9th Class

Number Systems  

• Rational numbers (Q): The numbers of the form,\[\frac{p}{q}\] where 'p' and 'q' are integers and are called rational     number A numfaer of the form r is a fraction. So all fractions are rational numbers.

            Note: A number of the form \[\frac{\mathbf{a}}{\mathbf{b}}\] is a fraction. So all are rational             numbers. in the fraction, ‘a’     is called the numbers and ‘b’ is called the denominator. e.g.             \[\frac{\mathbf{1}}{\mathbf{2}}\mathbf{-            }\frac{\mathbf{2}}{\mathbf{3}}\mathbf{,}\frac{\mathbf{7}}{\mathbf{6}}\mathbf{,}\frac{\mat            hbf{6}}{\mathbf{11}}\mathbf{,-}\frac{\mathbf{2}}{\mathbf{9}}\mathbf{,}....\]                                     (i) Zero is a rational number.             Note: 0 by 0 is undefined.               (ii) Every integer is a rational number.             (iii) A rational number, may or may not be an integer.             (iv) To write W distinct rational numbers between any two rational numbers 'a' and \['b'\],             we write\[a=\frac{{{P}_{1}}}{q}\] and\[b=\frac{{{P}_{2}}}{q}\] such more...

Polynomials  

• An expression of the form \[p(\operatorname{x})=+{{a}_{n}}{{\operatorname{x}}^{n}}+{{a}_{n-1}}......+{{a}_{2}}{{\operatorname{x}}^{2}}+{{a}_{1}}{{\operatorname{x}}^{2}}+{{a}_{0'}}\,\operatorname{where}{{a}_{0}},{{a}_{1}},a{{ & }_{2}},......,\]are real numbers \['n'\]is a non-negative integer and \[{{a}_{n}}\ne 0\] is called a polynomial of degree.

 

• Each of \[{{a}_{n}}{{\operatorname{x}}^{n}},{{a}_{n-1}},......{{a}_{2}},{{x}^{2}},{{a}_{1}}\operatorname{x}\,and\,{{a}_{n}}\ne 0\]and a with is called a term of the polynomial p(x).

              Note: The power of variable in a polynomial must be a whole number.               

• An expression of the form\[\frac{p\left( \operatorname{x} \right)}{q\left( \operatorname{x} \right)}\] where p(x) and q(x) are polynomials and \[q(\operatorname{x})\ne 0\]is called a rational expression.

              Note: Every polynomial is a rational expression, but every rational expression need not be a polynomial.  

• A polynomial d(x) is called a divisor of a polynomial p(x) if p(x) = d(x).q(x) for some polynomial q(x).

 

• Polynomials more...

Co-ordinate Geometry  

• Co-ordinate Geometry: The branch of mathematics in which geometric problems are solved through algebra by using the coordinate system is known as coordinate geometry.
• In coordinate geometry, every point is represented by an ordered pair, called coordinates of that point.
• A pair of numbers 'a' and V listed in a specific order with 'a' at the first place and 'b' at the second place is called an ordered pair (a, b).

              Note :(i) (a, b) \[\ne \] (b, a)             (ii) If (a, b) = (c, d) then a = a and b=d.            

• The position of a point in a plane is determined with reference to two fixed mutually perpendicular lines called the coordinate axes.

 

• The horizontal line is called X-axis and more...

Linear Equations in Two Variables  

• Equation: A statement of equality of two algebraic expressions involving a variable is called an equation.

 

• Simple linear equation: An equation which contains only one variable of degree 1 is called a simple linear equation.

 

• Solution of an equation: The value of the variable, which when substituted in the given equation, makes the two sides L.H.S (Left Hand Side) and R.H.S (Right Hand Side) of the equation equal is called the solution of that equation.

 

• Transposition: Any term of an equation may be taken to the other side with a change in its sign. This process is called transposition.

 

• Cross multiplication: If \[\frac{\operatorname{ax}+b}{\operatorname{cx}+d}=\frac{p}{q}\]then q (ax + b) = p (cx + d). more...

Introduction to Euclid's Geometry  

• Axioms: Axioms or postulates are the assumptions which are obvious universal truths and are not to be proved.

 

• Some of the axioms given by Euclid: (i) Things which are equal to the same thing are equal to one another. i.e., if a = c and b = c, then a = b.

              (ii) If equals are added to equals, the wholes are equal. i.e., if a = b and c = d, then a + c = b + d.             Also a = b \[\Rightarrow \]a+c=b+c.             Here, a, b, c and d are same kind of things.               (iii) If equals are subtracted from equals, the remainders are equal.               (iv) The things which coincide with one more...

Lines and Angles  

• Angle: An angle is the union of two rays with a common initial point. An angle is denoted by symbol\[\angle \]. It is measured in degrees,

               The angle formed by the two rays \[\overline{AB}\,\,and\,\,\overline{AC}\text{ }is\text{ }\angle BAC\text{ }or\text{ }\angle CAB.~\]called \[\overline{AB}\,\,and\,\,\overline{AC}\]are called the arms and the common initial point ‘A’ is called the vertex of the angle.    

• Bisector of an angle: A line which divides an angle into two equal a parts is alled the bisector of the angle.

more...

Triangles  

• A triangle is a closed figure bounded by three straight lines. It is denoted by the symbol\[\Delta \].

\[\Delta \]ABC has three sides denoted by AB, BC and CA; three angles denoted by \[\angle ~A,\angle B\text{ }and\text{ }\angle C\,;\]and three vertices denoted by A, B and C.              

• Two geometrical figures are said to be congruent if they have exactly the same shape and size. Congruence is denoted by the symbol =.
• Two triangles are congruent if the sides and angles of one triangle are equal to the corresponding sides and angles of the other triangle.

 

• The congruence of more...

Quadrilaterals  

• A quadrilateral in which the measure of each angle is less than 180° is called a convex quadrilateral.

 

• A quadrilateral in which the measure of at least one of the angles is more than \[{{180}^{o}}\]s known as a concave quadrilateral.

 

• The sum of the angles of a quadrilateral is \[{{360}^{o}}\] (or) 4 right angles.

 

• When the sides of a quadrilateral are produced, the sum of the four exterior angles so formed \[{{360}^{o}}\]

 

• Various types of quadrilaterals:

      (i) Trapezium: (a) A quadrilateral having exactly one pair more...

Areas of Parallelograms and Triangles  

• A polygonal region is the union of a polygon and it’s interior. For e.g., the union of a triangle and its interior is called the triangular region.

 

• Every polygonal region has area. Area of a figure is a number associated with the part of the plane enclosed by that figure.

 

• Two congruent figures have equal areas but figures with equal areas need not be congruent.

 

• If ABCD is a rectangle with AB = \[l\] m and BC = b m, then the area of the rectangular region ABCD is\[lb\text{ }sq.\text{ }m\text{ }or\text{ l}b\text{ }{{m}^{2}}\].

 

• If A and B are two regions having at the most more...

Circles  

• A circle is a closed figure in a plane formed by

            The collection of all the points in the plane which centre are at a constant distance from a fixed point in the plane. The fixed point is called the centre of length of radlus the circle and the constant distance is called the radius of the circle.                                                          

• The plane region inside the circle is called tne interior of the circle,

 

• If a circle is drawn in the plane X (infinite dimensions), then the part of more...