Find value of k if ^-1 [k (2^-1(√(3))/(2) ) ]=(π)/(3).

MathsCBSE Class 12 · Board exam 2024, set 65/5/1 · 2 marks · Very short answer

CBSE Class 12 Mathematics 2024: question 21

Set in the CBSE Class 12 board exam of 2024, paper 65/5/1, section B, question 21, for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q21 · 2 marks · Very short answerInverse Trigonometric Functions

Find value of k if $sin^{- 1} [k tan (2 cos^{- 1} \frac{3}{2})] = \frac{π}{3}$ .

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 65/5/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

First find the principal value of $cos^{- 1} \frac{3}{2}$ .

Hint 2

If $sin^{- 1} P = \frac{π}{3}$ , then $P = sin \frac{π}{3}$ .

Hint 3

Substitute $tan \frac{π}{3} = 3$ and solve the linear equation in k.

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

Take sine of both sides: $k tan (2 cos^{- 1} \frac{3}{2}) = sin \frac{π}{3}$ .
½ mark
Since $cos^{- 1} \frac{3}{2} = \frac{π}{6}$ [principal value in $[0, π]$ ], we get $2 cos^{- 1} \frac{3}{2} = \frac{π}{3}$ . So $k tan \frac{π}{3} = \frac{3}{2}$ .
1 mark
$k 3 = \frac{3}{2} \Rightarrow k = \frac{1}{2}$ .
½ mark

Answer: $k = \frac{1}{2}$

Where the marks are

Removes $sin^{- 1}$ to get $k tan (\dots) = sin \frac{π}{3}$ (½ mark)
Uses $cos^{- 1} \frac{3}{2} = \frac{π}{6}$ and writes $sin \frac{π}{3} = \frac{3}{2}$ (1 mark)
Obtains $k = \frac{1}{2}$ (½ mark)

Revise the chapter

Chapter: Inverse Trigonometric Functions (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Inverse Trigonometric Functions, one at a time, marked against the answer key and marking points.Back to the full paper2024 board paper, set 65/5/1: every question, by section.

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CBSE Class 12 Mathematics 2024: question 21

Hints

Step-by-step solution

Revise the chapter

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