Question

If $\alpha$, $\beta$ are roots of the equation $6x^2+11x+3=0$, then

• A. $\text{both}{\cos }^{-1}\alpha \text{and}{\cos }^{-1}\beta \text{are}\text{real}$
• B. $\text{both}\cos \text{e}{\text{c}}^{-1}\alpha \text{and}{\cos }^{-1}\beta \text{are}\text{real}$
• C. $\text{both}\cos \text{e}{\text{c}}^{-1}\alpha \text{and}{\cot }^{-1}\beta \text{are}\text{real}$
• D. Both $BandC$

Answer 1

Answer: (D)

Both $BandC$

Given $\alpha$ and $\beta$ are roots of $6x^2+11x+3=0$

$\Rightarrow (2x+3)(3x+1)=0$

$\Rightarrow x=\frac{-3}{2}$ & $x=\frac{-1}{3}$

Let $\alpha =\frac{-3}{2}$, $\beta =\frac{-1}{3}$

(i) ${\cos }^{-1}\alpha ={\cos }^1\left(\frac{-3}{2}\right)\text{/}\in [-1,1]$

(ii) ${\sin }^{-1}\beta ={\sin }^{-1}\left(\frac{-1}{3}\right)\in [-1,1]$

(iii) $cose{c}^{-1}\alpha =cose{c}^{-1}\left(\frac{-3}{2}\right)\in [-1,1]$

(iv) ${\cot }^{-1}\alpha ,{\cot }^{-1}\beta \in [-1,1]$

Option B and C are correct.