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EEJ MAIN Mathematics QUESTION #6964
Question 1

Using only principal values of inverse functions, describe the set $A=\left\{x\geq0:\tan^{-1}(2x)+\tan^{-1}(3x)=\dfrac{\pi}{4}\right\}$.

  • Contains two elements
  • Contains more than two elements
  • Is a singleton✔️
  • Is an empty set
Correct Answer Explanation

Apply the addition formula: $\tan^{-1}(2x)+\tan^{-1}(3x)=\tan^{-1}\!\left(\dfrac{5x}{1-6x^2}\right)$ when $6x^2<1$.

Setting equal to $\pi/4$: $\dfrac{5x}{1-6x^2}=1 \Rightarrow 5x=1-6x^2 \Rightarrow 6x^2+5x-1=0$

$x=\dfrac{-5\pm\sqrt{25+24}}{12}=\dfrac{-5\pm7}{12}$

$x=\dfrac{2}{12}=\dfrac{1}{6}$ or $x=\dfrac{-12}{12}=-1$.

Since $x\geq0$, only $x=\dfrac{1}{6}$ is valid. Check: $6\cdot(1/6)^2=1/6<1$ ✓. So $A=\{1/6\}$ — a singleton.