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Irrational Numbers

Show that the...

Question

Show that the following numbers are irrational.
(i) $\frac{1}{\sqrt{2}}$
(ii) $7 \sqrt{5}$
(iii) $6 + \sqrt{2}$
(iv) $3 - \sqrt{5}$

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Solution

(i) Let us assume that is rational .Then , there exist positive co primes a and b such that
(ii) Let us assume that is rational .Then , there exist positive co primes a and b such that
We know that is an irrational number
Here we see that is a rational number which is a contradiction
(iii) Let us assume that is rational. Then , there exist positive co primes a and b such that
Here we see that is a rational number which is a contradiction as we know that is an irrational number
(iv) Let us assume that is rational .Then, there exist positive co primes a and b such that
Here we see that is a rational number which is a contradiction as we know that is an irrational number

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Similar questions

Q. Show that the following numbers are irrational.
(i)

\frac{1}{\sqrt{2}}

7 \sqrt{5}

6 + \sqrt{2}

3 - \sqrt{5}

Q. Prove that following numbers are irrationals:
(i)

\frac{2}{\sqrt{7}}

\frac{3}{2 \sqrt{5}}

4 + \sqrt{2}

5 \sqrt{2}

Q. Giving reason in each case, show that each of the following numbers is irrational.
(i)

4 + \sqrt{5}

(- 3 + \sqrt{6})

5 \sqrt{7}

- 3 \sqrt{8}

\frac{2}{\sqrt{5}}

\frac{4}{\sqrt{3}}

Q. Write the degrees of each of the following polynomials:

(i) 7x³ + 4x² − 3x + 12

(ii) 12 − x + 2x³

(iii)

5 y - \sqrt{2}

Q. Find the general solutions of the following equations:
(i)

\sin θ = \frac{1}{2}

\cos θ = - \frac{\sqrt{3}}{2}

cosec θ = - \sqrt{2}

\sec θ = \sqrt{2}

\tan θ = - \frac{1}{\sqrt{3}}

\sqrt{3} \sec θ = 2

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