If $\alpha \beta 0$ are the roots of the equation $a{x}^{2}+bx+1=0$, and $\lim _{x\to \frac{1}{a}}{\left(\frac{1-\cos \left({x}^{2}+bx+a\right)}{2(1-\alpha x{)}^{2}}\right)}^{\frac{1}{2}}=\frac{1}{k}\left(\frac{1}{\beta }-\frac{1}{\alpha }\right)$, then k is equal to [JEE Main 2023, 8 Apr (Shift 2)]

Question

If $\alpha >\beta >0$ are the roots of the equation $a{x}^{2}+bx+1=0$, and $\lim _{x\to \frac{1}{a}}{\left(\frac{1-\cos \left({x}^{2}+bx+a\right)}{2(1-\alpha x{)}^{2}}\right)}^{\frac{1}{2}}=\frac{1}{k}\left(\frac{1}{\beta }-\frac{1}{\alpha }\right)$, then k is equal to [JEE Main 2023, 8 Apr (Shift 2)]

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If \(\alpha >\beta >0\) are the roots of the equation \(a{x}^{2}+bx+1=0\), and \(\lim _{x\to \frac{1}{a}}{\left(\frac{1-…

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