CBSE 2024 · Region 5 · Set 1 · Q37 · 4 marks
A rectangular visiting card is to contain $\displaystyle 24$ sq.cm. of printed matter. The margins at the top and bottom of the card are to be $\displaystyle 1$ cm and the margins on the left and right are to be $\displaystyle 1 \frac{1}{2} \mathrm{~cm}$ as shown below :
On the basis of the above information, answer the following questions :(i)Write the expression for the area of the visiting card in terms of $\displaystyle \mathrm{x}$.(ii)Obtain the dimensions of the card of minimum area.
A rectangular visiting card is to contain $\displaystyle 24$ sq.cm. of printed matter. The margins at the top and bottom of the card are to be $\displaystyle 1$ cm and the margins on the left and right are to be $\displaystyle 1 \frac{1}{2} \mathrm{~cm}$ as shown below :
On the basis of the above information, answer the following questions :
(i)
Write the expression for the area of the visiting card in terms of $\displaystyle \mathrm{x}$.
(ii)
Obtain the dimensions of the card of minimum area.
Marking-scheme solution
(i)
Let $\displaystyle \mathrm{A}(\mathrm{x})$ be the area of the visiting card then,
$$\text { As } \mathrm{x} y=$\displaystyle 24$, \mathrm{A}(\mathrm{x})=(\mathrm{x}+$\displaystyle 3$)(y+$\displaystyle 2$)=$\displaystyle 2$ \mathrm{x}+$\displaystyle 3$ y+\mathrm{x} y+$\displaystyle 6$=$\displaystyle 2$ \mathrm{x}+\frac{72}{\mathrm{x}}+$\displaystyle 30$
$$(ii) $\displaystyle \mathrm{A}^{\prime}(\mathrm{x})=2-\frac{72}{\mathrm{x}^{2}}$ and $\displaystyle \mathrm{A}^{\prime \prime}(\mathrm{x})=\frac{144}{\mathrm{x}^{3}}$,
solving $\displaystyle \mathrm{A}^{\prime}(\mathrm{x})=0 \Rightarrow \mathrm{x}=6$ is the critical point.
$$\mathrm{A}^{\prime \prime}($\displaystyle 6$)=\frac{144}{6}>$\displaystyle 0$, \therefore \text { Area of the card is minimum at } \mathrm{x}=$\displaystyle 6$, y=$\displaystyle 4$
$$The dimension of the card with minimum area is Length $\displaystyle =9 \mathrm{~cm}$, Breadth $\displaystyle =6 \mathrm{~cm}$
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CBSE Class 12 Mathematics past-paper question from the 2024board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.