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Mathematics · 2024 · 4 marks
CBSE 2024 · Region 1 · Set 1 · Q36
A rectangular floor area can be completely tiled with $\displaystyle 200$ square tiles. If the side length of each tile is increased by $\displaystyle 1$ unit, it would take only $\displaystyle 128$ tiles to cover the floor.(i)Assuming the original length of each side of a tile be $\displaystyle x$ units, make a quadratic equation from the above information.(ii)Write the corresponding quadratic equation in standard form.(iii)Find the value of $\displaystyle x$, the length of side of a tile by factorisation.Solve the quadratic equation for $\displaystyle x$, using quadratic formula.
A rectangular floor area can be completely tiled with $\displaystyle 200$ square tiles. If the side length of each tile is increased by $\displaystyle 1$ unit, it would take only $\displaystyle 128$ tiles to cover the floor.
(i)
Assuming the original length of each side of a tile be $\displaystyle x$ units, make a quadratic equation from the above information.
(ii)
Write the corresponding quadratic equation in standard form.
(iii)
Find the value of $\displaystyle x$, the length of side of a tile by factorisation.
Solve the quadratic equation for $\displaystyle x$, using quadratic formula.
Marking-scheme solution
(i) \(\displaystyle 200 x^{2}=128(x+1)^{2}\)
(ii) \(\displaystyle 25 x^{2}=16 x^{2}+32 x+16\)
\[\Rightarrow 9 x^{2}-32 x-16=0
\]
(iii) (a) \(\displaystyle 9 x^{2}-32 x-16=0\)
\[\begin{aligned}
& \Rightarrow(9 x+4)(x-4)=0 \\
& x \neq \frac{-4}{9} \text { so, } x=4
\end{aligned}
\]
OR
(iii) (b) \(\displaystyle x=\frac{32 \pm \sqrt{1024+576}}{18}=\frac{32 \pm 40}{18}\)
\[x \neq \frac{-4}{9} \text { so, } x=4
\]
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