Vidaara.orgClass 12 · Mathematics
CodeVID-M12-02-IPV-01
Principal Values of Inverse Trig — Assignment
Name: ____________________
Roll No.: __________
Date: ____________
General Instructions
- 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.
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.$\mathbb{R}$
3.
$\sin^{-1}\!\left(\sin\tfrac{2\pi}{3}\right)=$
- A.$\tfrac{2\pi}{3}$
- B.$\tfrac{\pi}{3}$
- C.$-\tfrac{\pi}{3}$
- D.$\tfrac{\pi}{6}$
4.
$\tan^{-1}(\sqrt3)=$
- A.$\tfrac{\pi}{6}$
- B.$\tfrac{\pi}{4}$
- C.$\tfrac{\pi}{3}$
- D.$\tfrac{\pi}{2}$
5.
$\cos^{-1}\!\left(\cos\tfrac{7\pi}{6}\right)=$
- A.$\tfrac{7\pi}{6}$
- B.$\tfrac{5\pi}{6}$
- C.$\tfrac{\pi}{6}$
- D.$-\tfrac{5\pi}{6}$
Section B — Short Answer (2 marks)
4 × 2 = 8 marks
6.
Find the principal value of $\csc^{-1}(2)$.
7.
Find the principal value of $\sec^{-1}(2)$.
8.
Find the principal value of $\cot^{-1}(1)$.
9.
Find the principal value of $\sin^{-1}\!\left(-\tfrac12\right)$.
Section C — Short Answer (3 marks)
4 × 3 = 12 marks
10.
Evaluate $\cos^{-1}\!\left(\tfrac12\right)+2\sin^{-1}\!\left(\tfrac12\right)$.
11.
Evaluate $\sin^{-1}\!\left(\sin\tfrac{3\pi}{5}\right)$.
12.
Find the principal value of $\tan^{-1}\!\left(-\tfrac{1}{\sqrt3}\right)$.
13.
Evaluate $\sin\!\left(\tfrac{\pi}{3}-\sin^{-1}\!\left(-\tfrac12\right)\right)$.
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.
Find the principal values of $\sin^{-1}\!\left(\tfrac{\sqrt3}{2}\right),\ \cos^{-1}\!\left(-\tfrac12\right)$ and $\tan^{-1}(-1)$.
Answer Key
Section A — Multiple Choice Questions
- (A) $\tfrac{\pi}{6}$
- (B) $[0,\pi]$
- (B) $\tfrac{\pi}{3}$
- (C) $\tfrac{\pi}{3}$
- (B) $\tfrac{5\pi}{6}$
Section B — Short Answer (2 marks)
- $\tfrac{\pi}{6}$.
- $\tfrac{\pi}{3}$.
- $\tfrac{\pi}{4}$.
- $-\tfrac{\pi}{6}$.
Section C — Short Answer (3 marks)
- $\tfrac{2\pi}{3}$.
- $\tfrac{2\pi}{5}$.
- $-\tfrac{\pi}{6}$.
- $1$.
Section D — Long Answer (5 marks)
- $\tfrac{7\pi}{6}$.
- $\tfrac{\pi}{3},\ \tfrac{2\pi}{3},\ -\tfrac{\pi}{4}$.
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