JEE Advanced 2018 · Paper 1 · Mathematics Q13

MathematicsNumerical+3Question 49 of 54

Let $a, b, c$ be three non-zero real numbers such that the equation

$\sqrt{3}\, a \cos x + 2b \sin x = c, \quad x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right],$

has two distinct real roots $\alpha$ and $\beta$ with $\alpha + \beta = \dfrac{\pi}{3}$ . Then, the value of $\dfrac{b}{a}$ is _____ .

Your answer

Answer and worked solution

Official answer: 0.5

Subtract the equations for $\alpha$ and $\beta$ :
$\sqrt3a(\cos\alpha-\cos\beta)+2b(\sin\alpha-\sin\beta)=0.$

Sum-to-product. Cancel the common factor $2\sin\frac{\alpha-\beta}2\ne0$ :
$-\sqrt3a\sin\frac{\alpha+\beta}2+2b\cos\frac{\alpha+\beta}2=0.$

Substitute $\dfrac{\alpha+\beta}2=\dfrac\pi6$ :
$-\frac{\sqrt3}2a+\sqrt3b=0\implies\frac ba=\frac12.$

Answer: 0.5

Topics:Trigonometric equations

JEE Advanced 2018 · Paper 1, Mathematics question 13