinearequationintwovariablesx andy.
4. Consistentsystemof linearequations:A systemof twolinear equationsintwounknownsissaidtobe consistent
if it has at leastone solution.
5. Inconsistentsystemof linearequations:if asystemhasno solution,thenitiscalledinconsistent.
The systemof a pairof linearequations
x+ y+ =0
x + y+ =0
(i) has no solution.
If / = / ≠ /
(ii) hasan infinitenumberof solutions
If / = / = /
(iii) hasexactlyone solution.
If / ≠ /
6. Algebraicmethods:
(i)Methodof substitution
(ii)Methodof eliminationbyaddition orsubtraction
(iii)Methodof crossmultiplication
x+ y+ =0
x y+ =0
X= - / - ,y= - / -
b1 c1 a1 b1
x y 1
b2 c2 a2 b2
LEVEL- 1
1. The pair 2x=3y-5 and 2y= 5x-4 of linearequationsrepresentstwolineswhichare
(a) Parallel (b) coincident(c) intersecting(d) eitherparallel orcoincident [Ans(c)]
2. The pair x=pand y=q of the linearequationsintwovariablesx andy graphicallyrepresentstwo lines
whichare
(a) Parallel (b) coincident(c) intersectingat(p,q) (d) intersectingat(q,p) [Ans(c)]
3. If the linesrepresentedbythe pairof linearequations2x+5y=3 and(k+1)x +2(k+2)y=2k are coincident,then
the value of k is
(a) -3 (b) 3 (c) 1(d) -2 [Ans(b)]
4. If the pairof linearequations(3k+1)x+3y-2=0and (k2
+1 )x+(k-2)y-5=0inconsistent,then
The value of k is
(a) 1 (b) -1 (c) 2 (d)-2 [Ans(b)]

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5. If the pair of linearequations2x+3y=11 and2px+(p+q)y=p+5qhas infinitelymanysolution
Then (a) p=2q (b)q=2p(c)p=-2q(d) q= -2p [Ans(b)]
LEVEL- II
1. Findthe value of k for whichthe givensystemof equationshasunique solution:
2x+3y-5=0, kx-6y-8=0 [Ansk≠ -4]
2. For whatvalue of k will the followingsystemof linearequationshave infinite numberof solution.
2x+3y-5=2; (k+2)x+(2k+1)y=2(k-1) [Ansk=4]
3. Findtwonumberswhose sumis18 and difference is6. [Ans12,6]
4. Solve forx and y.
X+6/y=6, 3x-8/y=5. [Ansx=3, y=2]
5. The sum of the numeratorandthe denominatorof a fractionis20 if we subtract 5 from the numerator
and 5 fromdenominator,thenthe ratioof the numeratorandthe denominatorwillbe 1:4 .Findthe fraction.
[ Ans: 7/13]
LEVEL- III
1 Solve the followingsystemof equationsbyusingthe methodof eliminationbyequatingthe coefficients:
x/10+y/5+1=15, x/8+y/6=15. [Ansx=80, y=30]
2. If twodigitnumberisfourtimes the sum of its digitsandtwice the productof digits.Findthe number.
[Ans36]
3.Solve the followingsystemof equations.
bx/a- ay/b+a +b=0 [Ansx=-3a, y= -b]
bx –ay +2ab=0
4. Solve graphicallythe systemof linearequations.
4x-3y+4=0, 4x +3y=20 alsofindthe area of the regionboundedbythe lines andx-axis.
[Ansx=2,y=4, Area=12 sq. unit]
5.The sumof twonaturalsnumberis8 and sumof theirreciprocalsis8/15. Findthe numbers
[Ans5 and3]
LEVEL- IV
1. Solve forx and y:
2/2x+y – 1/x-2y+5/9 =0
9/2x+y – 6/x-2y+4 =0 [Ansx=2 , y=1/2]
2. Draw the graph the followingequations:
2x+3y-12=0 and7x-3y-15=0.Determine the coordinatesof the verticesof the triangle formedbythe linesand
the y-axis. [Ans(0,4),(3,2),(0,-5)]
3. The sum of the digitsof a two- digitnumberis12 the numberobtainedbyinterchangingthe two
digitsexceedthe givennumberby18. Findthe number. [Ans57]
4.Abdul traveled300kmby trainand 200km by taxi,ittookhim5 hours 30 minutes.Butif he travels260 kmby train
and 240 km by bushe takes6 minuteslonger.Findthe speedof the trainandof the taxi.
[Ans100 km/hr.,80km/hr.]
5.Solve the followingpairsof equationforx andy.
15/x-y+22/x+y=5, 40/x-y+55/x+y =13. [Ansx=8, y=3]

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SELF EVALUATION
1. Findthe value of ‘p’if(-3,p) lieson7x+2y=14.
2. Solve the followingsystemof linearequationsusingthe methodof cross-multiplication:
ax +by =1
bx +ay = (a+b)2
/a2
+b2
=1
3. Solve forx and y.
bx +ay = a+b.
ax[1/a-b-1/a+b]+by [1/b-a-1/b+a]=2

13
TRIANGLES
IMPORTANT CONCEPTS
TAKE A LOOK
1. SimilarTriangles:- Two trianglesare saidto be similar,if (a) Theircorrespondinganglesare equal and(b) Their
correspondingsidesare inproportion(orare inthe same ratio).
2. Basic proportionalityTheorem[ orThalestheorem forsolutionreferNCERTTextBook].
3. Converse of BasicproportionalityTheorem.
4. Criteriaforsimilarityof Triangles.
(a) AA or AAA similaritycriterion.
(b) S.A.Ssimilaritycriterion.
(c) S.S.Ssimilaritycriterion.
5. Areasof similartriangles.
6. Pythagorastheorem.
7. Converse of Pythagorastheorem.
Level I
1. If ∆ ABC issimilarto∆ DEF B=600
and c=500
, thendegree measure of D.
Ans-700
2. In Fig-(1) if DE||BCfindthe value of x. A
D E
B C
Fig(i) Ans-10cm
3. In the givenfig-(ii)PQ24 cm, QR=26 cm , PAR=900
, PA=6cm andAR=8cm findthe value of QPR.
Ans-QPR=900
4. In givenfig-(iii) ∆ABCand∆ DEF are similar,BC=3cm, EF=4cm, and areaof triangle ABC=54cm2
findthe area of
∆ DEF.
Ans-96 sq.cm
x cm
4 cm
3 cm
2 cm

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5. If the area of two similartrianglesare inthe ratio16:25 thenthe ratio of theircorrespondingsidesis.
Ans-4:5
6. If ar(∆ABC):ar(∆DEF)=25:81 thenAB:DE is.
Ans-5:9
7. A righttriangle hashypotenuse Pcmand one side qcm. If p-q=1,Findthe lengthof the thirdsides.
Ans-√2p-1
LEVEL- II
1. In givenfigure(i), PQ||SRandPO:RO=QO:SOfindthe value of x.
Ans : 7
2. In the givenfig-(ii)expressx intermsof a, b andc.
Ans: x =
3. In fig(iii) ∆PQRisa triangle rightangledatPand M is pointonQR such that PM  QR. Show that
PM2
= QMx MR.
4. The diagonalsof a quadrilateralsintersecteachotheratthe point0 suchthat AO/OC=BO/DOshow thatABCD is
a trapezium.
5. A Man goes10 m due eastand then30m due north.Findthe distance fromthe startingpoints.
Ans-31.62m
6. Two polesof height6mand 11m standon a plane ground.If the distance betweentheirfeetis12mfindthe
distance betweentheirtops. Ans-13cm
7. Prove that the line joiningthe midpointsof anytwosidesof a triangle isparallel tothe thirdsides.

15
8. ABC isan isoscelestriangle angledatB. Twoequilateral trianglesare constructedonside BCand AC inFig-(iv),
prove that areaof ∆BCD=1/2 area of ACE.
LEVEL-III
1. In fig-(i) BDBC, DE  AB and ACBC, prove that = .
2. D is a point on the side of ∆ ABC such that ADC = BAC prove that = or CA2
=BC.CD.
3. If the areas of two similar triangles are equal then the triangles are congruent.
4. The areas of two similar triangles ∆ABC and ∆PQR are 25 cm2
and 49 cm2
respectively. If QR=9.8 cm find BC.
Ans-7cm
5. Prove that the area of an equilateral triangle described on one side of a square is equal to half the area of an
equilateral triangle described on one of it diagonals.
6. BL and CM are medians of a ∆ABC, right angled at A. Prove that 4(BL2
+CM2
)=5BC2
7. In an equilateral triangleprove thatthree timesthe square of one side is equal to four times the square of one
of its altitudes.
8. In fig(ii) A triangle ABC is right angled at B side BC is trisected at point D and E prove that
8AE2
=3AC2
+5AD2
LEVEL-IV
1. Prove that if a line is drawn parallel to one side of a triangle the other two sides are divided in the same
ratio. Ans-[For solution refer NCERT Text Book]

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2. ABCis a triangle, PQisa line segmentintersectingABinP andAC inQ suchthat PQ||BCand divides∆ABC
intotwo partsequal inarea findBD/AB. Ans-BD/AB=√2-1/√2
3. In the Fig(i) PA,QBandRC are perpendiculartoACprove that 1/x+1/y=1/z.
4. State andprove converse of Basicproportionalitytheorems[RefertoyourtextBook forsolution]
5. In Fig-(ii) DE||BCand AD/BD=3/5 if AC=4.8cm findthe lengthof AE.
Ans-AE=1.8cm
6. Prove that the ratio of the areasof twosimilartrianglesisequal tothe ratioof the squaresof theirs
correspondingsides[Refertoyourtextbookfor proof].
7. The areasof twosimilartrianglesare 81cm2
and49 cm2
respectively.If the altitude of the biggertriangle is
4. 5 cm findthe correspondingaltitudeof the similartriangle. Ans-3.5c m
8. State and prove Pythagorastheorem[Refertextbookforproof andstatement].
9. In an equilateral triangleABC,Disa pointonside BC,such that BD=1/3 BC. Prove that 9 AD2
=7AB2
.
10. State and prove Pythagorasandits converse theorems.[Refertextbookforproof andstatement].
11. ∆ ABCis an isoscelestriangle inwhichAC=BC.If AB 2
=2AC2
thenprove that ∆ ABCis a righttriangle.

17
SELF EVALUATION QUESTIONS
1. D is a pointon the side BCof a triangle ABCsuch that ADC=BACshow that CA2
=BC.CD.
2. In Fig-(i) if LM||BC andLN || CD prove that
3. S and T are pointsonsidesPRand QR of ∆PQR such that p=RTS show that ∆RPQ=∆RTS.
4. D and E are pointson the sidesCA andCB respectivelyof atriangle ABCrightanglesat C. Prove thatAE2
+ BD2
=AB2
+DE2
.
5. Prove that the sumof the squaresof the diagonalsof parallelogramisequal tothe sumof the squaresof its
sides.
6. Two polesof heightametersandb metersare p metersapart. Prove thatthe heightof the pointof intersection
of the linesjoiningthe topof eachpole to the footof the opposite pole isgivenby meters.
7. The perpendicularfromA on side BC of a ∆ ABC intersectsBCat D. Suchthat BD=3CD prove that
2AB2
=2AC2
+BC2
.

18
INTRODUCTION TO TRIGONOMETRY
IMPORTANT CONCEPTS
TAKE A LOOK:
1. Trigonometricratiosof anacute angle of a right angle triangle.
2. Relationshipbetweendifferenttrigonometricratios
3. TrigonometricIdentities.
(i) sin2
 + cos2
 =1
(ii) 1 + tan2
 = sec2

(iii) 1 +cot2
 = cosec2

4. TrigonometricRatiosof some specificangles.
 0o
30o
45o
60o
90o
sin  0 ½ 1/2 3/2 1
cos  1 3/2 1/2 1/2 0
tan  0 1/3 1 3 Not defined
cot  Not defined 3 1 1/3 0
sec  1 2/3 2 2 Not defined
cosec  Not defined 2 2 2/3 1
5. Trigonometricratiosof complementaryangles.
(i) sin(90o
- ) = cos 
(ii) cos (90o
- ) = sin 
(iii) tan (90o
- ) = cot 
(iv) cot (90o
- ) = tan 
Side
oppositeto
angle
Hypotenuse
Side adjacenttoangle 

AB
C

19
(v) sec (90o
- ) = cosec
(vi) cosec (90o
- ) = sec
LEVEL – 1
Questionscarrying1 marks
1. In the adjoiningfigure findthe valuesof (a) sinA (b) cos B
2. sin2A = 2 sinA istrue whenA = ________.
3. Evaluate sin60o
.cos 30o
+ sin30o
.cos 60o
.
4. If tan A = cot B, thenshowthat A+B = 90o
.
5. Evaluate sin2
35o
+ sin2
55o
.
Ans: (1)a.4/5 b.4/5 (2) A=0o
(3) 1 (5) 1
LEVEL – 2
Questionscarrying2 marks
1. Evaluate 3sin2
45o
+2cos2
30o
– cot2
30o
2. Evaluate tan7o
tan23o
tan 60o
tan 67o
tan83o
.
3. If tan 2A = cot (A – 18o
). Findthe value of A.
4. If A, B and C are interioranglesof ∆ABCthenshow that
5. Prove that
Ans: (1) 0 (2) 3 (3) A=36o
LEVEL – 3
QuestionsCarrying3marks
1. Expressthe trigonometricratiosof sinA, secA and tan A in termsof cot A.
2. Prove that ( 1 + cot  – cosec  ) (1 + tan  + sec  ) = 2
3. Prove that
4. If 16 cot A=12 Thenfindthe value of
5. Prove that Tan 1o
. tan 2o
. tan 3o
.----------------.tan89o
= 1
LEVEL – 4
QuestionsCarrying4marks
1. Prove that
2. If sec  + tan  =P , prove that sin  =
3. If tan  + sin = m and tan  - sin =n show that m2
– n2
= 4
4. Prove the following(cosecA – sinA) (secA - cos A) =
5. Prove that
SELF EVALUATION
1. If tan(A+B) = 3 and tan(A - B) = thenfindA andB. [ 0o
A+B 90o
, A> B]
2. If tan A = 3 Findothertrigonometricratiosof A.

20
3. If 7 sin2
+ 3cos2
 = 4 showthat tan =
4. Findthe value of sin60o
geometrically.
5. Evaluate
6. Evaluate:(sin2
25o
+ sin2
65o
) + 3 (tan5o
.tan15o
. tan30o
. tan 75o
. tan85o
)
7. Prove that cosecA + cot A
8. Prove that :
STATISTICS
IMPORTANT CONCEPTS
TAKE A LOOK
The three measuresof central tendencyare :

21
i. Mean
ii. Median
iii. Mode
 Mean Of groupedfrequencydistribution canbe calculatedbythe followingmethods.
(i) Direct Method
Mean = =
Where Xi isthe classmark of the ith
class interval andfi frequencyof thatclass
(ii) AssumedMean methodor Shortcut method
Mean = = a +
Where a = assumedmean
Anddi = Xi - a
(iii) Step deviationmethod.
Mean = = a +
Where a = assumedmean
h = classsize
Andui = (Xi – a)/h
 Medianof a groupedfrequencydistributioncanbe calculatedby
Median= l +
Where
l = lowerlimitof medianclass
n = numberof observations
cf = cumulative frequencyof classprecedingthe medianclass
f = frequencyof medianclass
h = classsize of the medianclass.
 Mode of groupeddata can be calculatedbythe followingformula.
Mode = l +
Where
l = lowerlimitof modal class
h = size of classinterval
f1 = Frequencyof the modal class
fo = frequencyof classprecedingthe modal class
f2= frequencyof classsucceedingthe modal class
 Empirical relationshipbetweenthe three measuresof central tendency.
3 Median= Mode + 2 Mean
Or, Mode = 3 Median – 2 Mean
 Ogive
Ogive isthe graphical representationof the cumulative frequencydistribution.Itisof twotypes:
(i) Lessthan type ogive.
(ii) More than type ogive
 Medianby graphical method
The x-coordinatedof the pointof intersectionof ‘lessthanogive’and‘more thanogive’givesthe median.
LEVEL – 1
Slno Question Ans
1 What isthe meanof 1st
tenprime numbers? 12.9

22
2 What measure of central tendencyisrepresentedbythe abscissaof the pointwhere lessthanogive
and more than ogive intersect?
Median
3 If the mode of a data is 45 and meanis27, thenmedianis___________. 33
4 Findthe mode of the following
Xi 35 38 40 42 44
fi 5 9 10 7 2
Mode
=40
5 Write the medianclassof the followingdistribution.
Class 0-10 10-20 20-30 30-40 40-50 50-60 60-70
Frequency 4 4 8 10 12 8 4
30-40
LEVEL – 2
Slno Question Ans
1 Calculate the meanof the followingdistribution
Class interval 50-60 60-70 70-80 80-90 90-100
Frequency 8 6 12 11 13
78
2 Findthe mode of the followingfrequency distribution
Marks 10-20 20-30 30-40 40-50 50-60
No. of students 12 35 45 25 13
33.33
3 Findthe medianof the followingdistribution
Class interval 0-10 10-20 20-30 30-40 40-50 50-60
Frequency 5 8 20 15 7 5
28.5
4 A classteacherhas the followingabsentee recordof 40 studentsof a classfor the whole term.
No. of days 0-6 6-10 10-14 14-20 20-28 28-38 38-40
No. of students 11 10 7 4 4 3 1
Write the above distributionaslessthantype cumulative frequencydistribution.
Answer:
No. of days Less Than
6
Less Than
10
Less Than
14
Less Than
20
Less Than
28
Less Than
38
Less Than
40
No. of
students
11 21 28 32 36 39 40
LEVEL – 3
Slno Question Ans
1 If the meandistributionis25
Class 0-10 10-20 20-30 30-40 40-50
Frequency 5 18 15 P 6
Then findp.
P=16
2 Findthe meanof the followingfrequencydistributionusingstepdeviationmethod
Class 0-10 10-20 20-30 30-40 40-50
25
3 Findthe value of p if the medianof the followingfrequency distributionis50
Class 20-30 30-40 40-50 50-60 60-70 70-80 80-90
P=10

23
Frequency 25 15 P 6 24 12 8
4 Findthe medianof the followingdata
Marks Less
Than 10
Less
Than 30
Less
Than 50
Less
Than 70
Less Than
90
Less
Than 110
Less
Than
130
Less
than
150
Frequency 0 10 25 43 65 87 96 100
76.36
LEVEL – 4
Slno Question Ans
1 The mean of the followingfrequencydistributionis57.6 and the sumof the observationsis50.Find
the missingfrequenciesf1 andf2.
Class 0-20 20-40 40-60 60-80 80-100 100-120 Total
Frequency 7 f1 12 f2 8 5 50
f1 =8
and
f2 =10
2 The followingdistributiongivesthe dailyincome of 65 workersof a factory
Dailyincome
(in Rs)
100-120 120-140 140-160 160-180 180-200
No. of
workers
14 16 10 16 9
Convertthe above toa more than type cumulative frequencydistributionanddraw itsogive.
3 Draw lessthantype and more than type ogivesforthe followingdistributiononthe same graph.Also
findthe medianfromthe graph.
Marks 30-39 40-49 50-59 60-69 70-79 80-89 90-99
No. of
students
14 6 10 20 30 8 12

24
SELF – EVALUATION
1. What isthe value of the medianof the data usingthe graph infigure of lessthanogive andmore than ogive?
2. If mean =60 and median=50, thenfindmode usingempirical relationship.
3. Findthe value of p, if the meanof the followingdistributionis18.
Variate (xi) 13 15 17 19 20+p 23
Frequency(fi) 8 2 3 4 5p 6
4. Findthe mean,mode andmedianforthe followingdata.
Classes 0-10 10-20 20-30 30-40 40-50 50-60 60-70
frequency 5 8 15 20 14 8 5
5. The medianof the followingdatais52.5. findthe value of x and y,if the total frequencyis100.
ClassInterval 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90 90-100
frequency 2 5 X 12 17 20 Y 9 7 4
6. Draw ‘lessthanogive’and‘more thanogive’forthe followingdistributionandhence finditsmedian.
Classes 20-30 30-40 40-50 50-60 60-70 70-80 80-90
frequency 10 8 12 24 6 25 15
7. Findthe meanmarks forthe followingdata.
Marks Below
10
Below
20
Below
30
Below
40
Below
50
Below
60
Below
70
Below
80
Below
90
Below
100
No. of
students
5 9 17 29 45 60 70 78 83 85
8. The followingtable showsage distributionof personsinaparticularregion.Calculate the medianage.
Age in years Below
10
Below
20
Below
30
Below
40
Below
50
Below
60
Below
70
Below
80
No. of
persons
200 500 900 1200 1400 1500 1550 1560
9. If the medianof the followingdatais32.5. Findthe value of x and y.
ClassInterval 0-10 10-20 20-30 30-40 40-50 50-60 60-70 Total
frequency x 5 9 12 y 3 2 40

25
SECTION – A (F A1)
MCQ(CARRING, 1 MARKS EACH)
Q1. Euclid’sdivisionlemmastatesthat forany twopositive integer‘a’and‘b’there exitsunique integerqandr such
that a=bq+r where r mustsatisfy.
a. 1<r<b
b. 0<r b
c. 0 r<b
d. 0<r<b
Q2. If are the zeroesof the polynomial 2y2
+7y+5 thenthe value of is.
a. -1
b. -2
c. 1
d. 2
Q3. The value of c for whichthe pairof equationscx-y=2and6x-2y=3 will have infinitelymanysolutionis
a. 3
b. -3
c. -12
d. No value
Q4. If in the two trianglesABCandPQR, then
a. ∆PQR ∆CAB
b. ∆PQR ∆ABC
c. ∆CBA ∆PQR
d. ∆BCA ∆PQR
A
Q5. In fig.1BAC=900
and ADBCthen,
a. BD.CD=BC2
b. AB.AC=BC2
c. BD.CD=AD2
d. AB.AC=AD2
B D C
SECTION B
(EACH QUESTION CARRY 2 MARKS)
Q6. Use Euclid’sDivisionAlgorithmtofindthe HCFof 135 and 225.
Q7. Write a quadratic polynomialthe sumandproductof whose zeroesare 3,-2.
Q8. Findthe numberof solutionof the followingpairof linearequations:
X+2y-8=0 2x+4y=16

26
Q9. In fig:2,BC||DE.FindEC
Q10. In adjoiningfig:A,BandCare pointsonOP,OQand OR respectively
Such that AB||PQandAC||PRshowthat BC||QR
Q11. Solve thatequationbysubstitutionmethod
8x+5y=9
3x+2y=4
SECTION C
(EACH QUESTION CARRIES 3 MARKS)
Q12. Findall the zeroesof the polynomial 3x4
+6x2
-2x2
-10x-5,if twoof itszeroesare and -
Q13. Prove that isirrational.
Q14. Solve the followingpairof equationsbyreducingthemtoa pairof linearequation:
= and =
Q15. In an equilateral triangleprove that three timesthe square of one side isequal tofourtimesthe square of one
of itsaltitudesi.e 4AD2
=3AB2
=3BC2
=3CA2.
Q16. ABC isan isoscelestriangle rightangledatC.Prove thatAB2
=2AC2
.
A
B
D E
C
1.5cm
3 cm
1 cm
A
B
C
P
Q R
O

27
SECTION D
Q17. State ‘Basic Proportionalitytheorem’ or‘ThalesTheorem’ andprove it.
OR
Areaof twosimilartrianglesare inratioof the squaresof the correspondingsides.
Q18. Five yearsagoNuri was thrice as oldas Sonu.Ten yearslaterNuri will be twice asold as Sonu.How oldare Nuri
and Sonu?
ANSWERS
1. (a)
2. (a)
3. (d)
4. (a)
5. (c)
6. 45
7. X2
-3x-2
8. Infinitelymanysolutions
9. 2 cm
10.
11. X= -2 , Y=5
12. -1,-1
13.
14. X=1 andY=1
18. Nuri’spresentage=50 years,Sonu=20 years

28
SAMPLE PAPER FOR SA-1
CLASS – X
MATHEMATICS
Time : 3 ½ hours Maximum Marks : 80
General Instructions:
1. All questionsarecompulsory.
2. The question paperconsistsof 34 questionsdivided into foursectionsA, B, C and D. Section A comprisesof 10
questionsof 1 markeach, Section B comprisesof 8 questionsof 2 markseach.Section C comprisesof 10
questionsof 3 markseach and Section D comprisesof 6 questionsof 4 markseach.
3. Question numbers1 to 10 in Section A are multiple choice questionswhereyou are to select onecorrect option
outof thegiven four.
4. There is no overall choice.However,internal choice hasbeen provided in 1 question of two marks,3 questionsof
three markseach and 2 questionsof four markseach.You haveto attemptonly oneof thealternativesin all
such questions.
5. Useof calculatoris notpermitted.
6. An additional15 minutestime hasbeen allotted to read thisquestion paperonly.
SECTION – A
1. The HCF of 135 and 225 is :
(a) 135 (b) 35 (c) 45 (d) 65
2. If the zeroesof the quadraticpolynomial x2
+(a+1)x +bare 2 and -3 then:
(a)a= -7,b= -1 (b) a= 5, b= -1 (c) a= 2, b= -6 (d) a= 0, b= -6
3. In the givenfigure (i) the ABCandDEF are similar,BC=3cm, EF= 4 cm and area of triangle ABC= 54cm2
. The
area of DEF is:
(a) 72 sq.cm (b) 36 sq.cm (c) 96 sq.cm (d) 144 sq.cm
4. If cos 9= sinand 9<90o
, thenthe value of tan 5 is :
(a) 1/3 (b) 3 (c) 0 (d) 1
5. 9sec2
A – 9 tan2
A is :
(a) 1 (b) 9 (c) 8 (d) 0
6. x2
-1 isdivisibleby8 if x is:
(a) an integer (b) natural number (c) an eveninteger (d) an odd integer
7. If cos(40o
+x) = sin30o
, thenvalue of x is :
(a) 30o
(b) 40o
(c) 20o
(d) 60o
8. The mean of first6 multiplesof 3 is:
(a) 11 (b) 63 (c) 46 (d) 10.5
9. The pair of equationsx=aandy=b, graphicallyrepresentlineswhichare :
(a) Parallel (b) Intersectingat(b,a) (c) intersectingat(a,b) (d) coincident
10. If sin A = ½ , then4cos3
A – 3cos A is
(a) 0 (b) 1 (c) 3 (d) 1/

29
SECTION – B
11. If 1 is a zeroof the polynomial p(x) =ax2
-3(a-1)x -1,thenfindthe value of a.
12. Findthe HCF of 867 and 255 withthe helpof Euclid’sdivisionalgorithm.
13. If 7Sin2
+ 3Cos2
 = 4, showthattan=1/3.
OR
If Cot=15/8, Evaluate
14. Three anglesof a triangle are x,y and 40o
. The difference betweenthe twoanglesx andyis 30o
. Findx and y.
15. In PQR,S is anypointon QR such that RSP=RPQ,Prove that RS x RQ = RP2.
16. Write the lowerlimitof the medianclassinthe followingdistribution:
Classes 0-10 10-20 20-30 30-40 40-50
17. Is 7 x 5 x 3 x 2 +3 isa composite number?Justifyyouranswer.
18. Mean of the followingdatais21.5. Findthe missingvalue of ‘k’.
x 5 15 25 35 45
f 6 4 3 k 2
SECTION – C
19. Prove that 23 isirrational.
OR
Prove that (5 - 2) isirrational.
20. Showthat any positive oddintegerisof the form4q +1, 4q +3, where qis a positive integer.
21. A personrowingaboat at the rate of 5km/hourinstill watertake thrice as muchtime in going40km upstream
as ingoing40km downstream.Findthe speedof the stream.
OR
In a competitive examinationone markisawardedforeachcorrect answerwhile ½mark isdeductedforeach
wronganswer.Jayanti answered120 questionsandgot90 marks.How manyquestionsdidshe answer
correctly?
22. If A, B and C are interioranglesof atriangle ABC,thenshow that
23. If  and  are twozerosof the quadraticpolynomial x2
–2x + 5, finda quadraticpolynomial whosezeroesare
+ and
24. Prove that : (Sin+ Cosec)2
+(Cos+ Sec)2
= 7 + tan2
 + cot2

25. Findthe meanof the followingfrequencydistribution,usingstepdeviationmethod.
Classes 0-10 10-20 20-30 30-40 40-50

30
26. In the figure (ii),AB||CD,findthe value of x.
27. ABC isan equilateral triangle withside 2a.Findthe lengthof one of itsaltitudes.
28. Findthe medianof the followingdata.
Classes 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90 90-100
Frequency 5 3 4 3 3 4 7 9 7 8
SECTION – D
29. State and prove converse of Pythagorastheorem.
OR
Prove that if a line isdrawnparallel toone side of a triangle tointersectthe othertwosidesindistinctpoints,
the othertwo sidesare dividedinthe same ratio.
30. The medianof the distributiongivenbelow is14.4. Findthe value of x and y. if the total frequencyis20.
C.I. 0-6 6-12 12-18 18-24 24-30
Frequency 4 x 5 y 1
31. Prove that
OR
Evaluate :
32. If Sec + tan = p,prove that sin=
33. Findotherzeroesof the polynomial p(x) =2x4
+ 7x3
-19x2
– 14x +30 if twoof itszeroesare 2 and -2.
34. Sumof a two digitnumberandthe numberobtainedbyreversingthe orderof itsdigitsis121. If the digitsdiffer
by 3, findthe number(bycrossmultiplicationmethod) forsolvingthe problem.
---xxx---

31
SECTION-A
MARCKING SCHEME
MATHEMATIC
CLASS X
1. C
2. D
3. C
4. D
5. B
6. D
7. C
8. D
9. C
10. A
SECTION –B
11. Ans->1is a zeroof the polynomial p(x)=ax2
-3(a-1)x-1,
p(x)=0 1
P(1)=0 ½
1)2
- 3(a-1).1-1=0 ½
½
a=1
12. Ans->867=255x3+102 ½
255=102x2+51 ½
102=51x2+0 ½
H C F (867,255)=51 ½
13. 7sin2
+ 3 cos2
=4
4 sin2
+3sin2
+ 3cos2
=4 ½
 4 sin2
+3(sin2
+cos2
=4 ½
 4 sin2
½
sin2
sin
=
300
Nowtan300
= ½
Or
13. = 1
= 2 1/2
1/2
=cot2
= =

32
14. x+y+40=1800
x+y=1400 1/2
x-y=300 1/2
x=850
andy =55o ½+1/2
15. In ∆RSP and∆RPQ
RSP=RPQ (Given) 1
R=R (Common)
∆RSP ∆RPQ
 = ½
 =RP2
½
16.
>Medianclass 1
Here n=50, => n/2=25 ½
Whichliesinthe C.I.20-30 ½
17. 7x5x3x2+3=3(7x5x2+1) 1
=(70+1)=3x71 ½
The givennumberiscomposite number ½
18.
Frequencytable-> 1
k
 ½

 21.5 = , to findk=5 ½
SECTION C
19. Let if possible be arational no. ½
= = where pand q are co primesandq≠0 p & q integers 1
 =
 rational numberbecause isa rational number ½
Thisgivescontractionbecause isnota rational no.our assumptioniswrong ½
Hence is an irrational no. ½
Or
C.I. fi c.f
0-10 9 9
10-20 12 21
20-30 5 26
30-40 16 42
40-50 8 50
x 5 15 25 35 45
f 6 4 3 k 2
fx 30 60 75 35k 90

33
Let if possible 5- isa rational no ½
5- =p/q where pand q are integersandcoprimesq≠0 1
=p/q-5
= ½
isrational numberbecause isa rational no.thisisa contradictionbecause isnota rational
no. ½
Hence our suppositioniswrong
5+ is an irrational number ½
20. Let a be positive oddinteger ½
by Euclid’sDivisionalgorithm
a=4q+r, where q,rare positive
Integerand0 r <4 1
 a=4q or , a=4q+2 or 4q+3 ½
But 4q and4q+2 are both even.
a isthe form4q+1 or 4q+3. 1
21. Let the speedof streambe x km/hour
speedof the boat rowingupstream=(5-x)km/h
speedof the boat rowingdownstream=(5+x)km/h 1
Accordingto the eq.we have
=3 1 ½
 x=2.5 km/h
Hence the speedof the streamis 2.5 km/h ½
OR
Let the numberof correct answerbe x,
Thennumberof wrong answer=(120-x) 1
ATQ
we have
 x-1/2 (120-x)=90 ½
 3x/2= 150 1
 X=100
Hence numberof correctlyansweredquestions=100 ½
22. A+B+C=1800
 B+C=1800
- A ½
 B+C=1800
- A ½
2 2
sin = sin(90o
- - ) 1
Hence,sin = cos - 1

34
23. Sumof zeroes= =
Product of zeroes =5 ½
For newQ.Psum of zeroes=( ) +(1/ +1/ ) ½
=12/15
Productof zeroes=( x ( ½.
=4/5
- = => a=5, b= -12 ½
= => c =4
Requiredquadpol is5x2
-12x+4 1
24. (Sin2
+2 Sin )+(Cos2
+2 ) 1
=2+2+( Sin2
+ Cos2
) + Sec2
Cosec2
½
=.5+( tan2
+1) + (Cot2
+1) 1
=7+ tan2
+ cot2
½
25.
Classes ClassMark(xi) Frequency(fi) Ui=xi –a/h fiui
0-10
10-20
20-30
30-40
40-50
5
15
25
35
45
7
12
13
10
8
-2
-1
0
1
2
-14
-12
0
10
16
For frequencydistributiontable 2
½
½
26. In ∆AOB and∆COD
AOB=COD (Verticallyopposite angles)
OAB=OCD (Alternate angles) ½
∆AOB ∆COD(AA Criterion)
 = ½
 = 1

 X=10.5 1

35
27. Let ADBC
In rightADB usingthe pythogorastheorem
We have
AD2
= AB2
– BD2
½
AD2
= 4a2
- a2
½
AD = 3a ½
Hence,
length of one of itsaltitude =3a ½
1
28.
Classes 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90 90-100
Frequency 5 3 4 3 3 4 7 9 7 8
cumulative 5 8 12 15 18 22 29 38 45 53
For frequencytable 1 ½
Median= l +(N/2 –CF) x h ½
f
= 60 + (26.5 -22) x 10
7 ½
Median= 66.43 ½
SECTION – D
29. Statement ½
Given: to prove fig.construction 1 ½
Correct proof 2
OR
Givento prove constructionfig 2
Correct proof 2
30.
C.I F CF
0-6
6-12
12-18
18-24
24-30
4
X
5
Y
1
4
4+x
9+x
9+x+y
10+x+y
For Writingtable 1 ½
Formula
Median= l + ½
L=12, f=5, cf= 4+x, h=6 andx+y=10 1
To findx=4
Andy=6 1

36
31.
LHS:
1
= 1
= 1
= 1
OR
Cosec(90o
-)=sec, cot(90o
-) = tan , sin55o
= Cos35o
1
tan 80o
= cot 10o
, tan 70o
= cot 20o
, tan 60o
= 3 1
Givenexpression= 1
= = 1
32. We have,
Sec  + tan  = p
= ½
= ½
= ½
= 1
= ½
= ½
=
=sin= ½

37
33. p(x) = 2x4
+ 7x3
-19x2
-14x +30
If two zeroesof p(x) are 2 and-2 then(x+2)(x-2) =x2
-2 isfactor of p(x) 1
P(x) (x2
– 2) = [2x4
+ 7x3
-19x2
-14x +30] (x2
-2) 1 ½
= 2x2
+7x -15
= 2x2
+10x -3x -15
= (2x -3) (x +5) 1
Hence,othertwozeroof p(x) are 3/2 and -5. ½
34. Let unit digit be = x
And 10th place digit be = y ½
No = x + 10 y
After reversing the digit the no. = 10 x + y ½
Now According to question,
X + 10 y + 10 x + y = 121
And x – y = 3
 11x + 11 y = 121 and x – y = 3 1
x + y = 11 and x – y = 3
x + y – 11 = 0
x – y – 3 = 0 ½
 = = ½

x = 7 and y = 4 1
Here required number
X + 10 y = 7 + 10 x 4 = 47

38
DETAILS OF THE CONCEPTSTO BE MASTERED BY EVERY CHILD OF CLASSX WITH EXCERCISES ANDEXAMPLES OFNCERT
TEXT BOOK
SUMMATIVE ASSESSMENT -II
SYMBOLS USED
* : ImportantQuestions, **: Veryimportantquestions, ***: Very,VeryImportantquestions
01 Quadratic
Equation
Standardform of quadratic
equation
* NCERT Textbook
Q.1.2, Ex 4.1
Solutionof quadraticequation
by factorization
*** Example 3,4,5, Q.1, 5 Ex. 4.2
by completingthe square
** Example 8,9
Q.1 Ex. 4.3
by quadraticformula
*** Example.10,11,13,14,15 ,
Q2,3(ii) Ex.4.3
Nature of roots *** Example 16
Q.1.2, Ex. 4.4
02 Arithmetic
progression
General formof an A.P. * Exp-1,2,Ex. 5.1 Q.s2(a),
3(a),4(v)
nth termof an A.P. *** Exp.3,7,8 Ex. 5.2
Q.4,7,11,16,17,18
Sumof firstn termsof an A.P. **
*
**
***
Exp.11,13,15
Ex. 5.3, Q.No.1(i,ii)
Q3(i,iii)
Q.7,10,12,11,6, Ex5.4, Q-1
03 Coordinate
geometry
Distance formula ** Exercise 7.1,Q.No 1,2,3,4,7,8
Sectionformula
Mid pointformula
**
***
Example No.6,7,9
Exercise 7.2,Q.No.1,2,4,5
Example 10.
Ex.7.2, 6,8,9. Q.No.7
Areaof Triangle **
***
Ex.12,14
Ex 7.3 QNo-12,4 Ex.7.4,Qno-2
04 Some
applicationof
Trigonometry
Heightsanddistances Example-2,3,4
Ex 9.1
Q 2,5,10,12,13,14,15,16
05 Circles Tangentsto a circle Q3(Ex10.1)
Q 1,Q6,Q7(Ex 10.2),4
Numberof tangentsfroma
pointto a circle
*** Theorem10.1,10.2
Eg 2.1
Q8,9,,10,12,13
(Ex 10.2)
06 Constructions Divisionof line segmentinthe
givenratio
* Const11.1
Ex 11.1 Qno 1
Constructionof triangle similar
to giventriangle aspergiven
scale
*** Ex 11.1 Qno-2,4,5,7
Constructionof tangentstoa
circle
*** Ex 11.2 Qno 1,4

39
07 Arearelatedto
circles
Circumference of acircle * Example 1
Exercise 12.1 Q.No1,2,4
Areaof a circle * Example 5,3
Lengthof an arc of a circle * Exercise 12.2 Q No 5
Areaof sectorof a circle ** Example 2
Exercise 12.2 QNo 1.2
Areaof segment of a circle ** Exercise 12.2
Qno 4,7,9,3
Combinationof figures *** Ex 12.3 Example 4.5
1,4,6,7,9,12,15
08 Surface area
and volumes
Surface area of a combination
of solids
** Example 1,2,3
Exercise 13.1
Q1,3,6,7,8
Volume of combinationof a
solid
** Example 6
Exercise 13.2
Q 1,2,5,6
Conversionof solidsfromone
shape to another
*** Example 8 & 10
Exercise 13.3
Q 1,2,6,4,5
Frustumof a cone *** Example 12& 14
Exercise 13.4
Q 1,3,4,5 Ex-13.5, Q. 5
09 Probability Events * Ex 15.1 Q4,8,9
Probabilityliesbetween0
and1
** Exp- 1,2,4,6,13
Performingexperiment *** Ex 15 1,13,15,18,24

40
QUADRATIC EQUATIONS
IMPORTANT CONCEPTS:-
TAKE A LOOK:
1. The general formof a quadraticequationisax2
+bx+c=0,a≠o. a, b and c are real numbers.
2. A real numberx is saidto be a root of the quadraticequationax2
+bx+c=0where a≠o if ax2
+bx+c=0. The zeroesof
the quadraticequationpolynomial ax2
+bx+c=0 andthe rootsof the correspondingquadraticequation
ax2
+bx+c=0 are the same.
3. Discriminant:- The expressionb2
-4aciscalleddiscriminantof the equationax2
+bx+c=0andis usuallydenotedby
D. Thus discriminantD=b2
-4ac.
4. Everyquadraticequationhastworoots whichmaybe real , co incidentorimaginary.
5. IF and are the roots of the equationax2
+bx+c=0then
And =
6. Sumof the roots , + = - and productof the roots, ,
7. Formingquadraticequation,whenthe rootsaand B are given.
x2
-( + )x+ . =0
8. Nature of roots of ax2
+bx+c=0
i. If D 0, thenrootsare real and unequal.
ii. D=0, thenthe equationhasequal and real roots.
iii. D<0, thenthe equationhasno real roots
LEVEL-I
1. IF ½ isa root of the equationx2
+kx-5/4=0,thenthe value of Kis
(a) 2 [Ans(d)]
(b) -2
(c) ¼
(d) ½
2. IF D>0, thenrootsof a quadraticequationax2
+bx+c=0are
(a) (b) (c) (d) None of these
[Ans(a)]
3. Discriminantof x2
+5x+5=0 is
(a)5/2 (b) -5 (c) 5 (d)-4 [Ans(c)]
4. The sum of rootsof a quadraticequation +4x-320=0is
[Ans(a)]
(a)-4 (b)4 (c)1/4 (d)1/2
5. The product of roots of a quaradaticequation +7x-4=0 is
[Ans(d)]
(a)2/7 (b)-2/7 (c)-4/7 (d)-2
6. Valuesof Kfor whichthe equation +2kx-1=0 hasreal roots are:
[Ans(b)]
k 3 (b)k 3or K -3 (c)K -3 (d) k 3

41
LEVEL-II
1. Findthe roots of the quadratic equation -2√6+2=0
[Ans- x= , ]
2. The sum of the squaresof twoconsecutive oddnumberis394. Findthe numbers.
[ Ans- 13,15 or -15,13]
3. Findthe root of +√2x-2=0 by factorization.
[Ans- x= , ]
4. The sum of twonumbersis8 . Determine the numbersif the sumof theirreciprocalsis8/15.
[Ans- 5 and 3 or 3 and 5]
5. For whatvalue of k does(k-12)x2
+2(k-12)x+2=0has equal roots?
[Ans- k=14]
6. Divide 51 intotwoparts whose productis378.
[Ans-942]
LEVEL-III
1. For whatvalue of k, will the equation -2(1+2k)x+(3+2k)=0have real butdistinctroots? Whenwill the roots
be equal?
[Ansk< -5/2 or k>√5/2, k= >√5/2]
2. Solve forx:4√3x2
+5x-2√3=0. [Ans√3/4,-2/√3]
3. Usingquadratic formulasolve the followingquadraticequationforx: -2ax+(a2
-b2
)=0
[Ansa+b,a-b]
4. The speedof a boat in still wateris11 km/hr . It can go 12 km upstream andreturn downstreamtothe original
pointin2 hours45 minutes.Findthe speedof the stream.
[5 km/h]
5. Solve forx:a2
b2
x2
+b2
x-a2
x-1=0 [Ans1/b2
,-1/a2
]
6. Solve the followingquadraticequationforx:
X2
-2(a+2)x+(a+1)(a+3)=0
[Ans-a+1,a+3]

42
LEVEL-IV
1. Solve for + + ; a 0, b 0 x 0
[Ans. –a,-b]
2. An aeroplane left30 minuteslaterthanitsscheduledtimeandinorderto reachits destination1500 km
away intime,ithas to increase itsspeedby250 km/hr fromitsusual speed.Determine itsusual speed.
[Ans750 km/hr]
3. Usingthe quadraticformulasolve the equationa2
b2
x2
–(4b4
-3a4
)x-12a2
b2
=0
[Ans ,
4. Solve forx: + =3 (x 2,4)
[Ans5/2,5]
5. If (-5) isa root of the quadraticequation2x2
+px-15=0and the quadraticequationp(x2
+x)+k=0hasequal
roots thenfindthe valuesof Pand K.
[Ans-24m, 8m]
6. The sum or the areas of two squaresis640m2
. If the difference of theirperimetersis64m . Findthe sidesof
the two squares. [Ans24m,8m]
SELF EVALUATION
1. If the root of the equation(b-c)x2
+(c-a)x+(a-b)=0are equal,thenprove that2b=a+c
2. Solve byusingquadraticformula
(x2
+3x+2)2
- 8(x2
+3x)-4=0
3. If and are the rootsof the equationlx2
-mx+n=0,Findthe equationwhose rootsare / andb / .
4. The difference of twonumbersin5and the difference of theirreciprocal is1/10.Findthe numbers.

43
ARITHMETIC PROGRESSION
Important concepts:
Take a look:
Sequence:- A sequence isan arrangement of numbers ina definite orderaccordingtosome rule.
Progression: A sequence thatfollowadefinite patterniscalledprogression.
ArithmeticProgression(A.P.): A sequence inwhicheachtermdiffersfromitsprecedingtermbya constantis calledan
arithmeticprogression. Thisconstantis called commondifference of the A.P.Itisdenotedby‘d’.
General formof an A.P.: The general formof an A.P.is a, a+d,a+2d,a+3d ……………………………………….
nth
termof an A.P. : If ‘a’ isthe firstterm and‘d’ isthe commondifference than[ ]
nth
termfrom the lastof an A.P. : [ ]
where l = lastterm.
d= c.d.
Sumof n termsof an A.P. : -
=
Or Sn= . Where l = last term.
Commondifference: [d= ]
 Commondifferencemaybe +ve , - ve or zero.
nth
term: If Sn isgiven then[ = ]
Level – I
1. Is the progression3,9,15,21 ……………… isinA.P.?
Ans yes
2. Findthe firsttermand commondifference of the A.P.
1,5,9,13,17. Ans: a=1 , d=4
3. Findthe 10th
term of the A.P.63,58,53,48 …………………………
Ans: 18
4. Findthe 8th
termfrom the endof the A.P.7,10,13………………184.
Ans : 163
5. In the givenA.P.findthe missingterm:
, [ ] , 5 Ans : 3
6. Findthe sum of first24th
termsof the A.P.:
5,8,11,14,……………………………….. Ans : 948
Level – 2

44
1. Which termof the A.P. 84,80,76,…………………….. iszero. Ans : n=22
2. Findthe sumof oddnumbersbetween 0and 50. Ans : 625
3. Which termof the sequence 48,43,38,33……………………….is the first –ve term. Ans : 11th
4. if the no.4p+1,26,10p-5 are inA.P..Findthe value of p. Ans : p=4
5. If 9th
termof an A.P.iszero,prove that its29th
termisdouble the 19th
term.
Level – 3
1. The 7th
term of an A.P.is32 andits 13th
term is62. Findthe A.P.
Ans : 2, 7,12,………….
2. Findthe sum of first25th
termof an A.P.whose nth
termisgivenby Tn =2-3n.
Ans : -925
3. If m times the mth
termof an A.P.is equal ton timesitsnth
term;find(m+n)th
term.Ans:0
4. Whichterm of the A.P.3,10,17,…………………….will be 84 more than its13th
term.
Ans : 25th
5. If the sumof firstn,2n and 3n termsof an A.P.be , and respectivelythenprove that
= 3( ).
Level – 4
1. How manymultiple of 4lie between10and 250? Alsofindtheirsum.
Ans : n=60 =7800
2. The firstand last termof an A.P.is8 and 350 respectively.If itscommondifference is9,how manytermsare
there andwhat istheirsum?
Ans : n=39, =6981
3. The sum of first15 termsof an A.P.is 105 and the sum of the next15 termsis780. Findthe first3 termsof the
A.P.:
Ans : -14,-11,-8.
4. If the sumof firstnth
termsof an A.P.is givenby , findthe nth
term of the A.P.
Self Evaluation
1. Findthe commondifference andwrite the nexttwotermsof the A.P.8,3,-2,-7.
2. Which termof the A.P.4,9,14 …………………… is 89?
Alsofindthe sum.
3. Findthe sum of all two digitspositivenumbersdivisibleby3.
4. The sum of n termsof an A.P.is3n2
+5n.findthe A.P..Alsofind16th
term.
5. The ratioof the sum of n and m termsof an A.P.ism2
:n2
.Show that the ratioof the mth
term andnth
term is
(2m-1):(2n-1).

45
CO-ORDINATE GEOMETRY
IMPORTANT CONCEPTS
TAKE A LOOK
1. Distance Formula:-
The distance betweentwopointsA(x1,y1) andB(x2,y2) isgivenbythe formula.
AB=√(X2-X1)2
+(Y2-Y1)2
COROLLARY:- The distance of the pointP(x,y) fromthe origin0(0,0) isgive by
OP= √(X-0)2
+ (Y-0)2
ie OP= √X2
+Y2
2. SectionFormula :-
The co-ordinatesof the pointP(x,y) whichdividesthe line segment joiningA(x1,y1) andB(x2,y2)
internallyinthe ratiom:nare givenby.
2 1mx nx
x
m n



2 1my ny
y
m n



3. MidpointFormula:-
If R is the mid-point,thenm1=m2 andthe coordinatesof Rare
R x1+x2 , y1+y2
2 2
4. Co-ordinatesofthe centroidof triangle:-
The co-ordinatesof the centroidof a triangle whose verticesare P(x1,y1),Q(x2,y2) and
R(x3,y3) are
x1+x2+x3 y1+y2+y3
3 , 3
5. Area of a Triangle:-
The area of the triangle formedbythe pointsP(x1,y1)Q(x2,y2) andR(x3,y3) isthe numerical
value of the expression.
ar (∆PQR)=1/2 x1(y2-y3)+x2(y3-y1)+x3(y1-y2)
LEVEL-I
1. Findthe distance of the points(6,-6) fromorigin. Ans-62units
2. Showthat the point(1,1)(-2,7) and(3,-3) are collinear.
3. Findthe distance betweenthe pointsR(a+b,a-b)andS(a-b, -1-b) Ans-2√a2
+b2
units
4. Findthe pointon x-axiswhichisequidistantfrom(2,-5) and(-2,9). Ans- x=-7
5. Findthe area of the triangle whose vertices(-5,-1),(3,-5)(5,2) Ans-32 squnits

46
LEVEL-II
1. Showthat the points(-2,5),(3,-4) and (7,10) are the verticesof a rightangledisoscelestriangle.
2. Finda relationbetweenx andyif the points(x,y),(1,2) and(7,0) are collinear.
Ans: x+3y =7
3. Findthe pointon y axiswhichisequidistance fromthe points(5,-2) and(-3,2) Ans-(0,-2)
4. If the pointsA(4,3) andB(x,5) are onthe circle withthe centre O(2,3) findthe value of x. Ans-2
5. Findthe value of ‘k’for whichthe points(7,-2),(5;1) and(3,k) are collinear. Ans-k=4
6. Findthe area of triangle whose verticesare (2,-4),(-1,0) and(2,4) Ans-12 sq.units
7. Findthe ratio inwhichline segmentjoiningthe points(6,4) and(1,-7) is dividedbyx-axix alsofindthe
coordinatesof the pointsof division. Ans 7:4 and (46/11, 0)
LEVEL-III
1. Showthat the points(7,10),(-2,5) and(3,-4) are the verticesof an isoscelesrighttriangle.
2. In whatratio doesthe line x-y-2=0divide the line segmentjoining(3,-1) and(8,9)?Alsofindthe coordinates
of the pointof intersection. Ans-(2:3)(5,3)
3. Three consecutive verticesof aparallelogramare (-2,-1),(1,0) and(4,3).Findthe coordinatesof the fourth
vertex. Ans-(1,2)
4. Showthat the pointsA(5,6);B(1,5);C(2,1) and D(6,2) are the verticesof asquare.
5. The verticesof a triangle are (-1,3),(1,-1) and(5,1).Findthe lengthof mediansthroughvertices(-1,3) and
(5,1) Ans-(5,5)
6. Findthe value of P for whichthe points(-5,1),(1,P) and(4,-2) are collinear.
AnsP=-1
SELF EVALUATION QUESTION
1. Findthe distance betweenpoints.
a. A(6,0) B(14,0)
b. A(0,-5) B(0,10)
c. A(0,p) B(P,0)
2. Showthat the points(-1,-1),(1,1) and(-√3, √3) are the verticesof an equilateraltriangle.
3. The line joiningthe pointsA(4,-5) andB(4,5) is dividedbythe pointPsuchthat AP/AB=2/5.Findthe
coordinatesof P.
4. Findthe coordinatesof the pointswhichtrisectthe line segmentjoining(1,-2) and(-3,4).
5. Determine the ratioinwhichthe line 2x+y=4dividesthe line segmentjoiningthe points(2,-2) and(3,7).
6. Findthe value of K such thatthe point(0,2) is equidistantfromthe points(3,k) and(k,5).
7. Prove that the points(4,5),(7,6),(6,3) and(3,2) are the verticesof a parallelogram.Isita rectangle?

47
APPLICATIONS OF TRIGONOMETRY
(HEIGHT AND DISTANCES)
IMPORTANT CONCEPTS
TAKE A LOOK
Line of sight
Line segmentjoiningthe objecttothe eye of the observer
iscalledthe line of sight.
Angle of elevation
Whenan observerseesanobjectsituatedinupward
direction,the angle formedbylineof sightwithhorizontal
line is calledangle of elevation.
Angle of depression
Whenan observerseesanobjectsituatedindownward
directionthe angle formedbyline of sightwithhorizontal
line iscalledangle of depression.
LEVEL – 1 (Questionscarrying one marks)
1. The heightof a toweris10m. What isthe lengthof itsshadow whensun’saltitude is45o
.
Ans: 10m
2. A ladder15m longjust reachesthe topof a vertical wall.If the laddermakesanangle of 60o
withthe wall.Find
the heightof the wall.
Ans:7.53m
3. A pole 6cm highcasts a shadow23 m longon the ground,thenfindthe sun’selevation.
Ans: 60o
4. A bridge acrossa rivermakesan angle of 45o
withthe riverbank.If the lengthof the bridge across the riveris
150m, thenfindthe widthof the river.
Ans: 752m
5. A 6m tall tree casts a shadowof length4m.If at the same time a flagpole castsa shadow 50m in length,then
findthe lengthof the flagpole.
Ans:75m
LEVEL – 2 (Questionscarrying 3 marks)
1. A tree breaksdue to stormand the brokenpart bendssothat the top of the tree touchesthe groundmakingan
angle of 30o
withthe ground.The distance betweenthe footof the tree tothe pointwhere the toptouchesthe
groundis 8m. Findthe heightof the tree. [Ans-8√3m]

48
2. A kite isflyingata heightof 60m above the ground.The stringattachedto the kite istemporarilytiedtoapoint
on the ground.The inclinationof the stringtothe groundis 60o
. Findthe lengthof the stringassumingthat
there isno slackin the string. [Ans-40√3m]
3. The angle of elevationof the topof a hill atthe footof the toweris 60o
andthe angle of elevationof the topof
the towerfromthe footof the hill is30o
. If the toweris50m high,findthe heightof the hill.[Ans-150m]
LEVEL – 3 (Questionscarrying 4 marks)
1. From the top of a 7m highbuilding,the angle of elevationof the topof a cable toweris60o
andthe angle of
depressionof the footof the toweris30o
. Findthe heightof the tower. [Ans-28m]
2. The angle of elevationof anaeroplane from a pointonthe ground is45o
. Afterflightfor15 secondsthe
elevationchangesto30o
.If the aeroplane isflyingata heightof 3000m. Findthe speedof the aeroplane.
[Ans-527.4km/h]
3. The angle of elevationof acloudfrom a pointh metresabove alake is  and the angle of derpressionof its
reflectioninthe lake is .Prove thatthe heightof the cloudis metres.
SELF EVALUATION
1. From the top of a lighthouse,the anglesof depressionof twoshipsonopposite sidesof itare observedtobe 
and . If the heightof the lighthouse ishmetresandthe line joiningthe shipspassesthroughthe footof the
lighthouse,showthatthe distance betweenshipsis metres.
2. The angle of elevationof the topof towersfrompointsp and q at distancesof a and b respectivelyfromthe
base and inthe same straightline withitare complementary.Prove thatthe heightof the toweris .
3. A man standingonthe deckof a shipwhichis10m above the waterlevel observesthe angle of elevationof the
top of a hill as60o
and the angle of depressionof the base of the hill is30o
. Calculate the distance of the hill
fromthe shipandthe heightof the hill.
4. The angle of elevationof the topof the toweras observedfromapointonthe groundis  and onmoving‘a’m
towardsthe tower,the angle of elevationis .Prove thatthe heightof the toweris
5. A personstandingonthe bankof a riverobservesthatthe angle of elevationof the topof a tree standingonthe
opposite bankis60o
. Whenhe moves40m away fromthe bank,he findsthatangle of elevationtobe 30o
. Find
the heightof the tree and the widthof the river.[use 3=1.732]

49
Circles
Important Concepts
Take a look:
Tangentto a circle :
A tangenttoa circle isa line thatintersectsthe circle atonlyone point.
P tangent
P= pointof contact
 There isonlyone tangentat a pointona circle.
 There are exactlytwotangentstoa circle througha pointlyingoutside the circle.
 The tangentat anypointof a circle isperpendiculartothe radiusthroughthe pointof contact.
 The lengthof tangentsdrownfroman external pointtoa circle are equal.
Level -1
1. Findthe lengthof the tangentfrom T whichisat a distance of 13 cm from the centre of a circle of radius5 cm.
Ans : 12cm.
2. In the adjoiningfigure TP&TQare twotangentstoa circle withcentre o.if <POQ=1100
thenfindthe angle PTQ.
P
T
Q Ans : 700
3. In the adjoiningfigure ABC iscircumscribingacircle.
Findthe lengthof BC
A
4
11cm
3
Ans : 10cm
B C
4. In the adjoiningfigure acircle touchesthe side BCof ∆ ABC.At a pointP andtouchesAB&ACproducedat Q and R
respectively.If AQ=5 cm findthe perimeterof ∆ABC.
A Ans-10cm
.o
1100

50
B p C
Q R
Level-2
1. In the adjoiningfigure AB=AC.ProvethatBE=EC
A
D F
B C
E
2. TP and TQ are tangents fromT to the circle withcentero. R is the pointon the circle.Prove that
TA+AR =TB+BR
P
A
R T
Q B
3. Prove that the tangentsdrownat the endsof a diameterforof a circle are parallel.
4.A circle istouchingthe side BC of ∆ABC at P andtouching.
AB and AC producedat Q andR Respectively. Prove that – A
AQ= (Perimeterof ∆ABC).
B C
Q R
Level-3
1. Prove that the parallelogramcircumscribingacircle isa rhombus.
2. In the adjoiningfigure XY&X’Y’are two parallel tangentstoa,circle withcentre o.and anothertangentABwith
pointof contact C intersectXYat A andX’Y’ at B is drawn.
Prove that <AOB= 900
.
O
R
P

51
3. 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 segment joining the point of contact
at the centre.
4. The tangentat a pointC of a circle anda diameterABwhenextendedintersectatP.
If <PCA= 1100
,find<CBA.
Ans:700
Level-4
1. Prove that the lengthof tangentsdrawnfroman external point toacircle are equal.
2. Prove that the tangentsat any pointof a circle is perpendiculartothe radiusthroughthe pointof contact.
SELF EVALUATION
1. In the adjoiningfigure if PQ=PR
Prove that QS= SR. P
Q R
S
2. A quadrilateral ABCDisdrawnto circumscribe acircle.Prove that
AB+CD= AD+BC.
3.Two concentriccirclesare of radii 5cm. and 3cm. Findthe lengthof the chord of the longercircle whichtouches
the smallercircle.
5cm 3cm
P Q
4. Prove that tangentsdrawnat the endsof a chord of a circle make equal angleswiththe chord.
oL

52
CONSTRUCTION
TAKE A LOOK
1. Divisionof line segmentinthe givenratio.
2. Constructionof triangles:-
a. Whenthree sidesare given.
b. Whentwo sidesandincludedangle given.
c. Whentwo anglesandone side given.
d. Constructionof rightangledtriangle.
3. Constructionof triangle similartoa giventriangle aspergivenscale.
4. Constructionof tangentstoa circle.
LEVEL - I
1. Divide aline segmentingivenratio.
2. Draw a line segmentAB=8cmand divide itinthe ratio4:3.
3. Divide aline segmentof 7cminternallyinthe ratio2:3.
4. Draw a circle of radius4 cm. Take a pointP on it.Draw tangentto the givencircle atp.
5. Constructan isoscelestriangle whosebase 7.5cm andaltitude is4.2 cm.
LEVEL –II
1. Constructa triangle of sides4cm, 5cm and6cm and thentriangle similartoitwhose side are 2/3 of
correspondingsidesof the firsttriangle.
2. Constructa triangle similartoa given∆ABCsuch that eachof itssidesis2/3rd
of the correspondingsidesof
∆ABC. It isgiventhatAB=4cm BC=5cm and AC=6cm also write the stepsof construction.
3. Draw a right triangle ABCinwhich B=900
AB=5cm, BC=4cm thenconstructanothertriangle ABCwhose sides
are 5/3 timesthe correspondingsidesof ∆ABC.
4. Draw a pair of tangentsto a circle of radius5cm whichare inclinedtoeachotherat an angle of 600
.
5. Draw a circle of radius5cm from a point8cm awayfromits centre constructthe pair of tangentstothe circle
and measure theirlength.
6. Constructa triangle PQRinwhichQR=6cm Q=600
andR=450
. Constructanothertriangle similarto∆PQR
such that itssidesare 5/6 of the correspondingsidesof ∆PQR.

53
AREAS RELATED TWO CIRCLES
TAKE A LOOK
1. Circle:The setof pointswhichare at a constant distance of r unitsfroma fixedpointoiscalleda circle with
centre o.
R
2. Circumference:The perimeterof acircle iscalleditscircumference.
3. Secant:A line whichintersectsacircle at twopointsiscalledsecantof the circle.
4. Arc: A continuouspiece of circle iscalledanarc of the circle..
5. Central angle:- Anangle subtendedbyanarc at the centerof a circle iscalleditscentral angle.
6. Semi Circle:- A diameterdividesacircle intotwoequal arcs. Each of these twoarcs iscalleda semi circle.
7. Segment:- A segmentof a circle is the regionboundedbyanarc anda chord,includingthe arcand the chord.
8. Sectorof a circle:The regionenclosedbyanarc of a circle andits twoboundingradii iscalledasectorof the
circle.
9. Quadrant:- One fourthof a circularregioniscalleda quadrant.The central angle of a quadrantis 900
.
r
o

54
a. Lengthof an arc AB= 2
A B
l
b. Areaof majorsegment=Areaof a circle – Areaof minorsegment
c. Distance movedbya wheel in
1 rotation=circumference of the wheel
d. Numberof rotationin1 minute
=Distance movedin1 minute / circumference
S.N NAME FIGURE PERIMETER AREA
1.
2.
3.
4.
5.
Circle
Semi- circle
Ring(Shaded
region)
Sectorof a circle
Segmentof a
circle
or
+ 2r
Outer= 2 R
Inner= 2 r
l+2r=
+2r Sin
2
½ 2
(R2
-r2
)
or
- sin
0

55
LEVEL-I
1. If the perimeterof a circle isequal tothat of square,thenthe ratioof theirareasis
i. 22/7
ii. 14/11
iii. 7/22
iv. 11/14 [Ans-ii]
2. The area of the square that can be inscribedina circle of 8 cm is
i. 256 cm2
ii. 128 cm2
iii. 64√2 cm2
iv. 64 cm2
[Ans-ii]
3. Areaof a sectorto circle of radius36 cm is 54 cm2
. Thenthe lengthof the correspondingarcof the circle is
i. 6
ii. 3
iii. 5
iv. 8
[Ans–ii]
4. A wheel hasdiameter84 cm. The numberof complete revolutionitwill take tocover792 m is.
i. 100
ii. 150
iii. 200
iv. 300 [Ans-iv]
5. The lengthof an arc of a circle withradius12cm is10 cm. The central angle of thisarc is.
i. 1200
[Ans-iv]
ii. 600
iii. 750
iv. 1500
6. The area of a quadrantof a circle whose circumference is22cm is
i. 7/2 cm2
ii. 7 cm2
iii. 3 cm2
iv. 9.625 cm2
[Ans-iv]
LEVEL-II
1. In figo isthe centre of a circle.The area of sectorOAPBis5/18 of the areaof the circle findx.
[Ans100]
A B
P
2. If the diameterof asemicircularprotractoris14 cm, thenfinditsperimeter. [Ans-36cm]
O
x

56
3. The radiusof twocircle are 3 cm and 4 cm . Findthe radiusof a circle whose areaisequal to the sumof the
areas of the two circles.
[Ans-5cm]
4. The lengthof the minute hand of a clockis 14 cm. Findthe area sweptbythe minute handin5 minutes.
[Ans-154/3 cm]
LEVEL-III
1. Findthe area of the shadedregioninthe figure if AC=24cm ,BC=10 cm and o isthe centerof the circle (use
A [Ans- 145.33 cm2
]
B C
2. The innercircumference of acirculartrack is440m. The track is14m wide.Findthe diameterof the outercircle
of the track. [Take =22/7]
[Ans-168]
3. Findthe area of the shadedregion.
[Ans-9.625 m2
]
4. A copperwire whenbentinthe formof a square enclosesanareaof 121 cm2
. If the same wire isbentintothe
formof a circle,findthe areaof the circle (Use =22/7)
[Ans154 cm2
]
5. A wire isloopedinthe formof a circle of radius28cm. It isrebentintoa square form.Determine the side of the
square (use
[Ans-44cm]
o

57
LEVEL-IV
1. In fig,findthe areaof the shadedregion[use [Ans:72.67
cm2
]
2. In fig. the shape of the topof a table ina restaurantisthat of a sectorof a circle withcentre 0 and BOD=900
. If
OB=OD=60cm find
i. The area of the top of the table [Ans8478 cm2
]
ii. The perimeterof the table top(Take [Ans402.60 cm]
3. An arc subtendsanangle of 900
at the centre of the circle of radius14 cm. Findthe area of the minorsector
thusformedintermsof .
[Ans49 cm2
]
4. The lengthof a minorarc is 2/9 of the circumference of the circle. Write the measure of the angle subtendedby
the arc at the centerof the circle.
[Ans800
]
5. The area of an equilateraltriangleis49√3 cm2
.Taking eachangularpointas center,circlesare drawnwithradii
equal tohalf the lengthof the side of the triangle.Findthe areaof triangle notincludedinthe circles.
[Take √3=1.73] [Ans777cm2
]
SELF EVALUATION
1. Two circlestouchexternally.The sumof theirareasis130 cm2
and distance betweentheircentersis14 cm.
Findthe radii of circles.
2. Two circlestouchinternally.The sumof theirareasis 116 cm2
and the distance betweentheircentersis6
cm. Findthe radii of circles.
3. A pendulumswingsthroughanangle of 300
anddescribesanarc 8.8 cm inlength.Findlengthof pendulum.
4. What isthe measure of the central angle of a circle?
5. The perimeterandareaof a square are numericallyequal.Findthe areaof the square.

58
SURFACE AREAS AND VOLUMES
IMPORTANT FORMULA
TAKE A LOOK
SNo NAME FIGURE LATERAL
CURVED
SURFACE AREA
TOTAL SURFACE
AREA
VOLUME NOMENCLATURE
1 Cuboid 2(l+b)xh 2(lxb + bxh +
hx l)
l x b x h L=length,
b=breadth,
h=height
2 Cube 4l2
6l2
l3
l=edge of cube
3 Right
Circular
Cylinder
2rh 2r(r+h) r2
h r= radius
h=height
4 Right
Circular
Cone
rl r(l+r)
r2
h
r=radius ofbase,
h=height,
l=slant height=
5 Sphere 4r2
4r2
r3
r=radius ofthe
sphere
6 Hemisphere 2r2
3r2
r3
r=radius of
hemisphere
7 Spherical
shell
2(R2
+ r2
) 3(R2
- r2
)
(R3
- r3
)
R=External
radius,
r=internal radius
8 Frustum of
a cone
l(R+r)
where
l2
=h2
+(R-r)2
[R2
+ r2
+
l(R+r)]
h/3[R2 + r2 +
Rr]
R and r = radii of
the base,
h=height,l=slant
height.
9. Diagonal of cuboid=
10. Diagonal of Cube = 3l

59
LEVEL-I
1. In a rightcircular cone the cross sectionmade bya plane parallel tothe base isa.
i. Circle
ii. Frustumof a cone
iii. Sphere
iv. Semi sphere [Ans-i]
2. The radiusand heightof cylinderare inthe ratio 5:7 andits volume is550 cm3
.Its radiusis
i. 1 cm
ii. 7 cm
iii. 5 cm
iv. 6 cm [Ans-iii]
3. A cylinder,acone and a hemisphere are of equal base andhave the same height.Whatis the ratioof their
volumes.
i. 1:2:3
ii. 3:1:3
iii. 3:1:2
v. 2/3:1/3:1 [Ans-iii]
4. If surface areas of two spheresare inthe ratio 4:9 thenthe ratio of theirvolumesis:
i. 16/27
ii. 4/27
iii. 8/27
iv. 9/27
[Ans-iii]
5. Determine the ratioof the volume of acube to that of a sphere whichwill exactlyfitinside the cube:
i. 4:
ii.
iii.
iv. 6: [Ans-iv]
LEVEL-II
1. The slant heightof a frustumof a cone is10 cm. If the heightof the frustumis8 cm, thenfindthe difference of
the radiusof itstwo circularends.
[Ansr,r2=6cm]
2. A solidmetallicsphere of radius12 cm ismeltedandrecastintoa numberof small cones eachof radius4 cm
and height3 cm . Findthe numberof conesso formed.
[Ans-144]
3. How manyspherical leadshotsof radius2 cm can be made out of a solidcube of leadwhose edge measures44
cm? [Ans-2541]
4. Three cubesof metal whose edgesare inthe ratio 3:4:5 are meltedandconvertedintoasingle cube of diagonal
24√3 cm. Findthe edgesof the three cubes.
[Ans-12cm,16,cm,20cm]
5. A heapof rice inthe form of a cone of radius3 m and height3 m. Findthe volume of the rice.How much clothis
requiredtojustcoverthe heap?
[Ans9 cm3
,9√2 m2
]

60
LEVEL-III
1. Three solidmetallicspheresof radii 3cm , 4cm and 5cm respectivelyare meltedtoforma single solidsphere.
Findthe diameterof the resultingsphere.
[Ans12 cm]
2. Findthe numberof coins1.5 cm in diameterand0.2 cm thickto be meltedtoforma right circularcylinderof
height10 cm anddiameter4.5 cm.
[Ans-450]
3. The rain waterfrom a roof 22m x 20m drainsintoa cylindrical vessel havingdiameterof base 2 m and height3.5
of the vessel isjustfull .Findthe rainfall incm.
[Ans2.5 cm]
4. The radiusof the base and the heightof a solidrightcircularcylinderare inthe ratioof 2:3 andits volume is
1617 cm2
. Findthe total surface area of the cylinder.
[Ans-770 cm2
]
5. A semispherical bowlof internal radius9cm isfull of liquid.The liquidistobe filledintocylindrical shapedsmall
bottleseachof diameter3 cm and height4 cm. How many bottlesare neededtoemptythe bowl?
[Ans-54]
LEVEL-IV
1. A sphere of diameter12 cm isdroppedina right circularcylindrical vessel,partlyfilledwithwater.If the sphere
iscompletelysubmergedinwater,the waterlevelinthe cylindricalvessel risesby3 5/9 cm. Findthe diameter
of the cylindrical vessel.
[Ans-180cm]
2. A bucketmade upof metal sheetinthe formof a frustumof a cone.Its depthis24 cm andthe diametersof the
top andbottom are 30cm and 10cm respectively.Findthe costof milkwhichcancompletelyfillthe bucketat
rate of Rs 20 per metre andthe costof the metal sheetused,if itcostsRs 10 per 100 cm2
.
(use [Ans- Rs163.28 Rs 171.13]
3. A vessel isinthe formof a semi spherical bowlmountedbyahollow cylinder.The diameterof the semisphere is
14cm and the total heightof the vessel is13 cm.Findthe capacity of the vessel.(Take x=22/7] [Ans-
1642.67 cm3
approx]
4. If the radii of the endsof a bucket,45 cm height,are 28 cm and 7 cm determine the capacityandtotal suface
area of the bucket,
[Ans-4850 cm3
,5616.6cm2
]
5. A toyis inthe form of a cone mountedona hemisphere of commonbase radius7cm.The total heightof the toy
is13 cm . Findthe total surface of the toy.( =22/7)
[Ans- 858 cm2
]

61
SELF EVALUATION QUESTION
1. The base radii of tworight circularcone of the same heightare in the ratio 3:5. Findthe ratio of theirvolumes.
2. If a,b,c are the dimensionsof acuboid,S be the total surface areaand v itsvolume thenprove that
1/v=2/s(1/a+1/b+1/c).
3. If h, c, v respectivelyare the height,the curvedsurface areaandvolume of acone prove that
3vh3
-c2
h2
+9v2
=0
4. A toyis inthe form of a cone mountedona hemisphere of radius3.5cm. If the total heightof the toy is15.5
cm. Findthe volume of the toy.(use  =22/7).

62
PROBABLITY
TAKE A LOOK
1. Probability:- The theoretical probabilityof aneventE,writtenasP(E) is definedas.
P(E)=Numberof outcomesfavourable toE
Numberof all possibleoutcomesof the experiment
Where we assume that the outcomesof the experimentare equallylikely.
2. The probabilityof asure event(orcertainevent) is1.
3. The probabilityof animpossibleeventis0.
4. The probabilityof anEventE isnumberP(E) such that0≤P(E)≤1.
5. Elementaryevents:- Aneventhavingonlyone outcome iscalledanelementaryevent.The sumof the
probabilitiesof all the elementary eventsof anexperimentis1.
6. For any eventE,P(E)+P( )=1,where standsfornotE, E and are calledcomplementaryevent.
7. Performingexperiments:-
a. Tossinga coin.
b. Throwinga die.
c. Drawinga card fromdeckof 52 cards.
8. Sample space:-The setof all possible outcomesinanexperimentiscalledsample space.
LEVEL-I
1. Write a sample space of
a. Tossinga coin.
b. Throwinga die. Ans-{H,T},{1,2,3,4,5,6}
2. Define probability(Theoreticalprobabilityof anevent).
3. A card is drawnfroma well-shuffledpackof 52 cards what isthe probabilitythatitisan ace? Ans-1/13
4. A dice isthrownonce.Findthe probabilityof gettinganumbergreaterthan3.
Ans: ½
5. What isthe probabilitythatanumberselectedfromthe number1,2,3………………16 is prime number?
Ans-3/8
6. A letterischosenatrandom fromthe Englishalphabet.Findthe probabilitythatthe letterchosenprecedes‘g’.
Ans-3/13
7. Findthe probabilityof gettingaredheart. Ans-1/4
8. A coinis tossedtwice.Findthe probabilityof gettingatleastone head. Ans-3/4
9. What isthe probabilityof asure event? Ans-1

63
LEVEL-II
1. One card is drawnfrom a well shuffleddeckof 52 cards.Findthe probabilityof getting.
i. An ace
ii. A face card Ans- i- 1/13
ii- 3/13
2. A bag contains5 red balls,4 greenball and7 white balls.A ball isdrawnat randomfromthe bag.Findthe
probabilitythatthe ball drawnis(i) White (II) neitherRednorWhite.
Ans- (i) 7/16
(II)1/4
3. Findthe probabilityof getting53Fridayina leapyear. Ans-2/7
4. Two dice are thrownsimultaneously.Whatisthe probabilitythat.
i. 5 will come uponat leastone?
ii. 5 will come upat bothdice?
Ans-(i)11/36
(ii)1/36
5. Two coinsare tossedonce.Findthe probabilityof getting.
i. Exactlyone head Ans-(i)1/2
ii. Almostone head (ii)3/4
6. In a lotterythere are 10 prizesand25 blank.Findthe probabilityof gettingaprize.
Ans-2/7
7. The king,the queenandthe jack of clubsare removedfroma deckof 52 playingcardsand the remainingcards
are shuffled.A cardis drawnfrom the remainingcards.Findthe probabilityof gettingacard of .
i. Heart Ans-(i)13/49(ii)3/49(iii)10/49
ii. Queen
iii. Clubs
LEVEL - III
1. Nidhi andNishaare twofriends.Whatisthe probabilitythatbothwill have
a. Same birthday Ans-(a) 1/365
b. Differentbirthday(ignore the leapyear) (b)364/365
2. A box contains20 ballsbearingnumber 1,2,3,4,……..20.A ball isdrawn at randomfrom the box.What isthe
probabilitythatthe numberonthe ball is.
a. An oddnumber Ans-(a)1/2(b)13/20 (c)2/5
b. divisible by2or 3
c. Prime number
3. A card is drawnat random froma well shuffleddeckof 52 cards. What isthe probabilityof drawing
a. Kingor a spade Ans-(a) 4/13 (b)3/4(c)5/52
b. A nonspade
c. Eithera kingor a 10 of heart
4. Tom was borninFebruary2000. What isthe probabilitythathe wasbornon 13th
Feb?
Ans-1/29
5. Are the followingoutcomesequallylikelyornot?A baby isborn “ It is a boyor a girl”.
Ans-Yesequallylikely
6. Findthe probabilityof getting53Mondays ina leapyear53 Tuesdayina nonleapyear.
Ans-2/7and 1/7
7. A letterisselectedfromthe letterof wordMATHEMATICS. What is the probabilitythatit isM?
Ans-2/11

64
8. In an N.C.Ccamp there are 20 boysand 15 girls.The bestcadet isto be chosen.Whatisthe probabilitythat
the bestcadet isa girl? Ans-3/7
9. Out of 400 bulbsina box 15 bulbsare defectiveone bulbis takenoutatrandom fromthe box.Findthe
probabilitythatthe drawnbulbisnotdefective. Ans-77/80
Self evaluationquestions
1. A bag contains5 white balls,7red ballsand2 blue balls.One ball isdrawnatrandomfrom the bag whatis probability
that bulbdrawnis
a. White or blue
b. Black or red
c. Notwhite
2. A childhas a die whose six facesshowthe lettersasgivenbelow.
A B C D E A
If a die isthrownonce findprobabilityof gettingA andD.
3. Findthe probabilityof getting53 Sundaysina leapyear.
4. From a well shuffledpackof 52 cards, a card is drawnat random.Findthe probabilitythatitisa:-
a. Spade
b. King
c. Club
d. Queen
e. A redcard
f. The black king
g. The queenof diamonds.
5. Two dice are thrownat the same time findthe probabilityof getting.
a. Same numberonboth dice.
b. Differentnumberonbothdice.
6. Someone isaskedtotake a numberfrom1 to 100. Find the probabilitythatitisnota prime number.
7. Card markedwithnumber5 to 50 are placedina box and mixedthroughout.A cardisdrawn fromthe
box at random.Findthe probabilitythatthe numberon the takenoutcard is
a. A Prime numberlessthan10
b. A numberwhichisa perfectsquare
8.There are 20 cards numbered1,2,3,………20 in a box.One card is drawn.Findthe probabilitythatthe numberonthe
card is
a.A numberdivisibleby6
b.A numberdivisibleby7

65
FORMATIVE ASSESSMENT – III
TIME: 1 ½ HR MARKS : 40
SECTION A
(EACH QUESTION CARRIES 1 MARK)
Q1. Whichof the following equationshastwodistinctreal roots?
A).2X2
+ 3√2 X+9/4 =0 b). x 2
+x -5 =0
C).x2
+3x +2√2 = 0 d).5x2
– 3x +1 = 0
Ans: (b)
Q2. The sum of first16 termsof the A.P.: 10,6,2, …………… is.
a). -320 b).320 c) -352 d). -400.
Ans:(a)
Q3: The point(-4,0),(4,0) ,(0,3) are the verticesof a.
a). rightangledtriangle b).Isoscelestriangle c). Equilateral triangle
d).scalene triangle
Ans: (b)
Q4: A pole 6m highcasts a shadow2√3m long an the ground,thenthe sun’s elevationis.
a).60 0
b).45 0
c).30 0
d).90 0
Ans:(a)
Q5. If radii of twoconcentriccirclesare 4cm and 5cm,
Thenthe lengthof each chordof one circle whichistangentto othercircle is:
a).3cm b).6cm c). 9cm d). 1cm
Ans: (b)
SECTION B
Q6.Finda relationbetweenx andy suchthat the point(x,y) isequidistantfromthe point(3,6)
and(-3,4).
Ans: 3x+y-5=0.
Q7.Checkwhetherthe followingequationisaquadraticequation.
x2
-4x+6=0 Ans: yes
Q8.The angle of elevationof the topof a towerfroma pointonthe ground,whichis30m away fromthe foot of the
toweris300
. Findthe heightof the tower.
Ans: 10√3 m.
Q9.Findthe sum of the followingA.P.:
-37,-33,-29,………………….. to 12 terms.
Ans:
Q10. The lengthof a tangentfrom a point A at distance 5cm from the centerof the circle is4cm.
Findthe radiusof the circle.
Ans:-3cm
Q11. Findthe distance betweenthe points[ -8/5,2] and [2/5,2].
Ans:2 units.
SECTION C
Q12.If (1,2),(4,y),(x,6) and(3,5) are the verticesof a gm takenin order,findx and y
Ans:x=6; y=3.
Q13. The angle of elevationof the top of a buildingfromthe footof the toweris300
andthe angle of elevationof
the top of towerfrom the footof the buildingis600
.If the toweris50 meterhighfindthe heightof the building.
Ans:16 2/3 m.
Q14. If triangle ABCisdrownto circumscribe a circle of radius4cm, suchthat the segmentBD
AndDC intowhichBC is dividedbythe pointof contactD are of the length8cm and 6cm respectively.Findthe sides
AB and AC(fig.)

66
A
F E
C B
6 cm D 8cm
Q15. Findthe value of k forthe followingquadraticequation,sothattheyhave twoequal roots.
KX(X-2)+6=0
Ans:k=6.
Q16. Whichterm of the A.P.3,15,27,39,…………………….will be 132 more thanits 54th
term.
Ans:65th
term
SECTION D
Q17. Twowater tapstogethercan fill atank in9 (3/8) hours.The tap of longerdiametertakes10 hourslessthan the
smallerone tofill the tankseparately.Findthe time inwhicheachtapcan separatelyfill the tank.
Ans: longertap=15 hours. Smallertap=25 hours.
Q18. If the angle of elevationof acloudfroma pointh metersabove a lake is and the depressionof itsreflection
inthe lake , prove that the heightof the cloudis
h(tan +tan )
tan - tan
OR
The length h of tangentsdrownfroman external pointtoa circle are equal,prove it.
o

67
MODEL QUESTION PAPER
SA-II
CLASS-X TIME:3 to 3.1/2 hr
SUBJECT- MATHS MM: 80
General Instructions:
i. All questionsare compulsory.
ii. The questionpaperconsists of 34 questionsdividedintofoursection- A,B,CandD.
iii. SectionA contains10 questionof 1 mark eachwhichare multiple choicetype questions.SectionB
contains8 questionof 2 marks each,SectionC contains10 questionof 3 marks eachand SectionD
contains6 questionof 4 marks each.
iv. There isno overall choice inthe paper.However,internal choice isprovidedinone questionof 2marks ,
three questionsof 3marks and twoquestionsof 4 marks.
v. Use of calculatoris notpermitted.
SECTION A
QuestionNumbers1to10 carries 1 mark each.For each of the questionnumber1to10, fouralternative choice have
beenprovided,of whichonlyone iscorrect.Selectthe correctchoice.
1. The perimeter(incm) of square circumscribingacircle of radiusa cm, is
A. 8 a
B. 4 a
C. 2 a
D. 16 a
2. If A andB are the points(-6,7) and(-1,-5) respectively,thenthe distance 2AB isequal to
A. 13
B. 26
C. 169
D. 238
3. If the commondifferenceof anA.Pis 3, then is.
A. 5
B. 3
C. 15
D. 20
4. If P ( a/2,4) isthe midpointof the line segmentjoiningthe pointsA(-6,5) andB(-2,3),thenthe value of
a is .
A. -8
B. 3
C. -4
D. 4
5. In Figure 1 .○ O isthe centre of a circle,PQ isa chord and PT isthe tangentat P.If <PQR=700
, then<TPQ
isequal to

68
Figure 1
A. 550
B. 700
C. 450
D. 350
6. In Figure 2, ABand AC are tangenttothe circle withcentre Osuch that <BAC =400
Then<BOC is equal to
B
A 400
C
Figure- 2
A. 400
B. 500
C. 1400
D. 1500
7. The roots of the equationx2
-3x-m(m+3)=0,where misa constant are
A.m, m+3
B. –m, m+3
C. m, -(m+3)
D.–m , -(m+3)
8. A towerstandsverticallyonthe ground.Froma pointon the groundwhichis25m away fromthe footof
the tower,the angle of elevationof the topof the toweris foundto be 450
. Thenthe height(inmeters)
of the toweris
A. 25√2
B. 25√3
C. 25
D. 12.5
9. The surface area of solidhemisphere of radiusrcm (incm2
) is
A.
B. 3
C. 4
D.

69
10. A card is drawnfroma well-shuffleddeckof 52 playingcards.The probabilitythatthe cardis nota red
kingis
A.
B.
C.
D.
SECTION – B
QuestionNumbers11 to 18 carry 2 marks each.
11. Two cubeseachof volume 27 cm3
are joinedendtoendto form a solid.Findthe surface areaof the resulting
cuboid.
OR
A cone of height20 cm and radiusof base 5 cm ismade up of modelingclay.A childreshapesitinthe formof a
sphere.Findthe diameterof the sphere.
12. Findthe value of y for whichthe distance betweenthe pointsA(3,-1) andB(11,y) is10 units.
13. A ticketisdrawnat randomfroma bag containingticketsnumberedfrom1to 40. Findthe probabilitythatthe
selectedtickethasanumberwhichisa multiple of 5.
14. Findthe value of m so that quadraticequationmx(x-7) +49 =0 has twoequal roots.
15. In figure 3 , a circle touchesall the fourside of a quadrilateral ABCDwhose sidesare AB=6cm, BC=9cm and
CD=8 cm. Findthe lengthof side AD.
D C
A B
Figure-3
16. Draw a line segmentABof length7 cm. Usingrulerand compasses,find a pointP onAB such that = .
17. Whichterm of the A.P3,14,25,36,… will be 99 more that its 25th
term?
18. In Figure 4, a semi circle isdrawnwithO as centerand ABas diameter.Semi-circleare drawnwithAOand OB as
diameters.If AB=28 m,Findthe perimeterof the shadedregion.[Use
Figure- 4

70
SECTION-C
Question Numbers 19 to 28 carry 3 marks each.
19. Draw a right triangle ABCinwhichthe sides(otherthanhypotenuse)are of length4 cm and 3 cm. Then
construct anothertriangle whose sidesare 3/5 timesthe correspondingsidesof the giventriangle.
20. Solve the followingquadraticequationforx. p2
x2
-(p2
- q2
) x –q2
=0
OR,
Findthe value of ‘k’so that the quadraticequation2kx 2
- 40x +25 = 0 has real andequal roots.
21. Findthe sum of first15 termsof an APwhose nth
termisgivenby9 -5n..
OR
Findthe sum of all 3digitnumberswhichleavesthe remainder3whendividedby5.
22. A bag contains4 redballs,5 blackball and 6 white balls.A ball isdrawnfromthe bag at random.Findthe
probabilitythatthe ball drawnis:
a. red
b. white
c. not black
d. redor black
23. If the area of ∆ ABCwhose verticesare (2,4),(5,k) and(3,10) is 15 sq units,findthe value of ‘k’
OR
Findrelationbetweenx andysuch that the point(x,y)isequidistantfromthe points(3,6) and (-3,4).
24. A kite isflyingatheightof 60m above the ground.The stringattachedto the kite istemporarilytiedtoa point
on the ground.The inclinationof the stringtothe groundis 600
. Findthe lengthof the stringassumingthat
there isno slackin the string.
25. Three spherical metal ball of radii 6cm,8cm andx cm are meltedandrecastedintosingle sphere of radius12
cm. Findthe radiusof 3rd
sphere.
26. XY and X’Y’are twoparallel tangentstoa circle withcenterO andanothertangentAB withpointof contact C,
intersectingXYatA and X’Y’at B, is drawn.Prove that AOB=900
27. ABCD isSquare Park of side 10 m. It isdividedinto4equal squaresparks.Semi circularflowerbedsare made as
shownbythe shadedregion.Findthe areaof shadedregion.

71
28. Twoverticesof a triangle ABCare A(-7,6) andB(8,5). If the midpointof the side ACisat -
5/2,2 , Findthe coordinate of vertex C.
SECTION-D
QUESTION NUMBERS 29 TO 34 CARRY 4 MARKS EACH
29. Prove that the lengthof tangentsdrawnfroman external pointtoacircle are equal.
30. A shopkeeperbuysanumberof booksfor RS 1200. If he had bought10 more booksfor the same amount,each
bookwouldhave costhim Rs.20 less.How manybooksdidhe buy?
OR,
. A passengertraintake one hour lessfora journeyof 360 km if itsspeedisincreasedby5 km/ hr.from itsusual
speed.Calculate the usual speedof the train.
31. A man standingonthe deckof a ship whichis10m above the waterlevel,observesthe angle of elevationof the
top of a hill as600
andangle of depressionof the base of the hill as300
. Calculate the distance of the hill from
shipand the heightof the hill.
32. A chord of a circle of diameter24 cm subtendsanangle 600
at the centerof the circle.Findthe areaof the
corresponding.
(a) Minor sector
(b) Minor segment
33. How manycoins1.75 cm indiameterand2 mm thickmust be meltedtoforma cuboidof dimensions11cm x 10
cm x 7 cm.
OR
If the radii of the circular endsof a bucketinthe shape of frustumof a cone whichis45 cm highare 28 cm and
7cm , findthe capacityof the bucket.
34. The sum of first15 termsof an AP is105 and the sumof the next15 termsis780. Findthe first 3 termsof the
arithmeticprogression.

72
SA-II
MARKING SCHEME
CLASS-X (MATHS)
EXPECTED ANSWERS/VALUE POINTS
SECTION-A
Q . No Marks
1.(A):8a 2. (B):26 3.(C):15 1X10 =10
4. (A) : -8 5. (D):350
6. (C):1400
7. (B): -m,m+3 8. (C):25 9. (B): 3
10.(D)
SECTION-B
11. Gettingside of cube=3 cm
1/2m
Dimensionsof the resultingcuboidsare 6cm x 3cm x 3cm 1/2m
 Surface area = 2(6x3+3x3+3x6) cm2
=90 cm2
1m
OR
Volume of cone =1/3 (5)2
.20cm2
½ m
 4/3 r2
=1/3 (5)2
.20 r=5cm 1m
Diameter=10cm 1/2m
12. AB= = 1/2m
=> 65+y2
+2y=100 or y2+2y-35=0 1/2m
 y=-7 or y=5 1m
13. Total numberof tickets=40

73
Numberof ticketwithnumberdivisible by5=8 1m
 Requiredprobability=8/40 or 1/5 1m
14. mx(x-7)+49=0 => mx2
-7mx+49=0
For equal rootsB2
-4AC=0 ½ m
½ m
(-7m)2
-4(m)(49)=0
=>m=0 or m=4 ½ m
But m≠0 m=4 ½ m
15. Let P,Q,R,Sbe the pointsof contact
D R  AP=AS
C PB=BQ
DR=DS
S Q
CR=CQ ½ m
A p B
 (AP+PB)+(DR+CR)=(AS+DS)+(BQ+CQ) ½ m
AB+CD=AD+BC
6+8=AD+9=>AD=5CM 1 m
16. For writing(orusingAP:PB=3:2 For correct construction
1 ½ m
17. a25=3+24x11=3+264=267 1/2m
267+99=366=3+(n-1)11 1m
n=34 1/2m
18. Perimeter(of shadedregion)=circumferenceof semicircle of radius14 cm 1m
+2(circumference of semicircle of radius7cm) 1/2m
= (14)+2 x7=28 =28 x 22/7 =88cm 1m

74
SECTION-C
19. Construction∆ABC 1 m
Equal angles,construction ║lines 1m
Constructionof similartriangle 1m
20. P2
x2
–P2
x + q2
x- q2
=0 ½m
P2
x(x-1)+q2
(x-1)=0 1m
(p2
x+q2
) (x-1)=0 1m
x=1, -q2
/P2
½ m
OR
For real and equal roots
b2
-4ac=0 1 m
a=2k,b=-40,c=25 ½ m
(-40)2
-4(2K)(25)=0 1 m
16000-200K=0
200K=1600
K=8 ½ m
21. =9-5n
=4 ; = -1, = -6 (A.P) ½ m
a=4;d=-1-4=-5 ½ m
Sn=n/2[ 2a+(n-1)d] ½ m
=15/2[2(4)+(15-1)(-5)] ½ m
=15/2 [8-70]=15/2 x -62=15x-31 =-465 1 m
OR
3 digitsnumbersae
100,101___________________ 999
Nos.that leave remainder3on dividing
by 5 are: 103:108__________998 ½ m
a= 103 d=5 ½ m

75
Sn=n/2[ 2a+(n-1)d]
n=?
a=a + (n-1) d 1 m
998=103 + (n-1)5
N=180
sn=180/2 [2(103)+(180-1)5]
=90[206+895] 1 m
=99090
22. Total no.of balls= R B W
4 + 5 + 6 =15 ½ m
P(Redball)= = 4 ½ m
15
P(White ball)= =
P(Notblack)= = ½ x 4=2 m
P(redor black)= =
23. ( , ) (2,4) ( , ) = (3,10) ½ m
( , ) (5,K)
Areaof ∆ =1/2 [ ( ) + ( ) + ( )] 1 m
=>½ [ s(k-10)+5(10-4) +3(4-k)] =15 ½ m
=>2k -20 +50 -20 +12 -3k =30
1 m
=> -k+22= + 30
K=-8 or 52

76
OR
Distance formulae ½ m
√ [(x-3)2
+(y-6)2
] = √ [(x+3)2
+(y-4)2
] ½ m
2 m
Solvingandgetting3x+y=5
A Correctfig.: ½ m
24.
string
60 m
B C
½ m
AB=ht of the kite above the
ground level =60
AC=String
<ACB=angle of elevation=600
1 m
= =cosec
= =cosec =
1m
AC= = =40 m
25. + + = 1m
+ =
63
+ 83
+x3
= 123
X3
=123
-63
- 83
1m
= 1728 –216 – 512 = 1000.
 X3
=1000
X= 10 cm 1m
600
666
0

77
26.Join OC(lengthof tangentsfromExtpoint)
∆AOP ∆ AOCand ∆ BOQ ∆BOC (RHS ) 1m
POA =COA and COB=BOQ (cpct) 1m
POA +COA+COB+BOQ= 1800
AOB=900
1m
27. Side of the square =10m
Diameterof semi circle =5m
Radius= 5/2 m
2 semi circlesof equal radii =1 circle
4 semi circlesof equal radii =2 circles 1m
Hence Area of shadedregion.= 2 1m
= 2x22/7x(5/2)2
= 275/7
= 39.3 m2
1m
28. ( , ) (-7, 6)
( , ) (8,5)
( , ) ?
Let the coordinatesof Cbe(x,y)&
A( -7,6)
Mid pointof ACis 1m
 = -5/2, 1m
 X=2, = 2 => y = -2 . 1m
Hence coordinate of C are (2,-2)
SECTION-D
29. (Correctfigure) 1m
(Giventoprove) 1m
(Correctproof) 2m
30.Total amountspent=Rs 1200
Let the numberof booksoriginallybought=x
Cost of 1 book= Rs
1m
If the no.of books=x+10

78
Cost of each books= Rs
- =20 m
+10x-600=0
(x+30)(x-20)=0
X=20, -30
Rejectingx=-30 2 m
We getnumberof books=20
OR
Let the usual speed=x km/hr ½ m
 - =1 1½ m
X2
+5x -1800=0
D=b2
-4ac=7225
X=-45(rejected 2m
Or,X=40 km/hr

79
A
31.
hill
P D
10m
Correctfigure 1m
C B
AB=hill
1m
P=positionof the manon the deck
PC=10m =DB
APD=600
; BPD=300
To findA B and BC
∆PDB, tan 300
= PD=10
∆PDB, tan 600
= AD=PD
= 10 x =30m
 Distance of the hill fromthe ship
=BC=PD=10 =17.3m Approx 1m
Ht of the hill=AB=AD+DB 1m
=30+10=40m

80
32.
A B p
Diameter=24cm
Radius=12cm.
Areaof the minorsector
= x
x x12 x 12 2m
=75.4 cm2
Areaof ∆ OAB =1/2 x r2
sin
= 1/2x 12x12x /2
=62.35 cm 2
1m
 area of minorsegment=areaof minorsector- Areaof ∆
=75.4 – 62.35 =13.05 cm2
1m
33.Let the numberof coins=n
D= 1.75 cm ; r= = 0.875 cm
H=2mm =0.2 cm
Vol of each coin=
= x 0.2 cm3
2m
vol of n coins= x 0.875 x 0.875 x0.2 x n
Vol of ‘n’ coins=volume of cuboid 1m
o

81
x 0.875 x 0.875 x0.2 x n =11x10 x 7
1m
n= 1600
OR,
Volume (frustum) = h ( + ) 1m
= x x 45[(28)2
+(7)2
+(28)(7)] 2m
= 48510cm3
1m
34.Sum of first15 terms=105
Sumof next15 terms=780 1m
Sumof first30 terms=780+105
=885
1m
 [ 2a+(15-1)d]=105 and
[ 2a+(30-1)d]=885
 2a+14d=14 and 1m
2a+29d=59
a=-14 andd=3
The firstthree termsare 1m
a,a+d,a+2d……….
i.e. -14,-11,-8

82
ACTIVITES (TERM-I)
(Any Eight)
Activity1: To findthe HCF of twoNumbersExperimentallyBasedonEuclidDivisionLemma
Activity2: To Draw the Graph of a Quadratic Polynomialandobserve:
i. The shape of the curve whenthe coefficientof x2
ispositive
ii. The shape of the curve whenthe coefficientof x2
isnegative
iii. Its numberiszero
Activity3: To obtainthe zero of a linerPolynomial Geometrically
Activity4: To obtainthe conditionforconsistencyof systemof linerEquationsintwovariables
Activity5: To Draw a Systemof SimilarSquares,UsingtwointersectingStripswithnails
Activity6: To Draw a Systemof similarTrianglesUsingYshapedStripswithnails
Activity7: To verifyBasicproportionalitytheoremusingparallel line board
Activity8: To verifythe theorem:Ratioof the Areasof Two SimilarTrianglesisEqual tothe Ratioof the Squaresof
theircorrespondingsidesthroughpapercutting.
Activity9: To verifyPythagorasTheorembypapercutting,paperfoldingandadjusting(Arranging)
Activity10: Verifythattwofigures(objects) havingthe same shape (andnotNecessarilythe same size) are similar
figures.Extendthe similaritycriteriontoTriangles.
Activity11: To findthe Average Height(incm) of studentsstudyinginaschool.
Activity12: To Draw a cumulative frequencycurve (oranogive) of lessthantype.
Activity13: To Draw a cumulative frequencycurve (oranogive ) of more than type.

83
ACTIVITES (TERM-II)
(Any Eight)
Activity1: To findGeometricallythe solutionof aQuadraticEquationax2
+bx+c=0, a 0 (where a=1) byusingthe
methodof completingthe square.
Activity2: To verifythatgivensequenceisanA.P(ArithmeticProgression) bythe paperCuttingandPaperFolding.
Activity3: To verifythat by Graphical method
Activity4: To verifyexperimentallythatthe tangentatanypointto a circle isperpendiculartothe Radiusthrough
that point.
Activity5: To findthe numberof Tangents froma pointtothe circle
Activity6: To verifythatlengthsof TangentsDrawnfroman External Point,toa circle are equal byusingmethodof
papercutting, paperfoldingandpasting.
Activity7: To Draw a Quadrilateral Similartoa givenQuadrilateral aspergivenscale factor(Lessthan1)
Activity8: (a) To make mathematical instrumentclinometer(orsextant) formeasuringthe angle of
elevation/depressionof anobject
(b) To calculate the heightof an objectmakinguse of clinometers(orsextant)
Activity9: To get familiarwiththe ideaof probabilityof aneventthroughadouble colorcard experiment.
Activity10: To verifyexperimentallythatthe probabilityof gettingtwotailswhentwocoinare tossed
simultaneouslyis¼=(o.25) (Byeightytossesof twocoins)
Activity11: To findthe distance betweentwoobjectsbyphysicallydemonstratingthe positionof the twoobjects
say twoBoys ina Hall, takinga setof reference axeswiththe cornerof the hall as origin.
Activity12: Divisionof line segment bytakingsuitable pointsthatintersectsthe axesatsome pointsandthen
verifyingsectionformula.
Activity13: To verifythe formulaforthe areaof a triangle bygraphical method.
Activity14: To obtainformulaforAreaof a circle experimentally.
Activity15: To give a suggestive demonstrationof the formulaforthe surface Areaof a circus Tent.
Activity16: To obtainthe formulaforthe volume of Frustumof a cone.

84
PROJECTS
Project1 : Efficiencyinpacking
Project2 : GeometryinDailyLife
Project3: Experimentonprobability
Project4: DisplacementandRotationof a Geometrical Figure
Project5: Frequencyof letters/wordsinalanguage text.
Project 6: PythagorasTheoremanditsExtension
Project 7: Volume andsurface areaof cube andcuboid.
Project 8: GoldenRectangle andgoldenRatio
Project9 : Male-Female ratio
Project10 : BodyMass Index(BMI)
Project11 : Historyof IndianMathematiciansandMathematics
Project12 : CareerOpportunities
Project 13 : (Pie)
ProjectWork Assignment(AnyEight)

85
ACTIVITY- 1
TOPIC:- Prime factorizationof compositenumbers.
OBJECTIVE:- To verifythe prime factorization150 inthe form
52
x3x2 i.e 150=52
x3x2.
PRE-REQUISITEKNOWLEDGE:- For a prime numberP,P2
can be representedbythe areaof a square
whose eachside of lengthPunits.
MATERIALS REQUIRED:-
i. A sheetof graphpaper ( Pink/ Green)
ii. Colored(black) ball pointpen.
iii. A scale
TO PERFORMTHE ACTIVITY:-
Steps:-
1. Draw a square on the graph paper whose eachside isof length5 cm andthenmake partitionof this
square into25 small squaresasshownin fig1.1 eachsmall square hasits side of length1cm.
Here,we observe thatthe area of the square havingside of length5 cm =52
cm2
= 25 cm2
2. As showninFig1.2 draw there equal squareswhere eachsquare isof same size asinfigure 1.1 then
the total areain the fig1.2
=52
+52
+52
cm2
=53
x3cm2
ie,75 cm2
Fig=1.1
Fig=1.2

86
3. As showninfig1.3 drawsix equal square where eachsquare isas same size asin Fig1.1 Here , three
squaresare in one row andthree squaresinthe secondrow.
We observe thatthe total area of six squares
=52
x3cm2
+52
x3cm2
=c52
x3x52
x3cm2
=32
x3x2 cm2
Alsoobserve thatthe total area
=75cm2
+75cm2
=150cm2
Hence,we have verifiedthat
150=52
x3x2
Fig-1.3

87
ACTIVITY-2
TOPIC:- Ratioof the areas of two similartriangles
STATEMENT:- The ratioof the area of two similartriangle isequal tothe ratioof the squaresof their
correspondingsides.
OBJECTIVE:- To verifythe above statementthroughactivity.
PRE-REQUISITE KNOWLEDGE:-
1. The concept of similartriangles.
2. Divisionof aline segmentintoequal parts.
3. Constructionof linesparallel togivenline.
MATERIAL REQUIRED:-
1. White papersheet
2. Scale /Rubber
3. Paintbox
4. Black ball pointpenorpencil
STEPS:-
1. On the posterpapersheet,drawtwosimilartriangle ABCandDEF.We have the ratio of their
correspondingsidessame andletashave
AB: DE= BC: EF=CA:FD=5:3
ie , AB/DE=5/3 , BC/EF=5/3 , CA/FD=5/3,
ie DE =3/5 AB,EF=3/5 BC,FD=3/5 CA
2. Divide eachside of ∆ABCinto5 equal partsand those of ∆DEF into3 equal partsas showninFig(i)
and (ii).
3. By drawingparallel linesasshowninFig (i) and(ii).,we have partition∆ABCinto25 smallertriangleof
same size andalso eachsmallertriangle infig(i) hassame size andasthat of the smallertriangle fig
(ii).
4. Paintthe smallertriangle asshowninFig(i) and(ii)..

88
OBSERVATION:-
1. Areaof ∆ABC= Areaof 25 smallertriangle infig(i)=25square unit
Where area of one smallertriangle infig(i)=P(squareunit)
2. Areaof ∆DEF=Area of a smallertriangle inFig(ii)=9pwhere areaof one smallertriangleinfig(ii)=P
square units.
3. Areaof ∆ ABC = 25 P =25
Areaof ∆DEF 9P 9
4. (AB)2
(AB)2
(AB)2
25
(DE)2 =
(3/5AB2
) =
9/25(AB)2 =
9
Similarly
(BC)2
25 (CA)2
25
(EF)2 =
9 and (FD)2
= 9
5. From steps(3) and (4) , we conclude that
Areaof ∆ ABC (AB)2
(BC)2
(CA)2
Areaof ∆DEF =
(DE)2 =
(EF)2 =
(FD)2
Hence the ratio of the areas of twosimilartrianglesisequal tothe ratioof the squaresof their
correspondingsides.

89
ACTIVITY-3
TOPIC:- Trigonometricidentities.
STATEMENT:- sin2
θ+ cos2
θ=1,00
< θ<900
OBJECTIVE:- To verifythe above identity
PRE-REQUISITEKNOWLEDGE:- In a rightangledtriangle.
Side opposite toangle θ
sinθ = Hypotenuse of the triangle
Side adjacenttoangle θ
cos θ = Hypotenuse of the triangle
MATERIAL REQUIRED:-
1. Drawingsheet
2. Black ball pointpen
3. Geometrybox
4. Scale
TO PERFORMTHE ACTIVITY
Step:-
1. On the drawingsheet,draw horizontal rayAx .
2. Constructany arbitrary(CAX=θcsay)
3. Cot AC=10 cm.
4. From c draw CB AX.
5. Measure the lengthsidesof sidesABandBC of the rightangled∆ ABC (see fig)
6. We measure thatAB=8.4 cm (approx) andBC=5.4 cm (approx)
OBSERVATION
1. Sinθ= BC/AC=5.4/10=.54 (Approx)
2. Cosθ=AB/AC=8.4/10=.84(approx)
3. Sin2
θ +cos2
θ=(.54)2
+(.84)2
=.2916+.7056
=.9972(Approx)
Ie.Sin2
θ+Cos2
θ is mearlyequal to1. Hence the identityisverified.
C
10 cm
5.4cm
A B x
8.4
8.4

90
ACTIVITY-4
Topics:- Measure of the central tendenciesof adata.
STATEMENT:- We have an empirical relationshipforstatistical dataas3x median=Mode+2
x mean.
OBJECTIVE:- To verifythe above statementfora data.
PRE-REQUISITEKNOWLEDGE:-
Methodto findcentral tendenciesforgroupeddata.
MATERIAL REQUIRED:-
1. A data aboutthe heightsof studentsof a classand arrangedin groupedform.
2. A ball pointpen.
3. A scale.
Step:-
1. Countthe numberof girl studentsinthe class.The numberis51
2. Recordthe data abouttheirheightincentimeter.
3. Write the data in groupedformas below:-
Heightin
cm
135-
140
140-
145
145-
150
150-
155
155-
160
160-
165
Total no
of girls
Numberof
girls
4 7 18 11 6 5 51
4.On three differentsheetsof paperfindmeanheightonthe sheetof paper, medianheightonthe second
and the modal heightonthe thirdsheetof paper.
5.Let usfindmeanby stepdeviationmethod:-
Classof heights
(incm)
Frequency
-p
Classmark
xi U1=
a= 147.5,h=5
F1xu1
135-140
140-145
145-150
150-155
155-160
160-165
4
7
18
11
6
5
137.5
142.5
147.5
152.5
157.5
162.5
-2
-1
0
1
2
3
-8
-7
0
11
12
13

91
Mean=a+h x =147.5+5 x 23/51 =147.5+115/51
=(147.5+2.255)cm=149.755cm
6. Let us findmedianof the data:-
n/2=25.5
we have medianclass(145-150) it givesi=145,h=5,f=18,cf=11
median=1+ x h=145 + x5
=145+14.5 x5
18
=145+4.028
=149.028cm
7. Let usfindmode of the data:-
(Modal class)
Modal classis 145-150
Thus 12=45,h=5,fm=18,f1=7,f2=11
Mode=H xh=145 + x 5
=145+55/18 =145+3.055
=148.055 cm
Classof height(in
cm)
Frequency numberof girls Cumulative
frequency
135-140
140-145
145-150
150-155
155-160
160-165
4
7
18=f
11
6
5
4
11=cf
29
40
46
51
Total
n
Classof heights(in cm) FREQUENCY (Noof Girls)
135-140
140-145
145-150
150-155
155-160
160-165
4
7=f1
18=fm
11=f2
6
5
Total 51

92
8.CONCLUSION:-
Mean=149.755, median=149.028 and mode=148.055
3x median=3x149.028=447.084
Mode+xmean=148.055+2x149.755
=148.055+299.510=447.565
Thus we have verifiedthat3x median=mode +mean(Approx)
ACTIVITY – 5
TOPIC: Angle of Elevation
OBJECTIVE: To findthe angle of elevationof the sunat a particulartime ona sunnyday.
PRE-REQUISITEKNOWLEDGE: knowledge of trigonometricratios.
MATERIAL REQUIRED :
1. A metre rod
2. Measuringtape
3. Table for tangentof angles.
TO PERFORMTHE ACTIVITY:
STEPS :
1. On the particularsunnydayat the giventime,putthe metre rodon the level groundwithone endonthe
groundand the otherverticallyupward.
2. Measure the lengthof the shadowof the metre rodfrom the beginningtothe end.Letthe lengthof the shadow
be 58cm = 0.58m.
3. The lengthof the metre rod = 1m or 100cm.
OBSERVATION:
1. If  be the angle of elevationof the sunatthe givenmoment,thenwe have the followingfigure onasheetof
paperby takinga suitable scale.
2. From the rightangle OMP drawnin figure,we have
Tan =
Tan  = 3 (approx.)
i.e.tan = tan 60o
 = 60o
Hence,the requiredangle of elevationof the sunis60o
. For betterresult,we cantake the helpof the table of
tangentof angles.

93
ACTIVITY – 6
TOPIC - Probabilityof eventsof arandomexperiment.
STATEMENT: For an eventE of a randomexperiment,P(notE) = 1 – P(E).
OBJECTIVE:To verifythe above statementbytossingthree coinof differentdenominationssimultaneouslyforheadand
tail.EventE happensif we getat leasttwo headsandthe event not-Ehappensif we donot gettwo or more than two
heads.
PRE-REQUISITEKNOWLEDGE:
1. Probabilityof anevent: Numberof outcome whichfavourthe happeningof the eventE
Total numberof outcome
2. Eventnot-Ehappenswhenthe outcome isnot favourable forthe eventEto happen.
TO PERFORMTHE ACTIVITY:
STEPS:
1. Take three faircoins of differentdenominationsandtossthese coinssimultaneously.
2. We imagingaboutthe possible outcomesasbelow.
HHH, HHT, HTH, THH, HTT, TTH, TTT
i.e.there canbe 8 possible outcomes
favourable outcomestothe eventEare
HHH, HHT, HTH, THH
ThenP(E) = 4/8= ½
Now,favourable outcomestothe eventnot-Eare HTT, THT, TTH, TTT
ThenP(not-E) = 1- ½ = 1-P(E)
3. Repeatingabove randomexperiment,we recordthe observationof 20 trialsas below:
Numberof Heads: 0 1 2 3
Numberof timesoutof 20 trials: 4 7 5 4
4. From table instep3, we observe thatfor 2 headsor for 3 heads,the eventEhappensi.e.there are 5+4=9
chancesout of 20 whichfavourE
Thus,we have P(E) = 9/20.
Alsowe observe thatfor0 heador for 1 headthe eventnot-Ehappens.There are 4+7=11 chancesout of 20
whichfavournot-E.
So,P(not-E) = 11/20 = 1- 9/20 = 1-P(E).

94
QUIZ
(REAL NUMBERS)
Answerthe followingquestions
1. What isa lemma?
2. State Euclid’sDivisionLemma?
3. What doesHCF standfor?
4. Give the full formof LCM.
5. State Euclid’sdivisionalgorithm.
ORAL TEST
(REAL NUMBERS)
Answerthe followingquestions:
1. Euclid’sdivisionalgorithmisatechnique tocompute the ___________ of two givenpositiveintegers.
2. HCF(124, 24) is ___________.
3. “Everycomposite numbercanbe expressed(factorised)asa productof primes,andthisfactorisationisunique,
apart fromthe orderinwhichthe prime factors occurs”.The above statementiscalled___________.
4. For any twopositive integersaandb,
a x b = HCF(a,b) x ______
5. If a numbercannotbe writteninthe form p/q,where pand q are integersandq  0, thenitis called
____________.
QUIZ
(POLYNOMIALS)
1. What isa quadraticpolynomial?
2. What isthe degree of a quadraticpolynomial?
3. What are the zerosof a polynomial?
4. What isthe shape of curve of a quadraticpolynomial graph?
5. State remaindertheorem.
ORAL TEST
1. If P(x) isa polynomial inx,the highestpowerof x inP(x) iscalledthe ________ of the polynomial P(x).
2. A polynomial of degree 2iscalleda __________.
3. The linearpolynomialax + b,a 0, has exactlyone zero,namely,the x-coordinateof the pointwhere the graph
of y= ax + b intersectsthe _________.
4. A polynomial P(x) of degreenhasatmost________ zeroes.
5. The sum and the productof the zeroesof a quadraticpolynomial x2
+7x + 10 is ____ and _______.
QUIZ
(Pair of linearequationsin two variables)
1. What isa pairof linearequationsintwovariables?
2. Give the general formof a pair of linearequation?
3. What are the methodsof solvingapairof linearequationintwovariables?
4. What isthe conditionforinconsistentsolution?
5. What isthe shape of curve in graphof a linearequation?
Oral Test
1. Everysolution(x,y) of a linearequationintwovariables,ax+by+c= 0 correspondstoa ____ onthe line
representingthe equation,andvice versa.
2. If the pairof linearequationsintwovariableshave onlyone commonpointonboththe lines,thenwe have a
_______ solution.
3. A pairof equationswhichhasnosolutioniscalleda/an________ pairof linearequations.
4. Half the perimeterof a rectangulargarden,whose lengthis4m more than itswidthis36 m. The dimensionof
the gardenare ________ and ___________.
5. A pairof linearequationsintwovariablescanbe representedandsolvedbythe graphical methodand_______
method.

95
QUIZ
(Triangles)
1. What isSAS similaritycriterion?
2. What isthe relationshipbetweencongruencyandsimilarityof figures?
3. What isthe criteriaforthe similarityof twotriangles?
4. For whattypesof trianglesisPythagorastheoremapplicable?
5. What isthe anothername of Basic ProportionalityTheorem?
ORAL TEST
1. All _________ trianglesare similar(equilateral/isosceles/Scalene)
2. The longestside of a rightangledtriangle iscalled_________.
3. In a __________ the square of the hypotenuse isequal tothe sumof squaresof the othertwosides.
4. In the givenfigure,if DE||BC,thenthe value of x is_________
5. State whetherthe followingquadrilateral are similarornot.
QUIZ
(IntroductiontoTrigonometry)
1. What istrigonometry?
2. What are trigonometricratiosof anacute angle ina righttriangle?
3. From the figure findthe value of cosA.
4. Write the trigonometricratiosof 60o
.
5. Evaluate tan70o
/ cot 20o
.
ORAL TEST
1. In a righttriangle ABC,rightanglesat B, sinA = ______.
2. Sec(90o
– A) = __________
3. Sec2
A - _________ = 1 , for0o
 A  90o
.
4. If cot = 7/8, then(1+ sin )(1– sin)/(1+ cos )(1– cos ) = _____________
5. (1 – tan2
45o
)/( 1 +tan2
45o
)= ___________

96
QUIZ
(STATISTICS)
1. Name the measuresof central tendency.
2. What iscumulative frequency?
3. How will yourepresentthe cumulative frequencydistributiongraphically?
4. How will youfindthe medianof agroupeddatagraphicallywiththe helpof one ogive?
5. How will youfindthe medianof agroupeddatagraphicallywiththe helpof bothogives(i.e of the lessthantype
and of more than type)?
ORAL TEST
1. __________ is the sum of the valuesof all the observationsdividedbythe total numberof observations.
2. Classmark = _____ /2.
3. The formulafor findingthe meanusingthe stepdeviationmethodis_________.
4. The formulafor findingthe mode inagroupedfrequencydistributionis_________.
5. The formulafor findingthe medianof groupeddatais___________.
FORMATIVEASSESSMENT
QUIZ
1. Define the fundamental theoremof arithmetic.
2. Define euclid’sdivisionlemma.
3. What isa quadraticpolynomial.
4. What isthe relationshipbetweenzerosandcoefficientsof aquadraticpolynomial.
5. Give the conditionfora pairof linearequationstobe inconsistent.
ORAL TEST
1. For any twopositive integersaandb, HCF(a,b) x LCM(a, b) = _________
2. 5 – 3 is a/an________ number.
3. A polynomial of degree 3iscalleda ______ polynomial.
4. A quadraticpolynomial havingthe sumandproductof itszeroesrespectively5and 6 is________ .
5. All _______ trianglesare similar.(equilateral/isosceles/scalene).
QUIZ
QUADRATICEQUATION
1. What isa quadraticequation?
2. How manyroots can a quadraticequationhave?
3. Give the formulaforfindingthe rootsof ax2
+ bx + c = 0 (a 0)
4. Give the nature of rootsof the equationax2
+ bx + c = 0 (a 0)
5. Findthe nature of the roots of the equation3x2
– 2x +1/3 =0
ORAL TEST
1. A real number is saidto be a root of the quadraticequationax2
+ bx + c = 0 , if a2
+ b + c = ______.
2. A quadraticequationax2
+ bx + c = 0 has tworoots,if b2
– 4ac > 0.
3. The quadratic equation3x2
– 43x + 4 = 0 has two_______ roots.
4. The roots of a quadraticequation2x2
– 7x +3 = 0 are _____ and _______.
5. Two numberswhose sumis27 and product is182 are _______ and _______.
QUIZ
(ARITHMETIC PROGRESSIONS)
1. What isan A.P.?
2. What ismeantby commondifference inanA.P.?
3. What isthe formulaforthe nth termof an A.P.?
4. What isthe formulaforthe sum of firstn termsof an A.P.?
5. What isthe formulaforthe sum of firstn natural numbers?

97
ORAL TEST
1. The common difference of asequence of multiplesof 7is ________.
2. The difference of consecutivetermsinanA.P. is always_______.
3. The sum of first20 natural numbersis_______.
4. The sum of firsteightoddnatural numbersis________.
5. The sum of firsttenevennatural numbersis_______.
QUIZ
(Coordinate geometry)
1. What isabscissa?
2. What isordinate?
3. What isdistance formula?
4. What isthe distance of a pointp(x,y) fromorigin?
5. Give the sectionformula.
ORAL TEST
1. If the area of a triangle is0 square units,thenitsverticesare _______.
2. The area of a triangle whose verticesare (1, -1), (-4,6) and (-3, -5) is________ square units.
3. The distance betweenthe points(-5,7) and (-1,3) is ______ units.
4. _______ has beendevelopedasanalgebraictoll forstudyinggeometryof figures.
5. The distance betweenthe points(a,b) and(-a, -b) is________ units.
QUIZ
(Some applicationsof trigonometryorheightsanddistance
1. Why trigonometrywasinvented?Give itsuses.
2. What isthe line of sight?
3. What isthe angle of elevation?
4. What isthe angle of depression?
5. What isa theodolite?
ORAL TEST
1. The other name of clinometeris_________.
2. If heightof clinometeris1m, distance betweenobjectandclinometeris40mand angle of elevationof objectis
45o
, thenthe heightof objectis____________.
3. A towerstandsverticallyonthe ground.Fromthe pointonthe ground,whichis25m awayfrom the footof the
tower,the angle of elevationof the topof the towerisfoundto be 60o
. The heightof the toweris_____.
4. The anglesof elevation of the topof a towerfromtwo pointsatdistancesa and b fromthe base and on the
same straightline withitare complementary.The heightof the toweris__________.
5. A ladder15m longjust reachesthe topof a vertical wall.If the laddermakesanangle of 60o
withthe wall,then
the heightof the wall is__________.
QUIZ
(CIRCLES)
1. Define tangenttoa circle.
2. How manytangent(s) is/are there atapointof circle?
3. How manytangentcan be drawn to a circle froma pointoutside the circle?
4. Define lengthof atangent.
5. What isthe relationbetweenthe lengthsof tangentsdrawnfromanexternal pointtoacircle?

98
ORAL TEST
1. A tangentto a circle intersectsitin_____________ point(s).
2. A line intersectingacircle intwopointsiscalleda___________.
3. A circle can have _________ parallel tangentsatthe most.
4. The common pointof a tangentto a circle and the circle is called__________.
5. The tangentat anypointof a circle is____________ to the radiusthrough the pointof contact.
QUIZ
(Constructions)
1. What isscale factor?
2. How will youdrawa tangentat a pointof a circle?
3. How will youlocate the centre of a circle,if itis notgiven?
4. How manytangentscan be drawn froma pointoutside the circle?
5. Is itpossible todrawa tangentfroma pointinside acircle?
ORAL TEST
1. To divide aline segmentABinthe ratiom:n (m, n are positive integers),draw aray Ax sothat  BAx isan acute
angle an thenmarkpointon ray Ax at equal distancessuchthatthe minimumnumberof these pointsis
_______.
2. To draw a pair of tangentsto a circle whichare inclinedtoeachotherat an angle of 45o
, itis requiredtodraw
tangentsat the endpointof those two radii of the circle,the angle betweenwhichis_____.
3. To divide aline segmentABinthe ration4:5, a ray Ax isdrawn firstsuchthat BAx isan acute angle andthem
pointsA1,A2, A3… are locatedat equal distance onthe ray Ax and the pointB is joinedto________.
4. To construct a triangle similartoagiven ABCwithitssides3/5 of the correspondingsidesof ABC,firstdrawa
ray BX such that CBX isan acute angle andX liesonthe opposite side of A withrespectto BC.The locate
pointsB1, B2, B3, ____ on BX at equal distancesandnextstepistojoin_______ to________.
5. State ‘True’or ‘False’
a. By geometrical construction,itispossibletodivide aline segmentinthe ratio3+5: 3-5.
b. A pairof tangentscan be drawnfrom a pointP to a circle of radius4.5 cm situatedata distance of 4 cm
fromthe centre.
c. By geometrical construction,itispossibletodivide aline segmentinthe ratio 5: 1/5.
d. A pairof tangentscan be constructedto a circle inclinedatan angle of 175o
.
e. From a pointP outside the circle we candraw onlyone tangent.
f. We cannotlocate the centre of a circle if it isnot given.
QUIZ
(AREASRELATED TO CIRCLES)
1. What iscircumference of acircle?Give itsformula.
2. Name the great Indianmathematicianwhogave anapproximate value of .
3. Give the formulaforthe area of a circle of radiusr cm.
4. Give the formulaforarea of a sectorof a circle havingradiusr and measuringanangle at the centre.
5. How will youfind the areaof a segmentof acircle?
ORAL TEST
1. If the area of a circle is 154 cm2
, thenitsperimeteris________.
2. Areaof the largesttriangle thatcan be inscribedina semicircle of radiusris________.
3. The diameterof a circle whose areais equal to the sum of the areas of the twocirclesof radii 24 cm and7 cm is
_____.
4. If the areasof twocirclesare equal,thentheircircumferencesare ________.
5. The circleswhichhave the same centre are called________ circles.

99
QUIZ
(SURFACEAREASAND VOLUME)
1. A cone of height24cm and radiusof base 6cm ismade upof modelingclay.A childreshapesitinthe formof a
sphere.Findthe radiusof the sphere.
2. A shuttle corkusedforplayingbadmintonhasthe shape of the combinationof whichbasicsolids?
3. What isa frustumof a right circularcone?
4. Doesa frustumhas twocircular endswithequal radii?
5. Give the formulaforthe volume of the frustumof a cone.
ORAL TEST
1. A plumbline(sahul) showninthe figure isthe combinationof a________ and a cone.
2. If the radii of the circularendsof a conical bucketwhichis45cm high,are 28cm and 7cm thenthe capacity of
the bucketis_______ cm3
.
3. The volume of the solidformedbyjoiningtwobasicsolidswill actuallybe the _____ of the volumesof the
constituents.
4. The curvedsurface are of the frustumof a cone is_________, where l=
5. If two cubeseachof volumes64cm3 are joinedendtoendthenthe surface areaof the resultingcuboidis
________.
QUIZ
(PROBABILITY)
1. Define the theoretical probabilityof aneventE.
2. What isthe probabilityof asure event?
3. What isan elementaryevent?
4. What are complementaryevents?
5. One card is drawnfrom a well shuffleddeckof 52 cards.Calculate the probabilitythatthe cardwill be a king.
ORAL TEST
1. The probabilityof animpossibleeventis________.
2. The probabilityof aneventliesbetween________ and __________.
3. The sum of the probabilitiesof all the elementaryeventsof anexperimentis________.
4. A die isthrownonce,the probabilityof gettingaprime numberis________.
5. Two coinsare tossedsimultaneously.The probabilityof atmostone tail is______.

100
PUZZLES
1. CatchingFish
If Five fishermencatch5 fishin5 minutes,how longwill ittake fiftyfishermentocatchfiftyfish?
2. Look at the Division
One day professorAgarwal wenttothe blackboardanddemonstratedtohisastonishedclassthatone half of
eightwasequal tothree!What didthe professordo?
3. How Big
Can youguesshowbig the number: ninthpowernine?
4. CountingStreetLights
On twosidesof a street,there are 35 streetlights,eachone isat a distance of 30 metresfromthe other.The
streetlightsonone side are arrangedso that eachlamp fillsagap betweenthe twootherstreetlightsonthe
opposite.Howlongisthe street?
5. Who discoveredmore distance
Two friendsVijayandAjaywalkwithconstantspeedof 100m/min.Vijaytakesrestfor1 min afterwalking
100metres while ajaytakesrestfor3 minafterwalking300 metreson a square pathof side 400m. Both of them
start fromthe same cornerin opposite direction.Whodiscoveredmore distance,whentheymeet?
6. The missingSix
Place the six numbersbelowintoemptycircles,sothatboththe equationare true.Use eachnumberonce and
onlyonce.
7. Magic Triangle
Place the numbers4 through9 inthe circlesinsucha waythat everyside of the triangle addupto 21.
1 2 3 4 5 7
+
-
=
=

101
8. Addup
Here is an equilateral triangle.Addanotherequilateral triangle toitinsucha waythat youget five equilateral
triangles.
9. Magic Sticks
Justby movingone stick,make anotherequation.
10. Identical Four
Divide the adjoiningfigure intofouridentical pieces.
---XXX---

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