NCERT Class 10 Chapter 1 Real Numbers Official CBSE Board Sample Problems Long Answer
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Question
Show that reciprocal of is an irrational number
Solution
We have to prove that is an irrational number.
Let us assume that is rational.
There exists coprime integers a and such that
Question
Apply Euclid’s division algorithm to find HCF of numbers 4052 and 420.
Solution
Given number, arc 4052 and 420. 420)4.1)52 (9
On applying Euclid’s division algorithm, we have
So, HCF of 4052 and .

Euclid's Division Algorithm
Question
Three bells toll at intervals of 12 minutes, 15 minutes and 18 minutes respectively. If they start tolling together, after what time will they next toll together?
Solution
LCM of
So, next time the bells will ring together after 180 minutes.

Ring Together
Question
By using, Euclid’s algorithm, find the largest number which divides 650 and 1170.
Solution
Given numbers are 650 and 1170.
On applying Euclid’s division algorithm,
We get
At the last stage, the divisor is 130.
The HCF of 650 and 1170 is 130.

Euclid’s Division Algorithm
Question
Show that any positive odd integer is of the form, where m is some integer.
Solution
Let ‘a’ be any positive integer and, then by Euclid’s division algorithm, we have
Where and
Now ‘a’ may be of the form of
When
Where
= even number
When
= odd number
When
= even number
When
= odd number
Clearly, it is seen that any positive odd integer is of the form or for some integer m.
Question
If HCF of 144 and 180 is expressed in the form, find the value of m.
Solution
On applying Euclid’s division algorithm,
At the last stage, the divisor is 36.
HCF0f l44and 180 is 36.
So,

Algorithm
Question
Show that ¡s an irrational number.
Solution
We have to prove that is an irrational number. I.e. to prove is an irrational number.
Let us suppose is a rational number.
There exists coprime integers (say) a and b, b O such that
being rational number so. 113 become rational, which is contradiction with the fact that /15 is irrational. We led to contradiction due to wrong supposition.
Hence, is irrational.
Question
Find whether decimal expansion of is a terminating or non-terminating decimal. If it terminates, find the number of decimal places its decimal expansion has.
Solution
The given rational number is
Now
The denominator of the given rational number is of the form
The decimal expansion of is of the form of terminating.
The decimal expansion of terminates after 6 places of decimal

Terminates After 6 Places of Decimal
Question
Show that any positive even integer is expressible in the form of where q is a positive integer.
Solution
Let a be a positive evn integer and let
By Euclid’s Lemma a where
So or or
Now
are even as they are divisible by 2. The others are not.
Question
Prove that is an irrational number.
Solution
Let if possible, be a rational number.
RHS is rational as p and q are integers but LHS is irrational, So An irrational = a rational no. This is absurd
So our assumption is wrong. Hence 3 + 4/is irrational