Chap 9 ALGEBRAIC EXPRESSIONS AND IDENTITIES

Chap 9

ALGEBRAIC EXPRESSIONS AND IDENTITIES

SOME IMPORTANT FACTS

1.      A combination of constants and variables connected by + , - , x , ÷ is called  an algebraic expression.

2.      The several parts of the expression separated by  + or – sign are called terms.

3.      A monomial is a one term expression which is either a constant or the product of constant and one or more variable with positive integral exponents.

4.      A Binomial contains two terms .

5.      A trinomial contains three terms.

6.      The terms having same literal factors are called like terms, otherwise they are called unlike terms.

7.      The sum or difference of several like terms is another like term whose coefficient is the sum or difference of those like terms.

8.      In adding or subtracting algebraic expressions, we collect different groups of like terms and find the sum or difference of like terms in each group.

9.      If x is any variable and a , b are positive integers, then ;

x ^a x ^b = x ^{(a + b)}

x ^a y ^a = (xy) ^a

(x ^a) ^b = x ^(ab)

x ^(a/b) = b^th root of (x ^a) = ( b^th  (x) ) ^a

x ^(-a) = 1 / x ^a

x ^{(a - b)} = x ^a / x ^b

10.  If P ,Q , R are three monomials, then

P x ( Q ± R ) = ( P x Q ) ± ( P x R )

( Q ± R ) x P  = ( Q x P ) ± ( R x P )

11.  ( a + b ) x ( c + d )  =  a x ( c + d ) + b x ( c + d ) = ac + ad + bc + bd.

12.  Some identities are :

a)     (a+b)² = a² + 2ab + b²

b)    ( a – b )² =  a² - 2ab + b²

c)     ( a+ b )( a + b ) = a² -  b²

d)    ( x + a )( x + b )  = x²  + ( a + b ) x + ab 

Chap 3 UNDERSTANDING QUADRILATERALS

Chap 3

UNDERSTANDING QUADRILATERALS

SOME IMPORTANT FACTS

1.       CURVE :Any drawing  done without lifting the pencil may be called a curve. In this sense , a line is also curve. A simple curve is one that does not cross  itself.

2.       CLOSED/ OPEN CURVE : A curve is said to be closed if its ends are joined ; otherwise it is said to be open.

3.       A polygon is a closed curve made up of line segments. Here ,

a)      The line segments are the sides of the polygon.

b)      Any two sides with common end points are called adjacent sides.

c)       The meeting point of a pair of sides is called a vertex.

d)      The end points of the same side are adjacent vertices.

e)      The line joining any two non adjacent vertices is a diagonal.

4.       CONVEX POLYGON : if each angle of a polygon is less than 180⁰ , it is called a convex polygon.

5.       CONCAVE OR RE-ENTRANT POLYGON : If at least one angle of a polygon is more than 180⁰  , it is called a concave or re-entrant polygon.

6.   Special types of Quadrilaterals :

a)   Square :  A Quadrilateral having all sides equal and each angle measuring 90⁰ is called a square.

b)      Rhombus : A parallelogram having all sides equal is called a Rhombus.

c)       Parallelogram : A Quadrilateral having its opposite sides parallel is called a Parallelogram

d)      Rectangle : A Quadrilateral having its opposite sides equal and each angle measuring 90⁰ is called rectangle.

e)      Trapezium : A Quadrilateral in which a pair of opposite sies is parallel is called a trapezium.

f)      Kite : A Quadrilateral having two pairs of equal adjacent sides, but unequal opposite sides, is called a kite.

g)     Special types of Triangles ( 3 Sides ) - right, equilateral, isosceles, scalene, acute, obtuse.

h)     Polygon Names
Generally accepted names

Sides	Name
n	N-gon
3	Triangle
4	Quadrilateral
5	Pentagon
6	Hexagon
7	Heptagon
8	Octagon
10	Decagon
12	Dodecagon

i)    Names for other polygons have been proposed.

Sides	Name
9	Nonagon, Enneagon
11	Undecagon, Hendecagon
13	Tridecagon, Triskaidecagon
14	Tetradecagon, Tetrakaidecagon
15	Pentadecagon, Pentakaidecagon
16	Hexadecagon, Hexakaidecagon
17	Heptadecagon, Heptakaidecagon
18	Octadecagon, Octakaidecagon
19	Enneadecagon, Enneakaidecagon
20	Icosagon
30	Triacontagon
40	Tetracontagon
50	Pentacontagon
60	Hexacontagon
70	Heptacontagon
80	Octacontagon
90	Enneacontagon
100	Hectogon, Hecatontagon
1,000	Chiliagon
10,000	Myriagon

j)        The number of diagonals in a polygon of n sides  =  

 [ n( n – 1) / 2  - n ], = 1/2 N(N-3)

k)      A polygon is said to a regular polygon , if all its ,

a)      Interior angles are equal;

b)      Sides are equal  and

c)      Exterior angles are equal.

l)        Each Interior Angle of a n sided regular polygon

= [ ( n – 2 ) x 180⁰]  

                         2

m)   Each Interior Angle of a n sided regular polygon  =   360⁰

  N

n)      Numbers of sides in a regular polygon   =       360⁰

    Exterior angle

o)      At each vertex of a polygon :

Interior angle + Exterior angle = 180⁰

p)      Angle sum property of a Quadrilateral : The sum of angles of a Quadrilateral is 360⁰.

q)      Sum of Exterior angles of a polygon : if the sides of a polygon are produced in order , the sum of exterior angles so formed is always 360⁰.

r)        

PolygonFormulas
(N = # of sides and S = length from center to a corner)

Area of a regular polygon = (1/2) N sin(360°/N) S²

Sum of the interior angles of a polygon = (N - 2) x 180°

The number of triangles (when you draw all the diagonals from one vertex) in a polygon = (N - 2)

Chap 6 SQUARES AND SQUARE ROOTS

Chap 6

SQUARES AND SQUARE ROOTS

SOME IMPORTANT FACTS

1.      The Square of a number Is that number raised to the power 2.

2.      A natural number n is a perfect  square , if n= m² for some natural number m.

3.      The cube of even natural number is even.

4.      The cube of odd natural number is odd.

5.      A number ending in 2, 3, 7, 8, is never a perfect square .

6.      A number ending in an odd number of zeros is never a perfect square.

7.      For any natural number n , we have ;  n² = Sum of first n odd natural numbers.

8.      There are 2n  non perfect sqare numbers between the squares of the numbers n and      ( n + 1 ).

9.      Formula for finding the number of digits in the square root of a perfect square :

Let there be a perfect square number containing  n digit. Then , its square root will contain ( n / 2 ) digits, when n is even, and ( n + 1/ 2 ) digits  when n is odd.

10.  For any natural number m > 1 , ( 2m, m2 -1 and m2+1 ) is Pythagoras triplet.

11.  The square root of a number x is that number which when multiplied by itself gives x as the product and the square root of x  is denoted by √x

12.                        For finding the square  root of a perfect  square, resolve it into prime factors, make pairs of similar factors and take the product of prime factors, choosing one out of every pair.

13.  For finding the square root of a decimal fraction, make even numbers of decimal places by affixing a zero, if necessary; mark off periods and extract the square root, putting the decimal point in the square root as soon as the integral part is exhausted . 

Chap 1 RATIONAL NUMBERS

Chap 1

RATIONAL NUMBERS

SOME IMPORTANT FACTS

1.       Natural numbers : The counting numbers are called Natural numbers.

1,2,3,4,5,6,7,8,…………………………………….

2.       Whole Number : All natural numbers together with zero (0) are called whole numbers.

0,1,2,3,4,5,6,7,8,…………………………………….

3.       Integers : The Whole numbers together with the negative of counting numbers are known as integers.

……………,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,………………..

4.       Fractions : The numbers of the form  a/b, where a and b are natural numbers , are known as fractions.

5.       Rational numbers : A number of the form  P/Q where P,Q are integers and Q ≠ 0 is called rational number.

6.       Every integer is a rational number but a rational number need not be an integer.

7.       Every fraction is a rational number but a rational number numbers need not be a fraction number.

8.       The operation of addition of rational numbers has the following properties ;

a)     Closure Property : The addition of any two rational numbers is always a rational number.

b)    Commutative :  x + y = y + x, for any two rational numbers x and y.

c)     Associative : ( x + y ) + z = x + ( y + z ), for all rational numbers x,y,z .

d)    Existence of additive identity : x + 0 = 0 + x = x for all rational numbers x. the rational number 0 is the additive identity.

e)     Existence of additive inverse :  ( -x) + x = 0 = x + ( - x ).

9.       The multiplication of rational numbers has following properties ;

a)     Closure Property : The multiplication of any two rational numbers is always a rational number.

b)    Commutative :  x x y = y x x, for any two rational numbers x and y.

c)     Associative : ( x x y ) x z = x x ( y x z ), for all rational numbers x,y and z .

d)    Existence of identity : x x 1 = 1 x x = x for all rational numbers x. the rational number 1 is the identity element for multiplication.

e)     Existence of multiplicative  inverse :  x x 1/x = 1  = 1/x  x  x .

f)      Multiplication by 0 (zero ) : for any rational number x , x x 0 = 0 = 0 x x.

g)     Distributivity of multiplication over addition : x x ( y + z ) = x x y + x x z,  for any three rational numbers x, y, z.

10.   Subtraction of rational numbers has following properties ;

a)      Closure property :  x – y is a rational number for all rational numbers x , y.

b)      The subtraction of rational numbers is neither commutative nor associative.

11.   Between two rational numbers  x and y , there is a rational number x + y /2.

We can find as many rational numbers between x and y as we want.

solution

Chap 5 Data Handling

Chap 5

Data Handling

SOME IMPORTANT FACTS

1.       Information in the form of numerical figures is called observations.

2.       Observations gathered initially are called Raw data.

3.       The number of times a particular observation occur is called its frequency.

4.      When the number of observation is large , the data is usually organized into groups called class intervals.

5.       Table showing the frequencies of various class intervals is called a frequency distribution table.

6.       Data in the class intervals is called a grouped data

7.      A bar graph is used to show comparison among categories.

8.      A Pie chart is used to compare parts of a whole. In a pie chart, the values of different components are represented by the sectors of a circle. The total angle of 360⁰  at the centre of a circle is divided according to the values of the components.                      Central angle for a component  =

                                     ( Value of the component x 360/Total value ) ⁰

9.      A Histogram is a bar graph that shows data in intervals.

10.  A  line graph displays data that changes continuously over periods of time.

11.   A line graph which is a whole unbroken line is called a linear graph

12.   Probability : probability is a concept which numerically measures the degree of uncertainty and therefore , of certainty of the occurrence of events.

Probability of the happening of A            =

No of favorable outcomes of A / Total no. of possible outcome

P( impossible events ) = 0 , p ( sure event ) = 1   and  0≤p(A) ≤1.