MathsCBSE Class 12 · Sample paper, session 2024-25 · 5 marks · Long answer

CBSE Class 12 Mathematics sample paper 2024-25: question 32

From the official CBSE sample question paper for 2024-25, section D, question 32, for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q32 · 5 marks · Long answerApplication of Integrals

Draw the rough sketch of the curve $y = 20 cos 2 x$ ; (where $\frac{π}{6} \leq x \leq \frac{π}{3}$ ).
Using integration, find the area of the region bounded by the curve $y = 20 cos 2 x$ from the ordinates $x = \frac{π}{6}$ to $x = \frac{π}{3}$ and the $x$ -axis.

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

Find where $cos 2 x = 0$ inside the interval — the curve changes sign there.

Hint 2

Integrate separately on each side of that point and take the absolute value of the negative part.

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.

Sketch: $y = 20 cos 2 x$ on $[\frac{π}{6}, \frac{π}{3}]$ passes through $(\frac{π}{6}, 10)$ , crosses the $x$ -axis at $(\frac{π}{4}, 0)$ and ends at $(\frac{π}{3}, - 10)$ ; it is above the $x$ -axis on $[\frac{π}{6}, \frac{π}{4}]$ and below it on $[\frac{π}{4}, \frac{π}{3}]$ . Shade the regions between the curve and the $x$ -axis.
1 mark
Required area $= \int_{π /6}^{π /4} 20 cos 2 x d x + \int_{π /4}^{π /3} 20 cos 2 x d x$ .
2 marks
$= 20 [\frac{s i n 2 x}{2}]_{π /6}^{π /4} + 20 [\frac{s i n 2 x}{2}]_{π /4}^{π /3} = 10 (1 - \frac{3}{2}) + 10 (\frac{3}{2} - 1)$ .
1 mark
$= 10 (1 - \frac{3}{2}) + 10 (1 - \frac{3}{2}) = 20 (1 - \frac{3}{2}) = (20 - 103)$ sq. units.
1 mark

Answer: $20 (1 - \frac{3}{2}) = 20 - 103$ sq. units

Where the marks are

Correct rough sketch showing crossing at $x = π /4$ (1 mark)
Sets up area as sum of two integrals, with modulus for the part below the axis (1+1 marks)
Integrates correctly (1 mark)
Final area $20 (1 - \frac{3}{2})$ sq units (1 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 paperSample paper 2024-25: every question, by section.

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CBSE Class 12 Mathematics sample paper 2024-25: question 32

Hints

Step-by-step solution

Revise the chapter

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