GATE 2018 IN GA Q.3: solution and answer

GATE 2018 · INGA Q.3 · 1 mark · MCQ

For $0 \leq x \leq 2 π$ , $sin x$ and $cos x$ are both decreasing functions in the interval ______.

(A)
$(0, \frac{π}{2})$
(B)
$(\frac{π}{2}, π)$
(C)
$(π, \frac{3 π}{2})$
(D)
$(\frac{3 π}{2}, 2 π)$

See the question in the paper

Answer and solution

ANSWER

(B) $(\frac{π}{2}, π)$ . This is the second quadrant, the only one where $sin x$ (falling from $1$ to $0$ ) and $cos x$ (falling from $0$ to $- 1$ ) both decrease.

Official answer: BSource: Official answer key, GATE 2018 IN (IIT Guwahati)

A differentiable function decreases on an interval where its derivative is negative there. $\frac{d}{d x} sin x = cos x$ and $\frac{d}{d x} cos x = - sin x$ .
$sin x$ decreases where $cos x < 0$ . In $[0, 2 π]$ that is the 2nd and 3rd quadrants, $(\frac{π}{2}, \frac{3 π}{2})$ . Over this interval $sin x$ falls from $1$ at $\frac{π}{2}$ to $- 1$ at $\frac{3 π}{2}$ .
$cos x$ decreases where $- sin x < 0$ , i.e. where $sin x > 0$ . In $[0, 2 π]$ that is the 1st and 2nd quadrants, $(0, π)$ . Over this interval $cos x$ falls from $1$ at $0$ to $- 1$ at $π$ .
Both functions decrease on the overlap of the two intervals: $(\frac{π}{2}, \frac{3 π}{2}) \cap (0, π) = (\frac{π}{2}, π)$
Check the other options. (A) $(0, \frac{π}{2})$ : $sin x$ rises from $0$ to $1$ . (C) $(π, \frac{3 π}{2})$ : $sin x$ falls, but $cos x$ rises from $- 1$ to $0$ . (D) $(\frac{3 π}{2}, 2 π)$ : $sin x$ rises from $- 1$ to $0$ and $cos x$ rises from $0$ to $1$ . Only (B) works.

KEY IDEA

Each function decreases where its derivative is negative: $sin x$ where $cos x < 0$ , which is $(\frac{π}{2}, \frac{3 π}{2})$ , and $cos x$ where $sin x > 0$ , which is $(0, π)$ . The answer is where the two intervals overlap. You can also check each quadrant on the unit circle: the second quadrant is the only one where both coordinates are shrinking.

WATCH OUT FOR

Picking (C) because $sin x$ and $cos x$ are both negative in the third quadrant. Being negative is not the same as decreasing: in $(π, \frac{3 π}{2})$ , $cos x$ climbs from $- 1$ to $0$ . Another slip is mixing up which derivative controls which function, for example deciding that $sin x$ decreases wherever $sin x > 0$ .

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GATE 2018 IN GA Q.3: solution and answer

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