79024
There are 2 value of a which makes determinant \(\Delta=\left|\begin{array}{ccc}1 & -2 & 5 \\ 2 & a & -1 \\ 0 & 4 & 2 a\end{array}\right|=86\), Then, the sum of these numbers is
1 4
2 5
3 -4
4 9
Explanation:
(C) : We have, \(\Delta=\left|\begin{array}{ccc}1 & -2 & 5 \\ 2 & \mathrm{a} & -1 \\ 0 & 4 & 2 \mathrm{a}\end{array}\right|=86\) \(\therefore \quad 1\left(2 a^{2}+4\right)+2(4 a-0)+5(8-0)=86\) \(2 \mathrm{a}^{2}+4+8 \mathrm{a}+40=86\) \(2 \mathrm{a}^{2}+8 \mathrm{a}+44-86=0\) \(2 \mathrm{a}^{2}+8 \mathrm{a}-42=0\) \(\mathrm{a}^{2}+4 \mathrm{a}-21=0\) \(\mathrm{a}^{2}+7 \mathrm{a}-3 \mathrm{a}-21=0\) \((a+7)(a-3)=0\) \(a=-7,3\) Hence, the sum of 2 values of \(\mathrm{a}\) is : \(-7+3=-4\)
[JCECE-2018]
Matrix and Determinant
79025
If \(A=\left[\begin{array}{rr}2 & -3 \\ 1 & -1\end{array}\right]\) then what is \(\left|\mathbf{A}^{\mathbf{1 0 0 3}}-\mathbf{5 A}^{1002}\right|\) equal to?
79026
If \(A\) is a \(3 \times 3\) matrix and \(\operatorname{det}(3 A)=k \operatorname{det}(A)\), then what is the value of \(K\) ?
1 3
2 9
3 27
4 81
Explanation:
(C) : It is given that- \(\operatorname{det}(3 \mathrm{~A})=\operatorname{Kdet}(\mathrm{A})\) \(\because|3 \mathrm{~A}|=3^{3}|\mathrm{~A}|\) (As \(\mathrm{A}\) is a \(3 \times 3\) matrix) \(=27|\mathrm{~A}|\) \(\therefore \mathrm{K}=27\)
SCRA-2012
Matrix and Determinant
79027
If \(f(x)=\left|\begin{array}{ccc}1 & x & x+1 \\ 2 x & x(x-1) & x(x+1) \\ 3 x(x-1) & x(x-1)(x-2) & (x+1) x(x-1)\end{array}\right|\), then what is \(f(100)\) equal to ?
79024
There are 2 value of a which makes determinant \(\Delta=\left|\begin{array}{ccc}1 & -2 & 5 \\ 2 & a & -1 \\ 0 & 4 & 2 a\end{array}\right|=86\), Then, the sum of these numbers is
1 4
2 5
3 -4
4 9
Explanation:
(C) : We have, \(\Delta=\left|\begin{array}{ccc}1 & -2 & 5 \\ 2 & \mathrm{a} & -1 \\ 0 & 4 & 2 \mathrm{a}\end{array}\right|=86\) \(\therefore \quad 1\left(2 a^{2}+4\right)+2(4 a-0)+5(8-0)=86\) \(2 \mathrm{a}^{2}+4+8 \mathrm{a}+40=86\) \(2 \mathrm{a}^{2}+8 \mathrm{a}+44-86=0\) \(2 \mathrm{a}^{2}+8 \mathrm{a}-42=0\) \(\mathrm{a}^{2}+4 \mathrm{a}-21=0\) \(\mathrm{a}^{2}+7 \mathrm{a}-3 \mathrm{a}-21=0\) \((a+7)(a-3)=0\) \(a=-7,3\) Hence, the sum of 2 values of \(\mathrm{a}\) is : \(-7+3=-4\)
[JCECE-2018]
Matrix and Determinant
79025
If \(A=\left[\begin{array}{rr}2 & -3 \\ 1 & -1\end{array}\right]\) then what is \(\left|\mathbf{A}^{\mathbf{1 0 0 3}}-\mathbf{5 A}^{1002}\right|\) equal to?
79026
If \(A\) is a \(3 \times 3\) matrix and \(\operatorname{det}(3 A)=k \operatorname{det}(A)\), then what is the value of \(K\) ?
1 3
2 9
3 27
4 81
Explanation:
(C) : It is given that- \(\operatorname{det}(3 \mathrm{~A})=\operatorname{Kdet}(\mathrm{A})\) \(\because|3 \mathrm{~A}|=3^{3}|\mathrm{~A}|\) (As \(\mathrm{A}\) is a \(3 \times 3\) matrix) \(=27|\mathrm{~A}|\) \(\therefore \mathrm{K}=27\)
SCRA-2012
Matrix and Determinant
79027
If \(f(x)=\left|\begin{array}{ccc}1 & x & x+1 \\ 2 x & x(x-1) & x(x+1) \\ 3 x(x-1) & x(x-1)(x-2) & (x+1) x(x-1)\end{array}\right|\), then what is \(f(100)\) equal to ?
79024
There are 2 value of a which makes determinant \(\Delta=\left|\begin{array}{ccc}1 & -2 & 5 \\ 2 & a & -1 \\ 0 & 4 & 2 a\end{array}\right|=86\), Then, the sum of these numbers is
1 4
2 5
3 -4
4 9
Explanation:
(C) : We have, \(\Delta=\left|\begin{array}{ccc}1 & -2 & 5 \\ 2 & \mathrm{a} & -1 \\ 0 & 4 & 2 \mathrm{a}\end{array}\right|=86\) \(\therefore \quad 1\left(2 a^{2}+4\right)+2(4 a-0)+5(8-0)=86\) \(2 \mathrm{a}^{2}+4+8 \mathrm{a}+40=86\) \(2 \mathrm{a}^{2}+8 \mathrm{a}+44-86=0\) \(2 \mathrm{a}^{2}+8 \mathrm{a}-42=0\) \(\mathrm{a}^{2}+4 \mathrm{a}-21=0\) \(\mathrm{a}^{2}+7 \mathrm{a}-3 \mathrm{a}-21=0\) \((a+7)(a-3)=0\) \(a=-7,3\) Hence, the sum of 2 values of \(\mathrm{a}\) is : \(-7+3=-4\)
[JCECE-2018]
Matrix and Determinant
79025
If \(A=\left[\begin{array}{rr}2 & -3 \\ 1 & -1\end{array}\right]\) then what is \(\left|\mathbf{A}^{\mathbf{1 0 0 3}}-\mathbf{5 A}^{1002}\right|\) equal to?
79026
If \(A\) is a \(3 \times 3\) matrix and \(\operatorname{det}(3 A)=k \operatorname{det}(A)\), then what is the value of \(K\) ?
1 3
2 9
3 27
4 81
Explanation:
(C) : It is given that- \(\operatorname{det}(3 \mathrm{~A})=\operatorname{Kdet}(\mathrm{A})\) \(\because|3 \mathrm{~A}|=3^{3}|\mathrm{~A}|\) (As \(\mathrm{A}\) is a \(3 \times 3\) matrix) \(=27|\mathrm{~A}|\) \(\therefore \mathrm{K}=27\)
SCRA-2012
Matrix and Determinant
79027
If \(f(x)=\left|\begin{array}{ccc}1 & x & x+1 \\ 2 x & x(x-1) & x(x+1) \\ 3 x(x-1) & x(x-1)(x-2) & (x+1) x(x-1)\end{array}\right|\), then what is \(f(100)\) equal to ?
79024
There are 2 value of a which makes determinant \(\Delta=\left|\begin{array}{ccc}1 & -2 & 5 \\ 2 & a & -1 \\ 0 & 4 & 2 a\end{array}\right|=86\), Then, the sum of these numbers is
1 4
2 5
3 -4
4 9
Explanation:
(C) : We have, \(\Delta=\left|\begin{array}{ccc}1 & -2 & 5 \\ 2 & \mathrm{a} & -1 \\ 0 & 4 & 2 \mathrm{a}\end{array}\right|=86\) \(\therefore \quad 1\left(2 a^{2}+4\right)+2(4 a-0)+5(8-0)=86\) \(2 \mathrm{a}^{2}+4+8 \mathrm{a}+40=86\) \(2 \mathrm{a}^{2}+8 \mathrm{a}+44-86=0\) \(2 \mathrm{a}^{2}+8 \mathrm{a}-42=0\) \(\mathrm{a}^{2}+4 \mathrm{a}-21=0\) \(\mathrm{a}^{2}+7 \mathrm{a}-3 \mathrm{a}-21=0\) \((a+7)(a-3)=0\) \(a=-7,3\) Hence, the sum of 2 values of \(\mathrm{a}\) is : \(-7+3=-4\)
[JCECE-2018]
Matrix and Determinant
79025
If \(A=\left[\begin{array}{rr}2 & -3 \\ 1 & -1\end{array}\right]\) then what is \(\left|\mathbf{A}^{\mathbf{1 0 0 3}}-\mathbf{5 A}^{1002}\right|\) equal to?
79026
If \(A\) is a \(3 \times 3\) matrix and \(\operatorname{det}(3 A)=k \operatorname{det}(A)\), then what is the value of \(K\) ?
1 3
2 9
3 27
4 81
Explanation:
(C) : It is given that- \(\operatorname{det}(3 \mathrm{~A})=\operatorname{Kdet}(\mathrm{A})\) \(\because|3 \mathrm{~A}|=3^{3}|\mathrm{~A}|\) (As \(\mathrm{A}\) is a \(3 \times 3\) matrix) \(=27|\mathrm{~A}|\) \(\therefore \mathrm{K}=27\)
SCRA-2012
Matrix and Determinant
79027
If \(f(x)=\left|\begin{array}{ccc}1 & x & x+1 \\ 2 x & x(x-1) & x(x+1) \\ 3 x(x-1) & x(x-1)(x-2) & (x+1) x(x-1)\end{array}\right|\), then what is \(f(100)\) equal to ?
