Determinants and their Properties
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Matrix and Determinant

79098 If \(\left|\begin{array}{lll}\mathbf{a}_{1} & \mathbf{b}_{1} & \mathbf{c}_{1} \\ \mathbf{a}_{2} & \mathbf{b}_{2} & \mathbf{c}_{2} \\ \mathbf{a}_{3} & \mathbf{b}_{3} & \mathbf{c}_{3}\end{array}\right|=\mathbf{0}\), then the lines \(\mathbf{a}_{i} \mathbf{x}+\)
\(\mathbf{0}(\mathbf{i}=\mathbf{1}, \mathbf{2}, \mathbf{3})\) represent

1 parallel lines if \(\frac{\mathrm{a}_{\mathrm{i}}}{\mathrm{a}_{\mathrm{j}}} \neq \frac{\mathrm{b}_{\mathrm{i}}}{\mathrm{b}_{\mathrm{j}}} \neq \frac{c_{i}}{c_{j}}(i \neq j)\)
2 coincident lines if \(\frac{a_{i}}{a_{j}}=\frac{b_{i}}{b_{j}}(i \neq j)\)
3 concurrent lines but not coincident if \(\frac{a_{i}}{a_{j}}=\frac{b_{i}}{b_{j}}=\frac{c_{i}}{c_{j}}(i \neq j)\)
4 concurrent lines if \(\frac{a_{i}}{a_{j}} \neq \frac{b_{i}}{b_{j}} \neq \frac{c_{i}}{c_{j}}(i \neq j)\)
TS EAMCET-2020-11.09.2020
Matrix and Determinant

79099 Let \(a, b, c, d, \in R\) be such that \(a d-b c \neq 0\) and \(e\) be a positive number other than 1 . If
\(\mathbf{x}^{\mathrm{a}} \mathbf{y}^{\mathrm{b}}=\mathbf{e}^{\mathrm{m}}, \mathbf{x}^{\mathrm{c}} \mathbf{y}^{\mathrm{d}}=\mathbf{e}^{\mathrm{n}}, \Delta_{1}=\left|\begin{array}{ll}\mathbf{m} & \mathbf{b} \\ \mathbf{n} & \mathbf{d}\end{array}\right|, \quad \Delta_{2}=\left|\begin{array}{ll}\mathbf{a} & \mathbf{m} \\ \mathbf{c} & \mathbf{n}\end{array}\right|\) and \(\Delta_{3}=\left|\begin{array}{ll}a & b \\ c & d\end{array}\right|\), then the values of \(x\) and \(y\) are respectively.

1 \(\mathrm{e}^{\frac{\Delta_{1}}{\Delta_{3}}}, \mathrm{e}^{\frac{\Delta_{2}}{\Delta_{3}}}\)
2 \(\mathrm{e}^{\frac{\Delta_{3}}{\Delta_{2}}}, \mathrm{e}^{\frac{\Delta_{1}}{\Delta_{2}}}\)
3 \(\mathrm{e}^{\frac{-\Delta_{1}}{\Delta_{3}}}, \mathrm{e}^{\frac{-\Delta_{2}}{\Delta_{3}}}\)
4 \(\mathrm{e}^{\frac{\Delta_{2}}{\Delta_{1}}}, \mathrm{e}^{\frac{\Delta_{3}}{\Delta_{1}}}\)
TS EAMCET-2020-14.09.2020
Matrix and Determinant

79100 If \(a\) and \(b\) are any two real numbers, then
\(\left|\begin{array}{ccc} 2 \mathbf{a}-2 \mathbf{b}-\mathbf{4} & \mathbf{4 a} & \mathbf{4 a} \\ 4 & 2-\mathbf{b}-\mathbf{a} & 4 \\ 2 \mathbf{b} & \mathbf{2 b} & \mathbf{b}-\mathbf{a}-\mathbf{2} \end{array}\right|=\)

1 \(4\left[(a+b)^{3}+8(a+b)^{2}+16(a+b)+8\right]\)
2 \(\frac{1}{2}(a+b+2)^{3}\)
3 \(2\left[(a+b)^{3}+6(a+b)^{2}+12(a+b)+8\right]\)
4 \((a+b+2)^{3}\)
TS EAMCET-2020-14.09.2020
Matrix and Determinant

79101 If \(f(x)=\left|\begin{array}{ccc}-\sin x & 2 \sin 2 x & 4 \cos ^{2} x \\ \cos x & 4 \sin ^{2} x & 2 \sin 2 x \\ 0 & -\cos x & \sin x\end{array}\right|\),
then \(f\left(\frac{5 \pi}{4}\right)+f\left(\frac{5 \pi}{4}\right)=\)

1 0
2 -1
3 -2
4 -4
TS EAMCET-2022-19.07.2022
Matrix and Determinant

79102 Let \(\alpha\) be a root of the equation \((a-c) x^{2}+(b-\) a) \(x+(c-b)=0\) where \(a, b, c\) are distinct real numbers such that the matrix
\({\left[\begin{array}{lll}
\alpha^{2} & \alpha & 1 \\ 1 & 1 & 1 \\ \mathbf{a} & \mathbf{b} & \mathbf{c} \end{array}\right] \text { is singular. Then the value of }}\)
\(\frac{(\mathbf{a}-\mathbf{c})^{2}}{(\mathbf{b}-\mathbf{a})(\mathbf{c}-\mathbf{b})}+\frac{(\mathbf{b}-\mathbf{a})^{2}}{(\mathbf{a}-\mathbf{c})(\mathbf{c}-\mathbf{b})}+\frac{(\mathbf{c}-\mathbf{b})^{2}}{(\mathbf{a}-\mathbf{c})(\mathbf{b}-\mathbf{a})} \text { is }\)

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Matrix and Determinant

79098 If \(\left|\begin{array}{lll}\mathbf{a}_{1} & \mathbf{b}_{1} & \mathbf{c}_{1} \\ \mathbf{a}_{2} & \mathbf{b}_{2} & \mathbf{c}_{2} \\ \mathbf{a}_{3} & \mathbf{b}_{3} & \mathbf{c}_{3}\end{array}\right|=\mathbf{0}\), then the lines \(\mathbf{a}_{i} \mathbf{x}+\)
\(\mathbf{0}(\mathbf{i}=\mathbf{1}, \mathbf{2}, \mathbf{3})\) represent

1 parallel lines if \(\frac{\mathrm{a}_{\mathrm{i}}}{\mathrm{a}_{\mathrm{j}}} \neq \frac{\mathrm{b}_{\mathrm{i}}}{\mathrm{b}_{\mathrm{j}}} \neq \frac{c_{i}}{c_{j}}(i \neq j)\)
2 coincident lines if \(\frac{a_{i}}{a_{j}}=\frac{b_{i}}{b_{j}}(i \neq j)\)
3 concurrent lines but not coincident if \(\frac{a_{i}}{a_{j}}=\frac{b_{i}}{b_{j}}=\frac{c_{i}}{c_{j}}(i \neq j)\)
4 concurrent lines if \(\frac{a_{i}}{a_{j}} \neq \frac{b_{i}}{b_{j}} \neq \frac{c_{i}}{c_{j}}(i \neq j)\)
TS EAMCET-2020-11.09.2020
Matrix and Determinant

79099 Let \(a, b, c, d, \in R\) be such that \(a d-b c \neq 0\) and \(e\) be a positive number other than 1 . If
\(\mathbf{x}^{\mathrm{a}} \mathbf{y}^{\mathrm{b}}=\mathbf{e}^{\mathrm{m}}, \mathbf{x}^{\mathrm{c}} \mathbf{y}^{\mathrm{d}}=\mathbf{e}^{\mathrm{n}}, \Delta_{1}=\left|\begin{array}{ll}\mathbf{m} & \mathbf{b} \\ \mathbf{n} & \mathbf{d}\end{array}\right|, \quad \Delta_{2}=\left|\begin{array}{ll}\mathbf{a} & \mathbf{m} \\ \mathbf{c} & \mathbf{n}\end{array}\right|\) and \(\Delta_{3}=\left|\begin{array}{ll}a & b \\ c & d\end{array}\right|\), then the values of \(x\) and \(y\) are respectively.

1 \(\mathrm{e}^{\frac{\Delta_{1}}{\Delta_{3}}}, \mathrm{e}^{\frac{\Delta_{2}}{\Delta_{3}}}\)
2 \(\mathrm{e}^{\frac{\Delta_{3}}{\Delta_{2}}}, \mathrm{e}^{\frac{\Delta_{1}}{\Delta_{2}}}\)
3 \(\mathrm{e}^{\frac{-\Delta_{1}}{\Delta_{3}}}, \mathrm{e}^{\frac{-\Delta_{2}}{\Delta_{3}}}\)
4 \(\mathrm{e}^{\frac{\Delta_{2}}{\Delta_{1}}}, \mathrm{e}^{\frac{\Delta_{3}}{\Delta_{1}}}\)
TS EAMCET-2020-14.09.2020
Matrix and Determinant

79100 If \(a\) and \(b\) are any two real numbers, then
\(\left|\begin{array}{ccc} 2 \mathbf{a}-2 \mathbf{b}-\mathbf{4} & \mathbf{4 a} & \mathbf{4 a} \\ 4 & 2-\mathbf{b}-\mathbf{a} & 4 \\ 2 \mathbf{b} & \mathbf{2 b} & \mathbf{b}-\mathbf{a}-\mathbf{2} \end{array}\right|=\)

1 \(4\left[(a+b)^{3}+8(a+b)^{2}+16(a+b)+8\right]\)
2 \(\frac{1}{2}(a+b+2)^{3}\)
3 \(2\left[(a+b)^{3}+6(a+b)^{2}+12(a+b)+8\right]\)
4 \((a+b+2)^{3}\)
TS EAMCET-2020-14.09.2020
Matrix and Determinant

79101 If \(f(x)=\left|\begin{array}{ccc}-\sin x & 2 \sin 2 x & 4 \cos ^{2} x \\ \cos x & 4 \sin ^{2} x & 2 \sin 2 x \\ 0 & -\cos x & \sin x\end{array}\right|\),
then \(f\left(\frac{5 \pi}{4}\right)+f\left(\frac{5 \pi}{4}\right)=\)

1 0
2 -1
3 -2
4 -4
TS EAMCET-2022-19.07.2022
Matrix and Determinant

79102 Let \(\alpha\) be a root of the equation \((a-c) x^{2}+(b-\) a) \(x+(c-b)=0\) where \(a, b, c\) are distinct real numbers such that the matrix
\({\left[\begin{array}{lll}
\alpha^{2} & \alpha & 1 \\ 1 & 1 & 1 \\ \mathbf{a} & \mathbf{b} & \mathbf{c} \end{array}\right] \text { is singular. Then the value of }}\)
\(\frac{(\mathbf{a}-\mathbf{c})^{2}}{(\mathbf{b}-\mathbf{a})(\mathbf{c}-\mathbf{b})}+\frac{(\mathbf{b}-\mathbf{a})^{2}}{(\mathbf{a}-\mathbf{c})(\mathbf{c}-\mathbf{b})}+\frac{(\mathbf{c}-\mathbf{b})^{2}}{(\mathbf{a}-\mathbf{c})(\mathbf{b}-\mathbf{a})} \text { is }\)

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Matrix and Determinant

79098 If \(\left|\begin{array}{lll}\mathbf{a}_{1} & \mathbf{b}_{1} & \mathbf{c}_{1} \\ \mathbf{a}_{2} & \mathbf{b}_{2} & \mathbf{c}_{2} \\ \mathbf{a}_{3} & \mathbf{b}_{3} & \mathbf{c}_{3}\end{array}\right|=\mathbf{0}\), then the lines \(\mathbf{a}_{i} \mathbf{x}+\)
\(\mathbf{0}(\mathbf{i}=\mathbf{1}, \mathbf{2}, \mathbf{3})\) represent

1 parallel lines if \(\frac{\mathrm{a}_{\mathrm{i}}}{\mathrm{a}_{\mathrm{j}}} \neq \frac{\mathrm{b}_{\mathrm{i}}}{\mathrm{b}_{\mathrm{j}}} \neq \frac{c_{i}}{c_{j}}(i \neq j)\)
2 coincident lines if \(\frac{a_{i}}{a_{j}}=\frac{b_{i}}{b_{j}}(i \neq j)\)
3 concurrent lines but not coincident if \(\frac{a_{i}}{a_{j}}=\frac{b_{i}}{b_{j}}=\frac{c_{i}}{c_{j}}(i \neq j)\)
4 concurrent lines if \(\frac{a_{i}}{a_{j}} \neq \frac{b_{i}}{b_{j}} \neq \frac{c_{i}}{c_{j}}(i \neq j)\)
TS EAMCET-2020-11.09.2020
Matrix and Determinant

79099 Let \(a, b, c, d, \in R\) be such that \(a d-b c \neq 0\) and \(e\) be a positive number other than 1 . If
\(\mathbf{x}^{\mathrm{a}} \mathbf{y}^{\mathrm{b}}=\mathbf{e}^{\mathrm{m}}, \mathbf{x}^{\mathrm{c}} \mathbf{y}^{\mathrm{d}}=\mathbf{e}^{\mathrm{n}}, \Delta_{1}=\left|\begin{array}{ll}\mathbf{m} & \mathbf{b} \\ \mathbf{n} & \mathbf{d}\end{array}\right|, \quad \Delta_{2}=\left|\begin{array}{ll}\mathbf{a} & \mathbf{m} \\ \mathbf{c} & \mathbf{n}\end{array}\right|\) and \(\Delta_{3}=\left|\begin{array}{ll}a & b \\ c & d\end{array}\right|\), then the values of \(x\) and \(y\) are respectively.

1 \(\mathrm{e}^{\frac{\Delta_{1}}{\Delta_{3}}}, \mathrm{e}^{\frac{\Delta_{2}}{\Delta_{3}}}\)
2 \(\mathrm{e}^{\frac{\Delta_{3}}{\Delta_{2}}}, \mathrm{e}^{\frac{\Delta_{1}}{\Delta_{2}}}\)
3 \(\mathrm{e}^{\frac{-\Delta_{1}}{\Delta_{3}}}, \mathrm{e}^{\frac{-\Delta_{2}}{\Delta_{3}}}\)
4 \(\mathrm{e}^{\frac{\Delta_{2}}{\Delta_{1}}}, \mathrm{e}^{\frac{\Delta_{3}}{\Delta_{1}}}\)
TS EAMCET-2020-14.09.2020
Matrix and Determinant

79100 If \(a\) and \(b\) are any two real numbers, then
\(\left|\begin{array}{ccc} 2 \mathbf{a}-2 \mathbf{b}-\mathbf{4} & \mathbf{4 a} & \mathbf{4 a} \\ 4 & 2-\mathbf{b}-\mathbf{a} & 4 \\ 2 \mathbf{b} & \mathbf{2 b} & \mathbf{b}-\mathbf{a}-\mathbf{2} \end{array}\right|=\)

1 \(4\left[(a+b)^{3}+8(a+b)^{2}+16(a+b)+8\right]\)
2 \(\frac{1}{2}(a+b+2)^{3}\)
3 \(2\left[(a+b)^{3}+6(a+b)^{2}+12(a+b)+8\right]\)
4 \((a+b+2)^{3}\)
TS EAMCET-2020-14.09.2020
Matrix and Determinant

79101 If \(f(x)=\left|\begin{array}{ccc}-\sin x & 2 \sin 2 x & 4 \cos ^{2} x \\ \cos x & 4 \sin ^{2} x & 2 \sin 2 x \\ 0 & -\cos x & \sin x\end{array}\right|\),
then \(f\left(\frac{5 \pi}{4}\right)+f\left(\frac{5 \pi}{4}\right)=\)

1 0
2 -1
3 -2
4 -4
TS EAMCET-2022-19.07.2022
Matrix and Determinant

79102 Let \(\alpha\) be a root of the equation \((a-c) x^{2}+(b-\) a) \(x+(c-b)=0\) where \(a, b, c\) are distinct real numbers such that the matrix
\({\left[\begin{array}{lll}
\alpha^{2} & \alpha & 1 \\ 1 & 1 & 1 \\ \mathbf{a} & \mathbf{b} & \mathbf{c} \end{array}\right] \text { is singular. Then the value of }}\)
\(\frac{(\mathbf{a}-\mathbf{c})^{2}}{(\mathbf{b}-\mathbf{a})(\mathbf{c}-\mathbf{b})}+\frac{(\mathbf{b}-\mathbf{a})^{2}}{(\mathbf{a}-\mathbf{c})(\mathbf{c}-\mathbf{b})}+\frac{(\mathbf{c}-\mathbf{b})^{2}}{(\mathbf{a}-\mathbf{c})(\mathbf{b}-\mathbf{a})} \text { is }\)

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Matrix and Determinant

79098 If \(\left|\begin{array}{lll}\mathbf{a}_{1} & \mathbf{b}_{1} & \mathbf{c}_{1} \\ \mathbf{a}_{2} & \mathbf{b}_{2} & \mathbf{c}_{2} \\ \mathbf{a}_{3} & \mathbf{b}_{3} & \mathbf{c}_{3}\end{array}\right|=\mathbf{0}\), then the lines \(\mathbf{a}_{i} \mathbf{x}+\)
\(\mathbf{0}(\mathbf{i}=\mathbf{1}, \mathbf{2}, \mathbf{3})\) represent

1 parallel lines if \(\frac{\mathrm{a}_{\mathrm{i}}}{\mathrm{a}_{\mathrm{j}}} \neq \frac{\mathrm{b}_{\mathrm{i}}}{\mathrm{b}_{\mathrm{j}}} \neq \frac{c_{i}}{c_{j}}(i \neq j)\)
2 coincident lines if \(\frac{a_{i}}{a_{j}}=\frac{b_{i}}{b_{j}}(i \neq j)\)
3 concurrent lines but not coincident if \(\frac{a_{i}}{a_{j}}=\frac{b_{i}}{b_{j}}=\frac{c_{i}}{c_{j}}(i \neq j)\)
4 concurrent lines if \(\frac{a_{i}}{a_{j}} \neq \frac{b_{i}}{b_{j}} \neq \frac{c_{i}}{c_{j}}(i \neq j)\)
TS EAMCET-2020-11.09.2020
Matrix and Determinant

79099 Let \(a, b, c, d, \in R\) be such that \(a d-b c \neq 0\) and \(e\) be a positive number other than 1 . If
\(\mathbf{x}^{\mathrm{a}} \mathbf{y}^{\mathrm{b}}=\mathbf{e}^{\mathrm{m}}, \mathbf{x}^{\mathrm{c}} \mathbf{y}^{\mathrm{d}}=\mathbf{e}^{\mathrm{n}}, \Delta_{1}=\left|\begin{array}{ll}\mathbf{m} & \mathbf{b} \\ \mathbf{n} & \mathbf{d}\end{array}\right|, \quad \Delta_{2}=\left|\begin{array}{ll}\mathbf{a} & \mathbf{m} \\ \mathbf{c} & \mathbf{n}\end{array}\right|\) and \(\Delta_{3}=\left|\begin{array}{ll}a & b \\ c & d\end{array}\right|\), then the values of \(x\) and \(y\) are respectively.

1 \(\mathrm{e}^{\frac{\Delta_{1}}{\Delta_{3}}}, \mathrm{e}^{\frac{\Delta_{2}}{\Delta_{3}}}\)
2 \(\mathrm{e}^{\frac{\Delta_{3}}{\Delta_{2}}}, \mathrm{e}^{\frac{\Delta_{1}}{\Delta_{2}}}\)
3 \(\mathrm{e}^{\frac{-\Delta_{1}}{\Delta_{3}}}, \mathrm{e}^{\frac{-\Delta_{2}}{\Delta_{3}}}\)
4 \(\mathrm{e}^{\frac{\Delta_{2}}{\Delta_{1}}}, \mathrm{e}^{\frac{\Delta_{3}}{\Delta_{1}}}\)
TS EAMCET-2020-14.09.2020
Matrix and Determinant

79100 If \(a\) and \(b\) are any two real numbers, then
\(\left|\begin{array}{ccc} 2 \mathbf{a}-2 \mathbf{b}-\mathbf{4} & \mathbf{4 a} & \mathbf{4 a} \\ 4 & 2-\mathbf{b}-\mathbf{a} & 4 \\ 2 \mathbf{b} & \mathbf{2 b} & \mathbf{b}-\mathbf{a}-\mathbf{2} \end{array}\right|=\)

1 \(4\left[(a+b)^{3}+8(a+b)^{2}+16(a+b)+8\right]\)
2 \(\frac{1}{2}(a+b+2)^{3}\)
3 \(2\left[(a+b)^{3}+6(a+b)^{2}+12(a+b)+8\right]\)
4 \((a+b+2)^{3}\)
TS EAMCET-2020-14.09.2020
Matrix and Determinant

79101 If \(f(x)=\left|\begin{array}{ccc}-\sin x & 2 \sin 2 x & 4 \cos ^{2} x \\ \cos x & 4 \sin ^{2} x & 2 \sin 2 x \\ 0 & -\cos x & \sin x\end{array}\right|\),
then \(f\left(\frac{5 \pi}{4}\right)+f\left(\frac{5 \pi}{4}\right)=\)

1 0
2 -1
3 -2
4 -4
TS EAMCET-2022-19.07.2022
Matrix and Determinant

79102 Let \(\alpha\) be a root of the equation \((a-c) x^{2}+(b-\) a) \(x+(c-b)=0\) where \(a, b, c\) are distinct real numbers such that the matrix
\({\left[\begin{array}{lll}
\alpha^{2} & \alpha & 1 \\ 1 & 1 & 1 \\ \mathbf{a} & \mathbf{b} & \mathbf{c} \end{array}\right] \text { is singular. Then the value of }}\)
\(\frac{(\mathbf{a}-\mathbf{c})^{2}}{(\mathbf{b}-\mathbf{a})(\mathbf{c}-\mathbf{b})}+\frac{(\mathbf{b}-\mathbf{a})^{2}}{(\mathbf{a}-\mathbf{c})(\mathbf{c}-\mathbf{b})}+\frac{(\mathbf{c}-\mathbf{b})^{2}}{(\mathbf{a}-\mathbf{c})(\mathbf{b}-\mathbf{a})} \text { is }\)

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JEE Main-2023-24.01.2023
NEET DIGITAL LIBRARY
Matrix and Determinant

79098 If \(\left|\begin{array}{lll}\mathbf{a}_{1} & \mathbf{b}_{1} & \mathbf{c}_{1} \\ \mathbf{a}_{2} & \mathbf{b}_{2} & \mathbf{c}_{2} \\ \mathbf{a}_{3} & \mathbf{b}_{3} & \mathbf{c}_{3}\end{array}\right|=\mathbf{0}\), then the lines \(\mathbf{a}_{i} \mathbf{x}+\)
\(\mathbf{0}(\mathbf{i}=\mathbf{1}, \mathbf{2}, \mathbf{3})\) represent

1 parallel lines if \(\frac{\mathrm{a}_{\mathrm{i}}}{\mathrm{a}_{\mathrm{j}}} \neq \frac{\mathrm{b}_{\mathrm{i}}}{\mathrm{b}_{\mathrm{j}}} \neq \frac{c_{i}}{c_{j}}(i \neq j)\)
2 coincident lines if \(\frac{a_{i}}{a_{j}}=\frac{b_{i}}{b_{j}}(i \neq j)\)
3 concurrent lines but not coincident if \(\frac{a_{i}}{a_{j}}=\frac{b_{i}}{b_{j}}=\frac{c_{i}}{c_{j}}(i \neq j)\)
4 concurrent lines if \(\frac{a_{i}}{a_{j}} \neq \frac{b_{i}}{b_{j}} \neq \frac{c_{i}}{c_{j}}(i \neq j)\)
TS EAMCET-2020-11.09.2020
Matrix and Determinant

79099 Let \(a, b, c, d, \in R\) be such that \(a d-b c \neq 0\) and \(e\) be a positive number other than 1 . If
\(\mathbf{x}^{\mathrm{a}} \mathbf{y}^{\mathrm{b}}=\mathbf{e}^{\mathrm{m}}, \mathbf{x}^{\mathrm{c}} \mathbf{y}^{\mathrm{d}}=\mathbf{e}^{\mathrm{n}}, \Delta_{1}=\left|\begin{array}{ll}\mathbf{m} & \mathbf{b} \\ \mathbf{n} & \mathbf{d}\end{array}\right|, \quad \Delta_{2}=\left|\begin{array}{ll}\mathbf{a} & \mathbf{m} \\ \mathbf{c} & \mathbf{n}\end{array}\right|\) and \(\Delta_{3}=\left|\begin{array}{ll}a & b \\ c & d\end{array}\right|\), then the values of \(x\) and \(y\) are respectively.

1 \(\mathrm{e}^{\frac{\Delta_{1}}{\Delta_{3}}}, \mathrm{e}^{\frac{\Delta_{2}}{\Delta_{3}}}\)
2 \(\mathrm{e}^{\frac{\Delta_{3}}{\Delta_{2}}}, \mathrm{e}^{\frac{\Delta_{1}}{\Delta_{2}}}\)
3 \(\mathrm{e}^{\frac{-\Delta_{1}}{\Delta_{3}}}, \mathrm{e}^{\frac{-\Delta_{2}}{\Delta_{3}}}\)
4 \(\mathrm{e}^{\frac{\Delta_{2}}{\Delta_{1}}}, \mathrm{e}^{\frac{\Delta_{3}}{\Delta_{1}}}\)
TS EAMCET-2020-14.09.2020
Matrix and Determinant

79100 If \(a\) and \(b\) are any two real numbers, then
\(\left|\begin{array}{ccc} 2 \mathbf{a}-2 \mathbf{b}-\mathbf{4} & \mathbf{4 a} & \mathbf{4 a} \\ 4 & 2-\mathbf{b}-\mathbf{a} & 4 \\ 2 \mathbf{b} & \mathbf{2 b} & \mathbf{b}-\mathbf{a}-\mathbf{2} \end{array}\right|=\)

1 \(4\left[(a+b)^{3}+8(a+b)^{2}+16(a+b)+8\right]\)
2 \(\frac{1}{2}(a+b+2)^{3}\)
3 \(2\left[(a+b)^{3}+6(a+b)^{2}+12(a+b)+8\right]\)
4 \((a+b+2)^{3}\)
TS EAMCET-2020-14.09.2020
Matrix and Determinant

79101 If \(f(x)=\left|\begin{array}{ccc}-\sin x & 2 \sin 2 x & 4 \cos ^{2} x \\ \cos x & 4 \sin ^{2} x & 2 \sin 2 x \\ 0 & -\cos x & \sin x\end{array}\right|\),
then \(f\left(\frac{5 \pi}{4}\right)+f\left(\frac{5 \pi}{4}\right)=\)

1 0
2 -1
3 -2
4 -4
TS EAMCET-2022-19.07.2022
Matrix and Determinant

79102 Let \(\alpha\) be a root of the equation \((a-c) x^{2}+(b-\) a) \(x+(c-b)=0\) where \(a, b, c\) are distinct real numbers such that the matrix
\({\left[\begin{array}{lll}
\alpha^{2} & \alpha & 1 \\ 1 & 1 & 1 \\ \mathbf{a} & \mathbf{b} & \mathbf{c} \end{array}\right] \text { is singular. Then the value of }}\)
\(\frac{(\mathbf{a}-\mathbf{c})^{2}}{(\mathbf{b}-\mathbf{a})(\mathbf{c}-\mathbf{b})}+\frac{(\mathbf{b}-\mathbf{a})^{2}}{(\mathbf{a}-\mathbf{c})(\mathbf{c}-\mathbf{b})}+\frac{(\mathbf{c}-\mathbf{b})^{2}}{(\mathbf{a}-\mathbf{c})(\mathbf{b}-\mathbf{a})} \text { is }\)

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2 3
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4 12
JEE Main-2023-24.01.2023