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Mathematics · 2023 · 4 marks
CBSE 2023 · Region 2 · Set 2 · Q38
The use of electric vehicles will curb air pollution in the long run. The use of electric vehicles is increasing every year and estimated electric…
The use of electric vehicles will curb air pollution in the long run.
The use of electric vehicles is increasing every year and estimated electric vehicles in use at any time $\displaystyle \mathrm{t}$ is given by the function $\displaystyle \mathrm{V}$ : \[\mathrm{V}(\mathrm{t})=\frac{1}{5} \mathrm{t}^{3}-\frac{5}{2} \mathrm{t}^{2}+25 \mathrm{t}-2 \] where t represents the time and $\displaystyle \mathrm{t}=1,2,3 \ldots$ corresponds to year $\displaystyle 2001$, $\displaystyle 2002$, $\displaystyle 2003$, ....... respectively. Based on the above information, answer the following questions:
(i)
Can the above function be used to estimate number of vehicles in the year $\displaystyle 2000$ ? Justify.
(ii)
Prove that the function $\displaystyle \mathrm{V}(\mathrm{t})$ is an increasing function.
Marking-scheme solution
(i)
For the year $\displaystyle 2000$, $\displaystyle \mathrm{t}=0 \& \mathrm{V}(0)=-2$ and the number of vehicles cannot be negative
∴ the given function $\displaystyle \mathrm{V}(\mathrm{t})$ cannot be used.
(ii)
$\displaystyle \mathrm{V}^{\prime}(\mathrm{t})=\frac{3}{5} \mathrm{t}^{2}-5 \mathrm{t}+25=\frac{3}{5}\left[\left(\mathrm{t}-\frac{25}{6}\right)^{2}+\frac{875}{36}\right]>0, \therefore \mathrm{V}(\mathrm{t})$ is an increasing function.
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CBSE Class 12 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.