LINEAR EQUATION IN ONE VARIABLE
FUNDAMENTALS

- Variable: A symbol which takes various values is known as a variable. Normally it is denoted by letters x, y etc.
- Constant: A symbol having a fixed numerical value is called a constant.

- Coefficient: In the product of a variable and a constant, each is called the coefficient of the other. Sometimes, symbols like a, b, 1, m etc..., are used to denote the coefficients.
- Expression: An expression can be defined as a combination of constant, variables and coefficients by some or all of the four fundamental mathematical operations \[(+,\,\,-,\,\,\times \,\,\text{and}\,\,\div ).\]

- Equation: A statement of equality of two algebraic expression 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: That 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 sided) of the equation equal is called the solution or root of that equation.

- Rules for solving an equation

ALGEBRAIC EXPRESSIONS
FUNDAMENTALS

- Algebra: It is a branch of mathematics in which we use literal numbers and statements symbolically. Literal numbers can be positive or negative. They are Variables.
- Variable: A symbol which takes various values is known as a variable. Normally it is denoted by x, y, z etc.
- Coefficient: Symbols like a, b, 1, m etc.., are used to denote the coefficients. Coefficients that are numbers are called numerical coefficients.
- Algebraic expression: A combination of constants and variables connected by some or all of the four fundamental operations \[+,\text{ }-,\text{ }\times \]and \[\div \] is called an algebraic expression.

- Like and unlike terms: In a given algebraic expression, the terms having the same literal factors are called like or similar terms, otherwise they are called unlike terms.

- Factors: Each term of an algebraic expression consists of a constant or product of constant and variables.

- Various types of algebraic expression:

- Addition of Algebraic Expression: While adding algebraic expressions, we collect the like terms and add them. The sum of several like terms is another like term whose coefficient is the sum of the coefficients of those like terms. The like terms are added and the unlike terms are left as they are. e.g.,

- Subtraction of Algebraic Expression>: The difference of two like terms is a like term whose coefficients is the difference of the numerical coefficients of the two like terms.

- Value of an algebraic expression: The value of an algebraic expression depends on the values of the variables forming the expression.
- Using algebraic expressions - Formulae and Rules: Rules and formulae in mathematics are written in a concise and general form using algebraic expression.

RATIONAL NUMBERS
FUNDAMENTALS

- Natural numbers (N): 1, 2, 3, 4, 5..... ..etc., are called natural numbers.
- Whole numbers (W): 0, 1, 2, 3, 4, etc.., are called whole numbers.
- Integers (Z): 1.......\[-4,-3,-2,-1,\,\,0,\,\,1,\,\,2,\,\,3,\,\,4\]........ etc.., are called integers. (denoted by I or Z) 1, 2, 3, 4, .. ...etc., are called positive integers denoted by \[{{Z}^{+}}\]or \[{{I}^{+}}\].

- Fractions: The numbers of the form \[\frac{x}{y}\], where \[x\] and \[y\]c are natural numbers, are known as fractions. e.g., \[\frac{2}{5},\,\,\frac{3}{1},\,\,\frac{1}{122},.....\]etc.

- A rational number \[\frac{p}{q}\] is positive if p and q are either both positive or both negative.

- A rational number \[\frac{p}{q}\] is negative if either of p and q is positive and the other term (q or p) is negative.

- Representation of Rational numbers on a number line:

- In general, any rational number is either of the following two types.

- Standard form of a rational number:

INTEGERS
FUNDAMENTALS
Usually, negative numbers are placed in brackets to avoid confusion arising due to two signs in evaluations simultaneously,
e.g., \[3+\left( -\,5 \right)=-2\]
2.0 is not included in either \[{{Z}^{+}}\]c or\[{{Z}^{-}}\]. Hence, it is non-negative integer
Common use of numbers
(i) To represent quantities like profit, income, increase, rise, high, north, east, above, depositing, climbing and so on, positive numbers are used.
(ii) To represent quantities like loss, expenditure, decrease, fall, low, south, west, below, withdrawing, sliding and so on, negative numbers are used.
(iii) On a number line, when we
Add a positive integer, we move to the right.
Add a negative integer, we move to the left.
Subtract a positive integer, we move to the left.
Subtract a negative integer, we move to the right
Notation e means belongs to; a, b \[\in \] means the numbers a and b belong to \[\] (set of integers)
Note:
0 is neither positive nor negative.
The + sign is not written before a positive number
\[\frac{1}{2}\] and 0.3 are more...

- In lower classes, you would have read about counting number. 1, 2, 3,.........
- They are called natural numbers (N).

- Representation of natural numbers on a number line. To represent natural numbers on a number line, we should draw a line and write the number at equal distances on it as shown below:

- Whole Number (W): The set of natural numbers together with zero is known as the set of whole numbers.

- In set notation, set of whole numbers (W) = set of natural numbers (N) + zero; { } is used for set notation.
- Integers (Z): The set containing negatives of natural numbers along with whole numbers is called the set of integers.

FRACTIONS AND DECIMALS
FUNDAMENTALS
Improper fraction: A fraction whose numerator is greater than or equal to its denominator is called an improper fraction. If number is written as\[\frac{N}{D}\], then \[N~\underline{>}\,D\]where N = numerator D = Denominator e.g.,\[\frac{100}{100},\] \[\frac{51}{50},\] \[\frac{45}{2},.....\]etc.
Proper fraction: A fraction whose numerator is less than its denominator is called a proper fraction. e.g.,\[\frac{3}{8},\,\,\frac{6}{7},\,\,\frac{9}{16},\,\,\frac{8}{18},......\]etc.
Decimal fraction: A fraction whose denominator is 10, 100, 1000 etc., is called a decimal fraction. e.g., \[\frac{2}{10},\,\,\frac{8}{100},\,\,\frac{26}{1000},\,\,\frac{1312}{1000},......\]etc.
Vulgar fraction: A fraction whose denominator is a whole number other than 10, 100, 1000, etc., is called a vulgar fraction. e.g., \[\frac{3}{8},\,\,\frac{6}{7},\,\,\frac{9}{16},\,\,\frac{83}{103},\,\,......\]etc.
Simple fraction: A fraction in its lowest terms is known as a simple fraction. e.g., \[\frac{3}{8},\,\,\frac{6}{7},\,\,\frac{9}{16},\,\,\frac{8}{17},\,\,......\]etc.
Mixed fraction: A number which can be expressed as the sum of a natural number and a proper fraction is called a mixed fraction.
e.g., \[5\frac{2}{3},\,\,6\frac{1}{6},\,\,1\frac{3}{4},\,\,107\frac{1}{2},\,\,.......\]etc.
\[5\frac{2}{3}\]can be written as \[5+\frac{2}{3}\]
Like fractions: Fractions having the same denominator but different numerators are called like fractions. e.g.\[\frac{6}{17},\,\,\frac{9}{17},\,\,\frac{11}{17},\,\,......\]etc.
Unlike fractions: Fractions having different denominators are called unlike fractions. e.,\[\frac{3}{4},\,\,\frac{6}{7},\,\,\frac{5}{6},\,\,\frac{4}{5},\,\,\frac{8}{13},\,\,.......\]etc.

- A fraction is a part of a whole.
- A number of the form\[\frac{p}{q}\], where p and q are whole numbers and \[q\ne 0\]is known as a fraction.
- In the fraction\[\frac{p}{q}\], p is called the numerator and q is called the denominator.

- An important property: If the numerator and denominator of a fraction are both multiplied by the same non zero number, its value is not changed. Thus, \[\frac{2}{7}=\frac{2\times 2}{7\times 2}=\frac{2\times 5}{7\times 5}......\]etc.
- Equivalent fractions: A given fraction and the fraction obtained by multiplying (or dividing) its numerator and denominator by the same non — zero number, are called equivalent fractions. (See above property) e.g., Equivalent fractions of \[\frac{6}{12}\]are \[\frac{8}{16},\frac{1}{2},\frac{4}{8},\frac{5}{10}\]......etc.

- Method of changing unlike fraction to like fraction:

- Irreducible fractions : A fraction \[\frac{a}{b}\] is said to be irreducible or in lowest terms,

- Comparing fraction: Let \[\frac{a}{b}\] and \[\frac{c}{d}\] be two given fractions. Then,

- Method of comparing more than two fractions;

DATA HANDLING AND GRAPHS
FUNDAMENTALS
Collection and Tabulation of data:

- Data obtained in the original form is called a raw data.
- Data means information in the form of numerical figures
- Each numerical figure in a data is called an observation.
- Arranging the data in ascending or descending order is called an array.
- Arranging the data in a systematic tabular form is called tabulation.
- The number of times a particular observation occurs is called its frequency.
- The difference between the highest and the lowest values of the observations in a given data is called its range.
- When the number of observations is large, we make use of tally marks to find the frequencies.
- Tallies are usually marked in a bunch of five for ease of counting. more...

PROPERTIES OF TRIANGLE
FUNDAMENTALS

- A triangle (denoted as A delta) is a closed figure bounded by three line segments, it has three vertices, three sides and three angles. The three sides and three angles of a triangle are called its six elements.

- A triangle is said to be

- Angle sum property: The sum of the angle of a triangle is \[180{}^\circ .\]

CONGRUENCE OF TRIANGLES
FUNDAMENTALS

- Two figures, having exactly the same shape and size are said to be congruent.
- Two triangles are said to be congruent, if pairs of corresponding sides and corresponding angles are equal.

- Two line segments are congruent, If they have the same length, i.e. \[\overrightarrow{AB}\cong \overrightarrow{CD}\] and is read as line segment \[\overrightarrow{AB}\] is congruent to the line segment\[\overrightarrow{CD}\].
- Two angles are congruent, if they have the same measure. "\[\angle A\] is congruent to \[\angle B\]" is written symbolically as \[\angle A\cong \angle B\] or \[\angle A=\angle B\].
- (S.S.S.) Congruence criteria: If the three sides of a triangle are equal to the three corresponding sides of another triangle, then the two triangles are congruent.

- (S.A.S.) congruence condition: If two sides and the included angle of a triangle are respectively equal to the two corresponding sides and the included angle of another triangle, then the two triangles are congruent.

PRACTICAL GEOMETRY
FUNDAMENTALS

- A ruler protractor and compass are used for constructions.
- Given a line 1 and a point P not on it, a line parallel to 1 can be drawn through the point P, using the idea of 'equal alternate angles' or 'equal corresponding angles'.
- Students should draw rough sketch before actually constructing the triangle. This is very important for students to get the feel of the triangle.
- Three independent measurements are required to construct a triangle.
- The sum of lengths of any two sides of a triangle is greater than its third side»
- The difference of lengths of any two sides of a triangle is lesser than its third side.
- The sum of angles in a triangle is\[180{}^\circ \].
- The exterior angle of a triangle is equal in measure (i.e. is equal to the sum) of interior opposite angles.
- more...

MENSURATION (Perimeter and Area)
FUNDAMENTALS

- Perimeter is the distance around a closed figure.
- Area is the part of plane occupied by the closed figure.

- Area of a trapezium \[=\frac{1}{2}\,(a+b)\,h\,sq.\] units, where 'a' and 'b' are lengths of parallel sides and 'h' is the height between them.

- more...

- Himachal Govt Sanctions Rs 25 Cr For Renovating Bantony Castle
- Statehood Day Of Goa: 30 May
- Madhya Pradesh Launches 'My MP Rojgar Portal'
- 'Go To Village' Mission Launched In Manipur
- Maharashtra Becomes 1st To Provide Digitally-Signed Documents
- SpaceX Launches Most Powerful Falcon 9 Rocket
- Skill India Organized 'Ajeevika And Kaushal Vikas Mela' In Bhubaneswar
- Sushant Singh Rajput To Promote 2 Major Initiatives Of NITI Aayog
- Smart Cities India 2018 Expo Underway In New Delhi
- Shivangi Pathak Becomes Youngest Indian Woman To Scale Mount Everest

You need to login to perform this action.

You will be redirected in
3 sec

Free

Videos

Videos