XTH STD --- MATHS
CHAPTER 1 ------REAL NUMBERS
April 1 to April 14
Previous knowledge :
 Terminating , non Terminating, recurring and non recurring decimals
 Definitions of rational and irrational numbers
 Prime and composite numbers
 Prime factorization
Basic concepts :
 Euclid’s division lemma
 Fundamental theorem of Arithmetic
 Reviewing the concept of rational and irrational numbers
 Reviewing the decimal form of rational and irrational numbers
 Conditions for rational number to be terminating / non terminating
recurring decimal form
 Find the H.C.F. and L.C.M. by Euclid’s lemma and by Fundamental
theorem of Arithmetic.
 Verifying the property : H.C.F. x L.C.M = product of two numbers.
Learning outcome :
 State the Euclid’s division lemma and Fundamental theorem of
Arithmetic
 Define Real numbers and state few examples. Determine whether
given rational number is terminating / non terminating recurring
decimal form
 Prove given rational number can be written as terminating / non
terminating recurring decimal form by observing the factors of
denominator
 Calculate the H.C.F. and L.C.M. by Euclid’s lemma and by
Fundamental theorem of Arithmetic
 Apply the above concepts while solving problems.
 Analyse the relation between H.C.F. and L.C.M. and two given
numbers.
 Compare the relation between H.C.F. and L.C.M. and three given
numbers.
 Justify real numbers to be rational or irrational .
Sample questions:
I. MCQ
1)
1
2
…. is
a) Rational number b) Irrational number
c) an integer
d) none of these
2) HCF of ‘a’ and ‘b’ where a & b are prime no’s is :
a) 1
b) a
c) b
d) a+b.
II. 1) Define Euclid’s lemma
2) State fundamental theorem of arithmetic
III.
1) Find HCF of 305, 657 and hence find L.C.M.(305 , 607 ).
2) Show that 12n cannot end with the digit ‘0’ or ‘5’.
3) Prove that 5 + 3 2 is an irrational number.
4) State the type of decimal for given rational number with reason.
129
2  57  7 5
2
Chapter: Pair of linear equations in two variables
APRIL 15 to MAY 7, JUNE 15 to JUNE 25, 2011, Number of periods: 30
Previous knowledge:
 Mathematical equations
 Variables
 Examples of linear equations
 Solving of simple linear equations in one variable
 Graph of linear equations in two variables
Key Concepts:
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Pair of linear equations in two variables
Standard form of equations
Intersecting, parallel and coincident lines
Ratio of coefficients
Relation between nature of solutions and ratio of coefficients
Graph of a pair of linear equations in two variables
Relation between number of solutions and points of intersection of lines
Consistency of equations
Graphical method of solving a pair of linear equations in two variables
Algebraic method of solving a pair of linear equations in two variables
 Elimination method
 Substitution method
 Cross multiplication method
Equations reducible to a pair of linear equations in two variables
Word problems
Learning Outcomes:
The learner will be able to
 write equations in standard form

state the conditions for consistency of a pair of linear equations in two variables
 find the nature of solutions
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find whether the lines are intersecting, parallel or coincident
 draw graphs of given linear equations in two variables
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solve by the graphical method
solve by the elimination method
solve by the substitution method
solve by the cross multiplication method and verify by any other method
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reduce the given equations to linear equations and solve them
convert word problems to linear equations and solve them
Activity:
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To find the conditions for consistency of pairs of linear equations in two variables
Type of Evaluation:
 MCQ
 Activity
 Written test
Sample questions:
1. For what values of k is the system of equations 4x + 6y = 1 and 2x + ky = 7 consistent?
a) k = 2
b) k = 3
c) k = 4
d) k = - 2
2. The solution of 2x + 3y = 8, 4x + 6y = 14 is
a) x = 4, y = 0
b) x = 1, y = 2
c) x = 0, y = 4
d) no solution
3. Which of the following pair of linear equations has unique solution, no solution or
infinitely many solutions?
(i) 2x + y = 5; 3x + 2y = 8
(ii) x - 3y – 7 = 0; 3x - 3y - 15 = 0
(iii) x - 3y - 3 = 0; 3x – 9y – 2 = 0
(iv)3x – 5y = 20; 6x – 10y = 40
4. Are the following pair of linear equations consistent? Justify your answer.
(i)
– 3x – 4y = 12; 4y + 3x = 12
(ii) 3x – y = 1; x – 3y = 6
5. Solve graphically; 2x + y = 4 and 2x – y = 4. Write the vertices of the triangle formed by
these lines and the y-axis. Also find the area of this triangle.
6. Use elimination method to find the solution of: x – 5y = 11; 2x + 3y = 4
7. Solve by substitution method: 8x + 5y = 9; 3x + 2y = 4
8. Solve by cross multiplication method: 2ax – 2by + a + 4b = 0;2bx + 2ay + b – 4a = 0
9. Solve:
5
𝑥−1
+
1
𝑦−2
= 2;
6
𝑥−1
−
3
𝑦−2
=1
10. The sum of a two digit number and the number obtained by reversing the digits is 66. If
the digits of the number differ by 2. Find the number. How many such numbers are
there?
Chapter: Polynomials
JUNE 27 to JULY 9, 2011, Number of periods: 14
Previous knowledge:
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Condition for an algebraic expression to be a polynomial.
Polynomials with one variable
The degree of a polynomial
The zero of a polynomial
Remainder theorem and factor theorem
The difference between factor and a divisor and their relationship.
Solution of a linear equation.
Constructing graph
Factorizing a polynomial by splitting the middle term.
Factorizing a polynomial using factor theorem.
Division of a polynomial.
Graph of a first degree equation.
Key Concepts:
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Graph of a linear polynomial
Graph of a quadratic polynomial
The quadratic polynomial as a parabola.( a>0 it is upward ‘u’ shape and a< 0 then it is
reverse ‘u’ shaped curve)
Geometrical meaning of the zeroes of a polynomial
Points of intersection of graph with x-axis
Number of zeroes of a polynomial
The relation between the number of zeroes of a polynomial and the points of
intersection of graph with x-axis
Relation between zeroes of a polynomial and the coefficients of the polynomial
Division Algorithm
Verification of division algorithm
Application of division algorithm
Learning Outcomes:
The learner will be able to
 state the standard form of a polynomial
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define linear, quadratic and cubic polynomials and give examples
define the zero and degree of a polynomial
draw the graphs of polynomials
Find the number of zeroes of polynomials from the graph
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compute the zeroes of quadratic polynomial using suitable methods
verify the relation between zeroes and the coefficients of polynomials
frame a quadratic polynomial with the sum and product of zeroes given.
state division algorithm
do the long division of a polynomial of degree one and two
verify division algorithm by dividing one polynomial with another
apply division algorithm to find the dividend / divisor / quotient / remainder
factorise given polynomials using Factor theorem and long division and determine
other zeroes of polynomials of degree 3 or 4
Type of Evaluation:

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MCQ
Written test
Sample questions:
1. Quadratic polynomial having zeroes 1 and -2 is
a) x2 – x + 2
b) x2 – x – 2
c) x2 + x – 2
d) x2 + x + 2
2. If 1 is a zero of the polynomial p(x) = a2x2 – 3ax + 3x – 1, then the value of a is
a) – 1
b) 2
c) – 2
d) 0
3. Define a zero of a polynomial
4. State the division algorithm for polynomials
5. Find the zeroes of the quadratic polynomial and verify the relation between zeroes and
the coefficients of polynomial: x2 + 7x + 10
6. Verify division algorithm after dividing p(x) = x3 – x2 + 4x – 8 by g(x) = x + 3
7. On dividing 2x3 + 4x2 + 5x + 7 by g(x), the quotient and the remainder are 2x and 7 – 5x
respectively. Find g(x)
8. Form a quadratic polynomial whose zeroes are 3 and – 4
9. Form a quadratic polynomial which has its sum and product of zeroes ½ and – 4
respectively
10. Find other zeroes of x4 – 6x3 + 26x2 + 138x – 35 if two of its zeroes are 2 ± √3
Chapter: Statistics
AUGUST 16 to SEPTEMBER 7, 2011, Number of periods: 22
Previous knowledge:
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Collection of data
Organization of data
Frequency table
Class intervals
Class size
Class limits
Class mark
 Measures of central tendency – mean, median, mode
 Frequency polygon
 Histogram
Key Concepts:

Mean of grouped data
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Direct method – x =
𝛴𝑓𝑖 𝑥𝑖
𝛴𝑓𝑖
 Assumed mean method – x = a +
𝛴𝑓𝑖𝑑𝑖
𝛴𝑓𝑖
𝛴𝑓𝑖𝑢𝑖

Step-deviation method – x = a + h [

Mode of grouped data - mode = l + [
] where ui =
𝛴𝑓𝑖
𝑓1 −𝑓0
2𝑓1 −𝑓0 −𝑓2
𝑥𝑖 −𝑎
ℎ
]Xh
l = lowest limit of the modal class
h = size of the class interval (all class sizes to be equal)
f1 = frequency of the modal class
fo = frequency of the class preceding the modal class.
f2 = frequency of the class succeeding the modal class.
 Median of grouped data – median = l + [
𝑛
−𝑐𝑓
2
𝑓
] X h where
l = lower limit of the median class
cf = cu. frequency of class preceding the median class
n= no. of observation
h = class size (to be equal)
 Empirical relationship – 3 median = mode + 2 mean
 Cumulative frequency
 Ogive curves
where
 ‘More than’ and ‘less than’ ogives
 Median as the x-coordinate of the point of intersection of the two ogives
Learning Outcomes:
The learner will be able to
 define the three measures of central tendency
 state the formulae for mean, median and mode
 identify the symbols in the formulae and the values to be substituted in their place
 apply the formulae to find the value of mean, median and mode
 organize the given data in the frequency table and find cumulative frequency
 draw the ogive curves
 calculate the median from the point of intersection of the ogive curves
 describe the use of measures of central tendency in real life situations
Type of Evaluation:
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Written test
Sample questions:
1. If the point of intersection of the two ogives for a given data is (17.5, 15), find the
median of the data.
2. . Median of the following data is 525. If the total frequency is 100, find x and y.
Class 0-100
Interval
Freq.
2
100200
5
200300
x
300400
12
400500
17
500600
20
600700
y
700800
9
800900
7
9001000
4
3. The marks obtained by 30 students of class IX of a certain school in Math paper
consisting of 100 marks are presented in table below. Find the mean and mode of marks
obtained by the students. Using empirical formula, find the median also.
Marks
10 - 25 25 - 40 40 - 55 55 - 70
70 - 85
85 - 100
Number of
2
3
7
6
6
6
students
4. The annual profits earned by 30 shops of a shopping complex in a locality give rise to the
following distribution.
Profit (in lakhs Rs.)
No. of shops (frequency)
More than or equal to 5
30
More than or equal to 10
28
More than or equal to 15
16
More than or equal to 20
14
More than or equal to 25
10
More than or equal to 30
7
More than or equal to 35
3
Draw both ogives for the data given above. Hence obtain the median profit from the graph and
verify the result by using the formula.
5. The following distribution gives the daily income of 50 workers of a factory.
Daily income(Rs)
100-120
120-140
140-160
160-180
180-200
Number of workers
12
14
8
6
10
Convert the distribution above to a less than type cumulative frequency distribution and draw
its ogive. Hence obtain the median weight from the graph and verify the result by using the
formula.
TRIANGLES
July 10 to July 31 ---2011
No. of periods: 22
PREVIOUS CONCEPTS
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Area of triangles between same parallel lines
Area of triangle
Congruent triangles; its criteria and its corresponding parts
Pythagoras theorem (statement)
Basic concepts
Similar figurs.similar polygons
Condition for the similarity between two triangles, ratio of corresponding sides in two
equilateral triangles
Basic proportionality theorem and its converse
Criteria for similarity of triangles
Pythagoras theorem and its converse
LEARNING OUTCOME
Learner will be able to
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Define and recognize similar and congruent triangles
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State the theorem
Identify the corresponding parts of triangles and state the suitable criteria for…
Explain the proof of the BPT with the help of areas of triangles between two parallel
lines and area of triangle
Prove the theorem based on areas of the triangles and pythagorus theorem by applying
the criteria for the triangles.
Apply the theorem while solving the problems after analyzing the questions,changing
figures and explain the proof and summarizing the solution.
Justify the proof of the theorem by activity method
Activity
1. Pythagoras theorem
2. Areas of similar triangles
3. Basic Proportionality Theorem
FA-II
Sample Questions
1. D and E trisect BC. Prove that 8. AE2 = 3.AC2 + 5.AD2.
2. In  ABC, C = 90 0, AC= 3 .BC. Prove that ABC = 90 0.
3. In Q no.2, D is the midpoint of BC. Prove that BC2 = 4 ( AD2 - AC2 ) ;
AB2 = 4 AD2 – 3 AC2.
4. In  ABC, C is an obtuse angle and AD  BC, AB2 = AC2 + 3. BC2. Prove that BC = CD.
5. A point ‘D’ is on the side BC of an equilateral triangle ABC such that DC =
1
BC. Prove that AD2 = 13. AD 2.
4
6. ABC is a right triangle , right angled at B. AD and CE are two medians drawn from A and C
respectively. If AC = 3 5 c.m., find the length of CE.
7.In fig. DE ll BC such that AE = ¼ BC. If AB = 6 c.m. find AD.
8. In  ABC, D and E are points on the sides AB and AC respectively such that
DE ll BC if AD= x, DB = x-2, AE= x+2 and EC= x-1. Find the value of x.
9. AE is the bisector of the exterior CAD meeting BC produced in E. If
AB= 10cm, AC=6cm and BC=12cm. Find CE.
10. ABC=900 and BD  AC. If BD=8cm and AD=4cm, find CD.
11.The areas of 2 similar triangles are: 25cm2 and 36 cm2 respectively if the altitude of the first
triangle is 2.4cm. Find the corresponding altitude of the other.
12. Two isosceles triangles have equal vertical angles and their areas are in the ratio 36:25. Find
ratio of the corresponding heights.
13. ABCD is a trapezium in which AB ll CD. The diagonals AC and BD intersect at O. Prove that
AOB ~
14. In

COD, find Ar(  AOD)/ Ar(  COD).
 ABC, BD ll AC, BC2 =2AC*CD, then prove that AB=AC.
15. In  ABC, N is a point on AC such that BN  AC. If BN2=AN*NC prove that
B=900.
16.`ABCD is a rectangle points M and N are on BD such that AM  BD and
CN  BD. Prove that BM2+BN2= DM2+DN2.
17.In
 ABC, AD is a median. Prove that AB2+AC2=2AD2+2 DC2.

Introduction to Trigonometry
August 1 to August 12
No of periods: 15
Previous knowledge:
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Right angled triangle
Pythagoras theorem
Complementary angles
 Identity
Key concepts:
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Hypotenuse, adjacent and opposite sides of a given angle
Trigonometric ratios
Trigonometric ratios of Complementary angles
Trigonometric Identity
Learning outcome:
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Learner will be able to
 Define right triangle, Hypotenuse
 Name adjacent and opposite sides of given angle
 Define Trigonometric ratios
 Identify Complementary angles
 State and relate Trigonometric ratios of Complementary angles
 State, prove the identity
 Apply basic identity with the help of known concepts
Sample Questions
1. If tan + sin = a and tan - sin =b, show that a2 – b2 = 4.
2. Without using tables, evaluate:
Sin (50+) – cos (40-) +tan10. tan100. tan200. tan700. tan800. tan890.
3. Prove that sin2A.tanA+cos2A.cotA+2sinA.cosA=tanA+cotA.
4. If x=r.sin1.cos2 , y= r.sin1 r.sin2, z= r.cos1 , show that
x2+y2+z2 =r2.
5. If 3 tan =3 sin, find the values of sin2 - cos2. ( 1/3)
6. Prove that sin6+cos6=1-3 sin2.cos2.
7. If (cosec - sin ) =’ m ‘and sec  - cos  =’ n’. show that m2 n2 (m2 +n2 + 3) = 1