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RADICE RADICE 4 SQUARE If a number is multiplied by itself, the product obtained is called the square of that number. Example : 62 = 6 x 6 = 36, In words, 6 squared is equal to 36 or 36 is the square of 6 Note : The square of a number is its second power. In the above example the second power of 6=62 =36 SQUARE ROOT The square root of a given number x is the number whose square is x. The symbol ‘ ’ is used to denote the square root. Example : Square root of 36 is 6, as square of 6 is 36 Square of 6 is 62 = 36, Square root of 36 = 36 = 6. Note :The square root of a number is its ½ th power. In the above example 6=(36)½ PERFECT SQUARE Perfect square is the square of an integer. Example : 9, 16, 49, etc. are perfect squares since 9 =3 16 = 4 49 = 7 Note : A perfect square always has a square root that has no decimal expansion. RADICE 5 METHODS TO FIND THE SQUARE ROOT OF A NUMBER The following table gives the square of some numbers which helps to find out the square roots. Number Square 1 1 2 4 3 9 4 16 5 25 6 36 7 49 8 64 9 81 10 100 Number 625 { 52 = 5x5 = 25 { 7225 Square 20 400 30 900 40 1600 50 2500 60 3600 70 4900 80 6400 90 8100 100 10000 } 625 } 7225 From the above table we shall see that the square root of 625 lies between the numbers 20 and 30 It is because 400 < 625 < 900. Similarly the square root of 7225 lies between the numbers 80 and 90. DIVISION METHOD (To find the square root of perfect squares) Illustration to find 54756 using division method Step 1 Group the digits of the number in pairs starting from right to left. 5, 47, 56 RADICE 6 Step 2 Note the number in the left most group. 5, 47, 56 Step 3 Use the symbol for division method. 5, 47, 56 Find the largest perfect square which is less than or equal to the noted number in step 2 22 = 4 < 5 Step 5 Note the square root of the number mentioned in step 4 4=2 Step 6 Write the number noted in step 5 as the quotient and also as the divisor. Note the remainder. Step 7 Bring down the second group to the right of the remainder. Step 4 RADICE 2 2 5, 47, 56 4 1 2 2 5, 47, 56 4 1 47 7 2 Step 8 Double the quotient and write it as the new divisor. 2 4 5, 47, 56 4 1 47 Double of 2 =2x2=4 2 3 Step 9 Put the largest possible digit on the right of the new divisor such that it becomes the next digit in the quotient and note the remainder. 2 43 5, 47, 56 4 1 47 1 29 18 43x3 = 129 < 147 we cannot take the digits 4, 5, etc. since 44 x 4 = 176 > 147 2 3 2 Step 10 Bring down the next group to the right of the remainder. 43 5, 47, 56 4 1 47 1 29 18 56 RADICE 8 2 3 Step 11 Repeat Step 8 (Double the quotient and write it as the new divisor.) 2 43 46 5, 47, 56 4 1 47 1 29 18 56 Double of 23 = 2 x 23 = 46 Step 12 Repeat Step 9 (Put the largest possible digit on the right of the new divisor such that it becomes the next digit in the quotient and note the remainder.) 2 3 4 2 5, 47, 56 4 4 3 1 47 1 29 18 56 464 18 56 0 464x4 = 1856 we cannot take the digits 5, 6, etc. since 465 x 5 > 1856 We cannot proceed further because in the last step the remainder is 0 and there is no other groups to bring down. Thus 54756 = 234 If the given number is not perfect then it won’t end with remainder zero. Try yourself Find the square root of 276676 Step 1 RADICE Group the digits of the number in pairs starting from right to left. 9 Step 2 Note the number in the left most group. Step 3 Use the symbol for division method. 27, 66, 76 Find the largest perfect square which is less than or equal to the noted number in step 2 ............ = ........... < 27 Step 4 Step 5 Note the square root of the number mentioned in step 4 Step 6 Write the number noted in step 5 as the quotient and also as the divisor. Note the remainder. Step 7 Bring down the second group to the right of the remainder 25 = ............ .... .... 27, 66, 76 25 2 .... .... 27, 66, 76 25 2 .... RADICE 10 Step 8 Double the quotient and write it as the new divisor. .... .... Step 9 Put the largest possible digit on the right of the new divisor such that it becomes the next digit in the quotient and note the remainder. .... 10.... .... Step 10 Bring down the next group to the right of the remainder. 10.... .... Step 11 Repeat Step 8 (Double the quotient and write it as the new divisor.) 10.... 104 Step 12 RADICE Repeat Step 9 (Put the largest possible digit on the right of the new divisor such that it becomes the next digit in the quotient and note the remainder.) .... 10.... 104...... .... 27, 66, 76 25 2 66 .... .... 27, 66, 76 25 2 66 2 04 ...... .... .... 27, 66, 76 25 2 66 2 04 62 .... .... .... 27, 66, 76 25 2 66 2 04 62 76 .... .... ..... 27, 66, 76 25 2 66 2 04 62 76 62 76 0 11 DIVISION METHOD (To find the square root of decimal numbers) Illustration to find 10485.76 using division method Step 1 Group the integral part of the given decimal number in pairs from right to left. 1, 04, 85 . 76 Step 2 Group the decimal part of the given decimal number in pairs from left to right. 1, 04, 85 . 76, Step 3 Note the number in the left most group. 1, 04, 85 . 76, Step 4 Use the symbol for division method. 1, 04, 85 . 76, Step 5 Find the largest perfect square which is less than or equal to the noted number in step 3 Step 6 Note the square root of the number mentioned in step 4 Step 7 Write the number noted in step 6 as the quotient & also as the divisor. Note the remainder. 12 = 1 1 =1 1 1 1, 04, 85. 76 1 0 RADICE 12 1 Step 8 Bring down the next group to the right of the remainder. 1 1, 04, 85. 76 1 0, 04 1 Step 9 Double the quotient and write it as the new divisor. 1 double of 1 =1x2=2 Step 10 2 Put the largest possible digit 1 on the right of the new divisor such that it becomes the next 20 digit in the quotient and note the remainder. 20x0 = 0 < 4 we cannot take the digits 1, 2, etc. since 21 x 1 = 21 > 4 1, 04, 85. 76 1 0, 04 10 1, 04, 85. 76, 1 0, 04 00 4 10 Step 11 RADICE Repeat Step 8 (Bring down the next group to the right of the remainder.) 1 20 1, 04, 85. 76, 1 0, 04 00 4 85 13 10 Step 12 Repeat Step 9 (Double the quotient and write it as the new divisor.) 1 20 20 1, 04, 85. 76, 1 0, 04 00 4 85 Double of 10 = 2 x 10 = 20 10 2 Step 13 Repeat Step 10 (Put the largest possible digit on the right of the new divisor such that it becomes the next digit in the quotient and note the remainder.) 1 20 202 1, 04, 85. 76, 1 0, 04 00 4 85 4 04 81 202 x 2 = 404 we cannot take the digits 3, 4, etc. since 203 x 3 = 609 > 485 10 2 . Step 14 Place a decimal point after the quotient of the above step since the next group comes after the decimal point. 1 20 202 1, 04, 85. 76, 1 0, 04 00 4 85 4 04 81 RADICE 14 10 2 . 1 Step 15 Repeat step 8 (Bring down the next group to the right of the remainder.) 20 202 1, 04, 85. 76, 1 0, 04 00 4 85 4 04 81 76 10 2 . 1 Step 16 Repeat step 9 (Double the quotient and write it as the new divisor.) 20 202 Double of 102 = 2 x 102 = 204 204 1, 04, 85. 76, 1 0, 04 00 4 85 4 04 81 76 10 2 .4 Step 17 Repeat step 10 (Put the largest possible digit on the right of the new divisor such that it becomes the next digit in the quotient and note the remainder.) 1 20 202 2044 1, 04, 85. 76, 1 0, 04 00 4 85 4 04 81 76 81 76 0 We cannot proceed further because in the last step the remainder is 0 and there is no other groups to bring down. Thus RADICE 10485.76 = 102.4 15 Try yourself Find the square root of 605.16 Step 1 Group the integral part of the given decimal number in pairs from right to left. Step 2 Group the decimal part of the given decimal number in pairs from left to right. Step 3 Note the number in the left most group. Step 4 Use the symbol for division method. Step 5 Find the largest perfect square which is less than or equal to the noted number in step 3 Step 6 Note the square root of the number mentioned in step 4 Step 7 Write the number noted in step 6 as the quotient & also as the divisor. Note the remainder. 6, 05, . 16 ..... = ...... < 6 ... = ... .. .... 6, 05, . 16 ... 2 RADICE 16 .. Step 8 Bring down the next group to the right of the remainder. Step 9 Double the quotient and write it as the new divisor. .... .... .... Step 10 Put the largest possible digit on the right of the new divisor such that it becomes the next digit in the quotient and note the remainder. 6, 05, . 16 ... 2 ... .. 6, 05, . 16 ... 2 ... .. .... .... .... .... 6, 05, . 16 ... 2 05 1 76 ........ decimal point Step 11 Place a decimal point after the quotient of the above step since the next group comes after the decimal point. .. .... . .... .... .... 6, 05, . 16 ... 2 05 1 76 ........ .. .... . .... Step 12 RADICE Repeat Step 8 (Bring down the next group to the right of the remainder.) .... .... 6, 05, . 16 ... 2 05 1 76 ........ ..... 17 .. .... . .... Step 13 Repeat Step 9 (Double the quotient and write it as the new divisor.) 6, 05, . 16 ... 2 05 1 76 29 16 .... .... .... Step 14 Repeat Step 10 (Put the largest possible digit on the right of the new divisor such that it becomes the next digit in the quotient and note the remainder.) .. .... . .... 6, 05, . 16 ... 2 05 1 76 29 16 29 16 0 .... .... .... .... .... PRIME FACTORISATION METHOD Illustration to find Step 1 900 using Prime factorisation method 2 900 Factorise the given number into primes. 2 450 5 225 A natural number, which is greater than one and is divisible by 1 and itself only, is called a prime number. 5 45 3 9 3 Step 2 Write the number as the product of primes. 900 = 2x2x5x5x3x3 RADICE 18 Step 3 Group the factors using brackets such that one group contains a pair of same prime. Step 4 Take one factor from each pair and find their product which is the square root of the given number. 900 = (2x2)x(5x5)x(3x3) 900 = 2 x 5 x 3 = 30 Try yourself Find the square root of 784 using prime factorisation. Step 1 Factorise the given number into primes. 2 784 .... ....... .... 196 2 ....... .... 49 ....... Step 2 Write the number as the product of primes. Step 3 Group the factors using brackets such that one group contains a pair of same prime. Step 4 RADICE Take one factor from each pair and find their product which is the square root of the given number. 784 = 2x......x......x2x......x...... 784 = (2x.....) x (......x......) x (......x......) 784 = ..... x ...... x ...... 19 USING IDENTITIES Illustration to find 625 using Identities Step 1 Find two perfect squares such that one is less than and the other is greater than the given number. Step 2 Use the square root sign in the inequation in step 1. 400 < Step 3 Find the numbers between which the square root of the given number lies. 20 < Step 4 Write an equation connecting the square root and one of the number involved in the inequation in step 3 using a variable. Step 5 Step 6 400 < 625 < 900 Represent the equation in step 4 in the form of identity (a+b)2. Substitute the values 1, 2, 3, ... for x in the equation in step 5. 625 < 900 625 < 30 20 + x = 625 or 30 - x = 625 (20 + x)2 = 625 (20+1)2 = 212 = 21 x 21 = 441 ≠ 625 (20+2)2 = 222 = 22 x 22 = 484 ≠ 625 (20+3)2 = 232 = 23 x 23 = 529 ≠ 625 (20+4)2 = 242 = 24 x 24 = 576 ≠ 625 (20+5)2 = 252 = 25 x 25 = 625 The value x = 5 satisfies the equation in step 5. ∴ 625 = 25 RADICE 20 Try yourself Find the square root of 529 using identities. Step 1 Find two perfect squares such that one is less than and the other is greater than the given number. Step 2 Use the square root sign in the inequation in step 1. ...... < 529 < Step 3 Find the numbers between which the square root of the given number lies. ...... < 529 < ...... Step 4 Write an equation connecting the square root and one of the number involved in the inequation in step 3 using a variable. Step 5 Represent the equation in step 4 in the form of identity (a+b)2. Step 6 Substitute the values 1, 2, 3, ... for x in the equation in step 5. ...... < 529 < ...... .... + x = 529 (.... + x)2 = 529 (... +1)2 = .... ≠ 529 (...+2)2 = .... ≠ 529 (...+3)2 = .... = 529 The value x = 3 satisfies the equation in step 5. ∴ 529 = ....... RADICE ...... 21 DESCRIPTIVE PROBLEMS 1. Find the square root of 4489 by division method. Ans. 2. Find 15625 by prime factorisation method. Ans. 67 5 15625 6 44, 89 36 127 8 8 9 889 0 5 3125 5 625 5 125 5 25 5 ∴ 4489 = 67 15625 = (5 x 5) x (5 x 5) x (5 x 5) ∴ 3 15625 = 5 x 5 x 5 = 125 Find the square roots of the following numbers (a) 150.0625 (b) 7.7284 (c) Ans: a) 12.25 b) 2. 78 0.9801 c) 0. 99 1 1, 50. 06, 25 1 22 0 50 44 242 606 484 12225 2445 12225 0 2 7. 72, 84 4 47 372 329 548 4384 4384 0 9 0. 98, 01 81 189 1701 1701 0 ∴ ∴ ∴ 150.0625 = 12.25 7.7284 = 2.78 0.9801 = 0.99 RADICE 22 4. A play ground is in the shape of a square. It’s area is 3906.25 m2. It is to be fenced. Then what would be the length of the fence? Ans. The area of a square of side having length ‘a’ = a2 Given that the area of the square shaped play ground = 3906.25 m2 i.e., a2 = 3906.25 m2 ∴ Length of one side of the play ground = a = 3906.25 m 62.5 6 39,06. 25 36 122 306 244 1245 6225 6225 0 ∴ 3906.25 = 62.5 Length of one side of the play ground = 62.5m Length of the fence = Perimeter of the square = 4 x 62.5 = 250.0 m 5. A basket contains 125 flowers. A man goes for worship and puts as many flowers in each temple as there are temples in the city. If the man needs 20 baskets of flower, find the number of flowers that he puts in each temple. Ans : Let the number of flowers that he puts in each temple be ‘x’ ∴ The number of temples in the city = x Thus the total number of flowers used = the number of temples in the city x the number of flowers that he puts in each temple RADICE } = xxx = x2 23 The number of flowers in a basket = 125 The number of baskets = 20 Total number of flowers in 20 baskets = 125 x 20 = 2500 ⇒ ⇒ = 50 x2 = 2500 x = 2500 So the number of flowers that he puts in each temple = 50 TRAINING FOR COMPETITIVE EXAMINATION Very challenging problems are marked with 1. 81 a) 9 2. b)10 b) 0 Find the square root of a) ±121 4. The value of 1 x c) -1 d) none of these c) 121 d) c) ± 1 d) none of these 121 ± 11 -1 = b) +1 Which of the following is a perfect square? a) -121 6. ± b) 11 a) -1 5. d) 7 If a real number has one and only one square root, then it is a) 1 3. c) 8 0.0196 0.2 a) 0.7 b) 121 c) -64 d) All of them b) 7 c) 0.07 d) 0.007 = RADICE 24 196 x 14 7. 8. 9. 17 x 289 78 = 169 a) 2 b) 4 c) 6 1 11 = 25 a) 3 5 b) 5 3 c) 19 5 d) 8.33 d) 1 1 5 The area of a square is 2304, then the length of its side is a) 48 10. 12. d) 18 b) 64 c) 16 d) 18 The square root of a perfect square containing ‘n’ digits has —— digits a) n + 1 b) n c) Either n + 1 or n d) none of these 2 2 2 2 x 16 = 49 49 a) 4 13. c) 28 The least perfect square which is divisible by 2, 4 and 6 is a) 36 11. b) 38 then x = b) 7 c) 16 d) 28 A man plants his 5625 orchid trees and arranges them so that there are as many rows as there are trees in a row. How many rows are there? a) 85 14. b) 5 c) 65 d) 75 Which of the following is not a perfect square? a) 100 RADICE b) 2025 c) 324 d) 112 25 15. 16. If k = 2 5 then k = 5 a) 50 b) 10 b) 125 20. b) 9 ? then ? = 2.56 = ? 0.1 a) 0.4 b) 0.060 d) 5x625 c) 8 d) 7 c) 0.06 d) 0.09 b) 363 m c) 336 m d) 330 m 25 b) 144 5 c) 12 d) None of these 1 1 + = 16 9 21. a) 7 12 0.081 0.484 Square root of 0.0064 x 6.25 a) 0.45 23. c) 6252 The area of a square field is 8190.25m2. The perimeter of square field is a) 362m 22. d) 155 A number added to its square gives 56. The number is a) 12 19. c) 255 25 times of square of 125 is a) 125 x 252 b) 5 x 1252 18. d) none of these The square root of (11)(12)(13)(14)+1 is a) 225 17. c) 25 b) 0.75 c) 0.95 x 2.5 12.1 d) 0.99 Which of the following is a Pythagorean triplet a) (6,8,10) b) (3, 4, 7) c) (5, 12, 18) d) none of these RADICE 26 24. 5+2 21 x 0.169 25 1.6 a) 0.91 25. m+9 If x*y = a) -7 29. m+1 c) m2 + 2m d) none of these b) 9 c) 6 d) 3 b) 2.449 c) 2.645 d) 2.828 x2 + y2 , the Value of (1*2 2) (1*-2 2) is b) 0 c) 2 d) 9 c)5 d) 6.4 c) 7 16 d) 41- 21+ 19 - 9 a) 3 30. b) m + 2 What is the square root of 6? a) 2.236 28. d) 0.95 31+ 21+ 13 + 8 + 1 a) 74 27. c) 0.94 If m is a square number, then the next immediate square number is a) 26. b) 0.92 b) 6 25 x 13 x 2 0.25 18 32 a) 48 35 RADICE b) 16 7 35 48 27 ANSWERS 1. (a) 2. (b) Except zero, all the other real numbers have 2 square roots. 3. (d) 4. (d) ± 121 = ± 11 since 121 = ± 11 1x -1 = 1x-1 = -1 since negative numbers have no real square roots the answer is not in the given list. 5. (b) Negative numbers are not perfect squares so the only perfect square in the list is 121, 121 = 11 6. (a) 0. 0196 0.14 = = 0.7 0.2 0.2 7. (c) 196 17 x 14 289 8. (d) 1 9. (a) 11 = 25 x 78 169 25+11 = 25 = 36 25 14 17 x 14 17 = 6 5 =1 x 78 13 = 6 1 5 Area of a square = side × side = 2304 side2 = 2304 side = 2304 = 48 10. (a) In the given list 36, 64 and 16 are perfect squares. But 16 is not divisible by 6 and 64 is not divisible by 6. 36 is divisible by 2, 4 and 6 which is also a perfect square 11. (c) Consider some squares and their square roots, Let the number of digits in the square = n If n is even then the square root of the number have n digits 2 If n is odd then the square root of the number have n+ 1 digits. 2 RADICE 28 12. (d) 16 x = 49 49 13. (d) x 4 = 7 49 x = 49 x 4 7 Let the number of rows = x Then the number of trees in a row = x ∴ total number of trees = x. x = x2 Given that x2 28 = 5625 x 14. (d) = = 5625 = 75 No perfect square ends in 2, 3, 7, 8 ∴ 112 is not a perfect square or on the other hand you can find out the square roots 100 = 10, 2025 = 45 324 = 18 but 112 is not a square of an integer. 15. (b) k 5 =2 5 ⇒ k =2 5x 5 = 2 ( 5)2 =2x5 = 10 16. (d) (11) (12) (13) (14) + 1 = 24024 + 1 = 24025 Square root of (11) (12) (13) (14) + 1 = 24025 = 155 (using division method) 17. (c) RADICE Square of 125 is 1252 25 times of square of 125 is 25 x 1252 = 25 (25 x 5)2 since 125 = 25x5 = 25 x 252 x 52 29 = = = 18. (d) Let x be the number Then the square of x is x2 ∴by given condition x2 + x x2 +x – 56 that is (x + 8) (x – 7) 25 x 252 x 25 (25 x 25) 252 625 x 625 = (625)2 = 56 = 0 = 0 factorised using the identity (x+a) (x+b) = x2+(a+b)x + ab i.e. x + 8 = 0, x - 7 = 0 i.e., x = -8, x = 7 -8 is not in the given list so the number is 7. 19. (a) 2.56 = ? ⇒ 1.6 ? 0.1 ? Let ? = = ? 0.1 x 1.6 x = x 0.1 ie, 1.6 (0.1) = x2 then 0.16 x 20 (a) = x2 = 0.16 = 0.4 Area of square = side x side i.e, side2 = 8190.25 side = perimeter = 4 x side 8190.25 = 4 x 90.5 ie, ? = 0.4 = 8190.25 (using division = 90.5 method) = 362m. RADICE 30 21. (c) 1 1 + = 16 9 = 22. (a) 9+16 16x9 25 16x9 25 16 x 9 = 0.081 x 0.484 x 2.5 = 0.0064 6.25 12.1 = 5 4x3 = 5 12 81 484 25 x x 1000 1000 10 64 x 625 x 121 1000 100 10 = 81 x 484 x 25 64 x 625 x121 = 9 x 22 x 5 8 x 25 x 11 = 9 20 = 0.45 23. (a) A Pythogorean triplet is a set of 3 numbers where the sum of the squares of the first two is equal to the square of the third number. Consider 62 + 82 = 36 + 64 = 100 = 102 so (6, 8, 10) is a Pythagorean triplet. 24. (a) 5+2 21 x 0.169 = 25 1.6 = = RADICE 5+ 71 x 0.169 = 25 1.6 125+71 x 0.169 25 1.6 196 x 169 x 10 = 25 1000 x 16 14 x 13 = 0.91 5 x 10 x 4 196 x 169 25 100 x 16 31 25. (b) It is given that m is a square number i.e., m = x2 for some x ⇒ x= m Then next immediate square is (x+1)2 (x+1)2 = x2+2x+1 = m+2 m + 1 26. (c) = 31+ 21+ 13 + 8 + 1 31+ 21+ 13 + 8 + 1 since 1 = 1 27.(b) = 31+ 21+ 13 + 9 = 31+ 21+ 13+3 = 31+ 21+ 16 = 31+ 21+4 = 31+ 25 = 31+ 5 = 36 since 9 = 3 since 16 = 4 since 25 = 5 = 6 The square root of 6, approximating to 3 digits is 2.449 (using division method) 28. (d) given x*y = x2 + y2 = 12+(2 2)2 = 1+4 x 2 = 1+8 = 1*- 2 2 = 12+(-2 2)2 = 1+4 x 2 = 9 = 3 1* 2 2 9 = 3 (1*+2 2) (1*- 2 2) = 3 x 3 = 9 RADICE 32 29. (b) 30. (d) 41- 21+ 19 - 9 25 x 2 13 x 0.25 = 32 18 = 41- 21+ 19-3 = 41- 21+ = = 16 = 41- 21+ 4 41- 25 = 41 - 5 36 = 6 25 x 49 x 25 32 18 100 = 25 x 49 x 25 32 9x2 100 since 18 = 9 x 2 = 25 x 49 x 25 = 9 100 64 5 x 7 x 5 = 35 10 x 3 x 8 48 ADVANCED TOPIC Properties of squares:• No perfect square ends in 2, 3, 7 or 8. • The number of zeroes at the end of a perfect square is even. • The square of odd and even numbers are odd and even respectively. • If n2 is a perfect square then 2n2 cannot be a perfect square. Similarly the product of a prime and a perfect square also is not a perfect square. • The square of every non-zero real number is positive. RADICE 33 Properties of Square Roots:• We cannot find the square root of a negative number as a real number. • If there is any number ends in 2, 3, 7, 8, then it has no integer square root • If a number ends in odd number of zeroes, then it is not a perfect square. • The square root of an even number is even and that of an odd number is odd. • • • xy = x y x = y n = n x y 2 DESCRIPTIVE PROBLEMS 1. Why the following are not perfect squares? Give reason a) 23453 b) 22222 c) 125000 d) -9876 Ans: (a) Here the unit digit is 3 which is not possible for a perfect square. (b) Here also the unit digit is 2 and since no square number ends in 2, 3, 7, or 8, the given number is not a perfect square. (c) Here the number of zeroes at the end is 3, which is an odd number. A perfect square cannot have odd numbers of zeroes at the end thus the given number is not a perfect square. (d) Since the square of a non zero real number is always positive, it is not a perfect square. RADICE 34 2. Classify the following numbers according to whether their squares are even or not? a) 431 b) 9999 c) 13984 d) 9847 e) 46432 f) 23718 Ans: We know the square of odd and even numbers are odd and even respectively. Thus we can classify them as Numbers having odd squares 3. Find out Numbers having even squares 431 13984 9999 46432 9847 23718 4.2025 4.2025 = 42025 . 10000 = 42025 10000 = 205 100 = 2.05 So since 42025 10000 4.2025 = x y 42025 = 205, = x y 10000 = 100 MULTIPLE CHOICE QUESTIONS Very challenging problems are marked with 1. If x is an even then x2 is a) an even number b)an odd number c)either even or odd d) neither even nor odd RADICE 35 2. If x2 is odd then x is a) an even number b) an odd number c) either even or odd d) neither even nor odd. 3. Which of the following is a perfect square? a) -121 b) 121 c) -64 d) none of these. 4. If a perfect square has 8 zeroes at its end how many zeroes will be there in its square root at its end a) 8 b) 4 c) 2 d) 6 5. -4 = ? a) 2 b)-2 c) ± 2 d) none of these ANSWERS 1. (a) Square of an even number is always even x is even ⇒ x2 is even. 2. (b) Square root of an odd number is always odd x2 is odd ⇒ x is odd. 3. (b) Negative numbers are not perfect squares. 4. (b) The numbers of zeroes at the end of a square root of a number is half the number of zeroes at the end of the given number. square has 8 zeroes at the end square root has 4 zeroes at the end. 5. (d) Negative numbers have no real square roots. RADICE 36 PROJECT Introduction:This project is to derive an easy method to find square of numbers having its unit digit 5 Aim : Find the square of numbers having unit digit 5 and derive an easy relation between the digits in the number and its square. Sl. Number The No. having square of the unit number digit 5 The new number after deleting the unit digits 5 from the considerig number in coloumn 2 The new number after deleting last digits 25 from the square in coloumn 3 Relationship between the new numbers in coloumns 5 & 4 1 5 25 0 0 0 = 0 x (0+1) 2 15 225 1 2 2 = 1 x (1+1) 3 25 625 2 6 6 = 2 x (2+1) 4 35 1225 3 12 12 = 3 x (3+1) Conclusion The easy steps to be followed to find the square of a number having its unit digit 5 • Find the new number after deleting the unit digit 5 • Multiply this new number with the natural number immediately after it. Let the product be P. • The square of the given number can be written in the form P 25 e.g.: To find square of 105 The new number after deleting unit digit 5 from the given number=10 The natural number immediately after 10=11 RADICE 37 P = 10 x 11 = 110, Replacing P with 110, the form P 25 = 11025 ∴ (105)2 11025 = PRACTICE TEST “Practice makes perfect”. Test the level of your accuracy and speed by answering the given problems. Please note the time you spend for each problem. Write all the necessary steps. But if you are thorough with the ideas you can skip some steps. Please send the answers to our office in the business envelope which is free of cost. We will provide you proper guidance. If you want to get a proper evaluation and assessment, you have to do it sincerely. Very challenging problems are marked with 1. The area of a square is 100 square inches. Find the perimeter of the square. a) 40 inches b) 20 inches c) 100 inches d) 10 inches 2. Evaluate x y when x = 15 and y = 15 a) ± 225 3. b) 11 c) 12 d) 13 b) 900 c) 400 d) 225 If 67* = (62+7) hundred + 72 = 4349 then the value of 69* is a) 3481 6. d) both b and c The least perfect square exactly divisible by each of the numbers 6, 9, 15 and 20 is a) 36000 5. c) 15 Find out the difference in number of zeroes in square and square root of ten crores. a) 10 4. b) -15 b) 4581 c) 3681 d) 3281 A Gardiner plants trees in a row and finds that each row contains as many trees as there are rows. If the number of trees be 2704, find the RADICE 38 number of trees in each row. a) 52 7. 8. 9. b) 104 c) 8 d) 5 Digit at unit place in the square root of 4624 is a) 52 b) 7 4.2025 a) 205 100 2025 is = b) a) 35 b) 45 c) 8 205 100 c) d) 5 205 10 d) none of these c) 55 d) 65 c) 11025 d) 44025 10. Square of 205 is 11. 12. a) 42025 b) 21025 If x = 1.69 - 0.01 then the value of x is a) 11.99 b) 1.68 d) 12 x + 36 = 25% of 400 then the value of x is a) 3 13. c) 1.20 b) 36 c) 8 d) None of these c) 12 d) 30 35 x 23 x 6 = ? a) 108 b) 36 14. Some persons contributed Rs. 1089. Each person gave as many rupees as they are in number. Find their number a) 33 15. c) 45 d) 25 c) 8 d) 10 2n = 32, then the value of n is a) 512 16. b) 66 -4 = ? RADICE b) 4 39 a) +2 b) -2 c) ± 2 d) None of these 17. The square of a negative number is. a) positive 18. b) 9904 c) 9804 d) 9809 A certain number of people collected Rs. 125. If each person contributed as many five paise as they are in number, the number of persons were. a) 25 20. c) both positive and negative d) Neither positive nor negative The largest four digit number which is a perfect square a) 9801 19. b) negative b) 50 81 , the value of x is 0.09 a) 3 b) 30 c) 100 d) 125 c) 300 d) 0.3. x= RADICE 40 PUZZLES Which is your favourite subject in school? Is it Maths? Some of you don’t like Maths. But every one likes the Mathematical magic. Let’s have some fun with Maths. Try the following and enjoy yourself I will tell your date of birth Suppose your date of birth (only date) Add 2 to it. Multiply by 100 Add 5 to it. Add the number corresponding to month (If Jan, add 1) Subtract 205 from the result If the result is 2606, (2606-205 =2401) Pair it from right 24 , 01 24 corresponds to date 01 corresponds to month Jan If you tell me the result I will tell your date of birth. RADICE 41 FOR BRAIN EXERCISE Now try to answer the following. Answer will be provided in the next volume. 1. Can you name the smallest integer that can be written with 2 digits? 2. Can you make 100 using eight 8’s and using standard mathematical operations? 3. Combine 3 numbers in each group to get the same result in each of the 3 groups. You can use addition, subtraction, multiplication and division Group 1: 15, 19, 24 2: 11, 30, 36 3: 20, 22, 36 4. How many squares are there in the following figure? 5. Can you make 2 squares and 4 right angled triangles using only 8 straight lines? 6. Can you name the biggest number that can be written with four 1’s? 7. Can you make 100 using 6 match sticks? 8. Which is the only number that is the perimeter and area of the same square? RADICE 42 MATHEMATICS QUIZ Did you know this? You have heard about the Nobel prizes. Nobel prizes are awarded in Physics, Chemistry, literature, peace Physiology or medicine and Economics. It is not awarded to Mathematics. Have you ever heard about field’s medal? It is often described as the “Nobel prize” of Mathematics. The field’s medal is a prize awarded to 2,3,or 4 mathematicians not over 40 years at International Mathematical union, a meeting that takes place every 4 years. FIRST! FIRST! FIRST! • • It is the greatest award in Mathematics .It is awarded to 2, 3 or 4 Mathematicians not over than 40 years. • • • • RADICE Which is the first calculating machine? Abacus. Which is the first composite number? 4 Which is the first irrational number to be discovered? 2 Who is the first Indian Mathematician to be elected as a fellow of Royal society, London? Ramanujan. Which is the first Mathematical book in Malayalam? Yukthi Bhasha Which is the first printed book on Mathematics? Treviso Arithmetic. 43 SRINIVASA RAMANUJAN (The great Indian Mathematician ) 1887 -1920 Srinivasa Ramanujan was a great Mathematician born in South India. He was from a poor Tamil family. He was a clerk in Madras. He lived off the charity of friends. The English mathematician Hardy discovered his Mathematical talents. With the help of Professor Hardy, he went to Cambridge University for further studies. Ramanujan’s years in England were mathematically productive. Cambridge granted him a Bachelor of Science degree “by research” in 1916, and he was elected a Fellow of the Royal Society (the 1st Indian to be so honoured) in 1918.But Ramanujan had always lived in a tropical climate. In 1917 he was hospitalized. At that time professor Hardy paid him a visit. Prof .Hardy told Ramanujan that he rode a taxi cab to the hospital, with a very unlucky number. When Ramanujan enquired what the number was, Prof. Hardy replied that: 1729 Ramanujan’s face lit up with a smile and he said that it was not an unlucky number at all, but a very interesting number, the smallest number that can be expressed as the sum of cubes of 2 different numbers in 2 different ways. 1729 = 103+93 1729 = 123+13 The number 1729 is known as Ramanujan’s number. In spite of his illness, he discovered the mathematical formulas. By late 1918 his health condition became worst; he returned to India in 1919. But his health failed again and he died in the next year. RADICE 44 RADICE 45 RADICE 46 INSTRUCTIONS TO THE STUDENT ♦ In this book, the last page is provided for the student to register as a Radice Family Member. ♦ Write your comments about this book in the space provided. ♦ Refer your five good friends whom you think would benefit from this book. ♦ If you want a proper guidance and assessment, you have to answer the practice test on a worksheet and send to us. This guidance and feedback will be beneficial only if you are sincere with your work. ♦ A pre-paid envelope is enclosed herewith for you to send your comments and worksheet. RADICE 47 YOUR COMMENTS ON MATH COMPANION Name : School : Syllabus : CBSE/State/ICSE Home Address : Phone Number : ___________________________________________________________ ___________________________________________________________ ___________________________________________________________ ___________________________________________________________ ___________________________________________________________ ___________________________________________________________ ___________________________________________________________ ___________________________________________________________ ___________________________________________________________ ___________________________________________________________ ___________________________________________________________ RADICE 48 YOUR FIVE GOOD FRIENDS 1 Name : School : Syllabus : Home Address : Phone Number : 2 Name : School : Syllabus : Home Address : Phone Number : 3 Name : School : Syllabus : Home Address : Phone Number : 4 Name : School : Syllabus : Home Address : Phone Number : 5 Name : School : Home Address : Phone Number : RADICE Syllabus :