Write the set of values of a for which the equation -3sin x - x cos x - a has no solution-

We have √(3) sin x x cos x=a The LHS of this equation is of the form a sin x+b cos x Where a=√(3),b= 1 ∴ r=√(a^2+b^2)=√(3+1)=2 So that a=r cos α b=r sin α ...

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Standard XII

Mathematics

Equations of the Form acosθ+bsinθ = c

Write the set...

Question

$Write the set of values of a for which the equation \sqrt{3} s i n x - x c o s x = a h a s n o s o l u t i o n .$

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Solution

$W e h a v e \sqrt{3} s i n x - x c o s x = a The LHS of this equation is of the form a sin x+b cos x W h e r e a = \sqrt{3}, b = - 1 ∴ r = \sqrt{a^{2} + b^{2}} = \sqrt{3 + 1} = 2 S o t h a t a = r c o s α b = r s i n α ∴ L H S = r c o s α s i n x + r s i n α c o s x = r [s i n (x + α)] = 2 s i n (x + α) S o, \sqrt{3} s i n x - c o s x = a \Rightarrow 2 s i n (x + α) a \Rightarrow s i n (x + α) = \frac{a}{2} T h i s e q u a t i o n h a s a s o l u t i o n o n l y i f - 2 \leq a \leq 2. ∴ T h e e q u a t i o n h a s n o s o l u t i o n i f a \in (- \infty, - 2) (2, \infty) .$

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Write the set of values of a for which the equation √3 sin x − x cos x=a has no solution.

$Write the set of values of a for which the equation \sqrt{3} s i n x - x c o s x = a h a s n o s o l u t i o n .$