Q53 / 90 · KVPY 2015 (SB/SX) · Mathematics

Let $x=(\sqrt{50}+7)^{1 / 3}-(\sqrt{50}-7)^{/ 3}$

  • Correct answer
    A
    $\mathrm{x}=2$
  • B
    $x=3$
  • C
    $x$ is a rational number, but not an integer $
  • D
    x$ is an irrational number

Solution

$(\sqrt{50}+7)^{\frac{1}{2}}=\mathrm{t}$ $\mathrm{x}=\mathrm{t}-\frac{1}{\mathrm{t}}$ $\mathrm{x}^{3}=\mathrm{t}^{3}-\frac{1}{\mathrm{t}^{3}}-3\left(\mathrm{t}-\frac{1}{\mathrm{t}}\right)$ $\mathrm{x}^{3}+3 \mathrm{x}=\sqrt{50}+7-(\sqrt{50}-7)$ $\mathrm{x}^{3}+3 \mathrm{x}=14$ $\mathrm{x}^{3}+3 \mathrm{x}-14=0$ $(\mathrm{x}-2)\left(\mathrm{x}^{2}+2 \mathrm{x}+7\right) \Rightarrow \mathrm{x}=2$

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

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