Squares and Square Roots

- Square: The square of a number is the product obtained when a number is multiplied by itself.

- Perfect Square: A perfect squares are the shares of whole numbers. Perfect squares are formed by multiplying a whole number by itself.

- Properties of Squares:

- There are no natural numbers 'm' and 'n' for which \[{{m}^{2}}=2{{n}^{2}}.\]There are 2n non-perfect square numbers between the squares of the numbers n and (n + 1).

- Square root: Square root is the inverse operation of square.

- Determining the square roots of positive numbers without using a calculator

- Estimating the number of digits in the square root of a given number: Place bars over every more...

Cubes and Cube Roots

- Cubes

- Estimating the cubes of numbers

- Perfect cube

- Cube roots of numbers

Comparing Quantities

- Compound interest: Amount at compound interest is given by \[A=P{{\left( 1+\frac{R}{100} \right)}^{n}}\], where,

- If the rates of increase in population P are p%, q% and r% during 1st, 2nd and 3rd years respectively, then the population after 3 years =

- If principal = R.s P, rate = R% per annum and time = n years, then

- If a certain amount becomes N times in T years, then it will be

- Profit and loss:

- Profit and loss percentage:

- Important formulae:

- Discount: In order to give a boost to the sales of an item or to clear the old stock, articles are sold at reduced prices. This reduction is given on the Marked Price (M.P.) of the article and is known as discount.

Algebraic Expressions and Identities

- A combination of constants and variables connected by +, -, x and - is known as an algebraic expression.

- Polynomial:

- Like terms: Terms formed from the same variables whose powers are same are called like

- Unlike terms: Terms formed from different variables whose powers may be same or different are called unlike terms. The coefficients of unlike terms may or may not be the same.

- Degree of a polynomial:

- In case of a polynomial in one variable, the highest power of the variable is called the degree of the polynomial.

- In case of polynomial in more than one variable, the sum of the powers of the variables in each term is taken up and the highest sum so obtained is called the degree of the polynomial.

- Types of polynomials:

- Multiplication of polynomials:

- Identity:

- An identity is an equality, which is true for all values of the variables in the equality.

- Some important identities:

Visualising Solid Shapes

- Geometrical shapes:

- Description of solid shapes:

S.NO. | Solid Shape | Number of Vertices | Number of edges | Number of faces |

1. | Cube | 8 | 12 | 6 |

2. | Cuboid | 8 | 12 | 6 |

3. | Cone | 1 | 1 Curved edge | 1 Curved face, 1 flat face |

4. | Cylinder | Nil | 2 Curved edges | 2 flat face 1 Curved face |

5. | Sphere | Nil | Nil | 1 Curved face |

- Polyhedron:

- Regular polyhedron: A polyhedron with regular polygons as its faces is called a regular polyhedron. The same number of faces meet at each vertex.

- This more...

Mensuration

- Perimeter: The length of the boundary of a plane figure is called its perimeter.

- Area: The amount of surface enclosed by a plane figure is called its area.

- Rectangle: Given a rectangle of length T units and breadth 'b' units,

- Area of four walls of room: Let there be a room with length T units, breadth 'b' units and height 'h' units.

- (ii) Diagonal of room \[=\sqrt{{{l}^{2}}+{{b}^{2}}+}{{h}^{2}}\]units

- Perimeter and area of a square: Let each side of a square be 'a' units. Then

- Perimeter and area of a triangle:

- Then the area of the triangle \[=\left( \frac{1}{2}\times b\times h \right)\]sq. units

- (iv) Area of an equilateral triangle with each side 'a' units \[=\left( \frac{\sqrt{3}}{4}\times {{a}^{2}} \right)\]sq. Units.

- (vi) Area of a right triangle \[=\frac{1}{2}\times \](product of legs) sq. units

- Area of a parallelogram: Let ABCD be a parallelogram with base 'b' units and height 'h' units.

- Area of a rhombus: Let ABCD be a rhombus in which diagonal \[AC={{d}_{1}}\]units and diagonal \[BD={{d}_{2}}\]units.

Exponents and Powers

- Exponential equation: An equation which has an unknown quantity as an exponent is called an exponential equation.

- Standard form of numbers: A number written in the form \[\left( m\times {{10}^{n}} \right)\]is said to be in standard form if 'm' is a decimal number between 1 and 9 and 'n' is either a positive or a negative integer.

- Laws of exponents (Integers): For any two non-zero integers 'a' and 'b', and any integers 'm' and 'n', the following laws hold good.

- Positive integral exponent of a rational number: For any rational number\[\frac{a}{b}\]and a positive integer\['n',{{\left( \frac{a}{b} \right)}^{n}}=\frac{{{a}^{n}}}{{{b}^{n}}}\].

- Negative integral exponent of a rational number: For any rational number. \[\frac{a}{b}\] and a

- Zero exponent of a rational number: For any rational number\[\frac{a}{b}\]and '0',\[{{\left( \frac{a}{b} \right)}^{0}}=1\].

- Special case of\[{{\mathbf{a}}^{\mathbf{n}}}\]:

- For a = - 1,

- Laws of exponents (Rational numbers): Let\[\frac{a}{b}\,and\,\frac{c}{d}\]be any two rational numbers, and 'm' and 'n' be any integers. Then,

Direct and Inverse Proportions

- Unitary method:

- Direct proportion:

- a case if \[{{\operatorname{y}}_{1}}\,and\,{{y}_{2}}\] are the values of y corresponding to the values \[{{\operatorname{x}}_{1\,}}\,and\,{{x}_{2}}\]of z respectively then \[\frac{{{x}_{1}}}{{{y}_{1}}}=\frac{{{x}_{2}}}{{{y}_{2}}},\]

- Examples:

- Inverse proportion:

- the values of y corresponding to the values \[{{\operatorname{x}}_{1}}\,and\,{{x}_{2}}\]of x respectively then

- Examples:

Factorisation
\[\,{{(a+b)}^{2}}\equiv {{a}^{2}}+2ab+{{b}^{2}}\]
\[\,{{(a-b)}^{2}}\equiv {{a}^{2}}-2ab+{{b}^{2}}\]
III. \[\left( a+b \right)\left( a-b \right)\equiv {{a}^{2}}-{{b}^{2}}\]
\[\left( x+a \right)\left( x-b \right)\equiv {{x}^{2}}+\left( a+b \right)x-ab\]

- Factorisation:

- Some important identities:

- In expressions which have factors of the type (x + a) (x + b), remember that the numerical term gives ab. Its factors, a and b, should be so chosen that their sum, with signs taken care of, is the coefficient of x.

- We know that in the case of numbers, division is the inverse of multiplication. This idea is applicable also to the division of algebraic expressions.

Introduction to Graphs

- Bar graph: A bar graph is used to show comparison among categories.

- Pie graph: A pie graph is used to compare parts of whole.

- Histogram: Representation that shows data in intervals.

- Line graph: It shows data that changes continuously over periods of time.

- Linear graph: A straight line graph is called a linear graph.

- The Cartesian system:

- A graph shows the relation between two variables, one of which is an independent variable (or control variable) and the other a dependent variable.

- Advantages and disadvantages of various graphs: