JAMB Mathematics Past Questions & Answers - Page 262

x = 3 - \(\sqrt{3}\)

x² = (3 - \(\sqrt{3}\))²

= 9 + 3 - 6\(\sqrt{34}\)

= 12 - 6\(\sqrt{3}\)

= 6(2 - \(\sqrt{3}\))

∴ x² + \(\frac{36}{x^2}\) = 6(2 - \(\sqrt{3}\)) + \(\frac{36}{6(2 - \sqrt{3})}\)

6(2 - \(\sqrt{3}\)) + \(\frac{6}{2 - \sqrt{3}}\) = 6(- \(\sqrt{3}\)) + \(\frac{6(2 + \sqrt{3})}{(2 - \sqrt{3})(2 + \sqrt{3})}\)

= 6(2 - \(\sqrt{3}\)) + \(\frac{6(2 + \sqrt{3})}{4 - 3}\)

6(2 - \(\sqrt{3}\)) + 6(2 + \(\sqrt{3}\)) = 12 + 12

= 24

5\(\sqrt{18}\) - 3\(\sqrt{72}\) + 4\(\sqrt{50}\) = 5(3\(\sqrt{2}\)) - 3(6\(\sqrt{2}\)) + 4(5\(\sqrt{2}\))

15\(\sqrt{2}\) - 18\(\sqrt{2}\) + 20\(\sqrt{2}\) = 35\(\sqrt{2}\) - 18\(\sqrt{2}\)

= 17\(\sqrt{2}\)

\(\frac{(1.25 \times 10^{-4}) \times (2.0 \times 10^{-1})}{(6.25 \times 10^5)}\) = \(\frac{1.25 \times 2}{6.25}\) x 10^{4 - 1 - 5}

\(\frac{2.50}{6.25}\) x 10^-2 = \(\frac{250}{625}\) x 10^-2

0.4 x 10^-2 = 4.0 x 10^-3

\(\frac{4^{2x}}{4^{3x}}\) = 2

4^{2x - 3x} = 2

4^-x = 2

(2²)^-x

= 2¹

Equating coefficients: -2x = 1

x = -\(\frac{1}{2}\)

Let \(\log_{b} a^{n} = x\)

\(\therefore a^{n} = b^{x}\)

\(a^{n} = (a^{\frac{1}{n}})^{x}\)

\(a^{n} = a^{\frac{x}{n}} \implies n = \frac{x}{n}\)

\(x = n^{2}\)