Question nos. 687 to 689Column-1 represents a condition to form trigonometric equation. Column-2 represents the value of \(\sin\theta + \cos\theta\) and Column-3 represents the general value of \(\theta\) satisfying the trigonometric equation.Column-1Column-2Column-3(I) If \(2^{\sin\theta}\), \(\sqrt{2}\) and \(2^{\cos\theta}\) are three terms of a decreasing G.P.(i) \(\dfrac{\sqrt{3}+1}{2}\)(P) \(\theta = 2n\pi - \dfrac{\pi}{2}\)(II) If \(\cos\theta\), \(\sec\theta\) and \(\cot\theta\) are three positive numbers in H.P.(ii) \(\sqrt{2}\)(Q) \(\theta = 2n\pi + \dfrac{\pi}{6}\)(III) If \(2\log\sec\theta\), \(\log 2\) and \(2\log\text{cosec}\,\theta\) are in A.P.(iii) \(-1\)(R) \(\theta = 2n\pi + \dfrac{\pi}{2}\)(IV) If G.M. of \((2+\sin\theta)\), \((3+\sin\theta)\) and \((4+\sin\theta)\) is equal to cube root of 6.(iv) \(1\)(S) \(\theta = 2n\pi + \dfrac{\pi}{4}\)688. Which of the following options is the only correct combination?
Let a, b, c be three non-zero real numbers such that the equation acosx + 2bsinx = c, x ∈ [−π/2, π/2], has two distinct real roots α and β with α + β = π/3. Then, the value of b/a is __________________________.
If $x = y = z$, $x, y, z$ are in AP, and $\tan^{-1}x$, $\tan^{-1}y$, $\tan^{-1}z$ are also in AP, find the relationship between $x$ and $z$.
In \(\triangle ABC\), if incircle touches the sides \(AB\), \(BC\) and \(CA\) at \(P\), \(Q\) and \(R\) respectively and \(s - a = 3\), \(s - b = 5\) and \(s - c = 7\), then area of the quadrilateral \(QCRI\) is, where \(I\) is incentre of \(\triangle ABC\):[Note: Symbols used have usual meaning in \(\triangle ABC\).]