Chapter 1 Real Numbers class 10th maths worksheets

July 10, 2022 er.ashu2011 0 Comments

EXAMPLE 1 Show that every positive even integer is of the form 2q, and that every positive odd integer is of the form 2q + 1, where q is some integer. [NCERT]

EXAMPLE 2 Show that any positive integer is of the form 3q or, 3q + 1 or, 3q + 2 for some integer q.

EXAMPLE 3 Show that any positive odd integer is of the form 4q + 1 or 4q + 3, where q is some integer. [NCERT]

EXAMPLE 4 Show that n – 1 is divisible by 8, if n is an odd positive integer,

EXAMPLE 5 Prove that if x and y are odd positive integers, then x2 + y2 is even but not divisible by 4.

EXAMPLE 6 Prove that n2-n is divisible by 2 for every positive integer n.

EXAMPLE 7 Show that the square of any positive integer is of the form 3m or, 3m + 1 for some integer m. [NCERT, CBSE 2008]

EXAMPLE 8 Use Euclid’s division Lemma to show that the cube of any positive integer is either of the form 9m, 9m+ 1 or, 9m + 8 for some integer m. [NCERT]
Ans. x3 is either of the form 9m or, 9m + 1 or, 9m + 8.

EXAMPLE 9 Show that one and only one out of n, n + 2 or, n + 4 is divisible by 3, where n is any positive integer.

EXAMPLE 10 Prove that one of every three consecutive positive integers is divisible by 3.

Practice Set 1

  1. Prove that the product of two consecutive positive integers is divisible by 2.
  2. If a and b are two odd positive integers such that a > b, then prove that one of the two numbers (a + b)/2 and (a – b)/2 is odd and the other is even.
  3. Show that the square of an odd positive integer is of the form 8q + 1, for some integer q.
  4. Show that any positive odd integer is of the form 6q + 1 or, 6q + 3 or, 6q + 5, where q is some integer. [NCERT]
  5. Prove that the square of any positive integer is of the form 3 m or, 3m +1 but not of the form 3m + 2.
  6. Prove that the square of any positive integer is of the form 4q or 4q +1 for some integer q.
  7. Prove that the square of any positive integer is of the form 5q, 5q + 1,5q + 4 for some integer q.
  8. Prove that if a positive integer is of the form 6q + 5, then it is of the form 3q + 2 for some integer q, but not conversely.
  9. Prove that the square of any positive integer of the form 5q + 1 is of the same form.
  10. Prove that the product of three consecutive positive integer is divisible by 6.
  11. For any positive integer n, prove that n3– n divisible by 6.

HINTS TO SELECTED PROBLEMS
Let n -1 and n be two consecutive positive integers. Then, their product is (n – 1) n = n^2 – n. Now, proceed as in example 6.
Since any odd positive integer n is of the form 4m +1 or 4m + 3.
If n – 4m +1, then
n^2- 〖(4m + 1)〗^2 = 16m^2+ 8m +1 = 8m (m +1) +1 = 8q +1 where q = m (m +1)
If n = 4m + 3 then
If n = 〖(4m + 3)〗^( 2) = 〖16m〗^2 + 24m + 9 = 8 (〖2m〗^2+ 3m +1) + 1 = 8q +1, where q = 〖2m〗^2 +3m + 1
Hence, n^2 is of the form 8q + 1.
Let a be any odd positive integer and b = 6. Then, there exist integers q and r such that
a = 6q + r, 0 ≤ r < 6 [By division algorithm] => α = 6g or, 6q + 1 or, 6q + 2 or, 6q + 3 or, 6q + 4 or, 6q + 5
But, 6q, 6q + 2 and 6q + 4 are even positive integers.
∴ a = 6q +1 or, 6q + 3 or, 6q + 5
Since positive integer n is of the form 3q, 3q + 1 or, 3q + 2.
If n = 3q, then
n^2 = 9 q^2 = 3(3q) = 3m,where m = 3q
If n = 3q +1, then
n^2 = 9 q^2+ 6q + 1 = 3q (3q + 2) + 1 = 3m + 1, where m = q(3q + 2)
If π = 3q + 2, then
n^2= 〖(3q^2+ 2)〗^2= 9q^2 + 12q + 4 = 3{3q^2 + 4q +1) + 1 = 3m +1, where m = 〖3q〗^2 + 4q +1.
Hence, n^2 is of the form 3m or, 3m + 1 but not of the form 3m + 2.
Since any positive integer n is of the form 2m or, 2m +1
If n = 2m, then
n^2= 〖4m〗^2 = 4q, where q = m^2
If n = 2m +1, then
n^2= 〖(2m + 1)〗^2 = 4m^2 + 4m + 1 = 4m (m +1) + 1 = 4q +1, where q = m (m + 1)
Since any positive integer n is of the form 5m or 5m + 1, or 5m + 2 or 5m + 3 or 5m + 4. If n = 5m, then
n^2= 〖25m〗^2 = 5 (5m) = 5q, where q = 5m
If n = 5m + 1, then
n^2= 〖(5m + 1)〗^2 = 5m (5m + 2) + 1 = 5q + 1, where q = m (5m + 2)
If n = 5m + 2, then
n^2 = 〖(5m + 2)〗^2 = 5m (5m + 4) + 4 = 5q + 4, where q = m (5m + 4)
If n = 5m + 3, then
n^2 = 〖(5m + 3)〗^2 = 5 (m^2 + 6m + 1) + 4 = 5q + 4, where q = 5m^2 + 6m + 1
If n = 5m + 4, then
n^2 = 5 (5m^2 + 8m + 3) +1 = 5q + 1, where q = 5m^2 + 8m + 3
Hence, n^2 is of the form 5q or, 5q + 1 or, 5q + 4.
Let n = 6q+ 5, where is a positive integer. We know that any positive integer is of the form 3k or, 3k +1 or, 3k + 2.
∴ q = 3k or, 3k + 1 or, 3k + 2
If q = 3k, then
n = 6q + 5 = 18k + 5 = 3 (6k + 1) + 2 = 3m + 2, where m = 6k + 1
If q = 3k + 1, then
n = 6q + 5 = 6(3k + 1) + 5 = 3(6k + 3) + 2 = 3m + 2, where m = 6k + 3
If q = 3k + 2, then
n = 6q + 5 = 6(3k + 2) + 5 = 3(6k + 5) + 2 = 3m + 2, where m = 6k: + 5.
Let n = 5q + 1. Then,
n^2 = 25q^2 + 10q+ 1 = 5 (5q^2 + 2q) +1 = 5m + 1, where m = 5q^2 +2q
=>n^2is of the form 5m +1.
Let n be any positive integer. Since any positive integer is of the form 6q or, 6q + 1 or, 6q + 2 or, 6q + 3 or, 6q + 4 or, 6q + 5.
If n = 6q, then
n (n + 1) (n + 2) = 6q (6q + 1) (6q + 2), which is divisible by 6
If n = 6q + 1, then
n (n + 1) (n + 2) = (6q + 1) (6q + 2) (6q + 3) = 6 (6q + 1) (3q + 1) (2q + 1),
which is divisible by 6.
If n = 6q + 2, then
n (n + 1)(n + 2) = (6q + 2)(6q + 3)(6q + 4) = 12(3q + 1)(2q + 1)(2q + 3),
which is divisible by 6.
Similarly, n(n+ 1) (n+2) is divisible by 6 if n = 6q + 3 or, 6q + 4 or, 6q + 5.
We have,
n^3-n = (n – 1) (n) (n + 1), which is product of three consecutive positive integers. -So, proceed as in Q. No. 10.

ILLUSTRATIVE EXAMPLES
EXAMPLE 1 Use Euclid’s division algorithm to find the HCF of 210 and 55.
Ans. 5.

EXAMPLE 2 Use Euclid’s division algorithm to find the HCF of 4052 and 12576.
Ans. 4.

EXAMPLE 3 Find the HCF of 81 and 237 and express it as a linear combination of 81 and 237.
Ans. 3.

EXAMPLE 4 Find the HCF of 65 and 117 and express it in the form 65m + 117n.
Ans. 13.

EXAMPLE 5 If the HCF of 210 and 55 is expressible in the form 210 x 5 + 55y, find y.
Ans. -19

EXAMPLE 6 If d is the HCF of 56 and 72, find x, y satisfying d = 56x + 72y. Also, show that x and y are not unique.
Ans. x = 4 and y = -3.

EXAMPLE 7 A sweet seller has 420 Kaju burfis and 130 Badam burfis she wants to stack them in such a way that each stack has the same number, and they take up the least area of the tray. What is the number of burfis that can be placed in each stack for this purpose?
[NCERT]
Ans. 10 burfis

EXAMPLE 8 Any contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march? [NCERT]
Ans. 8

EXAMPLE 9 Find the largest number which divides 245 and 1029 leaving remainder 5 in each case.
Ans. 16.

EXAMPLE 10 Find the largest number that divides 2053 and 967 and leaves a remainder of 5 and 7 respectively.
Ans. 64.

EXAMPLE 11 Find the largest number that will divide 398, 436 and 542 leaving remainders 7, 11 and 15 respectively.
Ans. 17.

EXAMPLE 12 Two tankers contain 850 litres and 680 litres of petrol respectively. Find the maxi-mum capacity of a container which can measure the petrol of either tanker in exact number of times.
Ans. 170 litres.

EXAMPLE 13 In a seminar, the number of participants in Hindi, English and Mathematics are 60,84 and 108, respectively. Find the minimum number of rooms required if in each room the same number of participants are to be seated and all of them being in the same subject.
Ans. 21.

EXAMPLE 14 Three sets of English, Hindi and Mathematics books have to be stacked in such a way that all the books are stored topic-wise and the height of each stack is the same. The number of English books is 96, the number of Hindi books is 240 and the number of Mathematics books is 336. Assuming that the books are of the same thickness, determine the number of stacks of English, Hindi and Mathematics books.
Ans. Number of stacks of English books = 2
Number of stacks of Hindi books = 5
Number of stacks of Mathematics books = 7

Practice Set 2

  1. Define HCF of two positive integers and find the HCF of the following pairs of numbers:
    (i) 32 and 54 (ii) 18 and 24 (iii) 70 and 30
    (iv) 56 and 88 (v) 475 and 495 (vi) 75 and 243.
    (vii) 240 and 6552 (viii) 155 and 1385 (ix) 100 and 190 [CBSE 2009]
    (x) 105 and 120 [CBSE 2009]
  2. Use Euclid’s division algorithm to find the HCF of
    (i) 135 and 225 (ii) 196 and 38220 (iii) 867 and 255. [NCERT]
  3. Find the HCF or the following pairs of integers and express it as a linear combination of them.
    (i) 963 and 657 (ii) 592 and 252 (iii) 506 and 1155
    (iv) 1288 and 575
  4. Express the HCF of 468 and 222 as 468x + 222y where x, y are integers in two different ways.
  5. If the HCF of 408 and 1032 is expressible in the form 1032 m – 408 × 5, find m.
  6. If the HCF of 657 and 963 is expressible in the form 657x + 963x -15, find x.
  7. An army contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march?
  8. Find the largest number which divides 615 and 963 leaving remainder 6 in each case.
  9. Find the greatest number which divides 285 and 1249 leaving remainders 9 and 7 respectively.
  10. Find the largest number which exactly divides 280 and 1245 leaving remainders 4 and 3, respectively.
  11. What is the largest number that divides 626, 3127 and 15628 and leaves remainders of 1, 2 and 3 respectively.
  12. Find the greatest number that will divide 445, 572 and 699 leaving remainders 4, 5 and 6 respectively.
  13. Find the greatest number which divides 2011 and 2623 leaving remainders 9 and 5 respectively.
  14. The length, breadth and height of a room are 8 m 25 cm, 6 m 75 cm and 4 m 50 cm, respectively. Determine the longest rod which can measure the three dimensions of the room exactly.
  15. 105 goats, 140 donkeys and 175 cows have to be taken across a river. There is only one boat which will have to make many trips in order to do so. The lazy boatman has his own conditions for transporting them. He insists that he will take the same number of animals in every trip and they have to be of the same kind. He will naturally like to take the largest possible number each time. Can you tell how many animals went in each trip?
  16. 15 pastries and 12 biscuit packets have been donated for a school fete. These are to be packed in several smaller identical boxes with the same number of pastries and biscuit packets in each. How many biscuit packets and how many pastries will each box contain?
  17. A mason has to fit a bathroom with square marble tiles of the largest possible size. The size of the bathroom is 10 ft. by 8 ft. What would be the size in inches of the tile required that has to be cut and how many such tiles are required?
  18. Two brands of chocolates are available in packs of 24 and 15 respectively. If I need to buy an equal number of chocolates of both kinds, what is the least number of boxes of each kind I would need to buy?
  19. 144 cartons of Coke Cans and 90 cartons of Pepsi Cans are to be stacked in a Canteen. If each stack is of the same height and is to contain cartons of the same drink, what would be the greatest number of cartons each stack would have?
  20. During a sale, colour pencils were being sold in packs of 24 each and crayons in packs of 32 each. If you want full packs of both and the same number of pencils and crayons, how many of each would you need to buy?
  21. A merchant has 120 litres of oil of one kind, 180 litres of another kind and 240 litres of third kind. He wants to sell the oil by filling the three kinds of oil in tins of equal capacity. What should be the greatest capacity of such a tin?

ANSWERS

  1. (i) 2 (ii) 6 (iii) 10 (iv) 8 (v) 5 (vi) 3 (vii) 24 (viii) 5 (ix) 10 (x) 15
  2. (i) 45, 45 = (-1) 225 + 2 × 135 (ii) 196, 196 = 38220 × 1 + (-194) × 196
    (iii) 51 = 51 = (-2) 867 + 7 × 255
  3. (i) 9 = (-15) × 963 + 22 × 657 (ii) 4 = 77 × 252 + (-20) 592
    (iii) 11 = 16 × 506 + (-7) × 1155 (iv) 23 = (-4) × 1288 + 9 × 575
  4. 6 = 468 × -9 + 222 × 19, 6 = 468 × 13 + 222 x (-449) 5. 2 6. 22
  5. 8 columns 8. 87 9. 138 10. 138 11. 625
  6. 63 13. 154 14. 75 cm 15. 35
  7. 4 biscuit packets, 5 pastries 17. 24 inches, 20 tiles
  8. 5 of first kind, 8 of second kind 19. 18
  9. 4 packets of colour pencils, 3 packets of crayons 21. 60 litres

ILLUSTRATIVE EXAMPLES
EXAMPLE 1 Express each of the following positive integers as the product of its prime factors:
(i) 140 (ii) 156 (iii) 234
Ans. (i) 140 = 2 × 2 × 5 × 7 = 2^2 × 5 × 7
(ii) 156 = 2 × 2 × 3 × 13
(iii) 234 = 2 × 3 × 3 × 13 = 2 × 3^2 × 13

EXAMPLE 2 Express each of the following positive integers as the product of its prime factors:
(i) 3825 (ii) 5005 (iii) 7429
Ans. (i) 3825 = 3 × 3 × 5 × 5 × 17 = 3^2 × 5^2 × 17
(ii) 5005 = 5 × 7 × 11 × 13
(iii) 7429 = 17 × 19 × 13

EXAMPLE 31 Determine the prime factorization of each of the following numbers:
(i) 13915 (ii) 556920
Ans. (i) 13915 = 5×11×11×3 = 5×〖11〗^2×23
(ii) 556920 = 2 × 2 × 2 × 3 × 3 × 5 × 7 × 13 ×17 = 2^3× 3^2× 5 × 7 × 13 × 17

EXAMPLE 4 Prove that there is no natural number for which 4n ends with the digit zero.
[NCERT]

EXAMPLE 5 Show that there are infinitely many positive primes. [HOTS]

EXAMPLE 6 Prove that every positive integer different from 1 can be expressed as a product of a non-negative power of 2 and an odd number. [HOTS]

EXAMPLE 7 Prove that a positive integer n is prime number, if no prime p less than or equal to √n divides n. [HOTS]

Practice Set 3

  1. Express each of the following integers as a product of its prime factors:
    (i) 420 (ii) 468 (iii) 945 (iv) 7325
  2. Determine the prime factorisation of each of the following positive integer:
    (i) 20570 (ii) 58500 (iii) 45470971
  3. Explain why 7 × 11 × 13 +13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.
  4. Check whether 6ncan end with the digit 0 for any natural number n. [NCERT]

ANSWERS
(i) 2^2 × 3 × 5 × 7 (ii) 2^2 × 3^2 × 13 (iii) 3^3 × 5 × 7 (iv) 5^2 × 293
(i) 2 × 5 × 〖11〗^2 × 17 (ii) 2^2 × 3^2 × 5^3× 13 (iii) 7^2 × 〖13〗^2× 〖17〗^2× 19
Since7 × 11 × 13 + 13 = (7 × 11 + 1) × 13
and, 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 = (7 × 6 × 4 × 3 × 2 × 1 + 1) × 5 4. No

HINTS TO SELECTED PROBLEM
We have, 6^n= 〖(2 × 3)〗^n= 2^n × 3^n. Therefore, prime factorisation of 6^ndoes not contain5 as a factor. Hence, 6^ncan never end with the digit 0 for any natural number.

ILLUSTRATIVE EXAMPLES
EXAMPLE 1 Find the HCF and LCM of 90 and 144 by the prime factorisation method.
Ans. HCF =18
LCM = 720

EXAMPLE 2 Find the HCF and LCM of 144,180 and 192 by prime factorisation method.
Ans. HCF =12
LCM = 2880

EXAMPLE 3 Find the HCF of 96 and 404 by prime factorization, method. Hence, find their LCM. [NCERT]
Ans. HCF = 4
LCM = 9696

EXAMPLE 4 Find the largest positive integer that will divide 398,436 and 542 leaving remainders 7,11 and 15 respectively.
Ans. 17

EXAMPLE 5 In a seminar, the number of participants in Hindi, English and Mathematics are 60,84 and 108 respectively. Find the minimum number of rooms required if in each room the same number of participants are to be seated and all of them being in the same subject.
Ans. 21

EXAMPLE 6 Three sets of English, Hindi and Mathematics books have to be stacked in such a way that all the books are stored topic wise and the height of each stack is the same. The number of English books is 96, the number of Hindi books is 240 and the number of Mathematics books is 336. Assuming that the books are of the same thickness, determine the number of stacks of English, Hindi and Mathematics books.
Ans. Number of stacks of English books = 2
Number of stacks of Hindi books = 5
Number of stacks of Mathematics books = 7

EXAMPLE 7 There is a circular path around a sports field. Priya takes 18 minutes to drive one round of the field, while Ravish takes 12 minutes for the same. Suppose they both start at the same point arid at the same time, and go in the same direction. After how many minutes will they meet again at the starting point? [NCERT]
Ans. Ravish and Priya will meet again at the starting point after 36 minutes.

EXAMPLE 8 In a school there are two sections – section A and section B of class X. There are 32 students in section A and 36 students in section B. Determine the minimum number of books required for their class library so that they can be distributed equally among students of section A or section B.
Ans. Required number of books is 288.

Practice Set 4

  1. Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = Product of the integers:
    (i) 26 and 91 (ii) 510 and 92 (iii) 336 and 54
  2. Find the LCM and HCF of the following integers by applying the prime factorisation method:
    (i) 12, 15 and 21 NCERT 17, 23 and 29 NCERT 8, 9 and 25 [NCERT]
    (iv) 40, 36 and 126 (v) 84, 90 and 120 (vi) 24, 15 and 36
  3. Find the greatest number of 6 digits exactly divisible by 24, 15 and 36.
  4. A rectangular courtyard is 18 m 72 cm long and 13 m 20 cm broad. It is to be paved with square tiles of the same size. Find the least possible number of such tiles.
  5. Find the least number that is divisible by all the numbers between 1 and 10 (both inclusive).
  6. What is the smallest number that, when divided by 35, 56 and 91 leaves remainders of 7 in each case?
  7. In a morning walk three persons step off together, their steps measure 80 cm, 85 cm and 90 cm respectively. What is the minimum distance each should walk so that he can cover the distance in complete steps?
  8. Determine the number nearest to 110000 but greater than 100000 which is exactly divisible by each of 8, 15 and 21.
  9. Find the smallest number which leaves remainders 8 and 12 when divided by 28 and 32 respectively.
  10. Find the smallest number which when increased by 17 is exactly divisible by both 520 and 468.
  11. A circular field has a circumference of 360 km. Three cyclists start together and can cycle 48, 60 and 72 km a day, round the field. When will they meet again?
  12. The LCM and HCF of two numbers are 180 and 6 respectively. If one of the numbers is 30, find the other number.
  13. The HCF of two numbers is 16 and their product is 3072. Find their LCM.
  14. The HCF of two numbers is 145 and their LCM is 2175. If one number is 725, find the other.\Can two numbers have 16 as their HCF and 380 as their LCM? Give reason.
  15. Given that HCF (306,657) = 9, find LCM (306,657). [NCERT]

ANSWERS

  1. (i) LCM = 182, HCF = 13 (ii) LCM = 23 460, HCF = 2
    (iii) LCM = 3024, HCF = 6
  2. LCM HCF 3. 999720
    (i) 420 3
    (ii) 1139 1
    (iii) 1800 1
    (iv) 2520 2
    (v) 2520 6
    (vi) 360 3
  3. 4290 5. 2520 6. 3647 7. 122 m 40 cm
  4. 109200 9. 204 10. 4663 11. 30 days
  5. 36 13. 192 14. 435 15. No 16. 22338

HINTS TO SELECTED PROBLEM

  1. Greatest number of 6 digits is 999999. Required number must be divisible by the LCM of 24, 15 and 36 i.e., by 360.
    Hence, required number = 999999 – Remainder when 999999 is divided by 360

ILLUSTRATIVE EXAMPLES
EXAMPLE 1 Prove that √2 is an irrational number.
[NCERT, CBSE 2010]
EXAMPLE 2 Prove that √3 is an irrational number. [NCERT, CBSE 2009,10]

EXAMPLE 3 Prove that 3√2 is irrational. [NCERT]

EXAMPLE 4 Prove that √5 is an irrational number. [NCERT, CBSE 2009, 10]

EXAMPLE 5 Prove that 5 – √5 is an irrational number. [NCERT]

EXAMPLE 6 Prove that 3 + 2√5 is irrational. [NCERT]

EXAMPLE 7 Prove that √2 + √5 is irrational.

EXAMPLE 8 Show that there is no positive integer n for which √(n – 1)+ √(n + 1) is rational. [HOTS]

EXAMPLE 9 Let a and b be positive integers. Show that √2 always lies between a/b and (a + 2b)/(a + b).
[HOTS]

EXAMPLE 10 Let a, b, c, d be positive rationals such that a + √b = c +√d, then either a = c and b = d or b and d are squares of rationals. [HOTS]

EXAMPLE 11 Let a, b, c, p be rational numbers such that p is not a perfect cube.
If a + bp^(1/3) + cp^(2/3) = 0, then prove that a = b = c = 0. [HOTS]

EXAMPLE 12 For any positive real number x, prove that there exists an irrational number y such that 0 < y < x.

Practice Set 5

  1. Prove that for any prime positive integer p, √p is an irrational number.
  2. Show that the following numbers are irrational.
    (i)1/√2 (ii) 7√5 (iii) 6 + √2 (iv) 3-√5
  3. Prove that √5 + √3 is irrational.
  4. If p, q are prime positive integers, prove that √p+ √q is an irrational number.
  5. Prove that √3 + √4 is an irrational number.
  6. Prove that following numbers are irrationals:
    (i)2/√7 (ii)3/(2√5) (iii) 4 + √2 (iv) 5√2
  7. Show that 2 – √3 is an irrational number. [CBSE 2008]
  8. Show that 3 + √2 is an irrational number. [CBSE 2009]
  9. Show that 5 -2√3 is an irrational number. [CBSE 2009]
  10. Prove that 2 – 3√5 is an irrational number. [CBSE 2010]
  11. Prove that 4 – 5√2 is an irrational number. [CBSE 2010]
  12. Prove that 2√3 – 1 is an irrational number. [CBSE 2010]

HINTS TO SELECTED PROBLEMS
Let us assume on the contrary that√pis rational. Then, there exist positive co-primes a and b such that
√p = a/b
=> p = a^2/b^2
=> b^2p = a^2
=> p|a^2 [∵ p|b^2p]
=> p|a
=> a = pc for some positive integer c.
Now, b^2p = a^2
=> b^2p = p^2 c^2 [∵ a = pc]
=> b^2= pc^2
=> p|b^2 [∵ p|pc^2]
=> p|b
∴ p|a and p|b
This contradicts that a and b are co-primes.
Hence, √pis irrational.
(i) If possible, let 1/√2 be rational. Then, there exist positive co-primes a and b such that
1/√2 = a/b
=> 2a^2 =b^2 [∵ 2|2a^2]
=> 2|b^2
=> 2|b
=> b = 2c for some positive integer c
∵ 2a^2 = b^2=> 2a^2 = 4c^2 =>a^2= 2c^2 => 2|a^2 [∵ 2|2c^2]
=> 2|a
This is a contradiction to the fact that a, b are co-primes.
Hence, 1/√2 is irrational.
(ii) Let 7√5 be rational. Then,
7√5 =a/b => √5 = a/7b => √5is rational, a contradiction.
∴ 7√5 is irrational.
(iii) Let 6 + √2 be a rational number equal to a/b, where a, b are positive co-primes. Then,
6 + √2 = a/b
=> √2 = a/b – 6
=> √2 = (a – 6b)/b
=> √2is rational.
This is a contradiction.
Hence, 6 + √2 is irrational
(iv) Let 3 – √5 be a rational equal to a/b Then,
3 – √5 = a/b
=>√5 = 3 – a/b
=>√5 = (3b – a)/b
=> √5 is rational.
This is a contradiction
Hence 3-√5 is irrational.
Let √5 +√3.is be rational equal to a/b. Then,
√5 +√3 = a/b
=> √5= a/b – √3
=> 〖(√5)〗^2= (a/b–√3)^2
=> 5 = a^2/b^2 –(2a√3)/b + 3
=> 2 = a^2/b^2 – 2√3 a/b
=> 2√3 a/b = (a^2– 2b^2)/b^2
=> √3 = (a^2– 2b^2)/2ab =>√3 is rational, a contradiction.
Hence √5 + √3 is irrational.

ILLUSTRATIVE EXAMPLES
EXAMPLE 1 Without actually performing the long division, state whether the following rational numbers will have terminating decimal expansion or a non-terminating repeating decimal expansion. Also, find the number of places of decimals after which the decimal expansion terminates.
(i) 17/8 NCERT64/455 NCERT 29/343 NCERT 15/1600 [NCERT]
(v) 13/3125 NCERT 23/(2^3 5^2 ) [NCERT]
Ans. (i) The decimal expansion of17/8 terminates after three places of decimals.
(ii) The decimal expansion of 64/455 is non-terminating repeating.
(iii) The decimal expansion of 29/343 is non-terminating repeating.
(iv) The decimal expansion of15/1600 terminates after 6 places of decimals.
(v) The decimal expansion of13/3125 terminates after 5 places of decimals.
(vi) The decimal expansion of 23/(2^3 5^2 ) terminates after 3 places of decimals.

EXAMPLE 2 What can yon say about the prime factorisations of the denominators of the following rationals:
(i) 34.12345 (ii) 34.¯5678

Practice Set 6

  1. Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion.
    (i) 23/8 (ii) 125/441 (iii) 35/50 NCERT 77/210 [NCERT]
    (v) 129/(2^2 × 5^7 × 7^17 )
  2. Write down the decimal expansions of the following rational numbers by writing their denominators in the form 2^m× 5^n, where m, n are non-negative integers.
    (i) 3/8 (ii) 13/125 (iii) 7/80 (iv) 14588/625 (v) 129/(2^2 × 5^7 ) [NCERT]
  3. What can you say about the prime factorisations of the denominators of the following rationale:
    (i) 43.123456789 (ii) 43.¯123456789
    (iii) 27.¯142857CBSE 2010 0.120120012000120000 …. [NCERT]

ANSWERS
(i) Terminating (ii) Non-terminating repeating (iii) Terminating
(iv) Non-terminating repeating (v) Non-terminating repeating.
(i) 0.375 (ii) 0.104 (iii) 0.0875 (iv) 23.3408 (v) 0.0004128
(i) Prime factorisation of the denominator is of the form 2^m×5^n, where m, n are non-negative integers.
(ii) Prime factorisation of the denominator contains factors other than 2 or 5.
(iii) Prime factorisation of the denominator contains factors other than 2 or 5.
(iv) Prime factorisation of the denominator contains factors other than 2 or 5.

Leave a Reply

Your email address will not be published. Required fields are marked *