Check the validity of the statements given below by the method given against it.

**(i)** p: The sum of an irrational number and a rational number is irrational (by contradiction method).

**(ii)** q: If n is a real number with n > 3, then n^{2} > 9 (by contradiction method).

Asked by Pragya Singh | 11 months ago | 125

**(i)** The given statement is as follows

p: The sum of an irrational number and a rational number is irrational.

Let us assume that the statement p is false. That is,

The sum of an irrational number and a rational number is rational.

Hence,

where

\( \sqrt{a}\) is irrational and b, c, d, e are integers.

\( \dfrac{d}{e}-\dfrac{b}{c}=\sqrt{a}\)

But here, \( \dfrac{d}{e}-\dfrac{b}{c}\) is a rational number and \( \sqrt{a}\) is an irrational number

This is a contradiction. Hence, our assumption is false.

The sum of an irrational number and a rational number is rational.

Hence, the given statement is true.

**(ii)** The given statement q is as follows

If n is a real number with n > 3, then n^{2} > 9

Let us assume that n is a real number with n > 3, but n^{2} > 9 is not true

i.e. n^{2} < 9

So, n > 3 and n is a real number

By squaring both sides, we get

n^{2} > (3)^{2}

This implies that n^{2} > 9 which is a contradiction, since we have assumed that n^{2} < 9

Therefore, the given statement is true i.e., if n is a real number with n > 3, then n^{2} > 9.

Determine whether the argument used to check the validity of the following statement is correct: p: “If x^{2} is irrational, then x is rational.” The statement is true because the number x^{2} = π^{2} is irrational, therefore x = π is irrational.

Which of the following statements are true and which are false? In each case give a valid reason for saying so

**(i)** p: Each radius of a circle is a chord of the circle.

**(ii) **q: The centre of a circle bisect each chord of the circle.

**(iii)** r: Circle is a particular case of an ellipse.

**(iv)** s: If x and y are integers such that x > y, then – x < – y.

**(v)** t: \( \sqrt{11}\) is a rational number.

By giving a counter example, show that the following statement is not true. p: “If all the angles of a triangle are equal, then the triangle is an obtuse angled triangle.”

Show that the following statement is true “The integer n is even if and only if n^{2} is even”

Show that the following statement is true by the method of the contrapositive p: “If x is an integer and x^{2} is odd, then x is also odd.”