If and are the roots of the quadratic equation , then is equal to

Question

If $α = \lim_{x \to 0^{+}} (\frac{e^{\sqrt{\tan x}} - e^{\sqrt{x}}}{\sqrt{\tan x} - \sqrt{x}})$ and $β = \lim_{x \to 0^{+}} {(1 + \sin x)}^{\frac{1}{2} \cot x}$ are the roots of the quadratic equation $a x^{2} + b x - \sqrt{e} = 0$ , then $24 \log_{e} (a + b)$ is equal to _______

Smart Achievers · Accepted Answer

(12)

$α = \lim_{x \to 0^{+}} \frac{e^{\sqrt{\tan x}} - e^{\sqrt{x}}}{\sqrt{\tan x} - \sqrt{x}} = 1, β = \lim_{x \to 0} {(1 + \sin x)}^{\frac{1}{2} \cot x} = e^{1 / 2}$

The equation having roots $1, \sqrt{e}$ is $x^{2} - (1 + \sqrt{e}) x + \sqrt{e} = 0$

It is given as $a x^{2} + b x - \sqrt{e} = 0$

By comparing we get $\frac{a}{1} = \frac{b}{- (1 + \sqrt{e})} = \frac{- 1}{1} \Rightarrow a = - 1, b = 1 + \sqrt{e}$

$∴$ $12 \log_{e} (a + b) = 12 \log_{e} (- 1 + 1 + \sqrt{e}) = 6$

Solution

Q.
If $α = \lim_{x \to 0^{+}} (\frac{e^{\sqrt{\tan x}} - e^{\sqrt{x}}}{\sqrt{\tan x} - \sqrt{x}})$ and $β = \lim_{x \to 0^{+}} {(1 + \sin x)}^{\frac{1}{2} \cot x}$ are the roots of the quadratic equation $a x^{2} + b x - \sqrt{e} = 0$ , then $24 \log_{e} (a + b)$ is equal to _______

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Solution

Q. If α=limx→0+(etanx-extanx-x) and β=limx→0+(1+sinx)12cotx are the roots of the quadratic equation ax2+bx-e=0, then 24loge(a+b) is equal to _______

Ans. (12) α=limx→0+etanx-extanx-x=1, β=limx→0(1+sinx)12cotx=e1/2 The equation having roots 1, e is x2-(1+e)x+e=0 It is given as ax2+bx-e=0 By comparing we get a1=b-(1+e)=-11⇒a=-1, b=1+e ∴ 12loge(a+b)=12loge(-1+1+e)=6

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Q.
If $α = \lim_{x \to 0^{+}} (\frac{e^{\sqrt{\tan x}} - e^{\sqrt{x}}}{\sqrt{\tan x} - \sqrt{x}})$ and $β = \lim_{x \to 0^{+}} {(1 + \sin x)}^{\frac{1}{2} \cot x}$ are the roots of the quadratic equation $a x^{2} + b x - \sqrt{e} = 0$ , then $24 \log_{e} (a + b)$ is equal to _______