Let - - - be three nonzero real numbers such that the equation -3- cos x-2 - sin x- x - - -2- -2 - has two distinct real roots a and b with a - b-3- If the range of values of 2-3- - 2 - is -q- r- then the value of q - r is

√(3)α cos a+2 β sin a=γ √(3)α cos b+2 β sin b=γ nbsp; adding √(3)α(cos a + cos b) +2 β(sin a+sin b)=2γ 2√(3)α cos ( a+b/2 ) cos ( a b/2 )+2 β. 2 sin ( a+b/2 ...

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Standard XIII

Mathematics

Trigonometric Ratios of Compound Angles

Let α, β, γ b...

Question

Let $α, β, γ$ be three nonzero real numbers such that the equation $\sqrt{3} α cos x + 2 β sin x = γ, x \in [- \frac{π}{2}, \frac{π}{2}]$ has two distinct real roots $a$ and $b$ with $a + b = \frac{π}{3}$ . If the range of values of $\frac{2 γ}{3 α + 2 β}$ is $[q, r)$ , then the value of $q + r$ is

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Solution

$\sqrt{3} α cos a + 2 β sin a = γ$
$\sqrt{3} α cos b + 2 β sin b - ---------------------- - = γ$
adding $\sqrt{3} α (cos a + cos b) + 2 β (sin a + sin b) = 2 γ$

$2 \sqrt{3} α cos (\frac{a + b}{2}) cos (\frac{a - b}{2}) + 2 β . 2 sin (\frac{a + b}{2}) cos (\frac{a - b}{2}) = 2 γ$

$(3 α + 2 β) cos (\frac{a - b}{2}) = 2 γ$

$cos (\frac{a - b}{2}) = \frac{2 γ}{3 α + 2 β}$

$\frac{a - b}{2} \in [- \frac{π}{3}, \frac{π}{3}] - {0}$

$\Rightarrow \frac{1}{2} \leq \frac{2 γ}{3 α + 2 β} < 1$

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Let α, β, γ be three nonzero real numbers such that the equation √3α cos x+2β sin x=γ, x∈[−π2,π2] has two distinct real roots a and b with a+b=π3. If the range of values of 2γ3α+2β is [q,r), then the value of q+r is