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

CBSE Class 10 Mathematics Basic sample paper 2023-24: question 32(a)

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

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

A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey. Find the speed of the train.

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

Time = distance ÷ speed. Write the two journey times in terms of $x$ .

Hint 2

The usual time minus the faster time equals 1 hour. Clear the fractions to get a quadratic, then split $5 x$ using factors of 1800 that differ by 5.

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 uniform speed of the train be $x$ km/h.
½ mark
Time taken for 360 km $= \frac{360}{x}$ h.
½ mark
With speed $(x + 5)$ km/h the time is 1 h less:
$\frac{360}{x} - \frac{360}{x + 5} = 1$
1 mark
$360 (x + 5) - 360 x = x (x + 5) \Rightarrow 1800 = x^{2} + 5 x \Rightarrow x^{2} + 5 x - 1800 = 0$
1 mark
$x^{2} + 45 x - 40 x - 1800 = 0 \Rightarrow x (x + 45) - 40 (x + 45) = 0 \Rightarrow (x + 45) (x - 40) = 0$
1 mark
$x = 40$ or $x = - 45$ . Speed cannot be negative, so the speed of the train is 40 km/h.
1 mark

Answer: Speed of the train = 40 km/h

Where the marks are

Assumes the speed as $x$ km/h (1/2 mark)
Writes the time as $\frac{360}{x}$ (1/2 mark)
Forms the equation $\frac{360}{x} - \frac{360}{x + 5} = 1$ (1 mark)
Reduces it to $x^{2} + 5 x - 1800 = 0$ (1 mark)
Factorises correctly (1 mark)
Rejects the negative root; speed = 40 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.

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CBSE Class 10 Mathematics Basic sample paper 2023-24: question 32(a)

Hints

Step-by-step solution

Revise the chapter

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