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

Euclid’S Division Algorithm

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

Ring Together

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

Euclid’S Division Algorithm

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

Algorithm

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

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

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