CBSE Class 10 Maths Chapter-1 Real Numbers – Free PDF Download
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CBSE Class 10 Maths Chapter-1 Real Numbers Important Questions
CBSE Class 10 Maths Important Questions Chapter 1 – Real Numbers
3 Mark Questions
1. Use Euclid’s division algorithm to find the HCF of:
(i) 135 and 225
(ii) 196 and 38220
(iii) 867 and 255
Ans. (i) 135 and 225
We have 225 > 135,
So, we apply the division lemma to 225 and 135 to obtain
225 = 135 × 1 + 90
Here remainder 90 ≠ 0, we apply the division lemma again to 135 and 90 to obtain
135 = 90 × 1 + 45
We consider the new divisor 90 and new remainder 45≠ 0, and apply the division lemma to obtain
90 = 2 × 45 + 0
Since that time the remainder is zero, the process get stops.
The divisor at this stage is 45
Therefore, the HCF of 135 and 225 is 45.
(ii) 196 and 38220
We have 38220 > 196,
So, we apply the division lemma to 38220 and 196 to obtain
38220 = 196 × 195 + 0
Since we get the remainder is zero, the process stops.
The divisor at this stage is 196,
Therefore, HCF of 196 and 38220 is 196.
(iii) 867 and 255
We have 867 > 255,
So, we apply the division lemma to 867 and 255 to obtain
867 = 255 × 3 + 102
Here remainder 102 ≠ 0, we apply the division lemma again to 255 and 102 to obtain
255 = 102 × 2 + 51
Here remainder 51 ≠ 0, we apply the division lemma again to 102 and 51 to obtain
102 = 51 × 2 + 0
Since we get the remainder is zero, the process stops.
The divisor at this stage is 51,
Therefore, HCF of 867 and 255 is 51.
2. Find the greatest number of 6 digits exactly divisible by 24, 15 and 36.
Ans. The greater number of 6 digits is 999999.
LCM of 24, 15, and 36 is 360.
3. Prove that the square of any positive integer is of the form 4q or 4q + 1 for some integer q.
Ans. Let a = 4q + r, when r = 0, 1, 2 and 3
Numbers are 4q, 4q + 1, 4q + 2 and 4q + 3
4. 144 cartons of coke can and 90 cartons of Pepsi can 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 greater number of cartons each stack would have?
Ans. We find the HCF of 144 and 90
5. Prove that product of three consecutive positive integers is divisible by 6.
Ans. Let three consecutive numbers be x, (x + 1) and (x + 2)
Let x = 6q + r; 0
6. Prove that (3 –
) is an irrational number.
Ans. 
7. Prove that if x and y are odd positive integers, then x2 + y2 is even but not divisible by 4.
Ans. Let x = 2p + 1 and y = 2q + 1
8. Show that one and only one out of n, (n + 2) or (n + 4) is divisible by 3, where n
N.
Ans. Let the number be (3q + r)
9. Use Euclid’s Division Lemma to show that the square of any positive integer of the form 3m or (3m + 1) for some integer q.
Ans. 
10. Prove that in
is not a rational number, if n is not perfect square.
Ans. 
11. Prove that the difference and quotient of 
and 
are irrational.
Ans. 
12. Show that (n2 – 1) is divisible by 8, if n is an odd positive integer.
Ans. Let n = 4q + 1(an odd integer)
13. Use Euclid Division Lemma to show that cube of any positive integer is either of the form 9m, (9m + 1) or (9m + 8).
Ans. Let a = 3q + r
