Q214 / 270 · NDA 2011 (Phase 2) · Mathematics

If $\tan A=1 / 2$ and $\tan B=1 / 3$, then what is the value of $4 \mathrm{~A}+4 \mathrm{~B}$ ?

  • A
    $\pi / 4$
  • B
    $\pi / 2$
  • Correct answer
    C
    $\pi$
  • D
    $2 \pi$

Solution

$\tan \mathrm{A}=\frac{1}{2}, \tan \mathrm{~B}=\frac{1}{3}$ $\tan (A+B)=\frac{\tan A+\tan B}{1-\tan A \cdot \tan B}$ $=\frac{\frac{1}{2}+\frac{1}{3}}{1-\frac{1}{2} \times \frac{1}{3}}=\frac{\frac{5}{6}}{\frac{5}{6}}$ $\begin{aligned} \tan (\mathrm{A}+\mathrm{B}) & =1 \\ \tan (\mathrm{~A}+\mathrm{B}) & =\tan \frac{\pi}{4} \\ \mathrm{~A}+\mathrm{B} & =\frac{\pi}{4} \\ 4(\mathrm{~A}+\mathrm{B}) & =\frac{\pi}{4} \times 4 \\ 4 \mathrm{~A}+4 \mathrm{~B} & =\pi .\end{aligned}$

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

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