Q69 / 180 · COMEDK 2024 (Afternoon Shift) · Mathematics
$\text { If } \alpha=\tan ^{-1}\left(\tan \dfrac{5 \pi}{4}\right) \text { and } \beta=\tan ^{-1}\left(-\tan \dfrac{2 \pi}{3}\right) \text { then }$
- A$\alpha+\beta=-\dfrac{7 \pi}{12}$
- B$3 \alpha=4 \beta$
- C$\alpha-\beta=\dfrac{7 \pi}{12}$
- Correct answerD$4 \alpha=3 \beta$
Solution
The expression for $\alpha$ is given by $\alpha = \tan^{-1}(\tan \dfrac{5\pi}{4})$. Since $\dfrac{5\pi}{4}$ is in the third quadrant, $\tan \dfrac{5\pi}{4} = \tan(\pi + \dfrac{\pi}{4}) = \tan \dfrac{\pi}{4} = 1$.
Thus, $\alpha = \tan^{-1}(1) = \dfrac{\pi}{4}$.
The expression for $\beta$ is given by $\beta = \tan^{-1}(-\tan \dfrac{2\pi}{3})$. Since $\tan \dfrac{2\pi}{3} = \tan(\pi - \dfrac{\pi}{3}) = -\tan \dfrac{\pi}{3} = -\sqrt{3}$, we have $-\tan \dfrac{2\pi}{3} = \sqrt{3}$.
Thus, $\beta = \tan^{-1}(\sqrt{3}) = \dfrac{\pi}{3}$.
Checking the options:
Option (1): $\alpha + \beta = \dfrac{\pi}{4} + \dfrac{\pi}{3} = \dfrac{7\pi}{12} \neq -\dfrac{7\pi}{12}$.
Option (2): $3\alpha = 3(\dfrac{\pi}{4}) = \dfrac{3\pi}{4}$ and $4\beta = 4(\dfrac{\pi}{3}) = \dfrac{4\pi}{3}$. These are not equal.
Option (3): $\alpha - \beta = \dfrac{\pi}{4} - \dfrac{\pi}{3} = -\dfrac{\pi}{12} \neq \dfrac{7\pi}{12}$.
Option (4): $4\alpha = 4(\dfrac{\pi}{4}) = \pi$ and $3\beta = 3(\dfrac{\pi}{3}) = \pi$. Since $4\alpha = 3\beta = \pi$, this option is correct.
Answer: $4 \alpha=3 \beta$
Thus, $\alpha = \tan^{-1}(1) = \dfrac{\pi}{4}$.
The expression for $\beta$ is given by $\beta = \tan^{-1}(-\tan \dfrac{2\pi}{3})$. Since $\tan \dfrac{2\pi}{3} = \tan(\pi - \dfrac{\pi}{3}) = -\tan \dfrac{\pi}{3} = -\sqrt{3}$, we have $-\tan \dfrac{2\pi}{3} = \sqrt{3}$.
Thus, $\beta = \tan^{-1}(\sqrt{3}) = \dfrac{\pi}{3}$.
Checking the options:
Option (1): $\alpha + \beta = \dfrac{\pi}{4} + \dfrac{\pi}{3} = \dfrac{7\pi}{12} \neq -\dfrac{7\pi}{12}$.
Option (2): $3\alpha = 3(\dfrac{\pi}{4}) = \dfrac{3\pi}{4}$ and $4\beta = 4(\dfrac{\pi}{3}) = \dfrac{4\pi}{3}$. These are not equal.
Option (3): $\alpha - \beta = \dfrac{\pi}{4} - \dfrac{\pi}{3} = -\dfrac{\pi}{12} \neq \dfrac{7\pi}{12}$.
Option (4): $4\alpha = 4(\dfrac{\pi}{4}) = \pi$ and $3\beta = 3(\dfrac{\pi}{3}) = \pi$. Since $4\alpha = 3\beta = \pi$, this option is correct.
Answer: $4 \alpha=3 \beta$
Hence, the correct answer is option: 'D'.