Answer:
Let Rs. x and Rs. y be the initial investment in 10% bond A and 15% bond B, respectively. Then, according to the given condition, we have \[(0.10)x+(0.15)y=4000\] and \[(0.10)\left( x+\frac{20}{100}x \right)+(0.15)\left( y+\frac{10}{100}y \right)=4500\] \[\Rightarrow \] \[2x+3y=80000\] ? (i) [Multiplying both sides by 20] and \[(0.10)\,(120x)+(0.15)\,(110y)=450000\] and \[8x+11y=300000\] ? (ii) [divide both sides by 1.5] Now, these equations can be written in matrix form as AX = B ? (iii) Where, \[A=\left[ \begin{matrix} 2 & 3 \\ 8 & 11 \\ \end{matrix} \right],\] \[X=\left[ \begin{matrix} x \\ y \\ \end{matrix} \right]\] and \[B=\left[ \begin{matrix} 80000 \\ 300000 \\ \end{matrix} \right]\] Here, \[|A|\,\,=\left[ \begin{matrix} 2 & 3 \\ 8 & 11 \\ \end{matrix} \right]=22-24=-\,2\ne 0\] \[\therefore \] \[{{A}^{-1}}\] exists. Now, \[{{A}^{-1}}=\frac{1}{|A|}\,\left[ \begin{matrix} 11 & -\,3 \\ -\,8 & 2 \\ \end{matrix} \right]\] \[\left[ \because \,i\text{f}\,A=\left[ \begin{matrix} a & b \\ c & d \\ \end{matrix} \right]\,\,\text{then}\,{{A}^{-1}}=\frac{1}{|A|}\,\left[ \begin{matrix} d & -b \\ -c & a \\ \end{matrix} \right] \right]\] \[=\frac{-1}{2}\,\left[ \begin{matrix} 11 & -\,3 \\ -\,8 & 2 \\ \end{matrix} \right]\] Now from Eq. (iii) we have \[X={{A}^{-1}}\,\,B=\frac{-1}{2}\left[ \begin{matrix} 11 & -\,3 \\ -\,8 & 2 \\ \end{matrix} \right]\,\,\left[ \begin{matrix} 80000 \\ 300000 \\ \end{matrix} \right]\] \[=\frac{-1}{2}\left[ \begin{matrix} 880000-900000 \\ -\,640000+600000 \\ \end{matrix} \right]\] \[=\frac{-1}{2}\left[ \begin{matrix} -\,20000 \\ -\,40000 \\ \end{matrix} \right]=\left[ \begin{matrix} 10000 \\ 20000 \\ \end{matrix} \right]\] \[\Rightarrow \] \[\left[ \begin{matrix} x \\ y \\ \end{matrix} \right]=\left[ \begin{matrix} 10000 \\ 20000 \\ \end{matrix} \right]\] \[\Rightarrow \] \[x=10000\] and \[y=20000\]| Hence, Mr. X invested Rs. 10000 in bond A and Rs. 20000 in bond B. Value: Yes, for a person investment is very necessary because investing is very much important to secure his future life. To build wealth, a person has to invest money.
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