Question

If $\left|\begin{array}{ccc}x+2& 2x+3& 3x+4\\ 2x+3& 3x+4& 4x+5\\ 3x+5& 5x+8& 10x+17\end{array}\right|=0$

then find value of $x$

• A. 1
• B. 2
• C. -1
• D. -2

Answer 1

Answer: (C), (D)

The correct options are
C -1
D -2

Let $\Delta =\left|\begin{array}{ccc}x+2& 2x+3& 3x+4\\ 2x+3& 3x+4& 4x+5\\ 3x+5& 5x+8& 10x+17\end{array}\right|$

Applying $R_2\to R_2-R_1,R_3\to R_3-R_1$

$\Delta =\left|\begin{array}{ccc}x+2& 2x+3& 3x+4\\ x+1& x+1& x+1\\ 2x+3& 3x+5& 7x+13\end{array}\right|=(x+1)\left|\begin{array}{ccc}x+2& 2x+3& 3x+4\\ 1& 1& 1\\ 2x+3& 3x+5& 7x+13\end{array}\right|$

Again applying $C_2\to C_2-C_1,C_3\to C_3-C_1$

$\Delta =(x+1)\left|\begin{array}{ccc}x+2& x+1& 2x+2\\ 1& 0& 0\\ 2x+3& x+2& 5x+10\end{array}\right|$

$=-(x+1)((x+1)(5x+10)-(x+2)(2x+2))$

$=-{(x+1)}^2(5x+10-2x-4)=-{(x+1)}^2(3x+6)$

For $\Delta =0\Rightarrow x=-1,x=-2$