Question Statement:
For 0 ≤ x ≤ 2𝜋, sin x and cos x are both decreasing functions in the interval ________.
(A) (0,
𝜋
2) (B) (
𝜋
2 , 𝜋) (C) (𝜋,
3𝜋
2 ) (D) (
3𝜋
2 , 2𝜋)
Options:
-
(A) (0,
𝜋
2)
-
(B) (
𝜋
2 , 𝜋)
-
(C) (𝜋,
3𝜋
2 )
-
(D) (
3𝜋
2 , 2𝜋)
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE BT 2018 Q.03)**
**1. Governing Concept & Question Classification**:
- **Section**: General Aptitude (GA)
- **Question Type**: **MCQ** (1 Mark)
- **Topic**: Calculus / Trigonometric Functions (Monotonicity)
**2. Detailed Step-by-Step Analysis**:
- A differentiable function $f(x)$ is decreasing on an open interval where its first derivative is negative, i.e., $f'(x) < 0$.
- For $f(x) = \sin x$:
$f'(x) = \cos x$
$\cos x < 0$ in Quadrant II and Quadrant III, i.e., $x \in (\pi/2, 3\pi/2)$.
- For $g(x) = \cos x$:
$g'(x) = -\sin x$
Since $\sin x > 0$ for $x \in (0, \pi)$, $-\sin x < 0$ in Quadrant I and Quadrant II, i.e., $x \in (0, \pi)$.
- For both $\sin x$ and $\cos x$ to be simultaneously decreasing, we take the intersection of their decreasing intervals:
$\left( rac{\pi}{2}, rac{3\pi}{2}
ight) \cap (0, \pi) = \left( rac{\pi}{2}, \pi
ight)$
- In the second quadrant $(\pi/2, \pi)$, as $x$ increases from $\pi/2$ to $\pi$:
- $\sin x$ decreases from $1$ to $0$.
- $\cos x$ decreases from $0$ to $-1$.
Both functions are strictly decreasing.
- **Option-by-Option Evaluation**:
- **Option (A) [INCORRECT]**: $(0, \pi/2)$ is Quadrant I, where $\sin x$ increases from $0$ to $1$.
- **Option (B) [CORRECT]**: $(\pi/2, \pi)$ is Quadrant II, where both $\sin x$ and $\cos x$ are strictly decreasing.
- **Option (C) [INCORRECT]**: $(\pi, 3\pi/2)$ is Quadrant III, where $\cos x$ increases from $-1$ to $0$.
- **Option (D) [INCORRECT]**: $(3\pi/2, 2\pi)$ is Quadrant IV, where $\sin x$ increases from $-1$ to $0$.
**3. Final Answer / Verification Summary**:
- **Official GATE Answer Key**: **B**
- **Scoring Criteria**: +1.0 Mark (Negative marking: -0.33 marks).