MathsCBSE Class 10 · Standard · Board exam 2023, set 30/5/1 · 5 marks · Long answer

CBSE Class 10 Mathematics Standard 2023: question 34(b)

Set in the CBSE Class 10 board exam of 2023, paper 30/5/1, section D, question 34(b), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q34(b) · 5 marks · Long answerCompetency-basedQuadratic Equations

Two pipes together can fill a tank in $\frac{15}{8}$ hours. The pipe with larger diameter takes 2 hours less than the pipe with smaller diameter to fill the tank separately. Find the time in which each pipe can fill the tank separately.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 30/5/1.

Hints

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

Hint 1

If a pipe fills a tank in t hours, it fills 1/t of the tank in one hour.

Hint 2

Add the one-hour work of both pipes and equate it to the one-hour work together, 8/15.

Hint 3

After solving the quadratic, check which root gives a positive time for the larger pipe.

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 smaller-diameter pipe fill the tank in $x$ hours; then the larger one takes $(x - 2)$ hours. In 1 hour they fill $\frac{1}{x}$ and $\frac{1}{x - 2}$ of the tank; together they fill $\frac{8}{15}$ of it:
$\frac{1}{x} + \frac{1}{x - 2} = \frac{8}{15}$
2 marks
$\Rightarrow 15 (x - 2 + x) = 8 x (x - 2) \Rightarrow 30 x - 30 = 8 x^{2} - 16 x$
$\Rightarrow 8 x^{2} - 46 x + 30 = 0 \Rightarrow 4 x^{2} - 23 x + 15 = 0$
1 mark
$\Rightarrow 4 x^{2} - 20 x - 3 x + 15 = 0 \Rightarrow (4 x - 3) (x - 5) = 0 \Rightarrow x = \frac{3}{4}$ or $x = 5$
1 mark
If $x = \frac{3}{4}$ , then $x - 2 < 0$ , which is impossible for time, so $x = 5$ .
Smaller-diameter pipe: 5 hours; larger-diameter pipe: $5 - 2 = 3$ hours.
1 mark

Answer: Smaller pipe: 5 hours; larger pipe: 3 hours

Where the marks are

Forms the equation 1/x + 1/(x − 2) = 8/15 (2 marks)
Simplifies to 4x² − 23x + 15 = 0 (1 mark)
Factorises to get x = 3/4 or 5 (1 mark)
Rejects 3/4 and states 5 h and 3 h (1 mark)

Revise the chapter

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

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 30/5/1 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard 2023: question 34(b)

Hints

Step-by-step solution

Revise the chapter

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