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Mathematics · 2023 · 5 marks
CBSE 2023 · Region 5 · Set 1 · Q34
(a) 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…
(a)
A train travels at a certain average speed for a distance of $\displaystyle 54$ km and then travels a distance of $\displaystyle 63$ km at an average speed of $\displaystyle 6$ km/h more than the first speed. If it takes $\displaystyle 3$ hours to complete the journey, what was its first average speed ?
OR
(b)
Two pipes together can fill a tank in $\displaystyle \frac{15}{8}$ hours. The pipe with larger diameter takes $\displaystyle 2$ hours less than the pipe with smaller diameter to fill the tank separately. Find the time in which each pipe can fill the tank separately.
Marking-scheme solution
Let first average speed of the train be x km/hr.
\[\begin{aligned}
& \frac{54}{x}+\frac{63}{x+6}=3 \\
& \Rightarrow 54 x+324+63 x=3 x^{2}+18 x
\end{aligned}
\]
\[\begin{aligned}
& \Rightarrow 3 x^{2}-99 x-324=0 \text { or } x^{2}-33 x-108=0 \\
& \Rightarrow(x-36)(x+3)=0 \\
& \Rightarrow x=36,-3 \text { (rejected) }
\end{aligned}
\]
Therefore, first average speed of the train was $\displaystyle 36$ km/hr.
Let the time taken by smaller diameter tap be x hrs.
Time taken by larger diameter tap is \(\displaystyle (\mathbf{x}-\mathbf{2}) \mathbf{h r s}\).
Therefore \(\displaystyle \frac{1}{x-2}+\frac{1}{x}=\frac{8}{15}\)
\[\begin{aligned}
& \Rightarrow 15(2 x-2)=8 x(x-2) \\
& \Rightarrow 8 x^{2}-46 x+30=0 \\
& \Rightarrow 4 x^{2}-23 x+15=0 \\
& \Rightarrow(4 x-3)(x-5)=0 \\
& \Rightarrow x=\frac{3}{4}, x=5
\end{aligned}
\]
\(\displaystyle \mathrm{x} \neq \frac{3}{4}\) as \(\displaystyle \mathrm{x}-2<0\)
Smaller diameter tap fills in $\displaystyle 5$ hrs.
Larger diameter tap fills in $\displaystyle 3$ hrs.
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CBSE Class 10 Mathematics past-paper question from the 2023 board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.