Q1. [1]
Directions: A statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option. Assertion (A): For an acute angle θ, $\sin\theta = \frac{3}{5} \Rightarrow \cos\theta = -\frac{4}{5}$. Reason (R): For any value of θ, $(0° \le \theta \le 90°)$, $\sin^2\theta + \cos^2\theta = 1$.
- A Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
- B Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of (A).
- C Assertion (A) is true, but Reason (R) is false.
- D Assertion (A) is false, but Reason (R) is true.
Previously asked in CBSE board exam
2025 30/5/1 Q19
Generated by claude-sonnet-4-6 · 2026-06-15 10:26 · grounding rag
Model Answer
Option D: Assertion (A) is false, but Reason (R) is true.
The Reason is correct: $\sin^2\theta + \cos^2\theta = 1$ for $0° \leq \theta \leq 90°$. However, the Assertion is false — since $\theta$ is acute, $\cos\theta$ must be positive. Using R: $\cos\theta = +\frac{4}{5}$, not $-\frac{4}{5}$.
Source: Chapter 8, Section 8.5 Summary (Identity 6)
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Explanation
- Why R is true: $\sin^2\theta + \cos^2\theta = 1$ is a fundamental identity valid for all $\theta$ in $[0°, 90°]$, as stated in the Summary.
- Why A is false: For an acute angle, all trigonometric ratios are positive. With $\sin\theta = \frac{3}{5}$, applying the identity gives $\cos\theta = \sqrt{1 - \frac{9}{25}} = \frac{4}{5}$ (positive). A negative cosine is impossible in the first quadrant.
- Key examiner tip: Always check the quadrant/range of $\theta$. The Assertion uses R incorrectly by taking the negative root, making A false while R remains true → Option D.
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