1.
Express 140 as a product of its prime
factors

2.
Find the LCM and HCF of 12, 15 and 21 by
the prime factorization method.

3.
Find the LCM and HCF of 6 and 20 by the
prime factorization method.

4.
State whether13/3125 will have a
terminating decimal expansion or a nonterminating repeating decimal.

5.
State whether 17/8 will have a terminating
decimal expansion or a nonterminating repeating decimal.

6.
Find the LCM and HCF of 26 and 91 and
verify that LCM × HCF = product of the two numbers.

7.
Use Euclid’s division algorithm to find the
HCF of 135 and 225

8.
Use Euclid’s division lemma to show that
the square of any positive integer is either of the form 3m or 3m + 1 for
some integer m

9.
Prove that √3 is irrational.

10. Show
that 5 – √3 is irrational

11. Show
that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5,
where q is some integer.

12. 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?

13. Express
156 as a product of its prime factors.

14. Find
the LCM and HCF of 17, 23 and 29 by the prime factorization method.

15. Find
the HCF and LCM of 12, 36 and 160, using the prime factorization method.

16. State
whether 6/15 will have a terminating decimal expansion or a nonterminating
repeating decimal.

17. State
whether35/50 will have a terminating decimal expansion or a nonterminating
repeatingdecimal.

18. Find
the LCM and HCF of 192 and 8 and verify that LCM × HCF = product of the two
numbers.

19. Use
Euclid’s algorithm to find the HCF of 4052 and 12576.

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

21. Use
Euclid’s division lemma to show that the square of any positive integer is
either of the form 3m or 3m + 1 for some integer m.

22. Prove
that 3√2 5 is irrational.

23. Prove
that 1/√2 is irrational. (3 marks)

24. In
a school there are tow 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.

25. Express
3825 as a product of its prime factors.

26. Find
the LCM and HCF of 8, 9 and 25 by the prime factorization method.

27. Find
the HCF and LCM of 6, 72 and 120, using the prime factorization method.

28. State
whether 29/343 will have a terminating decimal expansion or a nonterminating
repeating decimal.

29. State
whether 23/ 23 52 will have a terminating decimal expansion or a
nonterminating repeating decimal

30. Find the LCM and HCF of 336 and 54 and
verify that LCM × HCF = product of the two numbers

31. Use
Euclid’s division algorithm to find the HCF of 867 and 255

32. 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.

33. Use
Euclid’s division lemma to show that the cube of any positive integer is of
the form 9m, 9lm + 1 or 9m + 8.

34. Prove
that 7 √5 is irrational.

35. Prove
that √5 is irrational.

36. 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?

37. Express
5005 as a product of its prime factors.

38. Find
the LCM and HCF of 24, 36 and 72 by the prime factorization method.

39. Find
the LCM and HCF of 96 and 404 by the prime factorization method

40. State
whether 64/455 will have a terminating decimal expansion or a nonterminating
repeating decimal

41. State
whether15/ 1600 will have a terminating decimal expansion or a
nonterminating repeating decimal.

42. Find
the LCM and HCF of 510 and 92 and verify that LCM × HCF = product of the two
numbers.

43. Use
Euclid’s division algorithm to find the HCF of 196 and 38220 (3 marks)

44. Use
Euclid’s division lemma to show that the cube of any positive integer is of
the form 9m,9m + 1 or 9m + 8

45. Show
that every positive odd integer is of the form 2q, and that every positive
odd integer is of the form 2q + 1, where q is some integer

46. Show
that 3√ 2 is irrational.

47. Prove
that 3 + 2 √5 is irrational.

48. 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?

49. Use
Euclid’s division algorithm to find the HCF of : (i) 135 and 225 (ii) 196 and
38220 (iii) 867 and 255

50. Show
that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5,
where q is some integer.

51. 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? Sol. Hints: Find
the HCF of 616 and 32

52. Use
Euclid’s division lemma to show that the square of any positive integer is
either of the form 3m or 3m + 1 for some integer m. [Hint : Let x be any
positive integer then it is of the form 3q, 3q + 1 or 3q + 2. Now square each
of these and show that they can be rewritten in the form 3m or 3m + 1.]

53. Use
Euclid’s division lemma to show that the cube of any positive integer is of
the form 9m, 9m + 1 or 9m + 8.

54. Consider
the numbers 4n, where n is a natural number. Check whether there is any value
of n for which 4n ends with the digit zero.

55. Find
the LCM and HCF of 6 and 20 by the prime factorization method.

56. Find
the HCF of 96 and 404 by the prime factorization method. Hence, find their
LCM.

57. Find
the HCF and LCM of 6, 72 and 120, using the prime factorization method.

58. Find
the value of y if the HCF of 210 and 55 is expressible in the form 210 x 5 +
55y

59. Prove
that no number of the type 4K + 2 can be a perfect square.

60. Express
each number as a product of its prime factors:(i) 140 (ii) 156 (iii) 3825
(iv) 5005 (v) 7429

61. Find
the LCM and HCF of the following pairs of integers and verify that LCM × HCF
= product of the two numbers. (i) 26 and 91 (ii) 510 and 92 (iii) 336
and 54

62. Find
the LCM and HCF of the following integers by applying the prime factorization
method. 12, 15 and 21 (ii) 17, 23 and 29 (iii) 8, 9 and 25

63. Given
that HCF (306, 657) = 9, find LCM (306, 657).

64. Check
whether 6n can end with the digit 0 for any natural number n.

65. Explain
why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.

66. 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?

67. Use
Euclid’s division lemma to show that the square of any positive integer is of
the form 5q, 5q+1,5q+4 for some integer q.

68. Show
that any one of the numbers (n + 2), n and (n + 4) is divisible by 3.

69. If
793800 = 2 3 x 3 m x 5 n x 7 2, find the value of m and n.

70. If
the HCF of 210 and 55 is expressible in the form 210 × 5 + 55y then find y.

71. Given
that HCF (135, 225) = 45. Find LCM (135, 225).

72. Solve
√18x √50. What type of number is it, rational or irrational?

73. What
type of decimal expansion will 69/60 represent? After how many places will
the decimal expansion terminate?

74. Find
the H.C.F. of the smallest composite number and the smallest prime number.

75. If
a = 4q + r then what are the conditions for a and q. What are the values that
r can take?

76. What
is the smallest number by which √5  √ 3 be multiplied to make it a rational
no? Also find the no. so obtained.

77. What
is the digit at unit’s place of 96?

78. Find
one rational and one irrational no. between √3 and √5.

79. If
the no. p never to end with the digit 0 then what are the possible value (s)
of p?

80. State
Euclid’s Division Lemma and hence find HCF of 16 and 28.

81. State
fundamental theorem of Arithmetic and hence find the unique fraternization of
120.

82. Prove
that 1/(2  √5) is irrational number.

83. Check
whether 5 × 7 × 11 + 6 is a composite number.

84. Check
whether 7 × 6 × 3 × 5 + 5 is a composite number.

85. Find
all positive integral values of n for which n^{2}+96 is perfect
square

Tuesday, April 2, 2013
CBSE NCERT 10th Real Numbers test paper
Labels:
10th Maths Term 01
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Q. Prove that no number of the type 4k+2 can be a perfect square.
Ans: If p is a prime factor of a perfect square, p2 must also be a factor of that perfect square. 4k+2 = 2(2k+1)
2 is a factor of 4k+2 but 2k+1 is odd and cannot have factor 2, so 4k+2 is not divisible by 4, and therefore cannot be a perfect square.
Q. 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?
Ans: We need to calculate the LCM to find the answer. 18 = 2x3x3 ;12= 2x2x3 ; LCM = 36
Q. show that only one out of n, n+2 or n+4 is divisible by 3 where n is positive integer.
Solution: When n = 1, exactly one of 1, 1+2, and 1+4 is divisible by 3,
namely 1+2, since 3 is divisible by 3, and the other two 1 and 5 are not.
Suppose for n < k, only one out of n, n+2, n+4 is divisible by 3
For n = k, we consider k, k+2, k+4.
By the induction hypothesis, only one of k1, k+1 and k+3 is
divisible by 3. We look at the three possible cases.
Case 1: k1 is the one which is divisible by 3. Then k1 = 3m, for some positive integer m.
Then add 1 to both sides of k1 = 3m:
k1+1=3m+1
k = 3m+1 then k is NOT divisible by 3
Now add 3 to both sides of k1 = 3m:
k1+3=3m+3
k+2 = 3m+3
k+2 = 3(m+1)
then k+2 IS divisible by 3
Now add 5 to both sides of k1 = 3m:
k1+5=3m+5
k+4 = 3m+5
k+4 = 3m+3 + 2 = 3(m+1)+2
then k+4 is NOT divisible by 3.
So, we have proved case 1 for n = k
Case 2: k+1 is the one which is divisible by 3. Then k+1 = 3m, for some positive integer m.
Then add 1 to both sides of k+1 = 3m:
k+11=3m1
k = 3m1
then k is NOT divisible by 3
Now add 1 to both sides of k+1 = 3m:
k+1+1=3m+1
k+2 = 3m+1
then k+2 is NOT divisible by 3
Now add 3 to both sides of k+1 = 3m:
k+1+3=3m+3
k+4 = 3m+3
k+4 = 3(m+1)
so k+4 IS divisible by 3.
So, we have proved case 2.
Case 3: k+3 is the one which is divisible by 3. Then k+3 = 3m, for some positive integer m.
Then add 3 to both sides of k+3 = 3m:
k+33=3m3
k = 3(m1) then k IS divisible by 3
Now add 1 to both sides of k+3 = 3m:
k+31=3m1
k+2 = 3m1 then k+2 is NOT divisible by 3
Now add 1 to both sides of k+3 = 3m:
k+3+1=3m+1
k+4 = 3m+1 then k+4 is NOT divisible by 3.
So, we have proved case 3.
If the HCF of 210 and 55 is expressible in the form 210 × 5 + 55y then find y
Let us first find the HCF of 210 and 55.
Applying Euclid division lemna on 210 and 55, we get
210 = 55 × 3 + 45 ....(1)
Since the remainder 45 ≠ 0. So, again applying the Euclid division lemna on 55 and 45, we get
55 = 45 × 1 + 10 .... (2)
Again, considering the divisor 45 and remainder 10 and applying division lemna, we get
45 = 4 × 10 + 5 .... (3)
We now, consider the divisor 10 and remainder 5 and applying division lemna to get
10 = 5 × 2 + 0 .... (4)
We observe that the remainder at this stage is zero. So, the last divisor i.e., 5 is the HCF of 210 and 55.
∴ 5 = 210 × 5 + 55y
⇒ 55y = 5  1050 = 1045
⇒ y = 19
Q. finds the H.C.F. of 65 and 117 and express it in the form of 65m+117n.
Answer: Among 65 and 117; 2117 > 65
Since 117 > 65, we apply the division lemma to 117 and 65 to obtain
117 = 65 × 1 + 52 … Step 1
Since remainder 52 ≠ 0, we apply the division lemma to 65 and 52 to obtain
65 = 52 × 1 + 13 … Step 2
Since remainder 13 ≠ 0, we apply the division lemma to 52 and 13 to obtain
52 = 4 × 13 + 0 … Step 3
In this step the remainder is zero. Thus, the divisor i.e. 13 in this step is the H.C.F. of the given numbers
The H.C.F. of 65 and 117 is 13
From Step 2:
13 = 65 – 52× 1 … Step 4
From Step 1:
52 = 117 – 65 × 1
Thus, from Step 4, it is obtained
13 = 65 – (117 – 65 × 1)
⇒13 = 65 × 2 – 117
⇒13 = 65 × 2 + 117 × (–1)
In the above relationship the H.C.F. of 65 and 117 is of the form 65m + 117 n, where m = 2 and n = –1
Find all positive integral values of n for which n2+96 is perfect square.
Answer: Let n2 + 96 = x2
⇒ x2 – n2 = 96
⇒ (x – n) (x + n) = 96
⇒ both x and n must be odd or both even
on these condition the cases are
x – n = 2, x + n = 48
x – n = 4, x + n = 24
x – n = 6, x + n = 16
x – n = 8, x + n = 12
and the solution of these equations can be given as
x = 25, n = 23
x = 14, n = 10
x = 11, n = 5
x = 10, n = 2
So, the required values of n are 23, 10, 5, and 2.
Q. Prove that one of every three consecutive integers is divisible by 3.
Ans: n,n+1,n+2 be three consecutive positive integers We know that n is of the form 3q, 3q +1, 3q + 2
So we have the following cases
Case – I when n = 3q
In the this case, n is divisible by 3 but n + 1 and n + 2 are not divisible by 3
Case  II When n = 3q + 1
put n = 2 = 3q +1 +2 = 3(q +1) is divisible by 3. but n and n+1 are not divisible by 3
Case – III When n = 3q +2
put n = 2 = 3q +1 +2 = 3(q +1) is divisible by 3.
But n and n+1 are not divisible by 3
Hence one of n, n + 1 and n + 2 is divisible by 3
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