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Question

Let $$\alpha$$ and $$\beta$$ be the roots of $$\mathrm{a}\mathrm{x}^{2}+\mathrm{b}\mathrm{x}+\mathrm{c}= 0$$, then $$\displaystyle \lim_{x\displaystyle \rightarrow\alpha}\displaystyle \frac{1-\cos(ax^{2}+bx+c)}{(x-\alpha)^{2}}=$$


A
a2(αβ)22
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B
a22(αβ)2
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C
a2(αβ)2
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D
a22(αβ)2
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Solution

The correct option is B $$\displaystyle \frac{a^{2}(\alpha-\beta)^{2}}{2}$$
$$\displaystyle \lim_{x\rightarrow \alpha }\frac{1-\cos \left (
ax^{2}+bx+c \right )}{\left ( x-\alpha  \right )^{2}}=\lim_{x\rightarrow
\alpha }\frac{2\sin ^{2}\left ( \frac{ax^{2}+bx+c}{2} \right )}{\left (
x-\alpha  \right )^{2}}$$
$$\displaystyle =\lim_{x\rightarrow \alpha
}\frac{4\sin ^{2}\left [ \frac{a\left ( x-\alpha  \right )\left (
x-\beta \right )}{2} \right ]}{a^{2}\left ( x-\alpha  \right )^{2}\left (
x-\beta  \right )^{2}}\times \frac{a^{2}\left ( x-\beta  \right
)^{2}}{2}=\frac{a^{2}}{2}\left ( \alpha -\beta  \right )^{2}$$

Mathematics

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