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  • Provide solution for RD Sharma maths class 12 chapter Solution of Simultaneous Linear Equation exercise 7.1 question 14

Provide solution for RD Sharma maths class 12 chapter Solution of Simultaneous Linear Equation exercise 7.1 question 14

Answers (1)

Answer:

\begin{aligned} &x=500, y=2000, z=3500 \end{aligned}

Given:

As per the information the following equation is

\begin{aligned} &x+y+z=6000 \\ &x+0 y+3 z=1100 \\ &x-2 y+z=0 \end{aligned}

Hint:

X=A-1B is used to solve this problem. And the determinant and co-factor of matrix A, take it’s transpose, and that will be Adj A using Adj A calculate A-1.

Solution:

Let the award money given for honesty regularity and hard work be x, y, z respectively

Since total cash award is  Rs6000

\begin{aligned} &x+y+z=6000 \: \: \: \: \: \: .....(1)\end{aligned}

Three times the award money for hard work and honesty is Rs11000

\begin{aligned} x+0 y+3 z=1100\: \: \: \: \: \: .......(2) \end{aligned}

Award money for honesty and hardwork is double the one given for regularity

\begin{aligned}&x+z=2y \\ &x-2 y+z=0\: \: \: \: \: \: .....(3) \end{aligned}

\begin{aligned} &\left[\begin{array}{ccc} 1 & 1 & 1 \\ 1 & 0 & 3 \\ 1 & -2 & 1 \end{array}\right]\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{c} 6000 \\ 11000 \\ 0 \end{array}\right]\\ &A X=B\\ &A=\left|\begin{array}{ccc} 1 & 1 & 1 \\ 1 & 0 & 3 \\ 1 & -2 & 1 \end{array}\right|=6 \neq 0 \text { Thus, } A \text { is non singular } \end{aligned}

\begin{gathered} \operatorname{adj} A=\left[\begin{array}{ccc} 6 & -3 & 3 \\ 2 & 0 & -2 \\ -2 & 3 & 1 \end{array}\right] \\ A^{-1}=\frac{1}{|A|} a d j A=\frac{1}{6}\left[\begin{array}{ccc} 6 & -3 & 3 \\ 2 & 0 & -2 \\ -2 & 3 & 1 \end{array}\right] \end{gathered}

\begin{aligned} &X=A^{-1} B \\ &=\frac{1}{6}\left[\begin{array}{ccc} 6 & -3 & 3 \\ 2 & 0 & -2 \\ -2 & 3 & 1 \end{array}\right]\left[\begin{array}{c} 6000 \\ 11000 \\ 0 \end{array}\right] \end{aligned}

\begin{aligned} &{\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{c} 500 \\ 2000 \\ 3500 \end{array}\right]} \\ &\text { Hence, } x=500, y=2000, z=3500 \end{aligned}

Thus award money given for honesty, regularity and hardwork are Rs500, Rs2000 & Rs3500 respectively

Posted by

Gurleen Kaur

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