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Vidaara.orgClass 12 · Mathematics
CodeVID-M12-02-IPR-01
Properties of Inverse Trig — Assignment
Chapter: Inverse Trigonometric Functions
Topic: Properties of Inverse Trigonometric Functions
Maximum Marks: 35
Time: 75 minutes
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.
$\sin^{-1}x+\cos^{-1}x=$
  • A.$\pi$
  • B.$\tfrac{\pi}{2}$
  • C.$0$
  • D.$\tfrac{\pi}{4}$
2.
$\tan^{-1}x+\cot^{-1}x=$
  • A.$\tfrac{\pi}{2}$
  • B.$\pi$
  • C.$0$
  • D.$1$
3.
$\cos^{-1}(-x)=$
  • A.$-\cos^{-1}x$
  • B.$\pi-\cos^{-1}x$
  • C.$\tfrac{\pi}{2}-\cos^{-1}x$
  • D.$\pi+\cos^{-1}x$
4.
$2\tan^{-1}x=\tan^{-1}\dfrac{2x}{1-x^2}$ is valid for:
  • A.$|x|<1$
  • B.$|x|>1$
  • C.all $x$
  • D.$x=1$
5.
$\tan^{-1}1+\tan^{-1}2+\tan^{-1}3=$
  • A.$\tfrac{\pi}{2}$
  • B.$\pi$
  • C.$\tfrac{3\pi}{4}$
  • D.$2\pi$
Section B — Short Answer (2 marks) 4 × 2 = 8 marks
6.
Evaluate $\sin^{-1}\!\left(\tfrac{\sqrt3}{2}\right)+\cos^{-1}\!\left(\tfrac{\sqrt3}{2}\right)$.
7.
Simplify $\cos^{-1}(-x)$ in terms of $\cos^{-1}x$.
8.
Write $2\sin^{-1}x$ as a single inverse function (for $|x|\le1$).
9.
Evaluate $\tan^{-1}1+\tan^{-1}\tfrac12$.
Section C — Short Answer (3 marks) 4 × 3 = 12 marks
10.
Express $2\tan^{-1}\!\left(\tfrac13\right)$ as a single arctangent.
11.
Evaluate $\tan^{-1}2+\tan^{-1}3$.
12.
Prove $\sin^{-1}x+\cos^{-1}x=\tfrac{\pi}{2}$ and verify at $x=\tfrac12$.
13.
Simplify $\tan^{-1}\!\left(\dfrac{1-\cos x}{\sin x}\right)$, $0
Section D — Long Answer (5 marks) 2 × 5 = 10 marks
14.
Prove that $\tan^{-1}1+\tan^{-1}2+\tan^{-1}3=\pi$.
15.
Show that $2\tan^{-1}\!\left(\tfrac12\right)=\tan^{-1}\!\left(\tfrac43\right)$.

Answer Key

Section A — Multiple Choice Questions
  1. (B) $\tfrac{\pi}{2}$
  2. (A) $\tfrac{\pi}{2}$
  3. (B) $\pi-\cos^{-1}x$
  4. (A) $|x|<1$
  5. (B) $\pi$
Section B — Short Answer (2 marks)
  1. $\tfrac{\pi}{2}$.
  2. $\pi-\cos^{-1}x$.
  3. $\sin^{-1}\!\left(2x\sqrt{1-x^2}\right)$.
  4. $\tan^{-1}3$.
Section C — Short Answer (3 marks)
  1. $\tan^{-1}\tfrac34$.
  2. $\tfrac{3\pi}{4}$.
  3. Identity holds; at $x=\tfrac12$ value $=\tfrac{\pi}{2}$.
  4. $\tfrac{x}{2}$.
Section D — Long Answer (5 marks)
  1. $=\pi$.
  2. Verified ($=\tan^{-1}\tfrac43$).
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