Mathematics compendium for class ix

1 | P a g e MATHEMATICS COMPENDIUM CLASS – IX
NUMBER SYSTEM
Points to Remember :
1. Number used for counting 1, 2, 3, 4, ... are
known as Natural numbers.
2. All natural numbers together with zero i.e. 0, 1,
2, 3, 4, ..... are known as whole numbers.
3. All natural numbers, zero and negative numbers
together i.e. ...., –4, –3, –2, –1, 0, 1, 2, 3, 4, ... are
known as Integers.
4. Rational Numbers : Numbers of the form
where p, q both are integers and q ≠ 0. For e.g.
1/4 , -2/3 etc.
5. Every rational number have either terminating
or repeating (recurring) decimal representation.
Terminating Repeating (Recurring)
Any rational number (that is, a fraction in lowest
terms) can be written as either a terminating
decimal or a repeating decimal. Just divide the
numerator by the denominator. If you end up with
a remainder of 0, then you have a terminating
decimal. Otherwise, the remainders will begin to
repeat after some point, and you have a repeating
decimal.
6. There are infinitely many rational numbers
between any two given rational numbers.
7. Irrational Numbers : Numbers which cannot
be written in the form of p/q, where p, q are
integers and q ≠ 0.
For e.g. √2 , √5 ,0.202202220......, etc.
8. Real numbers : Collection of both rational and
irrational numbers. For e.g. -3, 7/5,0,√2 , √5 etc.
9. Every real number is represented by a unique
point on the number line. Also, every point on the
number
line represents a unique real number.
10. For every given positive real number x, we
can find x geometrically.
11. Identities related to square root :
Let p, q be positive real numbers. Then,
(i) = . (ii) =
√
√
; q≠ 0
(iii) ( - ) ( + ) = p-
POLYNOMIALS
1. A symbol having a fixed numerical value is
called a constant. For e.g.9, -7/3, √2 etc.
2. A symbol which may take different numerical
values is known as a variable. We usually
denotes variable by x, y, z etc.
3. A combination of constants and variables
which are connected by basic mathematical
operations, is known as Algebraic Expression.
For e.g. x2
– 7x + 2, xy2
– 3 etc.
4. An algebraic expression in which variable have
only whole numbers as a power is called a
Polynomial.
5. Highest power of the variable is called the
degree of the polynomial. For e.g. 7x3
– 9x2
+ 7x
– 3 is a polynomial in x of degree 3.
6. A polynomial of degree 1 is called a linear
polynomial. For e.g. 7x + 3 is a linear polynomial
in x.
7. A polynomial of degree 2 is called a Quadratic
Polynomial. For e.g. 3y2
– 7y + 11 is a Quadratic
polynomial in y.
8. A polynomial of degree 3 is called a Cubic
Polynomial. For e.g. 3t3
– 7t2
+ t – 3 is a cubic
polynomial in t.
9. According to number of terms, a polynomial
having one non-zero term is a monomial, a
polynomial having two non-zero terms is a
binomial and a polynomial have three non-zero
terms is a trinomial.
10. Remainder Theorem : Let f(x) be a
polynomial of degree n ≥ 1 and let a be any real
number. If f(x) is divided by linear polynomial (x
– a), then the remainder is f(a).
11. Factor Theorem : Let f(x) be a polynomial of
degree n > 1 and a be any real number.
(i) If f(a) = 0, then x – a is a factor of f(x).
(ii) If (x – a) is a factor of f(x) then f(a) = 0.
12. Algebraic Identities :
(i) (x + y)2
= x2
+ 2xy + y2
(ii) (x – y)2
= x2
– 2xy +
y2
(iii) x2
– y2
= (x – y) (x + y)
(iv) (x + y + z)2
= x2
+ y2
+ z2
+ 2xy + 2yz + 2xz
(v) (x + y)3
= x3
+ y3
+ 3xy (x + y)

(vi) (x – y)3
= x3
– y3
– 3xy (x – y)
(vii) x3
- y3
= (x – y) (x2
+ xy + y2
)
(viii) x3
+ y3
= (x + y) (x2 – xy + y2)
(ix) x3
+ y3
+ z3
– 3xyz = (x + y + z) (x2
+ y2
+ z2
–
xy – yz – xz)
13. If x + y + z = 0 then, x3
+ y3
+ z3
= 3xyz.
CO-ORDINATE GEOMETRY
1. Coordinate axes : Two mutually perpendicular
lines X´OX and YOY´ known as x-axis and y-axis
respectively, constitutes to form a co-ordinate
axes system. These axes interests at point O,
known as origin.
2. Co-ordinate axes divides the plane into four
regions, known as Quadrants.
3. The position of any point in a plane is
determined with reference to x-axis and y-axis.
4. The x-coordinate of a point is its perpendicular
distance from the y-axis measured along the x-
axis. The x-coordinate is known as abscissa.
5. The y-coordinate of a point is its perpendicular
distance from the x-axis measured along the y-
axis. The y-coordinate is known as ordinate.
6. Abscissa and ordinate of a point written in the
form of ordered pair, (abscissa, ordinate) is known
as the co-ordinate of a point.
7. If the point in the plane is given, we can find
the ordered pair of its co-ordinate and if the
ordered pair of real numbers is given, we can find
the point in the plane corresponding to this
ordered pair.
8. Sing Convention :
LINEAR EQUATIONS IN TWO
VARIABLES
1. An equation of the form ax + by + c = 0, where
a, b, c are real numbers, such that a and be are not
both zero, is known as linear equation in two
variables.
2. A linear equation in two variables has infinitely
many solutions.
3. The graph of linear equation in two variables is
always a straight line.
4. y = 0 is the equation of x-axis and x = 0 is the
equation of y-axis.
5. The graph of x = a is a straight line parallel to
the y-axis.
6. The graph of y = b is a straight line parallel to
the x-axis.
7. The graph of y = kx passes through the origin.
8. Every point on the graph of a linear equation in
two variables is a solution of the linear equation.
Also,
every solution of the linear equation is a point on
the graph of the linear equation.
INTRODUCTION TO EUCLID’S
GEOMETRY
1. A point, a line and a plane are concepts only
and these terms are taken as undefined.
2. Axioms (or Postulates) are assumptions which
are self evident truths.
3. Theorems are the statements which are proved,
using axioms, previously proved statements and
deductive reasoning.
4. Some of Euclid’s axioms were:
(a) Things which are equal to the same thing are
equal to one another.
(b) If equals are added (or subtracted) to / from
equals, the wholes / remainders are equal.
(c) The whole is greater than the part.
(d) Things which are double of the same things
are equal to one another.
5. Euclid’s Five Postulates:

Postulate 1: A straight line may be drawn from
any one point to any other line.
Postulate 2: A terminated line can be produced
indefinitely.
Postulate 3: A circle can be drawn with any
center and any radius.
Postulate 4: All right angles are equal to one
another.
Postulate 5: If a straight line falling on two
straight lines makes the interior angles on the
same side of it
taken together less than two right angles, then the
two straight lines, if produced indefinitely, meet
on that side on which the sum of angles is less
than two right angles.
6. Two equivalent versions of Euclid’s fifth
Postulate:
(a) Play fair axiom:
“Through a given point, not on the line, one and
only one line can be drawn parallel to a given
line.”
(b) Two distinct intersecting lines cannot be
parallel to the same line.
LINES AND ANGLES
1. If a ray stands on a line, then the sum of the two
adjacent angles so formed is 180° and vice-versa.
This property is known as the Linear Pair
Axiom.
2. If two lines intersect each other, then the
vertically opposite angles are equal.
3. If a transversal intersects two parallel lines,
then
(a) each pair of corresponding angles is equal.
(b) each pair of alternate interior angles is equal.
(c) each pair of interior angles on the same side of
the transversal is supplementary.
4. If a transversal intersects two line such that,
either
(a) any one pair of corresponding angles is equal,
or
(b) any one pair of alternate interior angles is
equal, or
(c) any one pair of interior angles on the same
side of the transversal is supplementary, then the
lines are parallel.
5. Two intersecting lines cannot both be parallel
to the same line.
6. Lines which are parallel to a given line are
parallel to each other.
7. The sum of three angles of a triangle is 180°.
8. If a side of a triangle is produced, the exterior
angle so formed is equal to the sum of the two
interior opposite angles
TRIANGLE
1. Two figures are congruent, if they are of same
shape and same size.
2. If two triangles ABC and XYZ are congruent
under the correspondence A  X, B  Y and
C Z, then symbolically, ABC XYZ
3. SAS Congruence Rule : If two sides and the
included angle of one triangle are equal to two
sides and the included angle of the other triangle,
then the two triangles are congruent.
4. ASA Congruence Rule : If two angles and the
included side of one triangle are equal to two
angles and the side of the other triangle, then the
two triangles are congruent.
5. AAS Congruence Rule : If two angles and one
side of one triangle are equal to two angles and
the corresponding side of the other triangle, then
the two triangles are congruent.
6. RHS congruence Rule : If in two right
triangles, hypotenuse and one side of a triangle
are equal to the hypotenuse and one side of other
triangle, then the two triangles are congruent.
7. SSS Congruence rule : If three sides of one
triangle are equal to the three sides of another
triangle, then the two triangles are congruent.
8. Angles opposite to equal sides of a triangle are
equal.
9. Sides opposite to equal angles of a triangle are
equal.
10. Each angle of an equilateral triangle is 60°.
11. Of all the line segments that can be drawn to a
given line from a point not lying on it, the
perpendicular line segment is the shortest.

12. In a triangle, angle opposite to the longer side
is greater.
13. In a triangle, side opposite the greater angle is
longer.
14. Sum of any two sides of a triangle is greater
than the third side.
15. Difference between any two sides of a triangle
is less than its third side.
QUADRILATERALS
1. The sum of the angles of a quadrilateral is 360°.
2. A diagonal of a parallelogram divides it into
two congruent triangles.
3. In a Parallelogram:
(i) opposite sides are equal
(ii) opposite angles are equal
(iii) diagonals bisect each other
4. A quadrilateral is a parallelogram, if
(i) opposite sides are equal, or
(ii) opposite angles are equal, or
(iii) diagonals bisect each other, or
(iv) a pair of opposite sides is equal and parallel.
5. Diagonals of a rhombus bisect each other at a
right angle and vice-versa.
6. Diagonals of a rectangle bisect each other and
are equal, and vice-versa.
7. Diagonals of a square bisect each other at right
angles and are equal, and vice-versa.
8. Mid-point Theorem : The line segment
joining the mid-points of any two sides of a
triangle is parallel to the third side and is half of
it.
9. A line through the mid-point of a side of a
triangle parallel to another side bisects the third
side.
10. The quadrilateral formed by joining the mid-
points of the sides of a quadrilateral, in order, is a
parallelogram.
AREAS OF PARALLELOGRAMS AND
TRIANGLES
1. Two congruent figures must have equal areas.
However, two figures having equal areas need not
to be congruent.
2. Two figures are said to be on the same base and
between the same parallels, if they have a
common base and the vertices (or the vertex)
opposite to the common base of each figure lie on
a line parallel to the base.
3. Parallelograms on the same base and between
the same parallels are equal in area.
4.Area of parallelogram = Base × corresponding
height.
5. Parallelograms on the same base (or equal
bases) and having equal areas lie between the
same parallels.
6. Two triangles on the same base (or equal bases)
and between the same parallels are equal in area.
7. Two triangles having the same base (or equal
bases) and equal areas lie betwen the same
parallels.
8. Area of Triangle = × Base × corresponding
height.
9. Area of a Rhombus = × product of diagonals.
10. Area of a Trapezium = × (sum of the
parallel sides) × (distance between them).
11. A median of a triangle divides it into two
triangles of equal area.
12. The diagonals of a parallelogram divides it
into four triangles of equal area.
CIRCLES
Points to Remember:
1. A circle is a collection of all the points in a
plane, which are equidistant from a fixed point in
the plane.
2. Equal chords of a circle (or of congruent
circles) subtend equal angles at the centre.
3. If the angles subtended by two chords of a
circle (or of congruent circles) at the centre
(corresponding centre) are equal, the chords are
equal.
4. The perpendicular from the centre of a circle to
a chord bisects the chord.
5. The line drawn through the centre of a circle to
bisect a chord is perpendicular to the chord.
6. There is one and only one circle passing
through three non-collinear points.

7. Equal chords of a circle (or of congruent
circles) are equidistant from the centre (or
corresponding
centres).
8. Chords equidistant from the centre (or
corresponding centres) of a circle (or of congruent
circles) are equal.
9. If two arcs of a circle are congruent, then their
corresponding chords are equal and conversely, if
two
chords of a circle are equal, then their
corresponding arcs (minor, major) are congruent.
10. Congruent arcs of a circle subtend equal
angles at the centre.
11. The angle subtended by an arc at the centre is
double the angle subtended by it at any point on
the remaining part of the circle.
12. Angles in the same segment of a circle are
equal.
13. Angle in a semicircle is a right angle.
14. If a line segment joining two points subtends
equal angles at two other points lying on the same
side of
the line containing the line segment, the four
points lie on a circle.
15. The sum of either pair of opposite angles of a
cyclic quadrilateral is 180°.
16. If the sum of a pair of opposite angles of a
quadrilateral is 180°, then the quadrilateral is
cyclic.
CONSTRUCTIONS
Points to Remember:
Using a graduated scale and a compass, we can
construct the following:
1. Perpendicular bisector of a line segment.
2. Bisector of an angle.
3. Angle of measures 60°, 90°, 45° etc.
4. A triangle given its base, a base angle and the
sum of the other two sides.
5. A triangle give its base, a base angle and the
difference of the other two sides.
6. A triangle given its perimeter and its base
angles.
SURFACE AREA & VOLUMES
Cuboid
Let length (), breadth (b) and height (h).
Surface area = 2 (b + bh + h)
Volume = base area × h
where base area = Breadth × length
So volume =  × b × h
Length of its diagonals =
and total length of its edges = 4 ( + b + h)
Cube
Let length of each side is ‘a’ then
Surface area = 6 a2
Volume = a3
Lateral surface area = 4 a2
Length of its diagonals =
Total length of its edges = 12 a
Right Circular Cylinder
Let radius of the base = r and height = h
Curved surface area = 2rh
Total surface area = 2r(h + r)
Volume = r2h where,  = or 3.14
Hollow Cylinder
Let external radius = R, Internal radius = r,
height = h. Then,
Outer curved surface area = 2Rh
Inner curved surface area = 2rh
Area of cross section = R2 – r2
Total surface area = 2Rh + 2rh + 2(R2 – r2)
Volume = (R2 – r2)h
Right Circular Cone
Let , h and r are the slant height, height
and radius of a cone then
2 = h2 + r2
Area of base = r2
Curved (lateral) surface area = r
Total surface area = r ( + r)
Volume = .
Sphere & Hemisphere
Let radius of sphere = r
Surface area = 4r2
Curved surface area of a hemisphere = 2r2
22
hb 2

3a
hr
3
1 2


Total surface area of a hemisphere = 3r2
Volume of the sphere =
Volume of the hemisphere = .
Right Triangular Prism
Volume of a right prism
= Area of the base × Height
Lateral surface area of a right prism
= Perimeter of the base ×
Height
Surface area of a right prism
= Lateral surface area + Area of ends
= Lateral surface area + 2(Area of the base)
Particular Case : If the base of a right prism
is an equilateral triangle of side a and height
h, then
Lateral surface area = 3a × h
Total surface area = 3a × h + a2
Volume = a2h.
HERON’S FORMULA
Points to Remember:
1. Area of right triangle = height .Base.
2. Area of an Equilateral triangle √3
4(side)2
3. Area of a isosceles triangle = √4 −
where, a is base and b represents equal sides.
4. Heron’s Formula : If a, b, c denote the lengths
of the sides of a triangle, then its Area,
A = ( − )( − )( − ) where, S =
5. Area of a quadrilateral can be calculated by
dividing the quadrilateral into triangles and using
heron’s formula for calculating area of each
triangle.
STATISTICS
1. Facts or figures, collected with a definite
purpose, are called Data.
2. Statistics is the area of study dealing with the
collection, presentation, analysis and
interpretation of data.
3. The data collected by the investigator himself
with a definite objective in mind are known as
Primary data.
4. The data collected by someone else, other than
the investigator, are known as Secondary data.
5. Any character which is capable of taking
reversal different values is called a variable.
6. Each group into which the raw data are
condensed is known as class-interval. Each class
is bounded by
two figures known as its limits. The figure on the
left is lower limit and figure on the right is upper
limit.
7. The difference between true upper limit and
true lower limit of a class is known as its class-
size.
8. Mid-value of a class (or class mark) =

9. Class size is the difference between any two
successive class marks (mid-values).
10. The difference between the maximum value
and the minimum value of the variable is known
as Range.
11. The count of number of observations in a
particular class is known as its Frequency.
12. The data can be presented graphically in the
form of bar graphs, histograms and frequency
polygons.
13. The three measures of central tendency for an
ungrouped data are :
(i) Mean : It is found by adding all the values of
the observations and dividing it by the total
number of observations. It is denoted by x .
(ii) Median : It is the value of the middle-most
observation(s).
3
r
3
4

3
r
3
2

2
3
4
3

If n is an odd number, then median = value of the
ℎ
observation. and, if n is an even number, then
median = mean of the values of ℎ +
1 ℎ observations.
(iii) Mode : The mode is the most frequently
occurring observation.
Empirical formula for calculating mode is given
by, Mode = 3 (Median) – 2 (Mean)
Probability
 Trial and Event :
An experiment is called a trial if it results in
anyone of the possible outcomes and all the
possible outcomes are called events.
For Example
(i) Participation of player in the game to win a
game, is a trial but winning or losing is an
event.
(ii) Tossing of a fair coin is a trial and turning up
head or tail are events.
(iii)Throwing of a dice is a trial and occurrence of
number 1 or 2 or 3 or 4 or 5 or 6 are events.
(iv)Drawing a card from a pack of playing cards
is a trial and getting an ace or a queen is an
event.
 Exhaustive Events :
Total possible outcomes of an experiment are
called its exhaustive events.
For Example
(i) Tossing a coin has 2 exhaustive cases i.e.
either head or tail may come upward.
(ii) Throwing of a die has 6 exhaustive cases
because any one of six digits 1, 2, 3, 4, 5, 6
may come upward.
(iii)Throwing of a pair of dice has 36 exhaustive
cases because any one of six digits 1, 2, 3, 4,
5, 6 may come upward on any dice so total
number of exhaustive cases = 6 x 6 = 36.
(iv)Tossing of two and three coins results in 4 and
8 exhaustive cases respectively because head
or tail may come upward on any coin. So in
case of two coins total number of cases = 2 x
2 = 4 and in case of three coins total number
of cases = 2 x 2 x 2 = 8
 Favourable Events :
Those outcomes of a trial in which a given
event may happen, are called favourable
cases for that event.
For Example -
(i) If a coin is tossed then favourable cases of
getting H is 1.
(ii) If a dice is thrown then favourable case for
getting 1 or 2 or 3 or 4 or 5 or 6, is 1.
(iii)If two dice are thrown, then favourable cases
of getting a sum of numbers as 9 are four i.e
(4,5), (5,4),
(3,6), (6,3).
 Equally likely events :
Two or more events are said to be equally
likely events if they have same number of
favourable cases.
For Example
(i) The result of drawing a card from a well
shuffled pack of cards, any card may appear in
a draw, so 52 different cases are equally
likely.
(ii) In tossing of a coin, getting of ‘H’ or ‘T’ are
two equally likely events.
(iii) In throwing of a dice, getting 1 or 2 or 3
or 4 or 5 or 6 are six equally likely events.
 Mutually Exclusive or Disjoint Events :
Two or more events are said to be mutually
exclusive, if the occurrence of one prevents or
precludes the occurrence of the others. In
other words they cannot occur together.
For example,
(i) In tossing of a coin, getting of ‘H’ or ‘T’ are
two mutually exclusive events because then
can not happen together.
(ii) In throwing of a dice, getting 1 or 2 or 3 or 4
or 5 or 6 are six mutually exclusive events.
(iii)In drawing a card from a pack of cards,
getting a card of diamond or heart or club or
spade are four mutually exclusive events.
 Simple and Compound Events :
If in any experiment only one event can
happen at a time then it is called a simple
event. If two or more events happen together
then they constitute a compound event.

For Example,
If we draw a card from a well shuffled pack of
cards, then getting a queen of spade is a
simple event and if two coins A and B are
tossed together then getting ‘H’ from A and
‘T’ from B is a compound event.
 Independent and Dependent Events :
Two or more events are said to be
independent if happening of one does not
affect other events. On the other hand if
happening of one event affects (partially or
totally) other event, then they are said to be
dependent events.
For Example,
(i) If we toss two coins, then the occurrence of
head on one coin does not influence the
occurrence of head or tail on the other coin in
any way. Hence these events are independent.
(ii) Suppose a bag contains 5 white and 4 black
balls. Two balls are drawn one by one. Then
two events that first ball is white and second
ball is black are independent if the first ball is
replaced before drawing the second ball. If the
first ball is not replaced then these two events
will be dependent because second draw will
have only 8 exhaustive cases.
 Sample Space :
The set of all possible outcomes of a trial is
called its sample space. It is generally
denoted by S and each outcome of the trial is
said to be a point of sample of S.
For example
(i) If a dice is thrown once, then its sample space
S = {1, 2, 3, 4, 5, 6}
(ii) If two coins are tossed together then its
sample space
S = {HT, TH, HH, TT}.
 Mathematical Definition of Probability
Let there are n exhaustive, mutually exclusive
and equally likely cases for an event A and m
of those are favourable to it, then probability
of happening of the event A is defined by the
ratio m/n which is denoted by P(A). Thus
P(A) =
=
Note : It is obvious that 0 m n. If an
event A is certain to happen, then m = n thus
P (A) = 1.
If A is impossible to happen then m = 0 and
so
P (A) = 0.
Hence we conclude that
0 P (A) 1
Further, if denotes negative of A i.e.
event that A doesn’t happen, then for
above cases m, n ; we shall have
P ( ) = = 1– = 1– P (A)
P (A) + P ( ) = 1
Playing Cards :
(i) Total : 52 (26 red, 26 black)
(ii) Four suits : Heart, Diamond, Spade, Club - 13
cards each
(iii)Court Cards : 12 (4 Kings, 4 queens, 4 jacks)
(iv) Honour Cards: 16 (4 aces, 4 kings, 4
queens, 4 jacks)
m
n
No of favourable cases to A
No of exhaustive cases to A
.
.
 
A
n m
n
 m
n
A

Mathematics compendium for class ix

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Mathematics compendium for class ix