Page 1 X â€“ Maths 3 CHAPTER 1 REAL NUMBERS KEY POINTS 1. Euclidâ€™s division lemma : For given positive integers â€˜aâ€™ and â€˜bâ€™ there exist unique whole numbers â€˜qâ€™ and â€˜râ€™ satisfying the relation a = bq + r, 0 ? r < b. 2. Euclidâ€™s division algorithms : HCF of any two positive integers a and b. With a > b is obtained as follows: Step 1 : Apply Euclidâ€™s division lemma to a and b to find q and r such that a = bq + r , 0 ? r < b. Step 2 : If r = 0, HCF (a, b) = b if r ? 0, apply Euclidâ€™s lemma to b and r. 3. The Fundamental Theorem of Arithmetic : Every composite number can be expressed (factorized) as a product of primes and this factorization is unique, apart from the order in which the prime factors occur. 4. Let ? ? , 0 p x q q to be a rational number, such that the prime factorization of â€˜qâ€™ is of the form 2 m 5 n , where m, n are non-negative integers. Then x has a decimal expansion which is terminating. 5. Let ? ? , 0 p x q q be a rational number, such that the prime factorization of q is not of the form 2 m 5 n , where m, n are non-negative integers. Then x has a decimal expansion which is non-terminating repeating. 6. ?p is irrational, which p is a prime. A number is called irrational if it cannot be written in the form p q where p and q are integers and q ? 0. Page 2 X â€“ Maths 3 CHAPTER 1 REAL NUMBERS KEY POINTS 1. Euclidâ€™s division lemma : For given positive integers â€˜aâ€™ and â€˜bâ€™ there exist unique whole numbers â€˜qâ€™ and â€˜râ€™ satisfying the relation a = bq + r, 0 ? r < b. 2. Euclidâ€™s division algorithms : HCF of any two positive integers a and b. With a > b is obtained as follows: Step 1 : Apply Euclidâ€™s division lemma to a and b to find q and r such that a = bq + r , 0 ? r < b. Step 2 : If r = 0, HCF (a, b) = b if r ? 0, apply Euclidâ€™s lemma to b and r. 3. The Fundamental Theorem of Arithmetic : Every composite number can be expressed (factorized) as a product of primes and this factorization is unique, apart from the order in which the prime factors occur. 4. Let ? ? , 0 p x q q to be a rational number, such that the prime factorization of â€˜qâ€™ is of the form 2 m 5 n , where m, n are non-negative integers. Then x has a decimal expansion which is terminating. 5. Let ? ? , 0 p x q q be a rational number, such that the prime factorization of q is not of the form 2 m 5 n , where m, n are non-negative integers. Then x has a decimal expansion which is non-terminating repeating. 6. ?p is irrational, which p is a prime. A number is called irrational if it cannot be written in the form p q where p and q are integers and q ? 0. 4 X â€“ Maths MULTIPLE CHOICE QUESTIONS 1. 5 × 11 × 13 + 7 is a (a) prime number (b) composite number (c) odd number (d) none 2. Which of these numbers always ends with the digit 6. (a) 4 n (b) 2 n (c) 6 n (d) 8 n where n is a natural number. 3. For a, b (a ? b) positive rational numbers ? ? ? ? ? ? a b a b is a ____ (a) Rational number (b) irrational number (c) ? ? ? 2 a b (d) 0 4. If p is a positive rational number which is not a perfect square then ?3 p is (a) integer (b) rational number (c) irrational number (d) none of the above. 5. All decimal numbers areâ€“ (a) rational numbers (b) irrational numbers (c) real numbers (d) integers 6. In Euclid Division Lemma, when a = bq + r, where a, b are positive integers which one is correct. (a) 0 < r ? b (b) 0 ? r < b (c) 0 < r < b (d) 0 ? r ? b 7. Which of the following numbers is irrational number (a) 3.131131113... (b) 4.46363636... (c) 2.35 (d) b and c both Page 3 X â€“ Maths 3 CHAPTER 1 REAL NUMBERS KEY POINTS 1. Euclidâ€™s division lemma : For given positive integers â€˜aâ€™ and â€˜bâ€™ there exist unique whole numbers â€˜qâ€™ and â€˜râ€™ satisfying the relation a = bq + r, 0 ? r < b. 2. Euclidâ€™s division algorithms : HCF of any two positive integers a and b. With a > b is obtained as follows: Step 1 : Apply Euclidâ€™s division lemma to a and b to find q and r such that a = bq + r , 0 ? r < b. Step 2 : If r = 0, HCF (a, b) = b if r ? 0, apply Euclidâ€™s lemma to b and r. 3. The Fundamental Theorem of Arithmetic : Every composite number can be expressed (factorized) as a product of primes and this factorization is unique, apart from the order in which the prime factors occur. 4. Let ? ? , 0 p x q q to be a rational number, such that the prime factorization of â€˜qâ€™ is of the form 2 m 5 n , where m, n are non-negative integers. Then x has a decimal expansion which is terminating. 5. Let ? ? , 0 p x q q be a rational number, such that the prime factorization of q is not of the form 2 m 5 n , where m, n are non-negative integers. Then x has a decimal expansion which is non-terminating repeating. 6. ?p is irrational, which p is a prime. A number is called irrational if it cannot be written in the form p q where p and q are integers and q ? 0. 4 X â€“ Maths MULTIPLE CHOICE QUESTIONS 1. 5 × 11 × 13 + 7 is a (a) prime number (b) composite number (c) odd number (d) none 2. Which of these numbers always ends with the digit 6. (a) 4 n (b) 2 n (c) 6 n (d) 8 n where n is a natural number. 3. For a, b (a ? b) positive rational numbers ? ? ? ? ? ? a b a b is a ____ (a) Rational number (b) irrational number (c) ? ? ? 2 a b (d) 0 4. If p is a positive rational number which is not a perfect square then ?3 p is (a) integer (b) rational number (c) irrational number (d) none of the above. 5. All decimal numbers areâ€“ (a) rational numbers (b) irrational numbers (c) real numbers (d) integers 6. In Euclid Division Lemma, when a = bq + r, where a, b are positive integers which one is correct. (a) 0 < r ? b (b) 0 ? r < b (c) 0 < r < b (d) 0 ? r ? b 7. Which of the following numbers is irrational number (a) 3.131131113... (b) 4.46363636... (c) 2.35 (d) b and c both X â€“ Maths 5 8. The decimal expansion of the rational number 4 21 7 2 5 ? ? ? will terminate after ___ decimal places. (a) 3 (b) 4 (c) 5 (d) never 9. HCF is always (a) multiple of L.C.M. (b) Factor of L.C.M. (c) divisible by L.C.M. (d) a and c both 10. The product of two consecutive natural numbers is always. (a) an even number (b) an odd number (c) a prime number (d) none of these 11. Which of the following is an irrational number between 0 and 1 (a) 0.11011011... (b) 0.90990999... (c) 1.010110111... (d) 0.3030303... 12. p n = (a × 5) n . For p n to end with the digit zero a = __ for natural no. n (a) any natural number (b) even number (c) odd number (d) none. 13. A terminating decimal when expressed in fractional form always has denominator in the form of â€” (a) 2 m 3 n , m, n > 0 (b) 3 m 5 n , m, n > 0 (c) 5 n 7 m , m, n > 0 (d) 2 m 5 n , m, n > 0 SHORT ANSWER TYPE QUESTIONS 14. What will be the value of 0.3 0.4 ? ? 15. If unitâ€™s digit of 7 3 is 3 then what will be the unitâ€™s digit of 7 11 . 16. Given that HCF (135, 225) = 45. Find LCM (135, 225). Page 4 X â€“ Maths 3 CHAPTER 1 REAL NUMBERS KEY POINTS 1. Euclidâ€™s division lemma : For given positive integers â€˜aâ€™ and â€˜bâ€™ there exist unique whole numbers â€˜qâ€™ and â€˜râ€™ satisfying the relation a = bq + r, 0 ? r < b. 2. Euclidâ€™s division algorithms : HCF of any two positive integers a and b. With a > b is obtained as follows: Step 1 : Apply Euclidâ€™s division lemma to a and b to find q and r such that a = bq + r , 0 ? r < b. Step 2 : If r = 0, HCF (a, b) = b if r ? 0, apply Euclidâ€™s lemma to b and r. 3. The Fundamental Theorem of Arithmetic : Every composite number can be expressed (factorized) as a product of primes and this factorization is unique, apart from the order in which the prime factors occur. 4. Let ? ? , 0 p x q q to be a rational number, such that the prime factorization of â€˜qâ€™ is of the form 2 m 5 n , where m, n are non-negative integers. Then x has a decimal expansion which is terminating. 5. Let ? ? , 0 p x q q be a rational number, such that the prime factorization of q is not of the form 2 m 5 n , where m, n are non-negative integers. Then x has a decimal expansion which is non-terminating repeating. 6. ?p is irrational, which p is a prime. A number is called irrational if it cannot be written in the form p q where p and q are integers and q ? 0. 4 X â€“ Maths MULTIPLE CHOICE QUESTIONS 1. 5 × 11 × 13 + 7 is a (a) prime number (b) composite number (c) odd number (d) none 2. Which of these numbers always ends with the digit 6. (a) 4 n (b) 2 n (c) 6 n (d) 8 n where n is a natural number. 3. For a, b (a ? b) positive rational numbers ? ? ? ? ? ? a b a b is a ____ (a) Rational number (b) irrational number (c) ? ? ? 2 a b (d) 0 4. If p is a positive rational number which is not a perfect square then ?3 p is (a) integer (b) rational number (c) irrational number (d) none of the above. 5. All decimal numbers areâ€“ (a) rational numbers (b) irrational numbers (c) real numbers (d) integers 6. In Euclid Division Lemma, when a = bq + r, where a, b are positive integers which one is correct. (a) 0 < r ? b (b) 0 ? r < b (c) 0 < r < b (d) 0 ? r ? b 7. Which of the following numbers is irrational number (a) 3.131131113... (b) 4.46363636... (c) 2.35 (d) b and c both X â€“ Maths 5 8. The decimal expansion of the rational number 4 21 7 2 5 ? ? ? will terminate after ___ decimal places. (a) 3 (b) 4 (c) 5 (d) never 9. HCF is always (a) multiple of L.C.M. (b) Factor of L.C.M. (c) divisible by L.C.M. (d) a and c both 10. The product of two consecutive natural numbers is always. (a) an even number (b) an odd number (c) a prime number (d) none of these 11. Which of the following is an irrational number between 0 and 1 (a) 0.11011011... (b) 0.90990999... (c) 1.010110111... (d) 0.3030303... 12. p n = (a × 5) n . For p n to end with the digit zero a = __ for natural no. n (a) any natural number (b) even number (c) odd number (d) none. 13. A terminating decimal when expressed in fractional form always has denominator in the form of â€” (a) 2 m 3 n , m, n > 0 (b) 3 m 5 n , m, n > 0 (c) 5 n 7 m , m, n > 0 (d) 2 m 5 n , m, n > 0 SHORT ANSWER TYPE QUESTIONS 14. What will be the value of 0.3 0.4 ? ? 15. If unitâ€™s digit of 7 3 is 3 then what will be the unitâ€™s digit of 7 11 . 16. Given that HCF (135, 225) = 45. Find LCM (135, 225). 6 X â€“ Maths 17. Solve ? 18 50. What type of number is it, rational or irrational. 18. Find the H.C.F. of the smallest composite number and the smallest prime number. 19. If a = 4q + r then what are the conditions for a and q. What are the values that r can take? 20. What is the smallest number by which ? 5 3 be multiplied to make it a rational no? Also find the no. so obtained. 21. What is the digit at unitâ€™s place of 9 n ? 22. Find one rational and one irrational no. between 3 and 5. 23. State Euclidâ€™s Division Lemma and hence find HCF of 16 and 28. 24. State fundamental theorem of Arithmetic and hence find the unique factorization of 120. 25. Prove that ? 1 2 5 is irrational number. 26. Prove that 2 5 3 7 ? is irrational number. 27. Prove that ? 2 7 is not rational number. 28. Find HCF and LCM of 56 and 112 by prime factorisation method. 29. Why 17 + 11 × 13 × 17 × 19 is a composite number? Explain. 30. Check whether 5 × 6 × 2 × 3 + 3 is a composite number. 31. Check whether 14 n can end with the digit zero for any natural number, n. 32. If the HCF of 210 and 55 is expressible in the form 210 × 5 + 55y then find y. LONG ANSWER TYPE QUESTIONS 33. Find HCF of 56, 96 and 324 by Euclidâ€™s algorithm. Page 5 X â€“ Maths 3 CHAPTER 1 REAL NUMBERS KEY POINTS 1. Euclidâ€™s division lemma : For given positive integers â€˜aâ€™ and â€˜bâ€™ there exist unique whole numbers â€˜qâ€™ and â€˜râ€™ satisfying the relation a = bq + r, 0 ? r < b. 2. Euclidâ€™s division algorithms : HCF of any two positive integers a and b. With a > b is obtained as follows: Step 1 : Apply Euclidâ€™s division lemma to a and b to find q and r such that a = bq + r , 0 ? r < b. Step 2 : If r = 0, HCF (a, b) = b if r ? 0, apply Euclidâ€™s lemma to b and r. 3. The Fundamental Theorem of Arithmetic : Every composite number can be expressed (factorized) as a product of primes and this factorization is unique, apart from the order in which the prime factors occur. 4. Let ? ? , 0 p x q q to be a rational number, such that the prime factorization of â€˜qâ€™ is of the form 2 m 5 n , where m, n are non-negative integers. Then x has a decimal expansion which is terminating. 5. Let ? ? , 0 p x q q be a rational number, such that the prime factorization of q is not of the form 2 m 5 n , where m, n are non-negative integers. Then x has a decimal expansion which is non-terminating repeating. 6. ?p is irrational, which p is a prime. A number is called irrational if it cannot be written in the form p q where p and q are integers and q ? 0. 4 X â€“ Maths MULTIPLE CHOICE QUESTIONS 1. 5 × 11 × 13 + 7 is a (a) prime number (b) composite number (c) odd number (d) none 2. Which of these numbers always ends with the digit 6. (a) 4 n (b) 2 n (c) 6 n (d) 8 n where n is a natural number. 3. For a, b (a ? b) positive rational numbers ? ? ? ? ? ? a b a b is a ____ (a) Rational number (b) irrational number (c) ? ? ? 2 a b (d) 0 4. If p is a positive rational number which is not a perfect square then ?3 p is (a) integer (b) rational number (c) irrational number (d) none of the above. 5. All decimal numbers areâ€“ (a) rational numbers (b) irrational numbers (c) real numbers (d) integers 6. In Euclid Division Lemma, when a = bq + r, where a, b are positive integers which one is correct. (a) 0 < r ? b (b) 0 ? r < b (c) 0 < r < b (d) 0 ? r ? b 7. Which of the following numbers is irrational number (a) 3.131131113... (b) 4.46363636... (c) 2.35 (d) b and c both X â€“ Maths 5 8. The decimal expansion of the rational number 4 21 7 2 5 ? ? ? will terminate after ___ decimal places. (a) 3 (b) 4 (c) 5 (d) never 9. HCF is always (a) multiple of L.C.M. (b) Factor of L.C.M. (c) divisible by L.C.M. (d) a and c both 10. The product of two consecutive natural numbers is always. (a) an even number (b) an odd number (c) a prime number (d) none of these 11. Which of the following is an irrational number between 0 and 1 (a) 0.11011011... (b) 0.90990999... (c) 1.010110111... (d) 0.3030303... 12. p n = (a × 5) n . For p n to end with the digit zero a = __ for natural no. n (a) any natural number (b) even number (c) odd number (d) none. 13. A terminating decimal when expressed in fractional form always has denominator in the form of â€” (a) 2 m 3 n , m, n > 0 (b) 3 m 5 n , m, n > 0 (c) 5 n 7 m , m, n > 0 (d) 2 m 5 n , m, n > 0 SHORT ANSWER TYPE QUESTIONS 14. What will be the value of 0.3 0.4 ? ? 15. If unitâ€™s digit of 7 3 is 3 then what will be the unitâ€™s digit of 7 11 . 16. Given that HCF (135, 225) = 45. Find LCM (135, 225). 6 X â€“ Maths 17. Solve ? 18 50. What type of number is it, rational or irrational. 18. Find the H.C.F. of the smallest composite number and the smallest prime number. 19. If a = 4q + r then what are the conditions for a and q. What are the values that r can take? 20. What is the smallest number by which ? 5 3 be multiplied to make it a rational no? Also find the no. so obtained. 21. What is the digit at unitâ€™s place of 9 n ? 22. Find one rational and one irrational no. between 3 and 5. 23. State Euclidâ€™s Division Lemma and hence find HCF of 16 and 28. 24. State fundamental theorem of Arithmetic and hence find the unique factorization of 120. 25. Prove that ? 1 2 5 is irrational number. 26. Prove that 2 5 3 7 ? is irrational number. 27. Prove that ? 2 7 is not rational number. 28. Find HCF and LCM of 56 and 112 by prime factorisation method. 29. Why 17 + 11 × 13 × 17 × 19 is a composite number? Explain. 30. Check whether 5 × 6 × 2 × 3 + 3 is a composite number. 31. Check whether 14 n can end with the digit zero for any natural number, n. 32. If the HCF of 210 and 55 is expressible in the form 210 × 5 + 55y then find y. LONG ANSWER TYPE QUESTIONS 33. Find HCF of 56, 96 and 324 by Euclidâ€™s algorithm. X â€“ Maths 7 34. Show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m. 35. Show that any positive odd integer is of the form 6q + 1, 6q + 5 where q is some integer. 36. Prove that the square of any positive integer is of the form 5q, 5q + 1, 5q + 4 for some integer, q. 37. Prove that the product of three consecutive positive integers is divisible by 6. 38. Show that one and only one of n, n + 2, n + 4 is divisible by 3. 39. Two milk containers contains 398 l and 436 l of milk. The milk is to be transferred to another container with the help of a drum. While transferring to another container 7 l and 11 l of milk is left in both the containers respectively. What will be the maximum capacity of the drum. ANSWERS 1. b 2. c 3. a 4. c 5. c 6. b 7. a 8. b 9. b 10. b 11. b 12. b 13. d 14. 7 9 15. 3 16. 675 17. 30, rational 18. 2 19. Opposite integer r, q whole no. 0 ? r < 4 20. ? ? ? 5 3 , 2 21. even power = 1 odd power = 9 23. 4 24. 2 × 2 × 2 × 3 × 5Read More

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