CLASS-IX – Mathemetics
1. NUMBER SYSTEM
RATIONAL NUMBER: A number r is called a rational number, if it can be written in the form , where p and q are integers and q ≠.
A number s is called an irrational number, if it cannot be written in the form, where p and q are integers and q ≠ 0.
Every whole number is not a natural number but every natural number is a whole number.
One rational number between two given rational numbers a and b is =
N rational numbers between two irrational numbers are
Example 9 : Show that 0.2353535... = 0.235 can be expressed in the form of where p and q are integers and q ≠ 0. (Ans.= )
The decimal expansion of a rational number is either terminating or nonterminating recurring. Moreover, a number whose decimal expansion is terminating or non-terminating recurring is rational.
The decimal expansion of an irrational number is non-terminating non-recurring. Moreover, a number whose decimal expansion is non-terminating non-recurring is irrational. (√2 = 1.4142135623730950488016887242096...)
Let x be a rational number whose decimal expansion terminates. Then x can be expressed in the form p/q where p and q are coprime, and the prime factorisation of q is of the form 2n5m , where n, m are non-negative integers.
Rational numbers and Irrational numbers can be shown on number lines.
To rationalise the denominator of we multiply this by , where a and b are integers.
If r is rational and s is irrational, then r + s and r – s are irrational numbers, and and rs are irrational numbers, r ≠ 0.
Let a > 0 be a real number and p and q be rational numbers. Then
ap .aq = ap+q
(ii) (ap)q = apq
ap / aq = ap-q
ap.bp = (ab)p
a0 = 1
2.Polynomial
1- An algebraic expression of the form…………………+ is called polynomial of degree n, where, …………………………………..are real numbers and n is non negative integer.
2. A polynomial of one term is called a monomial.
3. A polynomial of two terms is called a binomial.
4. A polynomial of three terms is called a trinomial.
5. A polynomial of degree one is called a linear polynomial.
6. A polynomial of degree two is called a quadratic polynomial.
7. A polynomial of degree three is called a cubic polynomial.
8. A real number ‘a’ is a zero of a polynomial p(x) if p(a) = 0. In this case, a is also called a
root of the equation p(x) = 0.
9. Every linear polynomial in one variable has a unique zero, a non-zero constant
polynomial has no zero, and every real number is a zero of the zero polynomial.
10. Remainder Theorem : If p(x) is any polynomial of degree greater than or equal to 1
and p(x) is divided by the linear polynomial x – a, then the remainder is p(a).
11. Factor Theorem : x – a is a factor of the polynomial p(x), if p(a) = 0. Also, if x – a is a
factor of p(x), then p(a) = 0.
12. (x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2zx
13. (x + y)3 = x3 + y3 + 3xy(x + y)
14. (x – y)3 = x3 – y3 – 3xy(x – y)
15. x3 + y3 + z3 – 3xyz = (x + y + z) (x2 + y2 + z2 – xy – yz – zx)
16. If x + y +z=0, then x3 + y3 + z3 =3xyz
3.Coordinate Geometry
1. To locate the position of an object or a point in a plane, we require two perpendicular lines. One of them is horizontal, and the other is vertical.
2. The plane is called the Cartesian, or coordinate plane and the lines are called the coordinate axes.
3. The horizontal line is called the x -axis, and the vertical line is called the y - axis.
4. The coordinate axes divide the plane into four parts called quadrants.
5. The point of intersection of the axes is called the origin.
6. The distance of a point from the y - axis is called its x-coordinate, or abscissa, and the distance of the point from the x-axis is called its y-coordinate, or ordinate.
7. If the abscissa of a point is x and the ordinate is y, then (x, y) are called the coordinates of the point.
8. The coordinates of a point on the x-axis are of the form (x, 0) and that of the point on the y-axis are (0, y).
9. The coordinates of the origin are (0, 0).
10. The coordinates of a point are of the form (+ , +) in the first quadrant, (–, +) in the second
quadrant, (–, –) in the third quadrant and (+, –) in the fourth quadrant, where + denotes a
positive real number and – denotes a negative real number.
4. LINEAR EQUATIONS IN TWO VARIABLES
Variable: The value which varies or changes with respect to place and time.
Constant: The value which does not change anywhere.
Equation: An equality relation which is true for only selected value.
An equation of the form ax + by + c = 0, where a, b and c are real numbers, such that a and b are not both zero, is called a linear equation in two variables.
A linear equation in two variables has infinite many solutions.
The graph of every linear equation in two variables is a straight line.
x = 0 is the equation of the y-axis and y = 0 is the equation of the x-axis.
The graph of x = a is a straight line parallel to the y-axis.
The graph of y = a is a straight line parallel to the x-axis.
An equation of the type y = mx and x = my represents a line passing through the origin.
Every point on the graph of a linear equation in two variables is a solution of the linear equation. Moreover, every solution of the linear equation is a point on the graph of the
5. Euclid’s Geometry
Euclid’s Definitions,
1. A point is that which has no part.
2. A line is breadth less length.
3. The ends of a line are points.
4. A straight line is a line which lies evenly with the points on itself.
5. A surface is that which has length and breadth only.
6. The edges of a surface are lines.
7. A plane surface is a surface which lies evenly with the straight lines on itself.
Euclid’s Axioms
(1) Things which are equal to the same thing are equal to one another.
(2) If equals are added to equals, the wholes are equal.
(3) If equals are subtracted from equals, the remainders are equal.
(4) Things which coincide with one another are equal to one another.
(5) The whole is greater than the part.
(6) Things which are double of the same things are equal to one another.
(7) Things which are halves of the same things are equal to one another.
Euclid’s Postulates
Postulate 1:- A straight line may be drawn from any one point to any other point.
Postulate 2:- A terminated line can be produced indefinitely.
Postulate 3:- A circle can be drawn with any centre 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.
Once Euclid defined these he assumed that certain properties can not be proved and these assumption is called OBVIOUS UNIVERSAL TROUTH he divides it into two parts
Axioms: Common to any discipline (used throughout Mathematics)
Postulates: Specific to Geometry
AXIOMS-” The basic facts which are taken for granted, without proof and which are used throughout Mathematics are called Axioms”.
Postulates-” The basic facts which are taken for granted, without proof and which are specific to geometry are called postulates”.
Theorem: Theorems are statements which are proved using definition, Axioms, Postulates, Previous proved theorem and deductive reasoning.
Two equivalent versions of Euclid’s fifth postulate are:
(i) ‘For every line l and for every point P not lying on l, there exists a unique line m passing through P and parallel to l’.
(ii) Two distinct intersecting lines cannot be parallel to the same line.
Note: Euclid complied everything in a book and that’s why this geometry is called Euclid Geometry and the book name is called “The Element”.
6.Triangles
•Concept of congruency.
Two figures are congruent, if they are of the same shape and of the same size.
1. Two coins of same value of same year
2.Two circles of the same radii are congruent.
3.Two squares of the same sides are congruent.
•TheΔABC ΔPQR, then the corresponding elements are as A(P, B (Q and C (R,
•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.
•ASA Congruence Rule- If two angles and the included side of one triangle is equal to two angles and the included side of the other triangle, then the two triangles is congruent.
•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 .
• SSS Congruence Rule-If three sides of one triangle are equal to three sides of the other triangle, then the two triangles are congruent.
• 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.
• Properties of Triangle
* Exterior angle sum property
* Angle sum property
•Theorems on Triangles
1.Angles opposite to equal sides of a triangle are equal.
2.Sides opposite to equal angles of a triangle are equal.
3. Sum of any two sides of a triangle is greater than the third side.
LINES AND ITS ANGLES
An angle is the inclination of one ray to other if they have common initial point.
An angle whose measure is 900 is called a right angle.
An angle whose measure is less than 900 is called an acute angle.
An angle whose measure is greater than 900 and less than 1800 is called an obtuse angle.
An angle whose measure is 1800 is called a straight angle.
An angle whose measure is more than 1800 is called a reflex angle.
Two angles with sum 900 is called complementary angles.
Two angles with sum 1800 is called supplementary angles.
Two angles having a common vertex and a common arm are called adjacent angles.
If two lines intersect, then vertically opposite angles are equal.
If a transversal intersects two parallel lines, then each pair of
(i)Corresponding angles are equal.
(ii)Alternate interior angles are equal.
(iii) Interior angles on the same side of the transversal are supplementary.
Lines which are parallel to a given lines are parallel to each other.
8 .Quadrilateral
Basic Facts
.• A plane figure bounded by four sides is called a quadrilateral.
• Sum of the angles of a quadrilateral is 360°.
In the given figure, we have.
(i) The points A, B, C and D are the vertices of quadrilateral ABCD.
(ii) The line segments AB, BC, CD and DA are the sides of quadrilateral ABCD.
(iii) The line segments AC and BD are called the diagonals of quadrilateral ABCD. A
Note:
(i) Two sides having a common end point are called adjacent sides.
(ii) Two sides having no common end point are called opposite sides.
(iii) Two angles of a quadrilateral having a common arm are called consecutive angles.
(iv) Two angles of a quadrilateral having no common arm are called its opposite angles.
TYPES OF QUADRILATERALS
Various types of quadrilateral are:
(i) Parallelogram
A quadrilateral in which opposite sides are parallel is called a parallelogram. In the figure, ABCD is a parallelogram. Here, AB || CD and AD || BC.
Also, opposite sides of a parallelogram are equal.
(ii) Rectangle:
A parallelogram, each of whose angle is 90°, is called a rectangle. In the figure, PQRS is a rectangle. We write it as rect. PQRS.
(iii) Square:
A rectangle having all sides equal is called a square. In the figure, LMNO is a square.
(iv)Rhombus:
A parallelogram having all sides equal is called a rhombus. In the figure, PQRS is a rhombus.
(v) Trapezium:
A quadrilateral in which two opposite sides are parallel and two opposite sides are non-parallel, is called a trapezium. In the figure, BCDE is a trapezium.
Note: If the two non-parallel sides of a trapezium are equal, then it is called an isosceles trapezium.
(vi) Kite:
A quadrilateral in which two pairs of adjacent sides are equal is known as kite. PQRS is a kite such that PQ = QR and PS = RS.
Remark (i) A square, rectangle and rhombus are all parallelograms.
(ii) A square is a rectangle and is also a rhombus, but a rectangle or a rhombus is not a square.
(iii) A parallelogram is a trapezium, but a trapezium is not a parallelogram.
Key Facts
• A line segment joining the mid-points of any two sides of a triangle is parallel to the third side.
• A line segment joining the mid-points of any two sides of a triangle is half of the third side.
• The quadrilateral formed by joining the mid-points of the sides of a quadrilateral, is a parallelogram
• A quadrilateral is a parallelogram, if
(i) opposite sides are equal,
(ii) opposite angles are equal,
(iii) diagonals bisect each other,
• A diagonal of a parallelogram divides it into two congruent triangles.
• Diagonals of a rhombus bisect each other at right angles, and vice versa.
• Diagonals of a square bisect each other at right angles and are equal and vice versa.
• Diagonals of a rectangle bisect each other and vice versa.
PROPERTIES OF A PARALLELOGRAM
REMEMBER
1. A diagonal of a parallelogram, divides it into two congruent triangles.
2. In a parallelogram, opposite sides are equal.
3. In a parallelogram, opposite angles are equal.
4. The diagonals of a parallelogram bisect each other.
Mid-Point Theorem
• A line through the mid-point of a side of a triangle, parallel to another side bisects the third side.
9.AREA OF PARALLELOGRAM AND TRIANGLES
1. Area of a figure is a number (in some unit) associated with the part of the plane enclosed by that figure.
2. Two congruent figures have equal areas but the converse need not be true.
3. If a planar region formed by a figure T is made up of two non-overlapping planar regions formed by figures P and Q, then ar (T) = ar (P) + ar (Q), where ar (X) denotes the area of figure X.
4. Two figures are said to be on the same base and between the same parallels, if they have a common base (side) and the vertices, (or the vertex) opposite to the common base of each figure lie on a line parallel to the base.
5. Parallelograms on the same base (or equal bases) and between the same parallels are equal in area. 6. Area of a parallelogram is the product of its base and the corresponding altitude.
7. Parallelograms on the same base (or equal bases) and having equal areas lie between the same parallels.
8. If a parallelogram and a triangle are on the same base and between the same parallels, then area of the triangle is half the area of the parallelogram.
9. Triangles on the same base (or equal bases) and between the same parallels are equal in area.
10. Area of a triangle is half the product of its base and the corresponding altitude.
11. Triangles on the same base (or equal bases) and having equal areas lie between the same parallels. 12. A median of a triangle divides it into two triangles of equal areas.
(class – IX)
Chapter- 10 (Circles)
Definitions, statements and summary
A circle is the collection of all points in a plane, which are equidistant from a fixed point in the plane.
Equal chords of a circle (or of congruent circles) subtend equal angles at the centre.
If the angles subtended by two chords of a circle (or of congruent circles) at the centre (corresponding centres) are equal, the chords are equal.
The perpendicular from the centre of a circle to a chord bisects the chord.
The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.
Chords equidistant from the centre (or corresponding centres) of a circle (or of congruent circles) are equal.
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.
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.
Angles in the same segment of a circle are equal.
Angle in a semicircle is a right angle.
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.
If sum of a pair of opposite angles of a quadrilateral is 1800, the quadrilateral is cyclic.
Key points of the topic
There is one and only one circle passing through three non-collinear points.
Equal chords of a circle (or of congruent circles) are equidistant from the centre (or corresponding centres).
Congruent arcs of a circle subtend equal angles at the centre.
The sum of either pair of opposite angles of a cyclic quadrilateral is 1800.
Only one circle can be drawn from two given points.
Infinite number of diameters can be drawn in any circle.
Important formula
Diameter = 2 x radius (d = 2r)
Circumference of circle = 2 r
There is one and only one circle passing through three given non-collinear points
Chapter- 11 (Construction)
Summary and key points
1. To bisect a given angle.
2. To draw the perpendicular bisector of a given line segment.
3. To construct an angle of 60° etc.
4. To construct a triangle given its base, a base angle and the sum of the other two sides.
5. To construct a triangle given its base, a base angle and the difference of the other two sides.
6. To construct a triangle given its perimeter and its two base angles.
Chapter- 12 (Heron’s formula)
Area of a triangle with its sides as a, b and c is calculated by using Heron’s formula, stated as Area of triangle =
where s =
2. Area of a quadrilateral whose sides and one diagonal are given, can be calculated by dividing the quadrilateral into two triangles and using the Heron’s formula.
13.SURFACE AREAS AND VOLUMES
1. Cube Lateral surface area (LSA ) = 4 a2
Total surface area (TSA) = 6a2
Volume = a3
2. Cubiod Lateral surface area (LSA)
or (Area of four walls) = 2(l + b) h
Total surface area (TSA) = 2lb + 2bh +2lh
Volume = lbh
3. Cylinder: Lateral surface area (LSA) = 2Ï€rh
Total surface area (TSA) = 2Ï€rh + 2Ï€r2
Volume = πr2h
4. Cone : Lateral surface area (LSA) = πrl
Total surface area (TSA) = πr(r + l)
Volume = πr2h
l2 = h2 +r2 (relation between height, slant height and radius)
5. Sphere :-
Lateral surface area (LSA) = Total surface area (TSA) = 4Ï€r2
Volume = πr3
6. Hemi-Sphere Lateral surface area (LSA) = 2Ï€r2
Total surface area (TSA) = 3Ï€r2
Volume = πr3
Chapter-14
STATISTICS
Facts or figures, collected with a definite purpose, are called data.
OR
Statistics is the area of study dealing with the presentation, analysis and interpretation of data
Data can be presented graphically in the form of bar graphs, histograms and frequency polygons.
The three measures of central tendency for ungrouped data are:
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 .
The median is that value of the given number of observations, which divides it into exactly two parts. So, when the data is arranged in ascending (or descending) order.
Mode: The mode is the most frequently occurring observation.
To present a large amount of data so that a reader can make sense of it easily, we condense it into groups like 20-29, 30-39, . . ., 90-99 (since our data is from 23 to 98). These groupings are called ‘classes’ or ‘class-intervals’, and their size is called the class-size or class width, which is 10 in this case. In each of these classes, the least number is called the lower class limit and the greatest number is called the upper class limit, e.g., in 20-29, 20 is the ‘lower class limit’ and 29 is the ‘upper class limit’.
Presenting data in this form simplifies and condenses data and enables us to observe certain important features at a glance. This is called a grouped frequency distribution table.
The mid-points of the class-intervals are called class-marks. To find the class-mark of a class interval, we find the sum of the upper limit and lower limit of a class and divide it by 2.
Consider a situation in which there is no same class size. In such cases, histogram is prepared as follows :
Since the widths of the rectangles are varying, we need to make certain modifications in the lengths of the rectangles so that the areas are again proportional to the frequencies. The steps to be followed are as given below: 1. Select a class interval with the minimum class size. If the minimum class-size is 10. 2. The lengths of the rectangles are then modified to be proportionate to the class size 10.
For, instance, when the class size is 20, the length of the rectangle is 7. So when
the class size is 10, the length of the rectangle will be 7/20*10 = 3.5.
Frequency polygons are used when the data is continuous and very large. It is very useful for comparing two different sets of data of the same nature, for example, comparing the performance of two different sections of the same class.
if the data has a few points which are very far from most of the other points, (like 1,7,8,9,9) then the mean is not a good representative of this data. Since the median and mode are not affected by extreme values present in the data, they give a better estimate of the average in such a situation.
15.PROBABILITY
Probability:- Probability is the numerical measurement of uncertainty.
In this portion we have to study only experimental Probability
Experiment/ Trial- Any work which gives some fruitful results (outcomes).
Random Experiment- An experiment whose outcome can not be predicted in advance.
Event:- An event for an experiment is the collection of some outcomes of the experiment.
The empirical (or experimental) probability P(E) of an event E is given by
P(E) = .
The probability of an event E is a number P(E) such that 0 ≤ P(E) ≤ 1.
The probability of a sure event is 1.
The probability of an impossible event is o (zero).
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