MathsCBSE Class 12 · Board exam 2025, set 65/1/1 · 3 marks · Short answer

CBSE Class 12 Mathematics 2025: question 31

Set in the CBSE Class 12 board exam of 2025, paper 65/1/1, section C, question 31, for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q31 · 3 marks · Short answerApplication of Integrals

Sketch the graph of $y = ∣ x + 3∣$ and find the area of the region enclosed by the curve, $x$ -axis, between $x = - 6$ and $x = 0$ , using integration.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 65/1/1.

Hints

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

Hint 1

Break $∣ x + 3∣$ into two linear pieces at $x = - 3$ .

Hint 2

Integrate each piece over its own interval, or use the symmetry of the V-shape about $x = - 3$ .

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.

Graph. $y = ∣ x + 3∣$ is V-shaped with its vertex at $(- 3, 0)$ . For $x \geq - 3$ it is the line $y = x + 3$ , through $(0, 3)$ . For $x < - 3$ it is the line $y = - (x + 3)$ , through $(- 6, 3)$ . The region between $x = - 6$ and $x = 0$ above the $x$ -axis consists of two congruent triangles, one on each side of $x = - 3$ , each of height 3. Shade both.
1 mark
Required area $= \int_{- 6}^{0} y d x = \int_{- 6}^{0} ∣ x + 3∣ d x$ .
½ mark
The graph is symmetric about $x = - 3$ , so the area $= 2 \int_{- 3}^{0} (x + 3) d x$ .
½ mark
Integrate: $= 2 [\frac{( x + 3 ) ^{2}}{2}]_{- 3}^{0}$ .
½ mark
Evaluate: $= 2 (\frac{9}{2} - 0) = 9$ sq units.
½ mark

Answer: Area = 9 sq units

Where the marks are

Correct sketch of $y = ∣ x + 3∣$ with the region shaded (1 mark)
Sets up the area as $\int_{- 6}^{0} y d x$ (½ mark)
Uses symmetry to write $2 \int_{- 3}^{0} (x + 3) d x$ , or splits at $x = - 3$ (½ mark)
Integrates correctly (½ mark)
Gets area 9 sq units (½ mark)

Revise the chapter

Chapter: Application of Integrals (notes, formulas, where marks are lost and every past question)
Topic: Area Under Lines and Modulus Graphs

Practise this chapterBoard-style questions from Application of Integrals, one at a time, marked against the answer key and marking points.Back to the full paper2025 board paper, set 65/1/1: every question, by section.

The question is transcribed from the official paper; figures are described in words. The solution, hints and chapter tag are ours. Spotted an error? Write tosupport@questionbuddy.in.

CBSE Class 12 Mathematics 2025: question 31

Hints

Step-by-step solution

Revise the chapter

More questions from Application of Integrals