Q264 / 270 · NDA 2016 (Phase 1) · Mathematics
Which of the following determinants have value 'zero'?
1. $\left|\begin{array}{lll}41 & 1 & 5 \\ 79 & 7 & 9 \\ 29 & 5 & 3\end{array}\right|$
2. $\left|\begin{array}{lll}1 & \mathrm{a} & \mathrm{b}+\mathrm{c} \\ 1 & \mathrm{~b} & \mathrm{c}+\mathrm{a} \\ 1 & \mathrm{c} & \mathrm{a}+\mathrm{b}\end{array}\right|$
3. $\quad\left|\begin{array}{ccc}0 & \mathrm{c} & \mathrm{b} \\ -\mathrm{c} & 0 & \mathrm{a} \\ -\mathrm{b} & -\mathrm{a} & 0\end{array}\right|$
Select the correct answer using the code given below.
- A1 and 2 only
- B2 and 3 only
- C1 and 3 only
- Correct answerD1, 2 and 3
Solution
1.
$\text { } \begin{aligned}
\left|\begin{array}{lll}
41 & 1 & 5 \\
79 & 7 & 9 \\
29 & 5 & 3
\end{array}\right| & =\left|\begin{array}{lll}
40 & 1 & 5 \\
72 & 7 & 9 \\
24 & 5 & 3
\end{array}\right| C_1 \rightarrow C_1-C_2 \\
& =8\left|\begin{array}{lll}
5 & 1 & 5 \\
9 & 7 & 9 \\
3 & 5 & 3
\end{array}\right|=0
\end{aligned}$
$\because$ two columns are same so value of determinant is zero.
2.
$\text { } \begin{aligned}
\left|\begin{array}{lll}
1 & a & b+c \\
1 & b & c+a \\
1 & c & a+b
\end{array}\right| & =\left|\begin{array}{lll}
1 & a+b+c & b+c \\
1 & a+b+c & c+a \\
1 & a+b+c & a+b
\end{array}\right| \quad C_2 \rightarrow C_2+C_3 \\
& =(a+b+c)\left|\begin{array}{lll}
1 & 1 & b+c \\
1 & 1 & c+a \\
1 & 1 & a+b
\end{array}\right|=0
\end{aligned}$
$\because$ two columns are same so value of determinant is zero.
3. $\left|\begin{array}{ccc}0 & c & b \\ -c & 0 & a \\ -b & -a & 0\end{array}\right|=0$
$\because$ diagonal is zero so value of determinant is zero.
Hence, the correct answer is option: 'D'.