Q41 / 120 · VITEEE 2019 · Mathematics

If $\tan \theta=\sqrt{\mathrm{n}}$ for some non-square natural number $\mathrm{n}$, then $\sec 2 \theta$ is

  • Correct answer
    A
    a rational number
  • B
    an irrational number
  • C
    a positive number
  • D
    none of the above

Solution

$\cos 2 \theta=\frac{1-\tan ^{2} \theta}{1+\tan ^{2} \theta}=\frac{1-\mathrm{n}}{1+\mathrm{n}} \Rightarrow \sec 2 \theta=\frac{1+\mathrm{n}}{1-\mathrm{n}}$ Since, $\mathrm{n}$ is non-square natural number $\therefore \mathrm{n} \neq 1$ $\therefore \sec 2 \theta=\frac{1+\mathrm{n}}{1-\mathrm{n}}=$ rational number

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

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