Download REAL NUMBERS CLASS 10 TEST PAPER

Cynesure Institute
REAL NUMBERS
Class 10
1. Show that any positive odd integer is of the form 6q + 1, 6q + 3, or 6q + 5 where q is any
integer.
2. Using Euclid’s division lemma, show that the square of any positive integer is either of the
form 3m or 3m + 1, for some integer m.
3. Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m,
9m + 1 or 9m + 8.
4. Check whether there is any value of n for which 4n ends with the digit zero.
5. Prove that there is no number of the type 4k + 2 can be a perfect square.
6. Using Euclid’s division lemma, show that the square of any positive integer is of the form 5q,
5q+1, 5q+4 for some integer q.
7. Show that any one of the numbers (n + 2), n and (n + 4) is divisible by 3.
8.
If a = 4q + r then what are the values that ‘r’ can take?
9. Show that any positive odd integer is of the form of 4q + 1 or 4q + 3, where q is some
integer.
10. 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.
11. Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m,
9lm + 1 or 9m + 8.
12. What is the condition in a=bq+r which r must satisfy.
13. Show that one and only one out of n,n+2,n+4 is divisible by 3.
14. Show that the square of any odd integer is of the form 4q+1 for some integer q.
15. Prove that √5+√3 is irrational.
16. Make the factorization tree for 1024.
17. If the sum of two numbers is 1660 and HCF is 20, find the two numbers.
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18. If
𝑎𝑏 be an irrational number prove that 𝑎+ 𝑏 is irrational.
19. Every even integer a is of the form ……….
20. Every odd integer b is of the form
21. If n2 – 1 is divisible by 8, then n is ……………….
22. Prove that the square of any positive integer is either of the form 4q or 4q + 1 for some
integer q.
23. Prove that the cube of any positive integer is of the form 4m, 4m + 1 or 4m + 3 for some
integer m.
24. Show that the square of any positive integer can not be of the form 5q + 2 or 5q + 3.
25. Show that the square of any positive integer cannot be of the form 6m + 2 or 6m + 5.
26. Show that the square of any odd integer is of the form 4q + 1.
27. If n is an odd integer, then show that 𝑛2 – 1 is divisible by 8.
28. Prove that if a and b are odd positive integers, then 𝑎2 + 𝑏 2 is even, but not divisible by 4.
29. Prove that 𝑝 + 𝑞 is an irrational number, where p, q are primes.
30. Prove that only one out of n, n + 2 and n + 4 is divisible by 3.
31. Prove that only one out of any three consecutive positive integers is divisible by 3.
32. For any positive integer n, prove that n3– n is divisible by 6.
33. Show that only one out of n, n + 4, n + 8, n + 12 and n + 16 is divisible by 5.
34. Show that the square of an odd positive integer can be of the form 6q + 1 or 6q + 3.
35. What is difference between lemma and algorithm?
36. Prove that the square of any positive integer of the form 5g+1 is of the same form.
37. Prove that 𝑛 is not a rational number if n is not perfect square.
38. Prove that the difference and quotient of (3+ 2 3) and (3- 2 3) are irrational.
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1. Find LCM and HCF of 6 and 20 by the prime factorization method.
2. Find HCF of 96 and 404 by the prime factorization method. Also find their LCM.
3. Find HCF and LCM of 6, 72 and 120 using the prime factorization method.
4. Find the value of y if the HCF of 210 and 55 is expressible in the form 210 x 5 + 55y.
5. 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?
6. Express each number as a product of its prime factors: (i) 140
3825
(iv) 5005
(v) 7429
(ii) 156
(iii)
7. Find LCM and HCF of the following pairs of integers and verify that LCM × HCF = product
of the two numbers:
(i) 26 and 91
8.
(ii) 510 and 92
(iii) 336 and 54
Find LCM and HCF of the following integers by applying the prime factorization method.
(i) 12, 15 and 21
(ii) 17, 23 and 29
(iii) 8, 9 and 25
9. Given that HCF (306, 657) = 9, find LCM (306, 657).
10. Check whether 6n can end with the digit 0 for any natural number n.
11. Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.
12. There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of
the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point
and at the same time, and go in the same direction. After how many minutes will they meet
again at the starting point?
13. If 793800 = 23 x 3m x 5n x 72, find the value of m and n.
14. If HCF of 210 and 55 is expressible in the form 210 × 5 + 55y then find y.
15. Given that HCF (135, 225) = 45. Find LCM (135, 225).
16. Find HCF of the smallest composite number and the smallest prime number.
17. State Euclid’s Division Lemma and hence find HCF of 16 and 28.
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18. State fundamental theorem of Arithmetic and hence find the unique fraternization of 120.
19. Check whether 5 × 7 × 11 + 6 is a composite number.
20. Check whether 7 × 6 × 3 × 5 + 5 is a composite number.
21. Express 143 as product of its prime factors.
22. Find LCM and HCF of 120, 250 and 355 by the prime factorization method.
23. Find the LCM and HCF of 16 and 240 by the prime factorization method.
24. Find the LCM and HCF of 260 and 910 and verify that LCM × HCF = product of the two
numbers.
25. Use Euclid’s division algorithm to find the HCF of 135 and 225.
26. Use Euclid’s algorithm to find the HCF of 4052 and 12576.
27. In a school, there are two sections A and B of class 10. There are 32 students in section A
and 36 students in section B. Find the minimum number of books required for their class
library so that they can be equally distributed among students of section A or section B.
28. There is a circular path around a sports field. Mohan takes 18 minutes to drive one round of
the field, while Rahil takes 12 minutes for the same. Suppose they both start at the same
point and at the same time, and go in the same direction. After how many minutes will they
meet again at the starting point?
29. 49. A sweet seller has 420 kaju barfis and 130 badam barfis. 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 maximum number of barfis that can be placed in each stack for this purpose?
30. HCF of two consecutive integers is ……….
Ans-1
31. Write the greatest common factor of 2730 and 9350.
32. If HCF (45, 200)=5 , find LCM (4,200)
Ans-10
33. Mention the largest number which divides 265 and 1205 leaving remainder 5 in each case.
34. If the HCF of 65 and 117 is expressible in the form 65b +117a , find the value of a and b.
35. Find the largest number which divides 70 and 125 leaving remainders 5 and 8 respectively.
36. What is the product of a rational and an irrational number ?
37. Find the least number that is divisible by all the numbers from 1 to 10 (both inclusive).
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38. Prove that the product of two consecutive positive integers is divisible by 2.
39. Prove that the product of three consecutive positive integers is divisible by 6.
40. Can two numbers have 18 as their HCF and 200 as their LCM? Explain.
41. Using Euclid’s division algorithm, find HCF of 441, 567, 693.
42. Show that 12n cannot end with the digit 0 or 5, for any natural number n.
𝑝
43. If q is a rational number and q≠0, what is the condition of q so that the decimal representation
of p/q is terminating?
44. Insert any three rational numbers between √2 and √3.
Ans-1.5321
45. The decimal expansion of the rational number 23
33
×5
will terminate.
13
46. State whether 23 53 will have a terminating decimal expansion or a non-terminating
repeating decimal
47. What type of decimal expansion does 69/60 represent? Also after how many places will the
decimal expansion terminate?
13
48. State whether 3125 will have a terminating decimal expansion or a non-terminating
repeating decimal.
117
49. State whether 250 will have a terminating decimal expansion or a non-terminating repeating
decimal.
50. What is the smallest number by which 7 - 3 be multiplied to make it a rational no? Also
find the no. so obtained.
51. If the number p has never to end with the digit 0, then what does its factors much not
contain?
52. If HCF of two numbers is 1, then the two number are called?
53. Is 2.03003000300003…… rational or irrational?
54. What is the smallest composite number?
55. Use Euclid’s division algorithm to find the HCF of 4052 and 12576.
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56. Find the largest number which divides 245 and 1029 leaving remainder 5 in each case.
1. Solve 32 x 50 . What type of number is it - rational or irrational?
2. Find one rational and one irrational number between 3 and 5 .
1
3. Prove that 2 −
5
is irrational number.
4. Prove that √3 is irrational.
5. Show that 5 – 2√3 is irrational.
6. Prove that
7. Prove that
13
5 2
1
5
is irrational.
is irrational.
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