If ^-1(3x+5)>(π)/(4), then find the range of the values of x.

MathsCBSE Class 12 · Sample paper, session 2024-25 · 2 marks · Very short answer

CBSE Class 12 Mathematics sample paper 2024-25: question 21

From the official CBSE sample question paper for 2024-25, 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

If $cot^{- 1} (3 x + 5) > \frac{π}{4}$ , then find the range of the values of $x$ .

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2024-25.

Hints

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

Hint 1

Express $\frac{π}{4}$ as $cot^{- 1}$ of some number.

Hint 2

Is $cot^{- 1} x$ increasing or decreasing? Decide what happens to the inequality sign when you drop $cot^{- 1}$ .

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.

Write the right side as an inverse cotangent: $cot^{- 1} (3 x + 5) > \frac{π}{4} = cot^{- 1} 1$ .
½ mark
$cot^{- 1} x$ is strictly decreasing on its domain $R$ , so the inequality reverses: $3 x + 5 < 1$ .
½ mark
$3 x < - 4 \Rightarrow x < - \frac{4}{3}$ , so $x \in (- \infty, - \frac{4}{3})$ .
1 mark

Answer: $x \in (- \infty, - \frac{4}{3})$

Where the marks are

Writes $\frac{π}{4} = cot^{- 1} 1$ (½ mark)
Uses that $cot^{- 1}$ is decreasing to get $3 x + 5 < 1$ (½ mark)
Obtains $x < - \frac{4}{3}$ , i.e. $x \in (- \infty, - \frac{4}{3})$ (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 paperSample paper 2024-25: every question, by section.

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CBSE Class 12 Mathematics sample paper 2024-25: question 21

Hints

Step-by-step solution

Revise the chapter

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