CBSE 2025 · Region 5 · Set 2 · Q33 · 5 marks
An amount of ₹ $\displaystyle 10,000$ is put into three investments at the rate of $\displaystyle 10 \%$, $\displaystyle 12 \%$ and $\displaystyle 15 \%$ per annum. The combined annual income of all three investments is ₹ $\displaystyle 1,310$ , however the combined annual income of the first and the second investments is ₹ $\displaystyle 190$ short of the income from the third. Use matrix method and find the investment amount in each at the beginning of the year.
Marking-scheme solution
Let $\displaystyle \mathrm{x}, \mathrm{y}$ and z (in ₹) be three investment amounts.
Then $\displaystyle \boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{y}}+\boldsymbol{z}=\mathbf{1 0 , 0 0 0}$
\[\frac{10}{100} \mathrm{x}+\frac{12}{100} \mathrm{y}+\frac{15}{100} z=1310
\]
Let $\displaystyle A=\left[\begin{array}{ccc} 1 & 1 & 1 \\ 10 & 12 & 15 \\ -10 & -12 & 15 \end{array}\right], X=\left[\begin{array}{l}\mathrm{x} \\
\mathrm{y} \\
z\end{array}\right], B=\left[\begin{array}{c}10,000 \\
1,31,000 \\
19,000\end{array}\right]$
\[|\mathbf{A}|=\mathbf{6 0} \neq \mathbf{0}
\]
$\displaystyle \therefore \mathbf{A}^{-\mathbf{1}}$ exists
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CBSE Class 12 Mathematics past-paper question from the 2025board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.