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Mathematics · 2023 · 4 marks
CBSE 2023 · Region 3 · Set 1 · Q38
A volleyball player serves the ball which takes a parabolic path given by the equation h ( t )=-7/2 t^2+13/2 t +1, where h ( t ) is the height of…
A volleyball player serves the ball which takes a parabolic path given by the equation $\displaystyle \mathrm{h}(\mathrm{t})=-\frac{7}{2} \mathrm{t}^{2}+\frac{13}{2} \mathrm{t}+1$, where $\displaystyle \mathrm{h}(\mathrm{t})$ is the height of ball at any time t (in seconds), ( $\displaystyle \mathrm{t} \geq 0$ ).
Based on the above information, answer the following questions :
(i)
Is $\displaystyle \mathrm{h}(\mathrm{t})$ a continuous function? Justify.
(ii)
Find the time at which the height of the ball is maximum.
Marking-scheme solution
(i)
$\displaystyle \mathrm{h}(\mathrm{t})=-\frac{7}{2} \mathrm{t}^{2}+\frac{13}{2} \mathrm{t}+1$
Clearly $\displaystyle \mathrm{h}(\mathrm{t})$ is a polynomial function, hence continuous.
Hence $\displaystyle \mathrm{h}(\mathrm{t})$ is a continuous function.
(ii)
For maximum height ,
$$\frac{d \mathrm{h}}{d \mathrm{t}}=$\displaystyle 0$ \Rightarrow-$\displaystyle 7$ \mathrm{t}+\frac{13}{2}=$\displaystyle 0$
$$$\mathrm{t}=\frac{13}{14}$
$\displaystyle \frac{d^{2} \mathrm{h}}{d \mathrm{t}^{2}}=-7<0 \quad \therefore$ height is maximum at $\displaystyle \mathrm{t}=\frac{13}{14}$
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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.