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#### provide solution for RD Sharma maths class 12 chapter 5 Determinants exercise  Fill in the blanks question 26

Hint: Here, we use basic concept of determinant of matrix

Given: $x=-y$ and $\left[\begin{array}{lll} x & 3 & 7 \\ 2 & x & 2 \\ 7 & 6 & x \end{array}\right]=0$

Solution:

$\left[\begin{array}{lll} x & 3 & 7 \\ 2 & x & 2 \\ 7 & 6 & x \end{array}\right]$

\begin{aligned} &=x\left[\begin{array}{ll} x & 2 \\ 6 & x \end{array}\right]-3\left[\begin{array}{cc} 2 & 2 \\ 7 & x \end{array}\right]+7\left[\begin{array}{cc} 2 & x \\ 7 & 6 \end{array}\right] \\ &=x\left(x^{2}-12\right)-3(2 x-14)+7(12-7 x) \\ &f(x)=x^{3}-12 x-6 x+42+84-49 x=0 \\ &f(x)=x^{3}-67 x+126=0 \end{aligned}

Let put x = 2 and x = 7

\begin{aligned} &f(2)=(2)^{3}-67(2)+126 \\ &=8-134+126 \\ &=0 \\ &f(7)=(7)^{3}-67(7)+126 \\ &=49 \times 7-67+126 \\ &=0 \end{aligned}

So, 2 and 7 are other roots.