Maths · Inverse trigonometrical functions and their properties
Assertion then Reason \)
Assertion \( f_{i=1}^{2 n} \sin ^{-1} x_{i}=n \pi \forall n \epsilon N \) then \( \sum_{i=1}^{2 n} x_{i}= \) \( \sum_{i=1}^{2 n} x_{i}^{2}=\sum_{i=1}^{2 n} x_{i}^{n}=2 n \) Reason \( -\frac{\pi}{2} \leq \sin ^{-1} x \leq \frac{\pi}{2} \forall x \epsilon[-1,1] \)
- A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
- B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
- C. Assertion is correct but Reason is incorrect
- D. Assertion false but Reason is true
Step-by-step solution
The range of sin⁻¹ x is [-π/2, π/2]. For the sum of 2n terms of sin⁻¹ xᵢ to equal nπ = (2n)(π/2), each term must be π/2, so sin⁻¹ xᵢ = π/2 ⇒ xᵢ = 1 for all i. Then ∑xᵢ = 2n, ∑xᵢ² = 2n, ∑xᵢⁿ = 2n. Thus the Assertion holds. The Reason correctly states the range, which is used to deduce each term equals π/2, so it is the correct explanation.
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