NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers Ex 1.1

NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers Ex 1.1 are part of NCERT Solutions for Class 10 Maths. Here we have given NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers Ex 1.1

  • Real Numbers Class 10 Ex 1.1
  • Real Numbers Class 10 Ex 1.2
  • Real Numbers Class 10 Ex 1.3
  • Real Numbers Class 10 Ex 1.4
BoardCBSE
TextbookNCERT
ClassClass 10
SubjectMaths
ChapterChapter 1
Chapter NameReal Numbers
ExerciseEx 1.1
Number of Questions Solved5
CategoryNCERT Solutions

NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers Ex 1.1

Ex1.1 Class 10 Maths Question 1.
Use Euclid’s Division Algorithm to find the HCF of:
(i) 135 and 225
(ii) 196 and 38220
(iii) 867 and 255
Solution:
(i) By Euclid’s Division Algorithm, we have
225 = 135 x 1 + 90 135
= 90 x 1 + 45 90
= 45 x 2 + 0
∴ HCF (135, 225) = 45.

(ii) By Euclid’s Division Algorithm, we have
38220 = 196 x 195 + 0
196 = 196 x 1 + 0
∴  HCF (38220, 196) = 196.

(iii) By Euclid’s Division Algorithm, we have
867 = 255 x 3 + 102
255 = 102 x 2 + 51
102 = 51 x 2 + 0
∴ HCF (867, 255) = 51.

Ex1.1 Class 10 Maths Question 2.
Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is some integer.
Solution:
Let a be a positive odd integer. Also, let q be the quotient and r the remainder after dividing a by 6.
Then, a = 6q + r, where 0 ≤ r < 6.
Putting r = 0, 1, 2, 3, 4, and 5, we get:
a = 6q, a = 6q + 1, a = 6q + 2, a = 6q + 3, a = 6q + 4, a = 6q + 5
But a = 6q, a = 6q + 2 and a = 6q + 4 are even.
Hence, when a is odd, it is of the form 6q + 1, 6q + 3, and 6q + 5 for some integer q.
Hence proved.

Ex1.1 Class 10 Maths Question 3.
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?
Solution:
Let n be the number of columns such that the value of n be maximum and it must divide both the numbers 616 and 32.
Then, n = HCF (616, 32)
By Euclid’s Division Algorithm, we have:
616 = 32 x 19 + 8 32 = 8 x 4 + 0
∴ HCF (616, 32) = 8
i. e., n = 8
Hence, the maximum number of columns is 8.

Ex1.1 Class 10 Maths Question 4.
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.
Solution:
Let a be a positive integer, q be the quotient and r be the remainder.
Dividing a by 3 using the Euclid’s Division Lemma,
we have:
a = 3q + r, where 0 ≤ r < 3
Putting r = 0, 1 and 2, we get:
a = 3q
⇒ a2 = 9q2
= 3 x 3q2
= 3m (Assuming m = q2)
Then, a = 3q + 1
⇒  a2 = (3q + l)2 = 9q2 + 6q + 1
= 3(3q 2 + 2q) + 1
= 3m + 1 (Assuming m = 3q2 + 2q)
Next, a = 3q + 2
⇒ a2 = (3q + 2)2 =9q2 + 12q + 4
= 9q2 + 12q + 3 + 1
= 3(3q2 + 4q + 1) + 1
= 3m + 1.   (Assuming m = 3q2 + 4q+l)
Therefore, the square of any positive integer (say, a2) is always of the form 3m or 3m + 1.
Hence, proved.

Ex1.1 Class 10 Maths Question 5.
Use Euclid’s Division Lemma to show that the cube of any positive integer is either of the form 9m, 9m + 1 or 9m + 8.
Solution:
Let a be a positive integer, q be the quotient and r be the remainder.
Dividing a by 3 using the Euclid’s Division Algorithm, we have,
a = 3q + r, where 0 ≤ r < 3
Putting r = 0, 1 and 2, we get:
a = 3q, a = 3q + 1 and a = 3q + 2
If a = 3q, then a3 = 27q3 = 9(3q3) = 9m. (Assuming m = 3q3.)
If a = 3q + 1, then
a3 = (3q + l)3 = 27q3 + 9q(3q + 1) + 1 = 9(3q3 + 3q2 + q) + 1 = 9m + 1,  (Assuming m = 3q3 + 3q2 + q)
If a = 3q + 2, then a3 = (3q + 2)3
= 27q3 + 18q(3q + 2) + (2)3
= 9(3q3 + 6q2 + 4q) + 8
= 9m + 8, (Assuming m – 3q3 + 6q2 + 4q)
Hence, a3 is of the form 9m, 9m + 1 or 9m + 8.

We hope the NCERT Solutions for Class 10 Mathematics Chapter 1 Real Numbers Ex 1.1 help you. If you have any query regarding NCERT Solutions for Class 10 Mathematics Chapter 1 Real Numbers Ex 1.1, drop a comment below and we will get back to you at the earliest.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

%d bloggers like this:
search previous next tag category expand menu location phone mail time cart zoom edit close