Definitions [7]
The terms that do not have the same literal coefficients are called unlike terms.
For example:
6a, 6ab and 6ac are unlike terms.
Terms having the same literal part (same variables with the same powers) are called like terms.
For example:
xy, 5xy, -4xy, etc. are like terms
Identity: An identity is an equality, which is true for all values of the variables in equality.
A balanced equation is one in which the value of the left-hand side (LHS) is equal to the value of the right-hand side (RHS).
A Pictograph is a chart that uses pictures or symbols to represent data. Each picture stands for a specific number of items, making the data easy to understand at a glance.
The two mutually perpendicular number lines intersecting each other at their zeroes are called rectangular axes or coordinate axes, or axes of reference.
The position of a point in a plane is expressed by a pair of numbers, one concerning the x-axis and the other concerning the y-axis. called co-ordinates.
-
x → distance from y-axis (abscissa)
-
y → distance from x-axis (ordinate)
Formulae [8]
(a + b)2 = a2 + 2ab + b2
(a - b)2 = a2 - 2ab + b2.
(a + b)(a - b) = a2 - b2
- (x + a)(x + b) = x2 + (a + b)x + ab
- (a + b)3 = a3 + 3a2b + 3ab2 + b3.
- (a - b)3 = a3 - 3a2b + 3ab2 - b3.
- (a + b)2 = a2 + 2ab + b2
- (a - b)2 = a2 - 2ab + b2
- (a + b)(a - b) = a2 - b2
| Sr. No. | Name | Condition | Formula |
|---|---|---|---|
| i. | Distance Formula | Two points P(x₁, y₁), Q(x₂, y₂) | \[\mathrm{d(PQ)}=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}\] |
| a. | Internal Division | P divides AB in the ratio m: n | \[\mathrm{P\equiv\left(\frac{mx_{2}+nx_{1}}{m+n},\frac{my_{2}+ny_{1}}{m+n}\right)}\] |
| b. | Midpoint Formula | P is the midpoint of AB | \[\mathrm{P}\equiv\left(\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2}\right)\] |
| c. | External Division | P divides AB externally in m: n | \[\mathrm{P\equiv\left(\frac{mx_{2}-nx_{1}}{m-n},\frac{my_{2}-ny_{1}}{m-n}\right)}\] |
| iii. | Centroid Formula | Triangle with vertices A, B, C | \[\left(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3}\right)\] |
Key Points
Sign Convention
-
Right of y-axis → +x
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Left of y-axis → −x
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Above x-axis → +y
-
Below x-axis → −y
Standard Line Results
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x = 0 → y-axis
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y = 0 → x-axis
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x = a → line parallel to the y-axis
-
y = b → line parallel to the x-axis
Quadrant Reminder
| Quadrant | Sign of (x, y) |
|---|---|
| I | (+, +) |
| II | (−, +) |
| III | (−, −) |
| IV | (+, −) |
| Condition | Nature of Lines | Number of Solutions | Type of Pair |
|---|---|---|---|
| \[\frac{a_1}{a_2}\neq\frac{b_1}{b_2}\] | Intersecting | One (unique) solution | Consistent |
| \[\frac{a_1}{a_2}=\frac{b_1}{b_2}\neq\frac{c_1}{c_2}\] | Parallel | No solution | Inconsistent |
| \[\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\] | Coincident | Infinitely many solutions | Dependent (consistent) |
Concepts [43]
- Algebraic Expressions
- Terms, Factors and Coefficients of Expression
- Classification of Terms in Algebra
- Addition of Algebraic Expressions
- Subtraction of Algebraic Expressions
- Multiplication of Algebraic Expressions
- Multiplying Monomial by Monomials
- Multiplying a Monomial by a Binomial
- Multiplying a Monomial by a Trinomial
- Multiplying a Binomial by a Binomial
- Multiplying a Binomial by a Trinomial
- Division of Algebraic Expressions
- Dividing a Monomial by a Monomial
- Dividing a Polynomial by a Monomial
- Concept of Identity
- Expansion of (a + b)2 = a2 + 2ab + b2
- Expansion of (a - b)2 = a2 - 2ab + b2
- Expansion of (a + b)(a - b) = a2-b2
- Expansion of (x + a)(x + b)
- Expansion of (a + b)3
- Expansion of (a - b)3
- Expansion of (x + a)(x + b)(x + c)
- Factorising Algebraic Expressions
- Factorisation by Taking Out Common Factors
- Factorisation by Taking Out the Common Binomial Factor from Each Term
- Factorisation by Regrouping Terms
- Factorisation Using Identities
- Factors of the Form (x + a)(x + b)
- Factorise Using the Identity (a + b)3
- Factorise Using the Identity (a – b)3
- Concept of Find the Error
- Expressions with Variables
- Equation in Mathematics
- Word Problems on Linear Equations
- Concept of Graph
- Cartesian Coordinate System
- Foundation of Coordinate Geometry
- Quadrants and Sign Convention
- Plotting a Point in the Plane If Its Coordinates Are Given.
- Geometrical Representation of a Linear Equation by Plotting a Straight Line
- Geometrical Representation of a Linear Equation by Line Parallel to the Coordinate Axes
- Linear Pattern
- Graphical Method with Different Cases of Solution
