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 n² > 9 (by contradiction method).

<p>This is a &nbsp;Mathematical Reasoning Solutions Type Questions as classified in NCERT Exemplar</p><p><strong> (i) </strong>The given statement is as follows.</p><p>p : the sum of an irrational number and a rational number is irrational.</p><p>Let us assume that, p is false.</p><p>i.e. The sum of an irrational number and rational number is rational.</p><p>Hence, √a<span title="Click to copy mathml"><math><mrow><mo>+</mo><mfrac><mrow><mi>b</mi></mrow><mrow><mi>c</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>d</mi></mrow><mrow><mi>e</mi></mrow></mfrac><mo>, </mo></mrow></math></span>&nbsp;where √a<math><mrow></mrow></math>&nbsp;is irrational and b, c, d, e are integers.</p><p><span title="Click to copy mathml"><math><mrow><mfrac><mrow><mi>d</mi></mrow><mrow><mi>e</mi></mrow></mfrac><mo>−</mo><mfrac><mrow><mi>b</mi></mrow><mrow><mi>c</mi></mrow></mfrac></mrow></math></span>&nbsp;is a rational number and √a<math><mrow></mrow></math>&nbsp;is an irrational number.</p><p>This is a contradiction. Therefore, our assumption is wrong.</p><p>Therefore, the sum of an irrational number and a rational number is rational.</p><p>Thus, the given statement is true.</p><p><strong> (ii)</strong> The given statement, q is as follows.</p><p>If n is a real number with n&gt;3, then <em>n</em>²&gt;9.</p><p>Let us assume than n is a real number with n&gt;3, but <em>n</em>²&gt;9 is not true.</p><p>i.e. <em>n</em>² &lt; 9</p><p>Then n &gt; 9 and n is real number.</p><p>Squaring on both sides, we get</p><p><em>n</em>²&gt; (3)²</p><p><span title="Click to copy mathml"><math><mrow><mo>⇒</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>&gt;</mo><mn>9</mn><mo>, </mo></mrow></math></span>&nbsp;which is contradiction.</p><p>Since we have assumed that <em>n</em>²&lt;9.</p><p>Thus, the given statement is true. That is, if n is a real number with n&gt;3, then <em>n</em>²&gt;9.</p>

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