79162
Let \(a, b, c\) be positive and not all are equal, the value of the determinant \(\left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|\) is
1 \(+\mathrm{ve}\)
2 -ve
3 zero
4 none of these
Explanation:
(B) : According to given summation, Let, \(\quad \Delta=\left|\begin{array}{lll}\mathrm{a} & \mathrm{b} & \mathrm{c} \\ \mathrm{b} & \mathrm{c} & \mathrm{a} \\ \mathrm{c} & \mathrm{a} & \mathrm{b}\end{array}\right|\) Operating : \(\mathrm{C}_{1} \rightarrow \mathrm{C}_{1}+\mathrm{C}_{2}+\mathrm{C}_{3}\) we get \(\Delta=\left|\begin{array}{lll} \mathrm{a}+\mathrm{b}+\mathrm{c} & \mathrm{b} & \mathrm{c} \\ \mathrm{a}+\mathrm{b}+\mathrm{c} & \mathrm{c} & \mathrm{a} \\ \mathrm{a}+\mathrm{b}+\mathrm{c} & \mathrm{a} & \mathrm{b} \end{array}\right|=(\mathrm{a}+\mathrm{b}+\mathrm{c})\left|\begin{array}{lll} 1 & \mathrm{~b} & \mathrm{c} \\ 1 & \mathrm{c} & \mathrm{a} \\ 1 & \mathrm{a} & \mathrm{b} \end{array}\right|\) Operating: \(\mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-\mathrm{R}_{1}\) and \(\mathrm{R}_{3} \rightarrow \mathrm{R}_{3}-\mathrm{R}_{1}\) we get \(=(\mathrm{a}+\mathrm{b}+\mathrm{c})\left|\begin{array}{ccc} 1 & \mathrm{~b} & \mathrm{c} \\ 0 & \mathrm{c}-\mathrm{b} & \mathrm{a}-\mathrm{c} \\ 0 & \mathrm{a}-\mathrm{b} & \mathrm{b}-\mathrm{c} \end{array}\right|\) \(=(\mathrm{a}+\mathrm{b}+\mathrm{c})\left\{-(\mathrm{c}-\mathrm{b})^{2}-(\mathrm{a}-\mathrm{b})(\mathrm{a}-\mathrm{c})\right\}\) \(=-(\mathrm{a}+\mathrm{b}+\mathrm{c})\left\{\mathrm{a}^{2}+\mathrm{b}^{2}+\mathrm{c}^{2}-\mathrm{ab}-\mathrm{bc}-\mathrm{ca}\right\}\) \(=\frac{-1}{2}(a+b+c)\left\{2 a^{2}+2 b^{2}+2 c^{2}-2 a b-a b c-2 c a\right\} \\ =\frac{-1}{2}(a+b+c)\left\{(a-b)^{2}+(b-c)^{2}\right\}\) , which is always negative
BCECE-2007
Matrix and Determinant
79163
If \(\left|\begin{array}{ccc}-12 & 0 & \lambda \\ 0 & 2 & -1 \\ 2 & 1 & 15\end{array}\right|=-360\), then the value of \(\lambda\) is
1 -1
2 -2
3 -3
4 4
Explanation:
(C) : It is given that \(\left|\begin{array}{ccc} -12 & 0 & \lambda \\ 0 & 2 & -1 \\ 2 & 1 & 15 \end{array}\right|=-360\) Now, \(-12(30+1)-4 \lambda=-360\) \(-372+360=4 \lambda\) \(\lambda=\frac{-12}{4} \Rightarrow \lambda=-3\)
79162
Let \(a, b, c\) be positive and not all are equal, the value of the determinant \(\left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|\) is
1 \(+\mathrm{ve}\)
2 -ve
3 zero
4 none of these
Explanation:
(B) : According to given summation, Let, \(\quad \Delta=\left|\begin{array}{lll}\mathrm{a} & \mathrm{b} & \mathrm{c} \\ \mathrm{b} & \mathrm{c} & \mathrm{a} \\ \mathrm{c} & \mathrm{a} & \mathrm{b}\end{array}\right|\) Operating : \(\mathrm{C}_{1} \rightarrow \mathrm{C}_{1}+\mathrm{C}_{2}+\mathrm{C}_{3}\) we get \(\Delta=\left|\begin{array}{lll} \mathrm{a}+\mathrm{b}+\mathrm{c} & \mathrm{b} & \mathrm{c} \\ \mathrm{a}+\mathrm{b}+\mathrm{c} & \mathrm{c} & \mathrm{a} \\ \mathrm{a}+\mathrm{b}+\mathrm{c} & \mathrm{a} & \mathrm{b} \end{array}\right|=(\mathrm{a}+\mathrm{b}+\mathrm{c})\left|\begin{array}{lll} 1 & \mathrm{~b} & \mathrm{c} \\ 1 & \mathrm{c} & \mathrm{a} \\ 1 & \mathrm{a} & \mathrm{b} \end{array}\right|\) Operating: \(\mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-\mathrm{R}_{1}\) and \(\mathrm{R}_{3} \rightarrow \mathrm{R}_{3}-\mathrm{R}_{1}\) we get \(=(\mathrm{a}+\mathrm{b}+\mathrm{c})\left|\begin{array}{ccc} 1 & \mathrm{~b} & \mathrm{c} \\ 0 & \mathrm{c}-\mathrm{b} & \mathrm{a}-\mathrm{c} \\ 0 & \mathrm{a}-\mathrm{b} & \mathrm{b}-\mathrm{c} \end{array}\right|\) \(=(\mathrm{a}+\mathrm{b}+\mathrm{c})\left\{-(\mathrm{c}-\mathrm{b})^{2}-(\mathrm{a}-\mathrm{b})(\mathrm{a}-\mathrm{c})\right\}\) \(=-(\mathrm{a}+\mathrm{b}+\mathrm{c})\left\{\mathrm{a}^{2}+\mathrm{b}^{2}+\mathrm{c}^{2}-\mathrm{ab}-\mathrm{bc}-\mathrm{ca}\right\}\) \(=\frac{-1}{2}(a+b+c)\left\{2 a^{2}+2 b^{2}+2 c^{2}-2 a b-a b c-2 c a\right\} \\ =\frac{-1}{2}(a+b+c)\left\{(a-b)^{2}+(b-c)^{2}\right\}\) , which is always negative
BCECE-2007
Matrix and Determinant
79163
If \(\left|\begin{array}{ccc}-12 & 0 & \lambda \\ 0 & 2 & -1 \\ 2 & 1 & 15\end{array}\right|=-360\), then the value of \(\lambda\) is
1 -1
2 -2
3 -3
4 4
Explanation:
(C) : It is given that \(\left|\begin{array}{ccc} -12 & 0 & \lambda \\ 0 & 2 & -1 \\ 2 & 1 & 15 \end{array}\right|=-360\) Now, \(-12(30+1)-4 \lambda=-360\) \(-372+360=4 \lambda\) \(\lambda=\frac{-12}{4} \Rightarrow \lambda=-3\)
79162
Let \(a, b, c\) be positive and not all are equal, the value of the determinant \(\left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|\) is
1 \(+\mathrm{ve}\)
2 -ve
3 zero
4 none of these
Explanation:
(B) : According to given summation, Let, \(\quad \Delta=\left|\begin{array}{lll}\mathrm{a} & \mathrm{b} & \mathrm{c} \\ \mathrm{b} & \mathrm{c} & \mathrm{a} \\ \mathrm{c} & \mathrm{a} & \mathrm{b}\end{array}\right|\) Operating : \(\mathrm{C}_{1} \rightarrow \mathrm{C}_{1}+\mathrm{C}_{2}+\mathrm{C}_{3}\) we get \(\Delta=\left|\begin{array}{lll} \mathrm{a}+\mathrm{b}+\mathrm{c} & \mathrm{b} & \mathrm{c} \\ \mathrm{a}+\mathrm{b}+\mathrm{c} & \mathrm{c} & \mathrm{a} \\ \mathrm{a}+\mathrm{b}+\mathrm{c} & \mathrm{a} & \mathrm{b} \end{array}\right|=(\mathrm{a}+\mathrm{b}+\mathrm{c})\left|\begin{array}{lll} 1 & \mathrm{~b} & \mathrm{c} \\ 1 & \mathrm{c} & \mathrm{a} \\ 1 & \mathrm{a} & \mathrm{b} \end{array}\right|\) Operating: \(\mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-\mathrm{R}_{1}\) and \(\mathrm{R}_{3} \rightarrow \mathrm{R}_{3}-\mathrm{R}_{1}\) we get \(=(\mathrm{a}+\mathrm{b}+\mathrm{c})\left|\begin{array}{ccc} 1 & \mathrm{~b} & \mathrm{c} \\ 0 & \mathrm{c}-\mathrm{b} & \mathrm{a}-\mathrm{c} \\ 0 & \mathrm{a}-\mathrm{b} & \mathrm{b}-\mathrm{c} \end{array}\right|\) \(=(\mathrm{a}+\mathrm{b}+\mathrm{c})\left\{-(\mathrm{c}-\mathrm{b})^{2}-(\mathrm{a}-\mathrm{b})(\mathrm{a}-\mathrm{c})\right\}\) \(=-(\mathrm{a}+\mathrm{b}+\mathrm{c})\left\{\mathrm{a}^{2}+\mathrm{b}^{2}+\mathrm{c}^{2}-\mathrm{ab}-\mathrm{bc}-\mathrm{ca}\right\}\) \(=\frac{-1}{2}(a+b+c)\left\{2 a^{2}+2 b^{2}+2 c^{2}-2 a b-a b c-2 c a\right\} \\ =\frac{-1}{2}(a+b+c)\left\{(a-b)^{2}+(b-c)^{2}\right\}\) , which is always negative
BCECE-2007
Matrix and Determinant
79163
If \(\left|\begin{array}{ccc}-12 & 0 & \lambda \\ 0 & 2 & -1 \\ 2 & 1 & 15\end{array}\right|=-360\), then the value of \(\lambda\) is
1 -1
2 -2
3 -3
4 4
Explanation:
(C) : It is given that \(\left|\begin{array}{ccc} -12 & 0 & \lambda \\ 0 & 2 & -1 \\ 2 & 1 & 15 \end{array}\right|=-360\) Now, \(-12(30+1)-4 \lambda=-360\) \(-372+360=4 \lambda\) \(\lambda=\frac{-12}{4} \Rightarrow \lambda=-3\)
79162
Let \(a, b, c\) be positive and not all are equal, the value of the determinant \(\left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|\) is
1 \(+\mathrm{ve}\)
2 -ve
3 zero
4 none of these
Explanation:
(B) : According to given summation, Let, \(\quad \Delta=\left|\begin{array}{lll}\mathrm{a} & \mathrm{b} & \mathrm{c} \\ \mathrm{b} & \mathrm{c} & \mathrm{a} \\ \mathrm{c} & \mathrm{a} & \mathrm{b}\end{array}\right|\) Operating : \(\mathrm{C}_{1} \rightarrow \mathrm{C}_{1}+\mathrm{C}_{2}+\mathrm{C}_{3}\) we get \(\Delta=\left|\begin{array}{lll} \mathrm{a}+\mathrm{b}+\mathrm{c} & \mathrm{b} & \mathrm{c} \\ \mathrm{a}+\mathrm{b}+\mathrm{c} & \mathrm{c} & \mathrm{a} \\ \mathrm{a}+\mathrm{b}+\mathrm{c} & \mathrm{a} & \mathrm{b} \end{array}\right|=(\mathrm{a}+\mathrm{b}+\mathrm{c})\left|\begin{array}{lll} 1 & \mathrm{~b} & \mathrm{c} \\ 1 & \mathrm{c} & \mathrm{a} \\ 1 & \mathrm{a} & \mathrm{b} \end{array}\right|\) Operating: \(\mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-\mathrm{R}_{1}\) and \(\mathrm{R}_{3} \rightarrow \mathrm{R}_{3}-\mathrm{R}_{1}\) we get \(=(\mathrm{a}+\mathrm{b}+\mathrm{c})\left|\begin{array}{ccc} 1 & \mathrm{~b} & \mathrm{c} \\ 0 & \mathrm{c}-\mathrm{b} & \mathrm{a}-\mathrm{c} \\ 0 & \mathrm{a}-\mathrm{b} & \mathrm{b}-\mathrm{c} \end{array}\right|\) \(=(\mathrm{a}+\mathrm{b}+\mathrm{c})\left\{-(\mathrm{c}-\mathrm{b})^{2}-(\mathrm{a}-\mathrm{b})(\mathrm{a}-\mathrm{c})\right\}\) \(=-(\mathrm{a}+\mathrm{b}+\mathrm{c})\left\{\mathrm{a}^{2}+\mathrm{b}^{2}+\mathrm{c}^{2}-\mathrm{ab}-\mathrm{bc}-\mathrm{ca}\right\}\) \(=\frac{-1}{2}(a+b+c)\left\{2 a^{2}+2 b^{2}+2 c^{2}-2 a b-a b c-2 c a\right\} \\ =\frac{-1}{2}(a+b+c)\left\{(a-b)^{2}+(b-c)^{2}\right\}\) , which is always negative
BCECE-2007
Matrix and Determinant
79163
If \(\left|\begin{array}{ccc}-12 & 0 & \lambda \\ 0 & 2 & -1 \\ 2 & 1 & 15\end{array}\right|=-360\), then the value of \(\lambda\) is
1 -1
2 -2
3 -3
4 4
Explanation:
(C) : It is given that \(\left|\begin{array}{ccc} -12 & 0 & \lambda \\ 0 & 2 & -1 \\ 2 & 1 & 15 \end{array}\right|=-360\) Now, \(-12(30+1)-4 \lambda=-360\) \(-372+360=4 \lambda\) \(\lambda=\frac{-12}{4} \Rightarrow \lambda=-3\)