1. [Maximum mark: 5\( ] \) (a) For the following statements, state if they are true or false. (i) Reciprocal of \( \sqrt{2}-3 \) is the number \( \sqrt{2}+3 \). (ii) For every real number \( a,|a|^{2017} a=a^{2017}|a| \). (iii) For every value of \( a \geq 0, \sqrt[10]{a^{20}}=a^{2} \). (b) State the value of the real numbers \( x \) and \( y \), so that \( (2 x-1)^{4}+3|y+5|=0 \).

For the first set of statements, let’s evaluate their truths: (i) The statement is **false**. The reciprocal of \( \sqrt{2}-3 \) is given by \( \frac{1}{\sqrt{2}-3} \), which is not equal to \( \sqrt{2}+3 \). Instead, it can be simplified to \( \frac{\sqrt{2}+3}{(\sqrt{2}-3)(\sqrt{2}+3)} \). (ii) The statement is **true**. For any real number \( a \), the absolute value raised to an odd power maintains the sign, validating that \( |a|^{2017} a = a^{2017}|a| \). (iii) The statement is **true**. For \( a \geq 0 \), \( \sqrt[10]{a^{20}} = a^{20/10} = a^2 \). Now for part (b), we evaluate the equation \( (2x-1)^{4} + 3|y+5| = 0 \). Since \( (2x-1)^{4} \) is always non-negative and \( 3|y+5| \) is also non-negative, both must equal zero. This leads to two separate equations: 1. \( (2x-1)^{4} = 0 \) which implies \( 2x - 1 = 0 \), so \( x = \frac{1}{2} \). 2. \( 3|y+5| = 0 \) which gives \( |y+5| = 0 \), leading to \( y + 5 = 0 \) or \( y = -5 \). Thus, the values are \( x = \frac{1}{2} \) and \( y = -5 \).