Assertion (A) : The range of the function f(x) = 2^-1x + (3π)/(2), where x [-1, 1], is [(π)/(2), (5π)/(2) ]. Reason (R) : The range of the principal value branch of ^-1(x) is [0, π]. (A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A). (B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A). (C) Assertion (A) is true and Reason (R) is false. (D) Assertion (A) is false and Reason (R) is true.

MathsCBSE Class 12 · Board exam 2023, set 65/1/1 · 1 mark · Assertion-reason

CBSE Class 12 Mathematics 2023: question 19

Set in the CBSE Class 12 board exam of 2023, paper 65/1/1, section A, question 19, for 1 mark. Try it first; the hints and our step-by-step solution are below.

Q19 · 1 mark · Assertion-reasonCompetency-basedInverse Trigonometric Functions

Assertion (A) : The range of the function $f (x) = 2 sin^{- 1} x + \frac{3 π}{2}$ , where $x \in [- 1, 1]$ , is $[\frac{π}{2}, \frac{5 π}{2}]$ .
Reason (R) : The range of the principal value branch of $sin^{- 1} (x)$ is $[0, π]$ .

(A)Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
(B)Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
(C)Assertion (A) is true and Reason (R) is false.
(D)Assertion (A) is false and Reason (R) is true.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 65/1/1.

Hints

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

Hint 1

Recall the principal range of $sin^{- 1}$ .

Hint 2

Scale by 2 and shift by $3 π /2$ .

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.

$sin^{- 1} x \in [- \frac{π}{2}, \frac{π}{2}]$ , so $2 sin^{- 1} x + \frac{3 π}{2} \in [\frac{π}{2}, \frac{5 π}{2}]$ : A true.
½ mark
Principal range of $sin^{- 1}$ is $[- \frac{π}{2}, \frac{π}{2}]$ , not $[0, π]$ : R false. Option (C).
½ mark

Answer: Option (C)

Where the marks are

A true, R false with reason (1 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 paper2023 board paper, set 65/1/1: every question, by section.

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CBSE Class 12 Mathematics 2023: question 19

Hints

Step-by-step solution

Revise the chapter

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