CBSE Class 10 Mathematics Important Questions for Exam 2023 (Chapter - Real Numbers)

     These Mathematics Questions of Chapter {Real Number} are important for CBSE Examination 2023. So all the candidates Revise and Practices all the questions carefully.

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1. Write the HCF and LCM of the smallest odd composite number and the smallest odd prime number. If an odd number p divides q² , then will divide q³ also? Explain.
2. In a school the duration of a period in junior section is 40 minutes and senior section is 1 hour. If the first bell for each section rings at 9:00 A.M, when will the two bells ring together again.
3.  How many irrational numbers lie between √2 and √3? Write any two of them.
4.  Given that HCF (306, 1314) = 18. Find LCM (306, 1314).
5.  Find the HCF of 1656 and 4025 by Euclid’s division algorithm.
6. A rational number in its decimal expansion is 327.7081.W can you say about the prime factors of q, when this number is expressed in the form p/q? Give reason.
7.  Show the 5√6 is an irrational number.
8.  Write the denominator of the rational number 257/500 in the form 2^m X 5ⁿ, where m and n are non negative integer. Hence write in decimal expansion without actual division.
9. Show that numbers 8ⁿ can never end with the digit 0 for any natural number n.
10.  Show that (√3 + √5)² is an irrational number.
11. Use Euclid division lemma to show that the square of any positive integer cannot be of the form 5m+2, 5m +3 for some integer m.
12.  Find HCF of 90 and 126 by Euclid’s division algorithm. Also find their LCM and verify that LCM X HCF = product of numbers.
13. The decimal expansion of the rational number 43/2⁴ .5³, will terminate after how many places of decimal?
14.  Find the HCF by Euclid’s division algorithm of the numbers 92690, 7378 and 7161.
15.  The LCM of two numbers is 14 times their HCF. The sum of LCM and HCF is 600. If one number is 280, then find the other number.
16. Find the HCF of 180, 252 and 324 using Euclid’s division lemma.
17.  Find the greatest number that will divide 445, 572 and 699 leaving remainders 4, 5 and 6 respectively.
18.  Show that one and only one out of n, n+2 or n+4 is divisible by 3. Where n is any positive integer.
19.  If p is a prime number, then prove that √p is irrational.
20.  Prove that √5 is an irrational number and hence show that 2 - √5 is also a irrational number.
21.   Prove that √2 is an irrational number.
22.  Express the number 0.317831783178…….. in the form of rationala/b.ber a/b.
23.  Check whether 6ⁿ can end with the digit 0 for any natural number n.
24.  Explain (7 X 11 X 13) + 13 and (7 X 6 X 5 X 4 X 3 X 2 X 1) + 5 are composite number.
25.  The traffic light at three different road crossing changes after every 48 seconds, 72 seconds and 108 seconds respectively. If they change simultaneously at 8 a.m. at what time will they change together again?
26.  Prove that √3 + √5 is an irrational.
27.  Prove that 4 - 3√2 is an irrational.
28.  Show that n² – 1 is divisible by 8, if n is an odd positive integer.
29. Prove that n² – n is divisible by 2 for every positive integer n.
30.  Show that there is no positive integer n, for which √n-1 + √n+1 is rational.
31.  Show that the square of any positive integer is of the form 4m, 4m + 1where m is any integer.
32. Find HCF of 81 and 237 and express it as a linear combination of 81 and 237 i.e., HCF (81, 237) = 81x +237yor some x and y.
33.  Foe any positive integer n, prove that n³ – n is divisible by 6.
34.  Show that any positive odd integer is of the form 6q +1 or 6q + 3or 6q + 5, where q is some integer.
35.  If the HCF of 210 and 55 is expressible in the form 210 × 5 + 55y, find y.
36. Prove that √7 is an irrational number.
37. Using prime factorization, find the HCF and LCM of 72, 126 and 168. Also show that HCF x LCM = product of the three numbers.
38. The HCF of two numbers is 145 and their LCM is 2175. If one number is 725, find the other.
39. The product of two numbers is 20736 and their HCF is 54, find their LCM.
40. State the fundamental theorem of Arithmetic.
41. Check whether 5ⁿ can end with the digit 0 for any natural number n.
42. Express each of the following as the product of primes. (a) 6435   (b) 8085   (c) 2184
43. Show that square of an odd positive integer is of the form 8q + 1, for some positive integer q.
44. Show that every positive even integer is of the form 2q and every positive odd integer is of the form 2q+1, where q is some integer.
45. Prove that 5 + 2 √ 3 is an irrational number.
46. Show that 514 and 224 is a composite number.

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