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Mathematics · 2024 · 4 marks
CBSE 2024 · Region 4 · Set 2 · Q38
A store has been selling calculators at ₹ 350 each. A market survey indicates that a reduction in price (p) of calculator increases the number of…
A store has been selling calculators at ₹ $\displaystyle 350$ each. A market survey indicates that a reduction in price (p) of calculator increases the number of units $\displaystyle (x)$ sold. The relation between the price and quantity sold is given by the demand function $\displaystyle \mathrm{p}=450-\frac{1}{2} x$.
Based on the above information, answer the following questions :
(i)
Determine the number of units ( $\displaystyle x$ ) that should be sold to maximise the revenue $\displaystyle \mathrm{R}(x)=x \mathrm{p}(x)$. Also, verify the result.
(ii)
What rebate in price of calculator should the store give to maximise the revenue?
Marking-scheme solution
(i)
Revenue by selling $\displaystyle x$ items $\displaystyle =\mathrm{R}(x)=x \cdot \mathrm{p}(x)=450 x-\frac{x^{2}}{2}$
$$\frac{d \mathrm{R}}{d x}=$\displaystyle 450$-x
$$For Maxima or Minima, $\displaystyle \frac{d \mathrm{R}}{d x}=0 \Rightarrow x=450$
$$\frac{d^{$\displaystyle 2$} \mathrm{R}}{d x^{$\displaystyle 2$}}=-$\displaystyle 1$<$\displaystyle 0$
$$(Revenue is maximum when $\displaystyle x=450$ units are sold)
(ii)
At $\displaystyle x=450, \mathrm{p}=450-\frac{450}{2}=225$
So, Rebate $\displaystyle =350-225=$ Rs. $\displaystyle 125$ per calculator
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CBSE Class 12 Mathematics past-paper question from the 2024 board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.