MathsCBSE Class 12 · Board exam 2023, set 65/5/1 · 2 marks · Very short answer

CBSE Class 12 Mathematics 2023: question 23

Set in the CBSE Class 12 board exam of 2023, paper 65/5/1, section B, question 23, for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q23 · 2 marks · Very short answerApplication of Integrals

Sketch the region bounded by the lines $2 x + y = 8$ , $y = 2$ , $y = 4$ and the y-axis. Hence, obtain its area using integration.

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

The region is bounded by the y-axis on the left, so integrate with respect to $y$ .

Hint 2

Write $x$ in terms of $y$ from $2 x + y = 8$ and integrate 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 line $2 x + y = 8$ meets the y-axis at $(0, 8)$ and the x-axis at $(4, 0)$ .
The horizontal lines $y = 2$ and $y = 4$ cut it at $E (3, 2)$ and $F (2, 4)$ .
The required region is the trapezium with vertices $C (0, 2)$ , $E (3, 2)$ , $F (2, 4)$ and $B (0, 4)$ , lying between the y-axis and the line. Shade this region.
½ mark
Using horizontal strips, $x = \frac{8 - y}{2}$ and $y$ runs from $2$ to $4$ :
Area $= \int_{2}^{4} x d y = \int_{2}^{4} \frac{1}{2} (8 - y) d y$
½ mark
$= \frac{1}{2} [8 y - \frac{y ^{2}}{2}]_{2}^{4} = \frac{1}{2} [(32 - 8) - (16 - 2)]$
½ mark
$= \frac{1}{2} (24 - 14) = 5$ sq units
½ mark

Answer: Area = 5 sq units

Where the marks are

Correct sketch of the trapezium region (½ mark)
Sets up the integral $\int_{2}^{4} \frac{1}{2} (8 - y) d y$ (½ mark)
Correct integration with limits (½ mark)
Area = 5 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 paper2023 board paper, set 65/5/1: 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 2023: question 23

Hints

Step-by-step solution

Revise the chapter

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