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Mathematics · 2025 · 2 marks
Pair of Linear Equations in Two VariablesConsistency and Nature of Solutions2 marksApplyvery short answer
CBSE 2025 · Region 5 · Set 1 · Q21
(a) The cost of 2 kg apples and 1 kg of grapes on a day was found to be ₹ 320. The cost of 4 kg apples and 2 kg grapes was found to be ₹ 600. If cost…
The cost of $\displaystyle 2$ kg apples and $\displaystyle 1$ kg of grapes on a day was found to be ₹ $\displaystyle 320$ . The cost of $\displaystyle 4$ kg apples and $\displaystyle 2$ kg grapes was found to be ₹ 600. If cost of $\displaystyle 1$ kg of apples and $\displaystyle 1$ kg of grapes is ₹ $\displaystyle x$ and ₹ y respectively, represent the given situation algebraically as a system of equations and check whether the system so obtained is consistent or not.
Solve for $\displaystyle x$ and y : \[\begin{aligned} & \sqrt{2} x+\sqrt{3} y=5 \text { and } \\ & \sqrt{3} x-\sqrt{8} y=-\sqrt{6} \end{aligned} \]
Marking-scheme solution
\(\displaystyle 2 x+y=320\)
\[4 x+2 y=600
\]
Here, \(\displaystyle \frac{\mathrm{a}_{1}}{\mathrm{a}_{2}}=\frac{2}{4}=\frac{1}{2}, \frac{\mathrm{~b}_{1}}{\mathrm{~b}_{2}}=\frac{1}{2}, \frac{\mathrm{c}_{1}}{\mathrm{c}_{2}}=\frac{320}{600}=\frac{8}{15}\)
As \(\displaystyle \frac{\mathrm{a}_{1}}{\mathrm{a}_{2}}=\frac{\mathrm{b}_{1}}{\mathrm{~b}_{2}} \neq \frac{\mathrm{c}_{1}}{\mathrm{c}_{2}} \therefore\) System of equations is not consistent.
\(\displaystyle (\sqrt{2} x+\sqrt{3} y=5) \times \sqrt{3} \Rightarrow \sqrt{6} x+3 y=5 \sqrt{3}\)
\[(\sqrt{3} x-\sqrt{8} y=-\sqrt{6}) \times \sqrt{2} \Rightarrow \sqrt{6} x-4 y=-2 \sqrt{3}
\]
Solving the equations, we get
\[x=\sqrt{2} \text { and } y=\sqrt{3}
\]
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