A, B, C and D are integers between 1 to 100. If A/B and C/D are proper fractions such that A/B is the greatest fraction and (A/B - C/D) is always a proper fraction such that D is the largest prime number, then the number of possible fractions (C/D) is
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Ans – Barbora Krejcikova(x-1/x)³ = x³-1/x³ - 3×x×1/x(1–1/x)
or (x-1/x)³ = x³-1/x³ - 3(1–1/x)
or (x-1/x)³ = 14 - 3(1–1/x) . . . . [Given, x³-1/x³ = 14]
Let x-1/x = a
Therefore a³ = 14 - 3a
or a³+3a-14 =0
or a³-2a²+2a²-4a+7a-14 =0
or a²(a-2) +2a(a-2) +7(a-2) =0
or (a-2)(a²+2a+7) =0
or a-2 = 0/(a²+2a+7)
or a = 2
So x-1/x = 2