•

Corporate Office : 45, 2nd Floor, Maharishi Dayanand Marg, Corner Market, Malviya Nagar, New Delhi-110017 Tel. : 011-49842349 / 49842350

D. P. Gupta Shikhaa Nagpal Krishna Srivastava

Typeset by Disha DTP Team

DISHA PUBLICATION ALL RIGHTS RESERVED

© Publisher No part of this publication may be reproduced in any form without prior permission of the publisher. The author and the publisher do not take any legal responsibility for any errors or misrepresentations that might have crept in. We have tried and made our best efforts to provide accurate up-to-date information in this book. For further information about the books from DISHA, Log on to www.dishapublication.com or email to [email protected]

INDEX •

Chapter Utility Score

01. Number Systems

1-38

02. Polynomials

39-72

03. Coordinate Geometry

73-96

04. Linear Equations in Two Variables

97-126

05. Introduction To Euclid’s Geometry

127-144

06. Lines and Angles

145-178

07. Triangles

179-220

08. Quadrilaterals

221-262

09. Areas of Parallelograms and Triangles

263-306

10. Circles

307-356

11. Constructions

357-380

12. Heron’s Formula

381-412

13. Surface Areas and Volumes

413-460

14. Statistics

461-504

15. Probability

505-530

NUMBER SYSTEMS

1 VARIOUS TYPES OF NUMBERS 1.

Set of Natural Numbers, N = {1, 2, 3, …} Representation of N on number line: 0

1 2 3 4 5

2.

Set of whole numbers, W = {0, 1, 2, 3, …} Number line of W (Whole numbers):

3.

0 1 2 3 4 Set of integers, Z = {…, –3, –2, –1, 0, 1, 2, 3, … } Number line of Z (integers):

RATIONAL NUMBERS The numbers of the form

p , where p and q are integers and q ¹ 0 are called rational numbers. Rational q

ìï p üï 1 -3 numbers, Q = í : p , q Î Z, q ¹ 0ý . For example, , , –5, 6, 10 etc. 2 4 îï q þï

Note that all the natural numbers, whole numbers and integers are rational numbers.

Equivalent Rational Numbers The rational numbers do not have a unique representation in the form and q ¹ 0. For example,

1 2 10 25 = = = and so on 2 4 20 50

p , where p and q are integers q

Rational Numbers lying Between Two Rational Numbers There lies infinitely many rational numbers between any two rational numbers.

æ a +b ö If a and b are two rational numbers, then there exists another rational number ç ÷ between them. è 2 ø For inserting n rational numbers between a and b, where a < b, divide (b – a) by (n + 1) and then the required æb-a ö æb-a ö æb-a ö æb-a ö ,a + 2ç , a + 3ç ,..., a + n ç rational numbers will be: a + ç ÷ ÷ ÷ ÷. è n +1 ø è n +1 ø è n +1 ø è n +1 ø

Mathematics

2

IRRATIONAL NUMBERS A number ‘s’ is called irrational, if it cannot be written in the form For example,

2, 3, 15, p, 0.10110111011110...

Note that when we use the symbol

p , where p and q are integers and q ¹ 0. q

, we assume that it is the positive square root of the number. So

4 = 2, though both 2 and – 2 are square roots of 4.

REAL NUMBERS The set of rational numbers and irrational numbers form a set of real numbers which is denoted by R. Note that every real number is represented by a unique point on the number line. Also, every point on the number line represents a unique real number. NOTE : (i) The square root of every perfect square number is rational. eg. 4 = 2, 9 = 3, 16 = 4 etc. are all rational numbers. (ii) The square root of any positive number which is not a perfect square is an irrational number. E.g.: 5, 3, 10, 12, 3.4, 0.748 etc. (iii) p is an irrational number, which is actually the ratio of circumference to the diameter of a circle i.e. c p = , where c and d are the circumference and diameter of a circle. Approximate value of p is taken d 22 or 3.14 as 7

Decimal Expansions of Real Numbers The decimal expansions of real numbers can be used to distinguish between rationals and irrationals. There are three types of decimal expansion. 1. Terminating Decimal Expansions: In this case, the decimal expansion terminates or ends after a finite number of steps. We call such a decimal expansion as terminating. 2. Non-terminating Recurring Expansions: In this case we have a repeating block of digits in the quotient. We say that this expansion is nonterminating recurring. 1 The usual way of showing that 3 repeats in the quotient of is to write it as 0.3 . Similarly, since 3 1 1 the block of digits 142857 repeats in the quotient of , we write as 0.142857, where the bar 7 7 above the digits indicates the block of digits that repeats. 3. Non-terminating and Non-repeating: In this case, no digit on the right of decimal point in the decimal expansion is repeated periodically. Result 1: The decimal expansion of a rational number is either terminating or non-terminating recurring. Moreover, a number whose decimal expansion is terminating or non-terminating recurring is rational. Result 2: The decimal expansion of an irrational number is non-terminating non-recurring. Moreover, a number whose decimal expansion is non-terminating non-recurring is irrational.

Number Systems

3

CONVERSION OF A RATIONAL NUMBER IN DECIMAL FORM TO ITS p SIMPLEST FORM q p Conversion of a Terminating Decimal Number to its Simplest q Form. Step 1 : Obtain the rational number. Step 2 : Determine the number of digits in its decimal part. Step 3 : Remove decimal point from the given number and write 1 as its denominator followed by as many zeros as the total number of digits in the decimal part of the given number. Step 4: Write the number obtained in step-3 in its simplest form (i.e. the form in which there is no common factor other than 1 in its numerator and denominator). p The number so obtained is the required form. q

ILLUSTRATION : 1 Convert rational number 2.348 in simplest

SOLUTION :

p form. q

Given rational number = 2.348 There are three digits in the decimal part. 2348 \ 2.348 = 1000 2348 Now, write in its lowest form. 1000 2348 1174 587 2.348 = = = , which is the required form. 1000 500 250

Conversion of Non-Terminating and Repeating Decimal into a Fraction Step 1: Suppose the given decimal as any variable like x, y, ............. etc. Step 2: Multiply the given decimal with 10 or power of 10 in such a way that only repeating digits remain on the right of the decimal or all non repeating terms which are on the right come to left of the decimal. Step 3 : Multiply the decimal obtained in step 2 with 10 or powers of 10 in such a way that repeated digit or a set of digit comes to the left of the decimal. i.e. We multiply by 10 if there is only one digit is repeated, multiply by 102 or 100 if two digits repeated and so on. Step 4 : Now subtract the decimal obtained in step 2 from the decimal obtained in step 3. Step 5 : Solve the equation whatever get in step 4 and the value of variable in simplified form is the required fraction.

ILLUSTRATION : 2 Express 0.52 in the

SOLUTION :

p form. q

Let x = 0.52 x = 0.525252 ...(i) There is no non-repeating digit after decimal point on the right hand side in equation (i). Number of repeating digits after the decimal point on the right hand side of equation (i) is 2. Hence, multiplying both sides of equation (i) by (10)2 i.e. 100, we get

Mathematics

4

100 x = 52.525252........(ii) Subtract (i) from (ii), we get 100 x = 52.5252..... x = 0.525252... – – 99 x = 52

Þ x=

52 52 \ 0.52 = 99 99

REPRESENTING REAL NUMBERS ON THE NUMBER LINE Representation of Rational Numbers on the Number Line Through Successive Magnification Let us try to repersent 3.47 on the number line. We know that 3.47 lies between 3 and 4. We divide the portion between 3 and 4 into 10 equal parts as below: 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3

4

Now, 3.47 lies between 3.4 and 3.5. Again we divide the portion between 3.4 and 3.5 into 10 equal parts. 3.41 3.42 3.43 3.44 3.45 3.46 3.47 3.48 3.49 3.4

3.5

Now, we can easily locate 3.47 on the number line. In the above method, we have successively magnified different portions to represent 3.47 on the number line. This method of representation of real number on the number line is known as method of successive magnification.

Representation of Irrational Numbers on Number Line p , we use the Pythagoras theorem of a right angle triangle, q according to which, in a right angled triangle, the square of the hypotenuse is equal to the sum of the square of the other two sides. i.e. (Hypotenuse)2 = (Base)2 + (perpendicular)2 Q Suppose ‘x’ ox be a horizontal line and let O be the origin. Take OP as 1 unit and draw PQ ^ OP so that PQ = 1 unit. With centre O and OQ as radius draw an arc meeting at A. x P A x' O Then OA = OQ = 2 unit (by pythagoras theorem)

To represent an irrational number in the form

Similarly diagrams given below shows

0

3, 5 S

R

T

Q

x'

O 0

P A B 2 3

C 5

x

2

Number Systems

5

IMPORTANT PROPERTIES OF RATIONAL AND IRRATIONAL MUMBERS 1.

Rational Numbers (i) Rational numbers are closed with respect to addition subtraction, multiplication and division that is if we add, subtract, multiply or divide (except by zero) any two rational numbers, then we get rational numbers. (ii) It satisfies commutative, associative and distribution laws for addition and multiplication.

2.

Irrational Numbers (i) Irrational numbers also satisfies commutative, associative and distributive laws for addition and multiplication (ii) The sum, difference, product and quotient of irrational numbers may not be irrational. For example,

( 2 + 3 ) + (2 - 3 ) = 4 ;

6 3

2

3

=6 2

(7 + 5)(7 - 5) = (7) - ( 5)

= 49 – 5 = 44.

Here, 4, 6 and 44 are rational numbers. NOTE : (i) The sum or difference of a rational number and an irrational number is irrational. (ii) The product or quotient of a non-zero rational number with an irrational numbers is irrational.

RATIONALISATION When the denominator of an expression contains a term with a square root, the procedure of converting it to an equivalent expression whose denominator is a rational number is called rationalising the denominator. When the product of two radical expressions is rational, then each one is called the rationalising factor of the other. For example, (i) a is the rationalising factor of a and vice-versa. (ii)

( a + b ) is the rationalising factor of ( a - b ) and vice-versa.

Square Root of Real Numbers Let ‘a’ be any positive real number. We can express a = b if and only if b > 0 and b2 = a The value of ‘b’ is called the positive square root of the positive real number ‘a’

SOME IDENTITIES RELATED TO SQUARE ROOTS Let a and b be positive real numbers. Then (i)

ab = a

b

(iii) ( a + b ) ( a - b ) = a - b (v)

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

(ii)

a = b

a

b (iv) (a + b ) (a - b ) = a 2 - b

(vi) ( a + b ) 2 = a + 2 ab + b.

LAWS OF EXPONENTS (i)

an = a × a × a × .........× a(n factors)

(iii) (am)n = amn

(ii) am . an = am + n am (iv) a m ¸ a n = = a m- n , m > n an

Mathematics

6

(v)

am

bm

(a)0

(vi)

= 1. m

am æaö (viii) ç ÷ = m b èbø Root of a Real Number : Let a > 0 be a real number and n be a positive integer.

(vii) a–n = (ix) nth

=

(ab)m

Then

1 . an n

a = b, if bn = a and b > 0. Here n a denotes the nth root of a.

ILLUSTRATION : 3 Prove that

c

æ xb ö ÷ =1 ¸ç ç a÷ b a -c x ( ) èx ø xa (b - c )

SOLUTION : x a (b -c) xb ( a - c )

æ xb ¸ç ç xa è

c

ö x ab - ac 1 ÷ = ¸ ( xb - a )c = x(ab–ac)–(ba–bc) ´ ba -bc ÷ ( b x x a) c ø

1

= xab – ac – ba + bc ´

x

bc - ac

0 = x ab – ba – ac + ac + bc – bc = x = 1

Important Formulae, Terms and Definitions Rationalising Factor : If a and b are positive integers, then (i) Rationalising factor of (ii) Rationalising factor of

1

is

a 1

a± b

a. is a m b .

1

is a m b . a± b Laws of Exponents : If a, b are positive real numbers and m, n are rational numbers. Then, we have (iii) Rationalising factor of

am

= a m- n

(i) am × an = am + n

(ii)

(iii) (am)n = amn

(iv) a–m =

(v) (am bm = (ab)m

(vi)

a

n

am am

1 am

æaö =ç ÷ èbø

m

m

m m æ 1ö ç ÷ n = a = a n or n a m = n a m = a n (vii) ç ÷ è ø Inserting n rational numbers : For inserting n rational numbers between a and b, where a < b, divide (b – a) by (n + 1) and then the required rational numbers will be: 1 m n a

( )

æb-a ö æb-a ö æb-a ö æb-a ö a +ç ,a + 2ç , a + 3ç ,..., a + n ç ÷ ÷ ÷ ÷. è n +1 ø è n +1 ø è n +1 ø è n +1 ø

Number Systems

7