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

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

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

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

A train travels at a certain average speed for a distance of 54 km and then travels a distance of 63 km at an average speed of 6 km/h more than the first speed. If it takes 3 hours to complete the journey, what was its first average speed ?

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

Let the first speed be x; the second is x + 6. Write the time for each part of the journey.

Hint 2

The two times add up to 3 hours; clear the denominators to get a quadratic.

Hint 3

Factorise and reject the root that does not make sense for a speed.

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 first average speed be $x$ km/h; the second speed is $(x + 6)$ km/h. Time = distance/speed, and total time is 3 h:
$\frac{54}{x} + \frac{63}{x + 6} = 3$
2 marks
$\Rightarrow 54 (x + 6) + 63 x = 3 x (x + 6) \Rightarrow 54 x + 324 + 63 x = 3 x^{2} + 18 x$
$\Rightarrow 3 x^{2} - 99 x - 324 = 0 \Rightarrow x^{2} - 33 x - 108 = 0$
2 marks
$\Rightarrow (x - 36) (x + 3) = 0 \Rightarrow x = 36$ or $x = - 3$ . Speed cannot be negative, so $x = - 3$ is rejected.
The first average speed of the train was 36 km/h.
1 mark

Answer: 36 km/h

Where the marks are

Forms the equation 54/x + 63/(x + 6) = 3 (2 marks)
Simplifies to x² − 33x − 108 = 0 (2 marks)
Factorises, rejects −3 and states 36 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 paper2023 board paper, set 30/5/1 (Standard): every question, by section.

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

Hints

Step-by-step solution

Revise the chapter

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