We have,x = 5 - 25 + 2⇒x = 5 - 25 + 2 × 5 - 25 - 2 = 5 + 2 - 455 - 4 = 7 - 45Now, y = 5 + 25 - 2⇒y = 5 + 25 - 2 × 5 + 25 + 2 = 5 + 2 + 455 - 4 = 7 + 45Now, x2 = 7 - 452 = 49 + 80 - 565 = 129 - 565Now, y2 = 7 + 452 = 49 + 80 + 565 = 129 + 565Now, xy = 7 - 457 + 45 = 49 - 80 = -31Now, x2 + y2 + xy = 129 - 565 + 129 + 565 - 31 = 227

Q 6

We have, x = \frac{\sqrt{5} - 2}{\sqrt{5} + 2} \Rightarrow x = \frac{\sqrt{5} - 2}{\sqrt{5} + 2} \times \frac{\sqrt{5} - 2}{\sqrt{5} - 2} = \frac{5 + 2 - 4 \sqrt{5}}{5 - 4} = 7 - 4 \sqrt{5} Now, y = \frac{\sqrt{5} + 2}{\sqrt{5} - 2} \Rightarrow y = \frac{\sqrt{5} + 2}{\sqrt{5} - 2} \times \frac{\sqrt{5} + 2}{\sqrt{5} + 2} = \frac{5 + 2 + 4 \sqrt{5}}{5 - 4} = 7 + 4 \sqrt{5} Now, x^{2} = {(7 - 4 \sqrt{5})}^{2} = 49 + 80 - 56 \sqrt{5} = 129 - 56 \sqrt{5} Now, y^{2} = {(7 + 4 \sqrt{5})}^{2} = 49 + 80 + 56 \sqrt{5} = 129 + 56 \sqrt{5} Now, xy = (7 - 4 \sqrt{5}) (7 + 4 \sqrt{5}) = 49 - 80 = - 31 Now, x^{2} + y^{2} + xy = 129 - 56 \sqrt{5} + 129 + 56 \sqrt{5} - 31 = 227

-2

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