MathsCBSE Class 10 · Basic · Board exam 2023, set 430/1/1 · 5 marks · Long answer

CBSE Class 10 Mathematics Basic 2023: question 32(a)

Set in the CBSE Class 10 board exam of 2023, paper 430/1/1, section D, question 32(a), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q32(a) · 5 marks · Long answerCompetency-basedQuadratic Equations

The diagonal of a rectangular field is 60 m more than the shorter side. If the longer side is 80 m more than the shorter side, find the length of the sides of the field.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 430/1/1.

Hints

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

Hint 1

Let the shorter side be $x$ and write the other side and diagonal in terms of $x$ .

Hint 2

Use Pythagoras theorem and check the discriminant of the resulting quadratic — and ask whether a diagonal can be shorter than a side.

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.

Let the shorter side be $x$ m. Then longer side $= (x + 80)$ m and diagonal $= (x + 60)$ m.
1 mark
By Pythagoras theorem: $x^{2} + (x + 80)^{2} = (x + 60)^{2}$ .
1 mark
$x^{2} + x^{2} + 160 x + 6400 = x^{2} + 120 x + 3600 \Rightarrow x^{2} + 40 x + 2800 = 0$ .
1 mark
Discriminant $D = 4 0^{2} - 4 (1) (2800) = 1600 - 11200 = - 9600 < 0$ , so the equation has no real roots.
1 mark
Hence no rectangle with the given data exists (in fact the diagonal must exceed the longer side, but here diagonal $x + 60$ is less than longer side $x + 80$ ). [With the NCERT data 'longer side 30 m more than shorter side': $x^{2} + (x + 30)^{2} = (x + 60)^{2} \Rightarrow x^{2} - 60 x - 2700 = 0 \Rightarrow (x - 90) (x + 30) = 0$ , so sides are 90 m and 120 m.]
1 mark

Answer: With the given data, $x^{2} + 40 x + 2800 = 0$ has no real roots, so no such field exists (question has an error; with the intended data '30 m more', sides are 90 m and 120 m)

Where the marks are

Assigns variables to sides and diagonal (1 mark)
Applies Pythagoras theorem (1 mark)
Forms the quadratic $x^{2} + 40 x + 2800 = 0$ (1 mark)
Shows discriminant is negative (1 mark)
Concludes no real solution exists (1 mark) — scheme awards full marks to any attempt

Revise the chapter

Chapter: Quadratic Equations (notes, formulas, where marks are lost and every past question)
Topic: Word Problems on Quadratic Equations

Practise this chapterBoard-style questions from Quadratic Equations, one at a time, marked against the answer key and marking points.Back to the full paper2023 board paper, set 430/1/1 (Basic): every question, by section.

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CBSE Class 10 Mathematics Basic 2023: question 32(a)

Hints

Step-by-step solution

Revise the chapter

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