10th Class

Real Numbers

Rational numbers: Numbers which can be written in the form of\[\frac{p}{q}(q\ne 0)\]where p and q are integers, are called rational numbers.

Note: Every terminating decimal and non-terminating repeating decimal can be expressed as a rational number.

Irrational numbers: Numbers which cannot be written in the form of\[\frac{p}{q}\]where p and q are integers and\[q\ne 0\]are called irrational numbers. In other words, numbers which are not rational are called irrational numbers.
Real numbers: The rational numbers and the irrational numbers together are called real numbers.

Note: Any number that can be represented on a number line is called a real number.

Lemma: A proven statement which is used to prove another statement is called a lemma.
Euclid's division lemma: For any more...

Catgeory: 10th Class

Polynomials

Polynomial: A function p(x) of the form\[p(x)={{a}_{0}}+{{a}_{1}}x+......+{{a}_{n}}{{x}^{n}}\], where\[{{a}_{0}},{{a}_{1}},...{{a}_{n}}\]are real numbers and 'n' is a non-negative (positive) integer is called a polynomial.

Note: \[{{a}_{0}},{{a}_{1}},...{{a}_{n}}\]are called the coefficients of the polynomial.

If the coefficients of a polynomial are integers, then it is called a polynomial over integers.
If the coefficients of a polynomial are rational numbers, then it is called a polynomial over rational.
If the coefficients of a polynomial are real numbers, then it is called a polynomial over real numbers.
A function\[p(x)={{a}_{0}}+{{a}_{1}}x,...+{{a}_{n}}{{x}^{n}}\]is not a polynomial if the power of the variable is either negative or a fractional number.
Standard form: A polynomial is said to be in a standard form if it is written either in more...

Catgeory: 10th Class

Pair of Linear Equations In Two Variables

Pair of Linear Equations in Two Variables

Linear equation in two variables: An equation of the form\[\text{ax}+\text{by}=\text{c}\], where \[a\ne 0,b\ne 0\] and a, b and c are real numbers is known as a linear equation in two variables x and y.
A pair of linear equations in two variables: Two linear equations in the same two variables are called a pair of linear equations in two variables.
General form of a pair of linear equations in two variables: The general form of a pair of linear equations in two variables is \[{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0\] and\[{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0\], where \[{{a}_{1}},{{a}_{2}},{{b}_{1}},{{b}_{2}},{{c}_{1}}\]and\[{{c}_{2}}\]are real numbers such that \[a_{1}^{2}+b_{1}^{2}\ne 0\]and\[a_{2}^{2}+b_{2}^{2}\ne 0\].

Note: A pair of linear equations in two variables is called a system of simultaneous linear equations.

Solution of a pair of linear equations more...

Catgeory: 10th Class

Quadratic Equations

Quadratic equation: An equation of the form \[a{{x}^{2}}+bx+c=0\] where a, b and\[c\in R\]and \[a\ne 0\]is called a quadratic equation.

If p(x) is a quadratic polynomial, then p(x) = 0 is called a quadratic equation. Note: (i) An equation of degree 2 is called a quadratic equation. (ii) The quadratic equation of the form \[a{{x}^{2}}+bx+c=0\]. Solution or roots of a quadratic equation: If p(x) = 0 is a quadratic equation, then the zeros of the polynomial p(x) are called the solutions or roots of the quadratic equation p(x)=0. Note: (i) Since the degree of a quadratic equation is 2, it has 2 roots or solutions. (ii) x = a is the root of p(x) = 0, if p(a) = 0. (iii) Finding the roots of a quadratic equation is more...

Catgeory: 10th Class

Arithmetic Progressions

Sequence: Numbers arranged in a definite order according to definite rule are said to be in a sequence.

Term: Each number of a sequence is called a term.

\[{{\mathbf{n}}^{\mathbf{th}}}\]term: The term occurring at the\[{{n}^{th}}\]place of a sequence is called its n"1 term, usually denoted by\[{{t}_{n}}\].

Progressions: Sequences that follow a definite pattern are called progressions.

Arithmetic progressions: sequence in which each term differs from its preceding term by a fixed number (constant) is called an arithmetic progression, denoted as A.P.

Common Difference: The fixed number by which any two successive terms of an A.P. differ is called the common difference of A.P. denoted by 'd'. more...

Catgeory: 10th Class

Triangles

Similar figures: Two figures having the same shape (not necessarily the same size) are called similar figures.
Congruent figures: Two figures having the same shape and the same size are called congruent figures.

Note: Congruent figures are similar but similar figures are not congruent.

Similar polygons: Two polygons of the same number of sides are similar; if their corresponding angles are equal and their corresponding sides are in the same ratio (or proportion)

Note: The same ratio of the corresponding sides is referred to as the scale factor (or the Representative Fraction) for the polygons.

Two triangles are similar if their corresponding angles are equal and their corresponding sides are proportional.

more...

Catgeory: 10th Class

Co-ordinate Geometry

The branch of geometry that sets up a definite correspondence between the position of a point in a plane and a pair of algebraic numbers called coordinates is called Coordinate geometry.
The distance of a point from the Y-axis is called the X-coordinate or Abscissa.
The distance of a point from the X-axis is called the Y-coordinate or ordinate.
The coordinates of a point on the X-axis are of the form (x, 0) and of a point on the Y-axis are of the form (0, y).
The abscissa and ordinate of a point taken together is known as coordinates of a point.
The point of intersection of the axes of coordinates is called the origin.
The quarter plane that results from the division of the plane by the coordinate axes more...

Catgeory: 10th Class

Introduction to Trigonometry

Trigonometry: The branch of mathematics that deals with the study of relationships between the sides and angles of a triangle is called trigonometry.

The word 'trigonometry' is derived from the greek words 'tri' meaning three, 'gon' meaning sides and 'metron' meaning measure.

Trigonometric ratios: In right AABC, AB is the hypotenuse, AB is the side opposite to \[\angle C(=\theta )\], and BC is the side adjacent to 6.

The trigonometric ratios for angle \[\theta \]:

(a) \[\sin \theta =\frac{side\,opposite\,to\,\theta }{Hypotenuse}=\frac{AB}{AC}\] (b) \[\cos \theta =\frac{side\,adjacent\,to\,\theta }{Hypotenuse}=\frac{BC}{AC}\] (c) \[\tan \theta =\frac{side\,opposite\,to\,\theta }{side\,adjacent\,to\,\theta more...

Catgeory: 10th Class

Some Applications of Trigono

Some Applications of Trigonometry

Line of sight: The imaginary line drawn from the eye of the observer to the object, when the observer is looking at it is called the line of sight,

Note: Line of sight is called the line of vision.

Angle of elevation: The angle formed by the line of sight with the horizontal when the object is above the horizontal level is called the angle of elevation,