MathsCBSE Class 10 · Standard · Sample paper, session 2023-24 · 5 marks · Long answer

CBSE Class 10 Mathematics Standard sample paper 2023-24: question 32(b)

From the official CBSE sample question paper for 2023-24, section D, question 32(b), for 5 marks. Try it first; the hints and our step-by-step solution are below.

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

Two water taps together can fill a tank in $9 \frac{3}{8}$ hours. The tap of larger diameter takes 10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2023-24.

Hints

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

Hint 1

If a tap fills the tank in x hours, it fills 1/x of the tank in one hour.

Hint 2

Add the one-hour work of the two taps and set it equal to 8/75. Check which root keeps x − 10 positive.

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 tap take $x$ hours to fill the tank. Then the larger tap takes $(x - 10)$ hours.
½ mark
Part of the tank filled in 1 hour by the smaller tap $= \frac{1}{x}$ , and by the larger tap $= \frac{1}{x - 10}$ .
1 mark
Together the taps fill the tank in $9 \frac{3}{8} = \frac{75}{8}$ hours.
½ mark
So the part filled by both taps in 1 hour $= \frac{8}{75}$ .
½ mark
$\frac{1}{x} + \frac{1}{x - 10} = \frac{8}{75}$
½ mark
$75 (2 x - 10) = 8 x (x - 10)$
$\Rightarrow 8 x^{2} - 230 x + 750 = 0$
$\Rightarrow 4 x^{2} - 115 x + 375 = 0$
$x = \frac{115 \pm 13225 - 6000}{8} = \frac{115 \pm 85}{8}$
So $x = 25$ or $x = \frac{30}{8} = 3.75$ .
1 mark
If $x = 3.75$ , the larger tap's time $x - 10$ would be negative, which is impossible. So $x = 25$ .
½ mark
The smaller tap fills the tank in 25 hours, and the larger tap in $25 - 10 = 15$ hours.
½ mark

Answer: Smaller tap: 25 hours; larger tap: 15 hours

Where the marks are

Lets the smaller tap's time be x and the larger tap's time be x − 10 (1/2 mark)
Writes the one-hour work of each tap as 1/x and 1/(x−10) (1 mark)
Converts 9 3/8 h to 75/8 h (1/2 mark)
Writes the combined one-hour work as 8/75 (1/2 mark)
Forms the equation 1/x + 1/(x−10) = 8/75 (1/2 mark)
Gets the quadratic 8x² − 230x + 750 = 0 with roots 25 and 30/8 (1 mark)
Rejects 30/8 with a reason (1/2 mark)
States 25 h and 15 h (1/2 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 paperSample paper 2023-24 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard sample paper 2023-24: question 32(b)

Hints

Step-by-step solution

Revise the chapter

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