i) Natural numbers: Counting numbers, i.e, 1, 2, 3, 4,. are called natural numbers.

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1 Chapter 1 Integers Types of Numbers i) Natural numbers: Counting numbers, i.e, 1, 2, 3, 4,. are called natural numbers. ii) Whole numbers: Counting numbers and 0, i.e., 0, 1, 2, 3, 4, 5,.. are called whole numbers. iii) Integers: All natural numbers, zero and negative of natural numbers, i.e., 3, 2, 1, 0, 1, 2,. Are called integers. iv) Rational numbers: All numbers of the form q 0 are called rational numbers. p, whose p and q are co-prime integers and q Numbers System Whole numbers Integers Rational numbers In this chapter, we will do a complete analysis on Integers. Integers The numbers. 3, 2, 1, 0, 1, 2, 3, 4,.. are called integers. The numbers 1, 2, 3, i.e., natural numbers, are called positive integers and the numbers 1, 2, 3,. are called negative integers. The number 0 is simply an integer. It is neither positive nor negative. Representation of integers on number line Positive integers Negative integers Note i) All negative integers are smaller than zero and all positive integers are greater than zero. ii) 1 is the smallest positive integer. iii) 1 is the largest negative integer. iv) Every positive integer is greater than every negative integer.

2 Operations on Integers We are going to learn the following operations on integers: i) Addition ii) Subtraction iii) Multiplication iv) Division Addition of two integers Rules to add two integers: Rule 1: If a and b are two positive integers, then add a and b and put + sign to the result. Example 1: Find the value of Rule 2: Both 8 and 3 are positive integers = 11 If a is positive integer and b is negative integer (a > b) then subtract the smaller one from bigger one and put the sign of bigger number to the result. Example 2: Find the value of 8 + ( 13). 8 is positive integer and 13 is negative integer. Also 13 > 8 Following rule 2, 8 + ( 13) = [13 8] = 5 Rule 3: If a is a negative integer and b is positive integer (a < b) then also subtract the smaller one from bigger one and put the sign of bigger number to the result. Example 3: Find the value of ( 8) is negative integer and 3 is positive integer by rule 3, ( 8) + 3 = (8 3) = 5 [By Rule 3]. Rule 4: If a and b are negative integers, then add a and b and put sign to the result. Example 4 Find the value of ( 8) + ( 3). ( 8) and ( 3) are negative integers. By rule 4 ( 8) + ( 3) = (8 + 3) = 11 Addition of three or more integers: Hint: This can be done by method of grouping. Group first two integers then next two integers and so on. Example 5: Find the value of ( 5) + ( 6) + (+7). Solved Examples ( 5) + ( 6) + 7 = {( 5) + ( 6)} + 7 (Group first two integers) = ( 11) + 7 (Using Rule 4) = 4 (Using Rule 3)

3 Example 6: Find the value of ( 5) + ( 4) + ( 3) + ( 2) + ( 1). ( 5) + ( 4) + ( 3) + ( 2) + ( 1) = {( 5) + ( 4)} + {( 3) + ( 2)} + ( 1) (Group first two and next two integers) = {( 9) + ( 5)} + ( 1) (Use rule 4 and then group first two integers) = ( 14) + ( 1) (Using rule 4) = 15 (Using rule 4) Example 7: Find the value of ( 25) + (13) + ( 49). ( 25) + (13) + ( 49) = {( 25) + (13)} + ( 49) (By grouping) = ( 12) + ( 49) (Using rule 2) = 61 (Use rule 4) Example 8: Find the value of ( 999) + ( 1) ( 999) + ( 1) = { ( 999)} + ( 1) (By grouping) = 1 + ( 1) (Using rule 2) = 0 (Using rule 2) Subtraction of Two Integers Rules to subtract two integers: Rule 1: If a and b are positive integers, (a > b), then to obtain a b, subtract the smaller number from bigger number and put the sign of bigger number to the result. Example 9: Find the value of = 5 Example 10: Find the value of 8 (13). 8 and 13 are positive integer and 13 > = (13 8) = 5 Rule 2: If a is positive integer and b is negative integer (a > b) then to obtain a b, add a and b and put + sign to the result. Example 11: Find the value of 8 ( 3). 8 ( 3) = + (8 + 3) = + 11 Rule 3: If a is negative integer and b is positive integer (a < b) then to obtain a b, add a and b and put sign to the result. Example 12: Find the value of ( 8) (3). ( 8) (3) = (8 + 3) = 11 Rule 4: If a and b are negative integer, then to obtain a b, subtract the smaller number from bigger number and put (i) + sign if (b > a). (ii) sign if (a > b).

4 Example 13: Find the value of ( 8) ( 13). ( 8) ( 13) = + (13 8) = 5 (Rule 4 : 13 8 = 5) (13 > 8) ( 8) ( 13) = + 5 Example 14: Find the value of ( 8) ( 3). ( 8) ( 3) = (8 3) = 5 (Rule 4 : 8 3 = 5) (8 > 3) ( 8) ( 3) = 5 Subtraction of two or more integers Hint: Grouping first two integers only. Solved Examples Example 15: Find the value of ( 5) (6) (7). [( 5) ( 6)] 7 (Grouping first two terms) = 11 7 (Using rule 4) = 18 (Using rule 4) Example 16: Find the value of ( 5) ( 4) ( 3) ( 2) ( 1). ( 5) ( 4) ( 3) ( 2) ( 1) = {( 5) ( 4)} ( 3) ( 2) ( 1) {Grouping first two} = [( 1) ( 3)] ( 2) ( 1) {Using rule 4} = [(+2) ( 2)] ( 1) {Using rule 4} = 4 ( 1) {Using rule 2} = 5 {Using rule 2} Example 17: Find the value of = (100 1) 99 {Grouping first two} = = 0 Properties of addition and subtraction of integers: 1. Closure property : If a and b are integers, then (i) a + b is also an integer (ii) a b is also an integer. Hence, closure property holds for both addition and subtraction of integers. 2. Associative Property : If a, b, and c are integers, then i) a + (b + c) = (a + b) + c ii) a (b c) (a b) c Hence associative property holds for addition but not for subtraction. 3. Commutative property : If a and b are integers, then i) a + b = b + a ii) a b b a Hence commutative property holds for addition but not for subtraction.

5 4. Inverse : If a is an integer, then (i) a + ( a) = 0 (ii) a a = 0 a is called additive inverse of a (or) negative of a 5. Role of Zero : If a is an integer, then (i) a + 0 = 0 + a = a 0 is an additive identity (ii) a 0 = a but 0 a a [as 0 a = a] Multiplication of integers: i) Positive integer Positive integer = Positive integer ii) iii) iv) Positive integer Negative integer = Negative integer Negative integer Positive integer = Negative integer Negative integer Negative integer = Positive integer v) If there are odd number of negative integers in multiplication, then the result will be negative integer. vi) If there are even number of negative integers in multiplication, then the result will be in positive integer. Example 18: Find the value of ( 8) 5 Solved Examples ( 8) 5 = 40 Example 19: Find the value of ( 8) ( 5) ( 8) ( 5) = + 40 Example 20: ( 1) ( 2) ( 3) ( 4) ( 5) ( 1) ( 2) ( 3) ( 4) ( 5) = 120 [Odd number of negative integers] [even numbers of negative integers] [odd number of negative integers] Example 21: ( 1) ( 2) ( 3) ( 4) ( 5) ( 6) 1) ( 2) ( 3) ( 4) ( 5) ( 6) = Example-22 ( 15) (40) ( 5) =? ( 15) (40) ( 5) = [even number of negative integers] Division of two integers: Rules to remember: i) Positive integer / Positive integer = Positive integer ii) Negative integer / Positive integer = Negative integer iii) Positive integer / Negative integer = Negative integer iv) Negative integer / Negative integer = Positive integer

6 Example 23: Find the value of Example 24: Find the value of 140 ( 20) Example 25: Find the value of ( 140) / ( 20) Example 26: The value of ( 100) Solved Examples Properties of Multiplication and Division of integers 1. Closure property : If a and b are integers, then i) a b is an integer ii) a b need not be an integer Example-27: 2 3 = 6 is a integer 2 3 = 2/3 is a fraction Closure property is true for multiplication but not for division 2. Commutative property : If a and b are integers, then i) a b = b a ii) a b b a Hence commutative property holds for multiplication but not for division. 3. Assosciative property : If a, b and c are integers, then i) (a b) c = a (b c) ii) (a b) c a (b c) Hence associative property hold for multiplication but not for division. 4. Role of 1 : If a is an integer, then i) a 1 = 1 a = a [1 is called multiplicative identity] ii) a/1 = a but 1/a a 5. Inverse : If a is an integer, then i) 1 1 a a 1 a a [1/a is called multiplicative inverse of a] ii) a 1; i. e. a a 1 a 6. Distributive property of multiplication over addition : If a, b, c are integers, then a (b + c) = a b + a c

7 Example : 10 ( 5) (3 + 7) = ( 5) 10 = 50 ( 5) 3 + ( 5) 7 = ( 15) + ( 35) = 50 Therefore, ( 5) (3 + 7) = ( 5) 3 + ( 5) 7 TRY YOURSELF 1. State the sign of the product if a, b, c, d, etc., are positive integers: i) ( a) ( b) ( c) ( d) ( e) f g ( h ) ii) a bc ( d ) ( e) ( f ) iii) 28 positive integers 14 negative integers. 2. Use the distributive law to simplify and work out the following: i) ii) (546) ( 22) + ( 546) (78) iii) 7250 ( 31) + ( 7250) 69 [Hint: ( 7250) 69 = 7250 ( 69)] iv) ( 26743) [Hint: subtraction is the additive inverse of addition, and = ] v) ( 4) 20 ( 4) 15 ( 4) 62 + ( 4) In a quiz (+5) marks are awarded for each correct answer and ( 2) for an incorrect answer. Find: i) Ruchi s total score if she answered 10 questions right and 8 went wrong. ii) The number of wrong answers, if she scored 22 in total and gave 6 correct answers. 4. Simplify: ( 2.0) + ( 8) + ( 2) 3 5. Find the value: i) ii) iii) iv) 32 (35) 4 v) 3 (5 6 3) vi) vii) ( 20) ( 1) ( 28) 7 viii) ( 2) ( 8) ( 4) ix) ( 3) ( 8) ( 4) 2 ( 2) (x) ( 3) ( 4) ( 2) ( 1) 6. Which is greater? i) ( 4) ( 22) 4 ( 3) or ( 2) ( 1) ( 1) (2) ii) 7 6 ( 4) or 8 ( 2) ( 8)( 1) iii) 49 7 or 49 7 iv) ( 9) (2) ( 2) (7) or Find the value of: i) ( 2) + ( 15625) 98 ii) ( 18946) iii)

8 8. i) ( 487) 327 ii) ( 28945) Note: If there are two or more of the fundamental operations +,, and in a numerical expression, the order of operation is Division Multiplication Addition S ubtraction This rule is abbreviated as DMAS. It holds good for integers also. Example 28: Simplify = (division) = (multiplification) = 24 4 (addition) = 20 (subtraction) Example 29: Simplify 36 ( 3) + ( 4) 3 ( 4) 36 ( 3) + ( 4) 3 ( 4) = 12 + ( 4) 3 ( 4) (division) = 12 + ( 12) ( 4) (multiplication) = 24 ( 4) (addition) = 20 (subtraction) Use of Bracket: When brackets are present in a problem, we simplify the terms inside the brackets first. Brackets are of following types: If two or more types of brackets are present in a problem, the order of working is (i) round (ii) curly (iii) square. There are three kinds of brackets ( ) : round brackets { } : double or curly brackets [ ] : square brackets We also sometimes use another grouping symbol called bar or vinculum. The terms inside the bar are simplified before the brackets. The order of operations is, therefore ' ' ( ) { } [ ] Note: If there are two or more fundamental operations along with brackets the order of operation in as follows: Bracket Division Multiplication Addition S ubtraction This is abbreviated as BODMAS

9 Absolute Value of an Integer The absolute value of an integer is the numerical value of the integer without regard to its sign the absolute value of an integer a is denoted by a and is given by a, a 0 a a, a < 0 Example 30: SOLVED PROBLEMS Problem 1: Find the value of ( 8) + {( 2) + [3 4] 2} ( 15) 8 ( 8) + {( 2) + [3 4] 2} ( 15) 8 = ( 8) + {( 2) } ( 15) 8 [First simplify the innermost bracket] = ( 8) + {( 2) + 6} ( 15) 8 [Secondly, follow the hierarchy as mentioned above] = ( 8) + {4} ( 15) 8 [By hierarchy] = ( 8) 60 8 [By taking first two integers] = {( 8) 60} 8 = ( 68) (8) = 76 Problem 2: Find the value of 7 [16 (12 8) 4 2] (16 8) 10 [4 2 (4 2)] [First simplify brackets starting from the innermost] = 7 [16 (12 8) 4 2] (16 8) 10 = 7 + [16 8] 8 10 = [Follow hierarchy] = = = 17 Problem 3: Find the value of [ (54) ( 80 10)] [(5 2) (8 5)] [ (54) ( 80 10)] [(5 2) (8 5)] = [ (54) ( 90)] [(3) 3] [On simplifying brackets] = [4860] [3 3] = = 540

10 THINGS TO REMEMBER 1. The numbers.. 4, 3, 2, 1, 0, 1, 2, 3, 4,., etc., are integers. 2. 1, 2, 3, 4, 5,. are positive integers and 1, 2. 3, are negative integers is an integer which is neither positive nor negative is less than every positive integer and greater than every negative integer. 5. The absolute value of an integer is the numerical value of the integer without regard to its sign. 6. The absolute value of an integer is denoted by a and is given by a, a 0 a a, a 0 7. a and a are negative or additive inverse of each other. 8. Any integer when multiplied or divided by 1 gives itself and when multiplied or divided by 1 gives its negative. 9. If there are odd number of negative integers in multiplication, then the result will be negative integer. 10. If there are even number of negative integers in multiplication, then the result will be in positive integer TIPS Same signs gives positive result 3. Opposite signs gives negative result If a, b, c are integer then: a) a b a c b c, if c 0 b) a b a c b c, if c 0 6. If a is a non-zero integer, then 0 a If a is an integer, then a 0 is not meaningful. 8. ( ) ( ) 9. ( ) ( ) 10. ( ) ( ) 11. ( ) ( )

11 1. Find the value of: PART I: MISCELLANEOUS DOMAIN i) ( 3) ( 8) ( 4) 2 ( 2) ii) ( 3) ( 4) ( 2) ( 1) ii) ( 40) ( 1) ( 28) 7 2. Simplify: 5 {28 (29 7)} 3. Calculate: i) 3 (5 6 3) ii) iii) Simplify each of the following: 1 i) ( ) 2 iii) ( ) ii) ( ) 5. Using brackets, write a mathematical expression for each of the following : i) Nine multiplied by the sum of two and five. ii) Twelve divided by the sum of one and three. 6. Which of the following statements are true? i) The product of a positive and a negative integer is negative. ii) The product of three negative integers is a negative integer. iii) Of the two integers, if one is negative, then their product must be positive. 7. Simplify each of the following : i) 63 ( 3) 8 3 3{5 ( 2)( 1)} ii) [29 ( 2) 6 (7 3)} 3 5 ( 3) ( 2) 8. Find each of the following products: i) ( 45) 55 ( 10) ii) ( 2) ( 4) ( 6) ( 6) 9. Simplify each of the following : i) ( ) 1 ii) 4 10 ( ( 5) 5 \ 10. State which is greater : i) (8 + 9) 10 and 9 10 ii) (8 9) 10 and iii) {( 2) 5} ( 6) and ( 2) ( 6) 11. Verify the following: i) 19 {7 + ( 3)} = ( 3) ii) ( 23) {( 5) + (+19)} = ( 23) ( 5) + ( 23) (19)

12 12. i) If a ( 1) = 30, is the integer a positive or negative? ii) If a ( 1) = 30, is the integer a positive or negative? 13. Find each of the following products: i) ( 2) 36 ( 5) ii) 18 ( 27) 30 iii) ( 45) 55 ( 10) 14. Calculate: i) 3 (5 6 3) ii) (5 3) 15. Find the value of : [ ( 6)] Simplify: (4 1) iii) ( ) 17. Find each of the following products: i) ( 115) 8 ii) 9 ( 3) ( 6) iii) ( 12) ( 13) ( 5) 18. Simplify each of the following : i) 3 (5 6 3) ii) iii) What will be the sign of the product if we multiply together: i) 8 negative integers and 1 positive integer? ii) 21 negative integers and 3 positive integers? iii) 199 negative integers and 10 positive integers? 20. Simplify: i) 15 ( 3) ( 3) ( 6) ii) (11 11) ( 4) Find the quotient in each of the following: i) ( 1728) 12 ii) ( 15625) ( 125) iii) ( 100) 22. Divide : i) ( 729) by ( 27) ii) by 10 iii) 0 by Simplify: i) 15 ( 3) ( 3) ( 6) ii) (11 11) ( 4) 3 9 2

13 24. Find the quotients? i) 1000 ( 10) ii) 147 ( 21) iii) ( 16) 4 iv) 601 ( 1) v) 3454 ( 11) vi) ( 76) ( 19) 25. Simplify: 10 {5 + ( 3) + 8 ( 11)} 26. Simplify: 20 [ 15 {4 ( 1) 3} 6] 27. Simplify: 23 {29 (17 9 3)} 28. Using the BODMAS rule, solve this [3 4{3 2 ( 8)}] 29. Can you see any pattern in the digits of the units places of these numbers? 10 3? ( 2)?? 60 2?? ( 1)? Finish? ( 4) (3) i) First of all guess what the sign of the FINISH integer will be? Why? ii) Fill in the complete the pattern. iii) By what should you multiply the START integer to get the FINISH integer? Can you answer this question without filling all the circles. 30. Draw a path starting from the START position. While you go along the path, keep multiplying the integers until you reach the integer at the finish position. Choose only those integers that will give you the correct Finish answer. Remember that there can be more than one path that gives the correct answer. Start Finish

14 VALUE BASED QUESTIONS 1. In a competitive exam, 3 marks are given for every correct answer and 1 mark is deduced for every incorrect answer. Raju copied some answers from Reema and answered all the questions. He scored 20 marks though he got 10 correct answers. How many incorrect answers had he attempted? What values are promoted in the question? 2. In a quiz, `300 are awarded for every correct answer and a penalty of `75 is put for every incorrect answer. Madhuri answered 15 questions out of which only 6 answers were correct. How much money is earned by Madhuri in the quiz? If she distributes the money earned by her to poor children in the neighbourhood, what values are being promoted? HIGHER ORDER THINKING SKILLS (HOTS) 1. A water tank has steps inside it. A monkey is sitting on the first step. The water level is at the ninth step: i) He jumps 3 steps down and then jumps back 2 steps up. In how many jumps will be reach the water level? ii) After drinking water, he wants to go back. For this, he jumps 5 steps up and then jumps back 3 steps down in every move. In how many jumps will he reach back the top of the tank? 2. A shopkeeper earns a profit of `2 by selling a pen and a loss of 50 paise per pencil and loss of 15 paise per eraser while selling pencils and erasers of old stock. On a particular day, he earns a profit of `10. If he sold 10 pens and the number of pencils and erasers he sold are in the ratio 7 : 10, then find the number of pencils and erasers he sold on the day. 3. In a competition 3 marks are given for every correct answer and ( 2) marks are given for every incorrect answer and no marks for not attempting any questions. i) Sachin scored 20 marks. If he got 12 correct answers, how many questions has he attempted incorrectly? ii) Mohini scores ( 5) marks in this competition, though she has got 7 correct answers. How many questions she has attempted incorrectly? 4. An elevator descends into a mine at the rate of 6 m/min. If the descend starts from 10 m above the ground level, how long will it take to reach the shaft 350 m below the ground level? 5. A bookstore manager earns a profit of `20 by selling one new book and incurs a loss of `10 by selling a second hand old book. In a particular month he earns neither profit nor loss. If he sold 25 new books, how many second hand old books did he sell?

15 PART II: MULTIPLE CHOICE QUESTIONS 1. If 29 is added to the difference of 28 and 57 then the answer is: (a) 1 (b) 0 (c) 2 (d) 2 2. If X is successor of 9897, Y is a predecessor of 4859.Then X Y is: (a) 5038 (b) 5032 (c) 5036 (d) Match the following: Column I Column II A. ( 19) ( 13) (p) +6 B. (+19) (+13) (q) +32 C. (+19) ( 13) (r) +6 D. (+19) (+13) (s) 6 (a) A s; B r; C q; D p (c) A r; B s; C p; D q (b) A p; B r; C q; D s (d) A q; B p; C s; D r 4. If a ( 7) 12, b ( 10) 19, c ( 12) 26, d ( 16) 15 then (a) b > a (b) c > b (c) c < d (d) All of these 5. A pointer moves on a integer scale. A student notes the values and subtracts the sum of all positive values from the sum of all negative integers The dotes indicate the values indicated by the pointer: (a) 37 (b) 39 (c) 38 (d) If X = ( 3) ( 8) (+ 4), Y = ( 10) ( 3) + ( 4) then: (a) X < Y (b) X > Y (c) Y = X (d) None of these 7. X is an integer: i) It is less than every positive integer. ii) It is greater than every negative integer. Using the given condition identify the integer X. (b) X = 1 (b) X = 1 (c) X = 0 (d) X = 2

16 8. a = 12, b = 7, x = 10, y = 12 i) If a > b then b > a ii) If x > y then y < x Which of the statement is/are true? (a) (ii) (b) (i) (c) (i) and (ii) (d) neither (i) nor (ii) 9. If X = ( 2) + ( 2). 20 times Y = (+3) + (+3) + ( 3). 40 times then X + Y = (a) 40 (b) 40 (c) 0 (d) None of these 10. If P = ( 8) + ( 3) + (+7) Q = ( 9) + (+3) + (+3) + (+2) +( 9) R = ( 6) + ( 8) + (+3) + (+2) + ( 9) then P + Q + R (+27) = (a) 1 (b) 2 (c) 0 (d) None of these 11. If 4 p 32, 2q 16, then ( 10) p ( 18) q (a) 6 (b) 4 (c) 8 (d) If A = (+7) + ( 10) B = ( 3) + ( 8) C = (+9) + ( 13) then arrange A, B, C in ascending order: (a) A, B, C (b) C, B, A (c) B, C, A (d) B, A, C 13. If ( 9) ( 3) = X, (+7) ( 4) = Y, (+6) + ( 2) = Z then X Y Z = (a) 25 (b) 25 (c) 24 (d) If ( 7) + (+ x) = ( 4) then x = (a) 3 (b) 4 (c) 3 (d) none of these 15. If ( 8) + ( 9) = x, (+10) + ( 2) = y, (+11), + ( 13) = z the x + y + z = (a) 11 (b) 12 (c) 13 (d) Which statement is correct: i) sum of three negative integers is +ve ii) Sum of four negative integers is ve iii) Sum of two positive integers is +ve (a) (ii) (b) (iii) (c) (i) (d) (i), (ii), (iii)

17 17. Match the following: Column I Column II A. ( 10) ( 3) (p) +7 B. (+10) (+3) (q) +13 C. ( 10) ( 3) (r) +7 D. (+10) (+3) (s) 13 (a) A p; B r; C q; D s (b) A p; B s; C r; D q (c) A r; B p; C q; D s (d) A s; B r; C p; D q 18. In addition and subtraction of the integers the sign of answers depends upon: (a) Smaller number (b) Their difference (c) Their sum (d) Greater numerical value 19. Absolute value of 11 is: (a) 10 (b) 1 (c) 11 (d) Sum of two positive integers is always: (a) negative (b) positive (c) 0 (d) Sum of a negative and a positive integer is: (a) always negative (c) always positive (b) either positive or negative (d) zero 22. The pair of integers whose sum is 5: (a) 1, 4 (b) 1, 6 (c) 3, 2 (d) 5, Identify the property used in the following: = (2 + 8) 13 (a) Commutative (b) Closure (c) Associative (d) Distributive 24. Which number is multiplicative identity for the whole numbers? (a) 0 (b) 1 (c) 2 (d) What will be multiplicative inverse of 8? (a) 8 (b) 1 8 (b) 1 (d) 0 8

18 26. Which property is reflected in the following: 7 5 = 5 7 (a) Closure (b) Commutative (c) Associative (d) Distributive 27. On dividing a negative integers by other negative integer the quotient will be: (a) Always negative (b) Always positive (c) Either positive or negative (d) Which of the following statement is true? (a) 7 0 = 7 (b) 7 0 = 0 (c) 7 0 = 0 7 (d) 0 7 = i) The product of two integers with same signs is a +ve integer. ii) The product of two integers with unlike sign is a ve integer. iii) The product of ten negative integers is a +ve integer. iv) The product of seven negative integers is a ve integer. Which of the statements is/are false? (a) (i), (ii), (iii) (b) (ii), (iii), (iv) (c) (i), (iii), (iv) (d) None of these 30. Match the following: Column I Column II A (p) 70 B (q) 96 C (r) 84 D (s) + 84 (a) A p; B s; C q; D r (b) A s; B p; C r; D q (c) A s; B r; C p; D q (d) A p; B r; C s; D q