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Question
If 2f (x) − \[3f\left( \frac{1}{x} \right) = x^2\] (x ≠ 0), then f(2) is equal to
Options
(a) \[- \frac{7}{4}\]
(b) \[\frac{5}{2}\]
(c) −1
(d) None of these
Solution
(a) \[- \frac{7}{4}\]
2f (x) − \[3f\left( \frac{1}{x} \right) = x^2\] (x ≠ 0) ....(1)
\[\text{ Replacing x by } \frac{1}{x}: \]
\[2f\left( \frac{1}{x} \right) - 3f(x) = \frac{1}{x^2} . . . (2) \]
\[\text{ Solving equations (1) & (2) } \]
\[ - 5f (x) = \frac{3} {x^2} + 2 x^2 \]
\[ \Rightarrow f(x) = \frac{- 1}{5} \left( \frac{3}{x^2} + 2 x^2 \right)\]
\[\text{ Thus } , f(2) = \frac{- 1}{5} \left( \frac{3}{4} + 2 \times 4 \right)\]
\[ = \frac{- 1}{5} \left( \frac{3 + 32}{4} \right) \]
\[ = \frac{- 7}{4}\]
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