Q65 / 200 · WBJEE 2014 · Mathematics

The solution of the equation $\log _{101} \log _{7}(\sqrt{x+7}+\sqrt{x})=0$ is

  • A
    3
  • B
    7
  • Correct answer
    C
    9
  • D
    49

Solution

Given, $\log _{101} \log _{7}(\sqrt{x+7}+\sqrt{x})=0$ $\therefore \quad \log _{7}(\sqrt{x+7}+\sqrt{x})=(101)^{0}$ $\Rightarrow \quad \log _{7}(\sqrt{x+7}+\sqrt{x})=1$ $\Rightarrow \quad(\sqrt{x+7}+\sqrt{x})=7^{1}$ $\Rightarrow \quad \sqrt{x+7}+\sqrt{x}=7$ On squaring both sides, we get $(x+7)+x+2 \sqrt{x^{2}+7 x}=49$ $\Rightarrow \quad 2 x-42=-2 \sqrt{x^{2}+7 x}$ $\Rightarrow \quad x-21=-\sqrt{x^{2}+7 x}$ Again, squaring both sides, we get $x^{2}+441-42 x=x^{2}+7 x$ $\Rightarrow \quad 49 x=441$ $\Rightarrow \quad x=\frac{441}{49} \Rightarrow x=9$

Hence, the correct answer is option: 'C'.

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