Inverse trigonometric functions Practice Questions

Question 1 of 10 Authentic UPSC NDA Question

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Question 2 of 10 Authentic UPSC NDA Question

What is the value of

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Question 3 of 10 Authentic UPSC NDA Question

Let sin⁻¹x + sin⁻¹y + sin⁻¹z = 3π/2 for 0 ≤ x, y, z ≤ 1. What is the value of x¹⁰⁰⁰ + y¹⁰⁰¹ + z¹⁰⁰² ?

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Question 4 of 10 Authentic UPSC NDA Question

tan⁻¹x + cot⁻¹x = π/2 holds, when

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Question 5 of 10 Authentic UPSC NDA Question

The equation sin⁻¹x − cos⁻¹x = π/6 has

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Question 6 of 10 Authentic UPSC NDA Question

Consider the following in respect of inverse circular functions:

I. sin⁻¹(−x) = −sin⁻¹x

II. cos⁻¹(−x) = cos⁻¹x

III. tan⁻¹(−x) = π − tan⁻¹x

IV. cot⁻¹(−x) = π − cot⁻¹x

How many of the above are correct?

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Solution

Option (b) is correct.

Explanation:

To determine the validity of the statements regarding inverse circular functions:

\[ \text{I. } \sin^{-1}(-x) = -\sin^{-1}(x) \quad \text{(True)} \]

\[ \text{II. } \cos^{-1}(-x) = \cos^{-1}(x) \quad \text{(False)} \]

\[ \text{III. } \tan^{-1}(-x) = -\tan^{-1}(x) \quad \text{(False)} \]

\[ \text{IV. } \cot^{-1}(-x) = \pi - \cot^{-1}(x) \quad \text{(True)} \]

Thus, statements I and IV are correct, while II and III are incorrect. Therefore, two statements are valid.

📌 Hints / Properties Used:

Odd function: \( f(-x) = -f(x) \) for sine and tangent.
To find \( \cos^{-1}(-x) \) and \( \tan^{-1}(-x) \), analyze the range of the functions.
Cotangent identity for negative input.

📊 Visual Diagram Suggestion:

Illustrate the unit circle highlighting angles corresponding to inverse functions, emphasizing odd/even properties visually.

📖 Factual Verification & Reference:

Verified against NCERT Class XI Mathematics, Chapter 3 (Trigonometric Functions).

Question 7 of 10 Authentic UPSC NDA Question

What is equal to?

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Question 8 of 10 Authentic UPSC NDA Question

What is cot²(sec^–1 2) + tan² (cosec^–1 3) equal to?

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Question 9 of 10 Authentic UPSC NDA Question

If 4sin^–1 x +cos^–1 x = p, then what is sin^–1 x + 4cos^{– 1} x equal to?

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Question 10 of 10 Authentic UPSC NDA Question

If cos^–1x = sin^–1x, then which one of the following is correct?

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Inverse trigonometric functions Questions

Solution