Q151 / 270 · NDA 2015 (Phase 1) · Mathematics

Let $\theta$ be a positive angle. If the number of degrees in $\theta$ is divided by the number of radians in $\theta$, then an irrational number $\frac{180}{\pi}$ results. If the number of degrees in $\theta$ is multiplied by the number of radians in $\theta$, then an irrational number $\frac{125 \pi}{9}$ results. The angle $\theta$ must be equal to

  • A
    $30^{\circ}$
  • B
    $45^{\circ}$
  • Correct answer
    C
    $50^{\circ}$
  • D
    $60^{\circ}$

Solution

From going by the options, option (A), $\theta=30^{\circ}$, as we know that $180^{\circ}=\pi$ radian $\therefore \quad 30^{\circ}=\frac{30 \pi}{180}$ radian Now according to question, $\frac{30^{\circ} \times 180^{\circ}}{30^{\circ} \pi}=\frac{180}{\pi}$ Now number of degree in $\theta$ is multiplied by number of radians in $\theta$. $\therefore \quad 30^{\circ} \times \frac{30 \pi}{180}=\frac{900 \pi}{180}=\frac{10 \pi}{2}=5 \pi \neq \frac{125 \pi}{9}$ From option (B), $\begin{aligned} & \theta=45^{\circ} \\ \therefore \quad & 45^{\circ}=\frac{45^{\circ} \pi}{180} \text { radian }\end{aligned}$ Now according to question, $\frac{45^{\circ} \times 180}{45^{\circ} \pi}=\frac{180}{\pi}$ Now number of degree in $\theta$ is multiplied by number of radian in $\theta$. $\therefore \quad 45^{\circ} \times \frac{45^{\circ} \pi}{180^{\circ}}=\frac{45 \pi}{4} \neq \frac{125 \pi}{9}$ From option (C), $\theta=50^{\circ}$ As we know that $180^{\circ}=\pi$ radian $\therefore \quad 50^{\circ}=\frac{50 \pi}{180}$ radian Now according to question $\frac{50^{\circ} \times 180^{\circ}}{50^{\circ} \pi}=\frac{180}{\pi}$ Now number of degree in ' $\theta$ ' is multiplied by number of radian in $\theta$. $\therefore \quad 50^{\circ} \times \frac{50 \pi}{180}=\frac{2500 \pi}{180}=\frac{125 \pi}{9}$ $\therefore \quad$ Option (C) is correct.

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

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