MathsCBSE Class 10 · Standard · Board exam 2023, set 30/4/1 · 4 marks · Case-based

CBSE Class 10 Mathematics Standard 2023: question 37

Set in the CBSE Class 10 board exam of 2023, paper 30/4/1, section E, question 37, for 4 marks. Try it first; the hints and our step-by-step solution are below.

Q37 · 4 marks · Case-basedCompetency-basedCoordinate Geometry

Jagdish has a field which is in the shape of a right angled triangle AQC. He wants to leave a space in the form of a square PQRS inside the field for growing wheat and the remaining for growing vegetables (as shown in the figure). In the field, there is a pole marked as O.

Based on the above information, answer the following questions:

(i)
Taking O as origin, coordinates of P are (-200, 0) and of Q are (200, 0). PQRS being a square, what are the coordinates of R and S?
1 mark
(ii)(a)
What is the area of square PQRS?
2 marks
OR
(ii)(b)
What is the length of diagonal PR in square PQRS?
2 marks
(iii)
If S divides CA in the ratio K:1, what is the value of K, where point A is (200, 800)?
1 mark

Figure in the paper: Coordinate grid with X'OX horizontal and YOY' vertical, O at origin. Right-angled triangle AQC: C on the negative x-axis, Q on the positive x-axis at (200, 0), A directly above Q at (200, 800). Hypotenuse CA is a straight line passing through B on the y-axis. Square PQRS sits on the x-axis with P (-200, 0), Q (200, 0), R and S at the top corners (R on the right, S on the left), so the square has side 400. Points labelled on the figure: P, Q, R, S, O, A, B, C. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 30/4/1.

Hints

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

Hint 1

All sides of a square are equal, and PQ lies along the x-axis.

Hint 2

The diagonal of a square of side $a$ is $a^{2} + a^{2}$ .

Hint 3

For (iii), first find C as the point where line AS meets the x-axis, then use the section formula for the x-coordinate.

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.

(i) Side $P Q = 200 - (- 200) = 400$ . R lies directly above Q and S directly above P, each at height 400.
So $R (200, 400)$ and $S (- 200, 400)$ .
1 mark
(ii)(a) Side of the square: $P Q = 200 + 200 = 400$ units.
1 mark
(ii)(a) Area of square PQRS $= 400 \times 400 = 160000$ sq. units.
1 mark
(ii)(b) Using the distance formula with $P (- 200, 0)$ and $R (200, 400)$ :
$P R = (200 + 200)^{2} + (400 - 0)^{2} = 40 0^{2} + 40 0^{2}$
1 mark
(ii)(b) $P R = 320000 = 4002$ units ( $\approx 565.6$ units)
1 mark
(iii) Line CA passes through $A (200, 800)$ and $S (- 200, 400)$ . Its slope is $\frac{800 - 400}{200 + 200} = 1$ , so the line is $y = x + 600$ .
C is on the x-axis ( $y = 0$ ), so $C (- 600, 0)$ .
Let S divide CA in the ratio $k : 1$ . By the section formula:
$- 200 = \frac{200 k - 600}{k + 1}$
$\Rightarrow - 200 k - 200 = 200 k - 600$
$\Rightarrow 400 k = 400$
$\Rightarrow k = 1$
Check with the y-coordinate: $\frac{800 \cdot 1 + 0}{2} = 400$ ✓
1 mark

Answer: (i) R(200, 400), S(−200, 400); (ii)(a) 160000 sq. units; (ii)(b) 400√2 units; (iii) K = 1

Where the marks are

(i) R(200, 400) and S(−200, 400) (½ + ½ mark)
(ii)(a) Side = 400 (1 mark)
(ii)(a) Area = 160000 sq. units (1 mark)
(ii)(b) Applies the distance formula / Pythagoras (1 mark)
(ii)(b) PR = $4002$ (1 mark)
(iii) Uses the section formula with C(−600, 0) to get K = 1 (1 mark)

Revise the chapter

Chapter: Coordinate Geometry (notes, formulas, where marks are lost and every past question)
Topic: Section Formula (Internal Division)

Practise this chapterBoard-style questions from Coordinate Geometry, one at a time, marked against the answer key and marking points.Back to the full paper2023 board paper, set 30/4/1 (Standard): 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 10 Mathematics Standard 2023: question 37

Hints

Step-by-step solution

Revise the chapter

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