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

CBSE Class 10 Mathematics Basic 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

A motor boat whose speed is 18 km/h in still water takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream.

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

The stream slows the boat going upstream and speeds it up going downstream.

Hint 2

Upstream time − downstream time = 1 hour. Clear the fractions to get a quadratic.

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 speed of the stream be $x$ km/h.
Upstream speed $= (18 - x)$ km/h. Downstream speed $= (18 + x)$ km/h.
1 mark
Using time = distance ÷ speed:
Time upstream $= \frac{24}{18 - x}$ h. Time downstream $= \frac{24}{18 + x}$ h.
1 mark
The upstream journey takes 1 hour more:
$\frac{24}{18 - x} - \frac{24}{18 + x} = 1$
1 mark
$24 (18 + x) - 24 (18 - x) = (18 - x) (18 + x)$
$\Rightarrow 48 x = 324 - x^{2}$
$\Rightarrow x^{2} + 48 x - 324 = 0$
$\Rightarrow (x + 54) (x - 6) = 0$ , so $x = 6$ or $x = - 54$ .
1 mark
A speed cannot be negative, so we reject $x = - 54$ .
The speed of the stream is 6 km/h.
1 mark

Answer: Speed of the stream = 6 km/h

Where the marks are

Writes the upstream and downstream speeds as 18 − x and 18 + x (1 mark)
Writes both times as distance ÷ speed (1 mark)
Forms the equation 24/(18−x) − 24/(18+x) = 1 (1 mark)
Reduces it to x² + 48x − 324 = 0 and solves to get x = 6 or −54 (1 mark)
Rejects the negative value and states 6 km/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 paperSample paper 2023-24 (Basic): 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 Basic sample paper 2023-24: question 32(b)

Hints

Step-by-step solution

Revise the chapter

More questions from Quadratic Equations