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  • Please solve RD Sharma class 12 chapter Solution of Simultaneous Linear Equation exercise 7.1 question 17 maths textbook solution

Please solve RD Sharma class 12 chapter Solution of Simultaneous Linear Equation exercise 7.1 question 17 maths textbook solution

Answers (1)

Answer:

\begin{aligned} x=200, y=300, z=400 \end{aligned}

Given:

According to question

\begin{aligned} &x+y+z=9000 \\ &3 x+2 y+z=1600 \\ &4 x+y+3 z=2300 \end{aligned}

Hint:

X=A-1B is used to solve this problem.

Ans the determinant and co-factor of matrix A, take it’s transpose that will be Adj A using Adj A calculate A-1.

Solution:

Let the award money given for sincerity, truthfulness and helpfulness be  Rs x, Rs y  ,Rs z  respectively.

Since, the total cash award is Rs900 

\begin{aligned} &x+y+z=900\; \; \; \; \; \; ......(1) \\ \end{aligned}

Award money given by school A is Rs1600

\begin{aligned} &3 x+2 y+z=1600\; \; \; \; \; \; ........(2) \end{aligned}

Award money given by school B is Rs2300

\begin{aligned} 4 x+y+3 z=2300\; \; \; \; \; \; .........(3) \end{aligned}

From(1),(2)&(3)

\begin{aligned} &{\left[\begin{array}{lll} 1 & 1 & 1 \\ 3 & 2 & 1 \\ 4 & 1 & 3 \end{array}\right]\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{c} 900 \\ 1600 \\ 2300 \end{array}\right]} \\ &A X=B \end{aligned}

\begin{aligned} |A|=\left|\begin{array}{lll} 1 & 1 & 1 \\ 3 & 2 & 1 \\ 4 & 1 & 3 \end{array}\right| &=1(6-1)-1(9-4)+1(3-8) \\ &=5-5-5=-5 \neq 0 \end{aligned}

\begin{aligned} &c_{11}=(-1)^{1+1}\left|\begin{array}{ll} 2 & 1 \\ 1 & 3 \end{array}\right|=5 \\ &c_{12}=(-1)^{1+2}\left|\begin{array}{ll} 3 & 1 \\ 4 & 3 \end{array}\right|=-5 \\ &c_{13}=(-1)^{1+3}\left|\begin{array}{ll} 3 & 2 \\ 4 & 1 \end{array}\right|=-5 \\ &c_{21}=(-1)^{2+1}\left|\begin{array}{ll} 1 & 1 \\ 1 & 3 \end{array}\right|=-2 \end{aligned}

\begin{aligned} &c_{22}=(-1)^{2+2}\left|\begin{array}{ll} 1 & 1 \\ 4 & 3 \end{array}\right|=-1 \\ &c_{23}=(-1)^{2+13}\left|\begin{array}{ll} 1 & 1 \\ 4 & 1 \end{array}\right|=3 \\ &c_{31}=(-1)^{3+1}\left|\begin{array}{ll} 1 & 1 \\ 2 & 1 \end{array}\right|=-1 \\ &c_{32}=(-1)^{3+2}\left|\begin{array}{ll} 1 & 1 \\ 3 & 1 \end{array}\right|=2 \\ &c_{33}=(-1)^{3+3}\left|\begin{array}{ll} 1 & 1 \\ 3 & 2 \end{array}\right|=-1 \end{aligned}

\begin{aligned} &\operatorname{adj} C=\left[\begin{array}{ccc} 5 & -5 & -5 \\ -2 & -1 & 3 \\ -1 & 2 & -1 \end{array}\right]^{T}=\left[\begin{array}{ccc} 5 & -2 & -1 \\ -5 & -1 & 2 \\ -5 & 3 & -1 \end{array}\right] \\ &C^{-1}=\frac{1}{|A|} a d j C \end{aligned}

\begin{aligned} &=\frac{1}{-5}\left[\begin{array}{ccc} 5 & -2 & -1 \\ -5 & -1 & 2 \\ -5 & 3 & -1 \end{array}\right] \\ x=& C^{-1} B \end{aligned}

\begin{aligned} \left[\begin{array}{l} x \\ y \\ z \end{array}\right]=& \frac{1}{-5}\left[\begin{array}{ccc} 5 & -2 & -1 \\ -5 & -1 & 2 \\ -5 & 3 & -1 \end{array}\right]\left[\begin{array}{c} 900 \\ 1600 \\ 2300 \end{array}\right] \\ &=-\frac{1}{5}\left[\begin{array}{c} 4500-3200-2300 \\ -4500-1600+4600 \\ -4500+4800-2300 \end{array}\right] \end{aligned}

\begin{aligned} =-\frac{1}{5}\left[\begin{array}{l} -1000 \\ -1500 \\ -2000 \end{array}\right] \\ \left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{l} 200 \\ 300 \\ 400 \end{array}\right] \\ x=200, y=300, z=400 \end{aligned}

Hence the award money for each value of sincerity, truthfulness and helpfulness is 200, 300 & 400 one more value which should be considered for award hard work

Posted by

Gurleen Kaur

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