MathsCBSE Class 12 · Board exam 2023, set 65/5/1 · 3 marks · Short answer

CBSE Class 12 Mathematics 2023: question 29

Set in the CBSE Class 12 board exam of 2023, paper 65/5/1, section C, question 29, for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q29 · 3 marks · Short answerApplication of Integrals

Find the area of the following region using integration :

${(x, y) : y^{2} \leq 2 x and y \geq x - 4}$

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 65/5/1.

Hints

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

Hint 1

Find where the line meets the parabola by substituting $x = y + 4$ into $y^{2} = 2 x$ .

Hint 2

Integrating with respect to $y$ avoids splitting the region: take (right curve − left curve) from $y = - 2$ to $y = 4$ .

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.

Figure: The parabola $y^{2} = 2 x$ has its vertex at O and opens to the right.
The line $y = x - 4$ passes through $(4, 0)$ and $(0, - 4)$ .
The region is the part inside the parabola that lies above the line. It is enclosed between the points $(2, - 2)$ and $(8, 4)$ . Shade this region.
1 mark
Solving $y^{2} = 2 x$ and $x = y + 4$ : $y^{2} = 2 y + 8$ , so $y^{2} - 2 y - 8 = 0$ , i.e. $(y - 4) (y + 2) = 0$ .
So $y = 4$ or $y = - 2$ , giving the points $(8, 4)$ and $(2, - 2)$ .
½ mark
Using horizontal strips, the line ( $x = y + 4$ ) lies to the right of the parabola ( $x = \frac{y ^{2}}{2}$ ):
Area $= \int_{- 2}^{4} [(y + 4) - \frac{y ^{2}}{2}] d y = [\frac{y ^{2}}{2} + 4 y - \frac{y ^{3}}{6}]_{- 2}^{4}$
1 mark
$= (8 + 16 - \frac{64}{6}) - (2 - 8 + \frac{8}{6}) = \frac{40}{3} - (- \frac{14}{3}) = 18$ sq units
½ mark

Answer: Area = 18 sq units

Where the marks are

Correct figure with shaded region (1 mark)
Finds intersection at $y = 4$ and $y = - 2$ (½ mark)
Sets up and integrates $\int_{- 2}^{4} [(y + 4) - \frac{y ^{2}}{2}] d y$ (1 mark)
Area = 18 sq units (½ mark)

Revise the chapter

Chapter: Application of Integrals (notes, formulas, where marks are lost and every past question)

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 paper2023 board paper, set 65/5/1: every question, by section.

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CBSE Class 12 Mathematics 2023: question 29

Hints

Step-by-step solution

Revise the chapter

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