The product of all possible values of , for which is: [2026]

Question

The product of all possible values of $α$ , for which $\lim_{x \to 0} (\frac{1 - \cos (α x) \cos ((α + 1) x) \cos (α + 2) x)}{\sin^{2} ((α + 1) x)}) = 2$ is: [2026]

Smart Achievers · Accepted Answer

(3)

We have, $\lim_{x \to 0} (\frac{1 - \cos (α x) \cos ((α + 1) x) \cos (α + 2) x)}{\sin^{2} ((α + 1) x)}) = 2$

$\Rightarrow \lim_{x \to 0} \frac{1 - \cos (α x) \cos (α + 1) x \cos (α + 2) x}{{(α + 1)}^{2} x^{2}} = 2$

$\Rightarrow \lim_{x \to 0} \frac{1 - {1 + \frac{α^{2} x^{2}}{2} + \frac{{(α + 1)}^{2} x^{2}}{2} + \frac{{(+ 2)}^{2} x^{2}}{2} . . .}}{{(α + 1)}^{2} x^{2}} = 2$

$\Rightarrow \lim_{x \to 0} \frac{α^{2}}{2} + \frac{{(α + 1)}^{2}}{2} + \frac{{(α + 2)}^{2}}{2} = 2 {(α + 1)}^{2}$

$\Rightarrow 3 α^{2} + 6 α + 5 = 4 α^{2} + 8 a + 4$

$\Rightarrow α^{2} + 2 α - 1 = 0$

$∴$ Products of roots = –1

Solution

Q.
The product of all possible values of $α$ , for which $\lim_{x \to 0} (\frac{1 - \cos (α x) \cos ((α + 1) x) \cos (α + 2) x)}{\sin^{2} ((α + 1) x)}) = 2$ is: [2026]

1 –2

2 1

3 –1

4 5/4

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