Question

If $\alpha=\tan \left(\frac{1}{3} \tan ^{-1} \frac{4}{3}\right)$ and $\beta=\sin \left(\frac{1}{2} \sin ^{-1} \frac{4 \sqrt{2}}{9}\right)$, then the value of $\left|\left(\alpha^3-4 \alpha^2-3 \alpha+\beta\right)\right|$ is equal to

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (A)

Let $ \tan ^{-1}\left(\frac{4}{3}\right)=3 \theta ; \alpha=\tan \theta \Rightarrow \tan 3 \theta=\frac{4}{3}$
$\Rightarrow\left(\alpha^3-4 \alpha^2-3 \alpha\right)=\frac{-4}{3}$....(1)
Also, put $\sin ^{-1}\left(\frac{4 \sqrt{2}}{9}\right)=2 \phi ; \beta=\sin \phi \Rightarrow \sin 2 \phi=\frac{4 \sqrt{2}}{9} \Rightarrow \beta=\frac{1}{3}$....(2)
$\therefore\left|\alpha^3-4 \alpha^2-3 \alpha+\beta\right|=\left|\frac{-4}{3}+\frac{1}{3}\right|=1$