MathsCBSE Class 10 · Basic · Sample paper, session 2024-25 · 5 marks · Long answer

CBSE Class 10 Mathematics Basic sample paper 2024-25: question 32(B)

From the official CBSE sample question paper for 2024-25, 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 answerQuadratic Equations

A train travels at a certain average speed for a distance of 132 km and then travels a distance of 140 km at an average speed of 4 km/h more than the initial speed. If it takes 4 hours to complete the whole journey, what was the initial average speed? Determine the time taken by train to cover the distances separately.

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2024-25.

Hints

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

Hint 1

Write the time for each part as distance ÷ speed. The two times add up to 4 hours.

Hint 2

Clear the denominators to get a quadratic in $x$ . Reject the root that is not possible.

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 initial average speed be $x$ km/h. Using time = distance ÷ speed:
$\frac{132}{x} + \frac{140}{x + 4} = 4$
1 mark
$132 (x + 4) + 140 x = 4 x (x + 4) \Rightarrow 272 x + 528 = 4 x^{2} + 16 x \Rightarrow 4 x^{2} - 256 x - 528 = 0$
1 mark
Dividing by 4: $x^{2} - 64 x - 132 = 0 \Rightarrow (x - 66) (x + 2) = 0$
1 mark
$x = 66$ or $x = - 2$ . Speed cannot be negative, so the initial average speed is 66 km/h.
1 mark
Time for 132 km $= \frac{132}{66} = 2$ h.
Time for 140 km $= \frac{140}{70} = 2$ h.
1 mark

Answer: Initial speed 66 km/h; 2 hours for each part

Where the marks are

Forms the equation $\frac{132}{x} + \frac{140}{x + 4} = 4$ (1 mark)
Simplifies to $4 x^{2} - 256 x - 528 = 0$ (1 mark)
Gets $x^{2} - 64 x - 132 = 0$ and factorises (1 mark)
Rejects $- 2$ , so speed = 66 km/h (1 mark)
Finds both times: 2 h and 2 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 2024-25 (Basic): every question, by section.

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CBSE Class 10 Mathematics Basic sample paper 2024-25: question 32(B)

Hints

Step-by-step solution

Revise the chapter

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