MathsCBSE Class 10 · Standard · Sample paper, session 2025-26 · 5 marks · Long answer

CBSE Class 10 Mathematics Standard sample paper 2025-26: question 32

From the official CBSE sample question paper for 2025-26, section D, question 32, for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q32 · 5 marks · Long answerCompetency-basedQuadratic Equations

A train travels at a certain average speed for a distance of 63km and then travels at a distance of 72km at an average speed of 6km/hr more than its original speed. If it takes 3 hours to complete the total journey, what is the original average speed?

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2025-26.

Hints

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

Hint 1

Time = distance ÷ speed. Write each part's time in terms of x.

Hint 2

Add the two times and set the total equal to 3.

Hint 3

Clear the fractions, simplify to a quadratic and factorise it. Remember that speed must be 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 original average speed be $x$ km/h. Time taken for 63 km is $t_{1} = \frac{63}{x}$ h.
1 mark
The new speed is $(x + 6)$ km/h, so the time for 72 km is $t_{2} = \frac{72}{x + 6}$ h. Total time: $t_{1} + t_{2} = 3$ .
1 mark
$\frac{63}{x} + \frac{72}{x + 6} = 3$ , so $63 (x + 6) + 72 x = 3 x (x + 6)$ .
1 mark
$135 x + 378 = 3 x^{2} + 18 x$ , which gives $3 x^{2} - 117 x - 378 = 0$ , i.e. $x^{2} - 39 x - 126 = 0$ .
1 mark
$x^{2} - 42 x + 3 x - 126 = 0$ , so $(x - 42) (x + 3) = 0$ and $x = 42$ or $x = - 3$ . Speed cannot be negative, so $x = 42$ .
Check: $\frac{63}{42} + \frac{72}{48} = 1.5 + 1.5 = 3$ h ✓. The original average speed is 42 km/h.
1 mark

Answer: 42 km/h

Where the marks are

Writes the time for the first part as 63/x (1 mark)
Writes the time for the second part as 72/(x+6) and uses total = 3 h (1 mark)
Forms the equation $\frac{63}{x} + \frac{72}{x + 6} = 3$ (1 mark)
Reduces it to $x^{2} - 39 x - 126 = 0$ (1 mark)
Solves it, rejects −3 and states 42 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 2025-26 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard sample paper 2025-26: question 32

Hints

Step-by-step solution

Revise the chapter

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