A rectangular floor area can be completely tiled with 200 square tiles. If the side length of each tile is increased by 1 unit, it would take only 128 tiles to cover the floor. (Figure: photograph of a floor covered with grey rectangular tiles.) (i) Assuming the original length of each side of a tile be x units, make a quadratic equation from the above information. (ii) Write the corresponding quadratic equation in standard form. (iii) (a) Find the value of x, the length of side of a tile, by factorisation. (iii) (b) Solve the quadratic equation for x, using quadratic formula.

MathsCBSE Class 10 · Standard · Board exam 2024, set 30/1/1 · 4 marks · Case-based

CBSE Class 10 Mathematics Standard 2024: question 36

Set in the CBSE Class 10 board exam of 2024, paper 30/1/1, section E, question 36, for 4 marks. Try it first; the hints and our step-by-step solution are below.

Q36 · 4 marks · Case-basedCompetency-basedQuadratic Equations

A rectangular floor area can be completely tiled with 200 square tiles. If the side length of each tile is increased by 1 unit, it would take only 128 tiles to cover the floor.

(Figure: photograph of a floor covered with grey rectangular tiles.)

(i)
Assuming the original length of each side of a tile be $x$ units, make a quadratic equation from the above information.
1 mark
(ii)
Write the corresponding quadratic equation in standard form.
1 mark
(iii)
(a) Find the value of $x$ , the length of side of a tile, by factorisation.
2 marks
OR
(iii)
(b) Solve the quadratic equation for $x$ , using quadratic formula.
2 marks

Figure in the paper: Photograph of a floor laid with grey rectangular tiles in a staggered pattern. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 30/1/1.

Hints

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

Hint 1

The floor area stays the same: number of tiles × area of one tile.

Hint 2

Equate 200x² with 128(x+1)², then divide by 8 to simplify.

Hint 3

For factorising, find two numbers with product −144 and sum −32. Reject any negative length.

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) Area of one original tile $= x^{2}$ , so floor area $= 200 x^{2}$ .
New side $= (x + 1)$ , so floor area $= 128 (x + 1)^{2}$ .
The floor is the same, so:
$200 x^{2} = 128 (x + 1)^{2}$
1 mark
(ii) Divide both sides by 8:
$25 x^{2} = 16 (x^{2} + 2 x + 1) = 16 x^{2} + 32 x + 16$
$\Rightarrow 9 x^{2} - 32 x - 16 = 0$
1 mark
(iii)(a) Split the middle term ( $9 \times (- 16) = - 144 = - 36 \times 4$ ):
$9 x^{2} - 36 x + 4 x - 16 = 0$
$9 x (x - 4) + 4 (x - 4) = 0$
$(9 x + 4) (x - 4) = 0$
1 mark
(iii)(a) So $x = 4$ or $x = - \frac{4}{9}$ .
A length cannot be negative, so $x = 4$ units.
1 mark
(iii)(b) For $9 x^{2} - 32 x - 16 = 0$ : $a = 9$ , $b = - 32$ , $c = - 16$ .
$D = b^{2} - 4 a c = 1024 + 576 = 1600$
$x = \frac{32 \pm 1600}{18} = \frac{32 \pm 40}{18}$
1 mark
(iii)(b) So $x = \frac{72}{18} = 4$ or $x = \frac{- 8}{18} = - \frac{4}{9}$ .
Since a length cannot be negative, $x = 4$ units.
1 mark

Answer: (i) 200x² = 128(x+1)²; (ii) 9x² − 32x − 16 = 0; (iii)(a) x = 4 units; (iii)(b) x = 4 units (rejecting −4/9)

Where the marks are

(i) Equates the two floor areas: $200 x^{2} = 128 (x + 1)^{2}$ (1 mark)
(ii) Simplifies to $9 x^{2} - 32 x - 16 = 0$ (1 mark)
(iii)(a) Factorises as $(9 x + 4) (x - 4) = 0$ (1 mark)
(iii)(a) Rejects the negative root and gets $x = 4$ (1 mark)
(iii)(b) Substitutes correctly in the quadratic formula: $x = (32 \pm 40) /18$ (1 mark)
(iii)(b) Gets $x = 4$ , rejecting $- 4/9$ (1 mark)

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 paper2024 board paper, set 30/1/1 (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 2024: question 36

Hints

Step-by-step solution

Revise the chapter

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