Q56 / 160 · TS EAMCET 2013 · Mathematics

$\left|\begin{array}{ccc}x+2 & x+3 & x+5 \\ x+4 & x+6 & x+9 \\ x+8 & x+11 & x+15\end{array}\right|$ is equal to

  • A
    $3 x^2+4 x+5$
  • B
    $x^3+8 x+2$
  • C
    $0$
  • Correct answer
    D
    $-2$

Solution

Let $\Delta=\left|\begin{array}{lll}x+2 & x+3 & x+5 \\ x+4 & x+6 & x+9 \\ x+8 & x+11 & x+15\end{array}\right|$ Apply operations $R_2 \rightarrow R_2-R_1, R_3 \rightarrow R_3-R_1$, we get $$ \Delta=\left|\begin{array}{ccc} x+2 & x+3 & x+5 \\ 2 & 3 & 4 \\ 6 & 8 & 10 \end{array}\right| $$ Again, apply operation $C_2 \rightarrow C_2-C_1$, $C_3 \rightarrow C_3-C_1$, we get $$ \Delta=\left|\begin{array}{ccc} x+2 & 1 & 3 \\ 2 & 1 & 2 \\ 6 & 2 & 4 \end{array}\right| $$ Expand along $R_1$, we get $$ \begin{aligned} \Delta & =(x+2)(4-4)-1(8-12)+3(4-6) \\ & =0+4+3(-2)=4-6=-2 \end{aligned} $$

Hence, the correct answer is option: 'D'.

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