First slide

Functions (XII)

Question

If [2 sin x] + [cos x] = –3, then the range of the function $f (x) = \sin x + \sqrt{3} \cos x in [0, 2 π]$ (where [⋅] denotes the greatest integer function) is

Moderate

A

[–2, –1]
B

$(- 1, - \frac{1}{2})$
C

(–2, –1)
D

None of these

Solution

We have, [2 sin x] + [cos x] = –3
only if [2 sin x] = –2 and [cos x] = –1
⇒ –2 ≤ 2 sin x < –1 and –1 ≤ cos x < 0
$\begin{array}{l} \Rightarrow - 1 \leq \sin x < - \frac{1}{2} and - 1 \leq \cos x < 0 \\ \Rightarrow \frac{7 π}{6} < x < \frac{11 π}{6} and \frac{π}{2} < x < \frac{3 π}{2} \\ \Rightarrow \frac{7 π}{6} < x < \frac{3 π}{2} \end{array}$
For the above values of x, $\sin x + \sqrt{3} \cos x = 2 \sin (\frac{π}{3} + x)$ lies between –2 and –1.
$∴$ Range of f (x) is (–2, –1).

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