JEE MAIN - Mathematics (2017 - 8th April Morning Slot - No. 7)
Let f(x) = 210.x + 1 and g(x)=310.x $$-$$ 1. If (fog) (x) = x, then x is equal to :
$${{{3^{10}} - 1} \over {{3^{10}} - {2^{ - 10}}}}$$
$${{{2^{10}} - 1} \over {{2^{10}} - {3^{ - 10}}}}$$
$${{1 - {3^{ - 10}}} \over {{2^{10}} - {3^{ - 10}}}}$$
$${{1 - {2^{ - 10}}} \over {{3^{10}} - {2^{ - 10}}}}$$
Explanation
(fog) (x) = x
$$ \Rightarrow $$$$\,\,\,$$ f (g(x)) = x
$$ \Rightarrow $$$$\,\,\,$$ f (310. x $$-$$ 1) = x [ as g(x) = 310. x $$-$$ 1]
$$ \Rightarrow $$$$\,\,\,$$ 210 . (310 . x $$-$$ 1) + 1 = x
$$ \Rightarrow $$$$\,\,\,$$ 310 . x $$-$$ 1 + 2$$-$$10 = x . 2$$-$$10 [dividing by 210]
$$ \Rightarrow $$$$\,\,\,$$310 . x $$-$$ 2$$-$$10 . x = 1 $$-$$ 2$$-$$10
$$ \Rightarrow $$$$\,\,\,$$ x (310 $$-$$ 2$$-$$ 10) = 1$$-$$ 2$$-$$10
$$ \Rightarrow $$ $$\,\,\,$$ x = $${{1 - {2^{ - 10}}} \over {{3^{10}} - {2^{ - 10}}}}$$
$$ \Rightarrow $$$$\,\,\,$$ f (g(x)) = x
$$ \Rightarrow $$$$\,\,\,$$ f (310. x $$-$$ 1) = x [ as g(x) = 310. x $$-$$ 1]
$$ \Rightarrow $$$$\,\,\,$$ 210 . (310 . x $$-$$ 1) + 1 = x
$$ \Rightarrow $$$$\,\,\,$$ 310 . x $$-$$ 1 + 2$$-$$10 = x . 2$$-$$10 [dividing by 210]
$$ \Rightarrow $$$$\,\,\,$$310 . x $$-$$ 2$$-$$10 . x = 1 $$-$$ 2$$-$$10
$$ \Rightarrow $$$$\,\,\,$$ x (310 $$-$$ 2$$-$$ 10) = 1$$-$$ 2$$-$$10
$$ \Rightarrow $$ $$\,\,\,$$ x = $${{1 - {2^{ - 10}}} \over {{3^{10}} - {2^{ - 10}}}}$$
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