Case Study While designing the school year book, a teacher asked the student that the length and width of a particular photo is increased by x units each to double the area of the photo. The original photo is 18 cm long and 12 cm wide. Based on the above information, answer the following questions: (I) Write an algebraic equation depicting the above information. (II) Write the corresponding quadratic equation in standard form. (III) What should be the new dimensions of the enlarged photo? (III) Can any rational value of x make the new area equal to 220\ cm²?

(I) (18+x)(12+x) = 2(18×12); (II) x² + 30x - 216 = 0; (III)(a) 24 cm×18 cm; (III)(b) No, since D = 916 is not a perfect square

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

CBSE Class 10 Mathematics Standard 2023: question 36

Set in the CBSE Class 10 board exam of 2023, 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

Case Study

While designing the school year book, a teacher asked the student that the length and width of a particular photo is increased by $x$ units each to double the area of the photo. The original photo is 18 cm long and 12 cm wide.

Based on the above information, answer the following questions:

(I)
Write an algebraic equation depicting the above information.
1 mark
(II)
Write the corresponding quadratic equation in standard form.
1 mark
(III)
What should be the new dimensions of the enlarged photo?
2 marks
(III)
Can any rational value of $x$ make the new area equal to $220 c m^{2}$ ?
2 marks

Figure in the paper: A school photo (inner rectangle) of length 18 cm and width 12 cm, surrounded by a larger border rectangle. The border width x is marked on the left side, at the top, and at the bottom right; the width of the border on the top and left is x, and the enlarged outer rectangle has the photo's length 18 cm along the top and 12 cm along the right side, each extended by x. See it in the official paper.

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

Hints

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

Hint 1

Each side increases by $x$ . Write the new area as a product and set it equal to twice $18 \times 12$ .

Hint 2

Expand and bring everything to one side to get $a x^{2} + b x + c = 0$ . Look for two numbers with product $- 216$ and sum $30$ .

Hint 3

For the 220 cm² part, check the discriminant: the roots of a quadratic with integer coefficients are rational only if $D$ is a perfect square.

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) New length $= (18 + x)$ cm and new width $= (12 + x)$ cm. The new area must be double the original area $18 \times 12 = 216 cm^{2}$ :
$(18 + x) (12 + x) = 2 (18 \times 12)$
1 mark
(II) Expanding: $216 + 30 x + x^{2} = 432$ , so $x^{2} + 30 x - 216 = 0$
1 mark
(III)(a) Solve $x^{2} + 30 x - 216 = 0$ by splitting the middle term: $x^{2} + 36 x - 6 x - 216 = 0$ , so $(x + 36) (x - 6) = 0$ .
This gives $x = 6$ or $x = - 36$ . A length increase cannot be negative, so $x = 6$ .
1 mark
(III)(a) New dimensions: length $= 18 + 6 = 24$ cm, width $= 12 + 6 = 18$ cm. The enlarged photo is $24 cm \times 18 cm$ (area $432 = 2 \times 216$ ✓).
1 mark
(III)(b) For a new area of $220 cm^{2}$ : $(18 + x) (12 + x) = 220$ , so $x^{2} + 30 x + 216 = 220$ , giving $x^{2} + 30 x - 4 = 0$ .
1 mark
(III)(b) Discriminant $D = b^{2} - 4 a c = 3 0^{2} - 4 (1) (- 4) = 900 + 16 = 916$ . Since 916 is not a perfect square, $D$ is irrational and so the roots are irrational. Hence **no rational value of $x$ ** can make the new area $220 cm^{2}$ .
1 mark

Answer: (I) $(18 + x) (12 + x) = 2 (18 \times 12)$ ; (II) $x^{2} + 30 x - 216 = 0$ ; (III)(a) $24 cm \times 18 cm$ ; (III)(b) No, since $D = 916$ is not a perfect square

Where the marks are

(I) Correct equation $(18 + x) (12 + x) = 2 \times 18 \times 12$ (1 mark)
(II) Standard form $x^{2} + 30 x - 216 = 0$ (1 mark)
(III)(a) Factorises and takes $x = 6$ , rejecting $- 36$ (1 mark)
(III)(a) New dimensions $24 cm \times 18 cm$ (1 mark)
(III)(b) Forms $x^{2} + 30 x - 4 = 0$ (1 mark)
(III)(b) $D = 916$ is not a perfect square, so there is no rational $x$ (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 paper2023 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 2023: question 36

Hints

Step-by-step solution

Revise the chapter

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