Let α and β be roots of the equation [(t+2)^(1/7)-1]x^2+[(t+2^(1/6)-1]x+(t+2)^(1/21)-1)=0

Question

Let \(\alpha\) and \(\beta\) be roots of the equation \(\left[(t+2)^{\frac{1}{7}}-1\right] x^{2}+\left[\left(t+2^{\frac{1}{6}}-1\right] x+(t+2)^{\frac{1}{21}}-1\right)=0\) If \(\lim _\limits{t \rightarrow-1} \alpha=a\) and \(\lim _{t \rightarrow-1} \beta=b\) then \(72(a+b)^{2}\) is equal to

(1) 49

(2) 98

(3) 36

(4) 75

Answer 1

Correct option is: (2) 98 

Notice that 

\(\lim _\limits{t \rightarrow-1}(\alpha \beta)=\frac{\frac{1}{21}}{\frac{1}{7}}=\frac{7}{21}=\frac{1}{3}=a b\) 

\(\Rightarrow(a+b)^{2}=\frac{41}{36} \Rightarrow 72(a+b)^{2}=98\)