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Mathematics

If [mn]x−1=[nm]x−5\Big[\dfrac{m}{n}\Big]^{x-1} = \Big[\dfrac{n}{m}\Big]^{x-5}, the value of x is:

  1. 3

  2. -3

  3. 13\dfrac{1}{3}

  4. −13-\dfrac{1}{3}

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Answer

[mn]x−1=[nm]x−5⇒[mn]x−1=[mn]5−x\Big[\dfrac{m}{n}\Big]^{x-1} = \Big[\dfrac{n}{m}\Big]^{x-5}\\[1em] \Rightarrow \Big[\dfrac{m}{n}\Big]^{x-1} = \Big[\dfrac{m}{n}\Big]^{5-x}

According to the property,am=an⇒m=na^m = a^n \Rightarrow m = n

⇒(x−1)=(5−x)⇒x+x=5+1⇒2x=6⇒x=62⇒x=3\Rightarrow (x - 1) = (5 - x)\\[1em] \Rightarrow x + x = 5 + 1\\[1em] \Rightarrow 2x = 6\\[1em] \Rightarrow x = \dfrac{6}{2}\\[1em] \Rightarrow x = 3

Hence, option 1 is the correct option.

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