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Mathematics

If p, q are rational numbers and p15q=2354335p - \sqrt{15}q = \dfrac{2\sqrt{3} - \sqrt{5}}{4\sqrt{3} - 3\sqrt{5}}, find the values of p and q.

Rational Irrational Nos

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Answer

Since, it is given that

2354335=p15q\dfrac{2\sqrt{3} - \sqrt{5}}{4\sqrt{3} - 3\sqrt{5}} = p - \sqrt{15} q

On solving,

2354335×43+3543+35=23×43+23×355×435×35(43)2(35)2=24+61541515(43)2(35)2=2415+6154154845=9+2153=93+2153=3+23153(23)15=pq15\dfrac{2\sqrt{3} - \sqrt{5}}{4\sqrt{3} - 3\sqrt{5}} × \dfrac{4\sqrt{3} + 3\sqrt{5}}{4\sqrt{3} + 3\sqrt{5}} \\[1.5em] = \dfrac{2\sqrt{3} × 4\sqrt{3}+ 2\sqrt{3} × 3\sqrt{5} - \sqrt{5} × 4\sqrt{3} - \sqrt{5} × 3\sqrt{5} }{(4\sqrt{3})^2 -(3\sqrt{5})^2} \\[1.5em] = \dfrac{24 + 6\sqrt{15} - 4\sqrt{15} - 15 }{(4\sqrt{3})^2 -(3\sqrt{5})^2} \\[1.5em] = \dfrac{24 - 15 + 6\sqrt{15} - 4\sqrt{15} }{48 - 45} \\[1.5em] = \dfrac{9 + 2{\sqrt{15}}}{3} \\[1.5em] = \dfrac{9}{3} + \dfrac{2\sqrt{15}}{3} \\[1.5em] = 3 + \dfrac{2}{3}{\sqrt{15}} \\[1.5em] \therefore 3 - \Big(-\dfrac{2}{3}\Big){\sqrt{15}} = p - q\sqrt{15}

Hence, p = 3 and q = 23-\dfrac{2}{3}.

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