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CodeVID-M12-02-CT
Inverse Trigonometric Functions — Full Chapter Test
Name: ____________________
Roll No.: __________
Date: ____________
General Instructions
- This is a full-length test covering the whole chapter — every topic is included.
- All questions are compulsory.
- Section A carries 1 mark each, Section B 2 marks, Section C 3 marks and Section D 5 marks.
- Show all working for Sections B, C and D. Only final answers are given at the end — for full solutions, raise your doubts with your teacher.
Section A — Multiple Choice Questions
5 × 1 = 5 marks
1.
The principal value of $\sin^{-1}\!\left(\tfrac12\right)$ is:
- A.$\tfrac{\pi}{6}$
- B.$\tfrac{\pi}{3}$
- C.$\tfrac{\pi}{4}$
- D.$\tfrac{\pi}{2}$
2.
$\sin^{-1}x+\cos^{-1}x=$
- A.$\pi$
- B.$\tfrac{\pi}{2}$
- C.$0$
- D.$\tfrac{\pi}{4}$
3.
The range (principal branch) of $\cos^{-1}x$ is:
- A.$\left[-\tfrac{\pi}{2},\tfrac{\pi}{2}\right]$
- B.$[0,\pi]$
- C.$(0,\pi)$
- D.$\mathbb{R}$
4.
$\tan^{-1}x+\cot^{-1}x=$
- A.$\tfrac{\pi}{2}$
- B.$\pi$
- C.$0$
- D.$1$
5.
$\sin^{-1}\!\left(\sin\tfrac{2\pi}{3}\right)=$
- A.$\tfrac{2\pi}{3}$
- B.$\tfrac{\pi}{3}$
- C.$-\tfrac{\pi}{3}$
- D.$\tfrac{\pi}{6}$
Section B — Short Answer (2 marks)
4 × 2 = 8 marks
6.
Find the principal value of $\csc^{-1}(2)$.
7.
Evaluate $\sin^{-1}\!\left(\tfrac{\sqrt3}{2}\right)+\cos^{-1}\!\left(\tfrac{\sqrt3}{2}\right)$.
8.
Find the principal value of $\sec^{-1}(2)$.
9.
Simplify $\cos^{-1}(-x)$ in terms of $\cos^{-1}x$.
Section C — Short Answer (3 marks)
4 × 3 = 12 marks
10.
Evaluate $\cos^{-1}\!\left(\tfrac12\right)+2\sin^{-1}\!\left(\tfrac12\right)$.
11.
Express $2\tan^{-1}\!\left(\tfrac13\right)$ as a single arctangent.
12.
Evaluate $\sin^{-1}\!\left(\sin\tfrac{3\pi}{5}\right)$.
13.
Evaluate $\tan^{-1}2+\tan^{-1}3$.
Section D — Long Answer (5 marks)
2 × 5 = 10 marks
14.
Evaluate $\sin^{-1}\!\left(\sin\tfrac{2\pi}{3}\right)+\cos^{-1}\!\left(\cos\tfrac{7\pi}{6}\right)$.
15.
Prove that $\tan^{-1}1+\tan^{-1}2+\tan^{-1}3=\pi$.
Answer Key
Section A — Multiple Choice Questions
- (A) $\tfrac{\pi}{6}$
- (B) $\tfrac{\pi}{2}$
- (B) $[0,\pi]$
- (A) $\tfrac{\pi}{2}$
- (B) $\tfrac{\pi}{3}$
Section B — Short Answer (2 marks)
- $\tfrac{\pi}{6}$.
- $\tfrac{\pi}{2}$.
- $\tfrac{\pi}{3}$.
- $\pi-\cos^{-1}x$.
Section C — Short Answer (3 marks)
- $\tfrac{2\pi}{3}$.
- $\tan^{-1}\tfrac34$.
- $\tfrac{2\pi}{5}$.
- $\tfrac{3\pi}{4}$.
Section D — Long Answer (5 marks)
- $\tfrac{7\pi}{6}$.
- $=\pi$.
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