Determinants in 2-D
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Matrix and Determinant

79186 If \(x, y\) and \(z\) are greater than 1 , then the value of
\(\left|\begin{array}{ccc}1 & \log _x y & \log _x z \\ \log _y x & 1 & \log _y z \\ \log _z x & \log _z y & 1\end{array}\right|\) is

1 \(\log x \cdot \log y \cdot \log z\)
2 \(\log x+\log y+\log z\)
3 0
4 \(1-\{(\log \mathrm{x}) \cdot(\log \mathrm{y}) \cdot(\log \mathrm{z})\}\)
WB JEE-2016
Matrix and Determinant

79187 Let \(A=\left(\begin{array}{lll}1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1\end{array}\right)\) then for positive integer \(\mathrm{n}, \mathrm{A}^{\mathrm{n}}\) is

1 \(\left(\begin{array}{ccc}1 & n & n^{2} \\ 0 & n^{2} & n \\ 0 & 0 & n\end{array}\right) \quad\)
2 \(\left(\begin{array}{ccc}1 & n & n\left(\frac{n+1}{2}\right) \\ 0 & 1 & n \\ 0 & 0 & 1\end{array}\right)\)
3 \(\left(\begin{array}{ccc}1 & \mathrm{n}^{2} & \mathrm{n} \\ 0 & \mathrm{n} & \mathrm{n}^{2} \\ 0 & 0 & \mathrm{n}^{2}\end{array}\right)\)
4 \(\left(\begin{array}{ccc}1 & \mathrm{n} & 2 \mathrm{n}-1 \\ 0 & \frac{\mathrm{n}+1}{2} & \mathrm{n} \\ 0 & 0 & \frac{\mathrm{n}+1}{2}\end{array}\right)\)
WB JEE-2017
Matrix and Determinant

79188 The value of det \(A\), where
\(A=\left(\begin{array}{ccc}1 & \cos \theta & 0 \\ -\cos \theta & 1 & \cos \theta \\ -1 & -\cos \theta & 1\end{array}\right)\),lies

1 in the closed interval \([1,2]\)
2 in the closed interval \([0,1]\)
3 in the open interval \((0,1)\)
4 in the open interval \((1,2)\)
WB JEE-2017
Matrix and Determinant

79189 If the polynomial
\(f(x)=\left|\begin{array}{ccc}(1+x)^{a} & (2+x)^{b} & 1 \\ 1 & (1+x)^{a} & (2+x)^{b} \\ (2+x)^{b} & 1 & (1+x)^{a}\end{array}\right|\), then the

1 \(2-3 \cdot 2^{\mathrm{b}}+2^{3 \mathrm{~b}}\)
2 \(2+3 \cdot 2^{\mathrm{b}}+2^{3 \mathrm{~b}}\)
3 \(2+3 \cdot 2^{\mathrm{b}}-2^{3 \mathrm{~b}}\)
4 \(2-3 \cdot 2^{\mathrm{b}}-2^{3 \mathrm{~b}}\)
[ \(a\) and \(b\) are positive integers]
WB JEE-2018
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Matrix and Determinant

79186 If \(x, y\) and \(z\) are greater than 1 , then the value of
\(\left|\begin{array}{ccc}1 & \log _x y & \log _x z \\ \log _y x & 1 & \log _y z \\ \log _z x & \log _z y & 1\end{array}\right|\) is

1 \(\log x \cdot \log y \cdot \log z\)
2 \(\log x+\log y+\log z\)
3 0
4 \(1-\{(\log \mathrm{x}) \cdot(\log \mathrm{y}) \cdot(\log \mathrm{z})\}\)
WB JEE-2016
Matrix and Determinant

79187 Let \(A=\left(\begin{array}{lll}1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1\end{array}\right)\) then for positive integer \(\mathrm{n}, \mathrm{A}^{\mathrm{n}}\) is

1 \(\left(\begin{array}{ccc}1 & n & n^{2} \\ 0 & n^{2} & n \\ 0 & 0 & n\end{array}\right) \quad\)
2 \(\left(\begin{array}{ccc}1 & n & n\left(\frac{n+1}{2}\right) \\ 0 & 1 & n \\ 0 & 0 & 1\end{array}\right)\)
3 \(\left(\begin{array}{ccc}1 & \mathrm{n}^{2} & \mathrm{n} \\ 0 & \mathrm{n} & \mathrm{n}^{2} \\ 0 & 0 & \mathrm{n}^{2}\end{array}\right)\)
4 \(\left(\begin{array}{ccc}1 & \mathrm{n} & 2 \mathrm{n}-1 \\ 0 & \frac{\mathrm{n}+1}{2} & \mathrm{n} \\ 0 & 0 & \frac{\mathrm{n}+1}{2}\end{array}\right)\)
WB JEE-2017
Matrix and Determinant

79188 The value of det \(A\), where
\(A=\left(\begin{array}{ccc}1 & \cos \theta & 0 \\ -\cos \theta & 1 & \cos \theta \\ -1 & -\cos \theta & 1\end{array}\right)\),lies

1 in the closed interval \([1,2]\)
2 in the closed interval \([0,1]\)
3 in the open interval \((0,1)\)
4 in the open interval \((1,2)\)
WB JEE-2017
Matrix and Determinant

79189 If the polynomial
\(f(x)=\left|\begin{array}{ccc}(1+x)^{a} & (2+x)^{b} & 1 \\ 1 & (1+x)^{a} & (2+x)^{b} \\ (2+x)^{b} & 1 & (1+x)^{a}\end{array}\right|\), then the

1 \(2-3 \cdot 2^{\mathrm{b}}+2^{3 \mathrm{~b}}\)
2 \(2+3 \cdot 2^{\mathrm{b}}+2^{3 \mathrm{~b}}\)
3 \(2+3 \cdot 2^{\mathrm{b}}-2^{3 \mathrm{~b}}\)
4 \(2-3 \cdot 2^{\mathrm{b}}-2^{3 \mathrm{~b}}\)
[ \(a\) and \(b\) are positive integers]
WB JEE-2018
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Matrix and Determinant

79186 If \(x, y\) and \(z\) are greater than 1 , then the value of
\(\left|\begin{array}{ccc}1 & \log _x y & \log _x z \\ \log _y x & 1 & \log _y z \\ \log _z x & \log _z y & 1\end{array}\right|\) is

1 \(\log x \cdot \log y \cdot \log z\)
2 \(\log x+\log y+\log z\)
3 0
4 \(1-\{(\log \mathrm{x}) \cdot(\log \mathrm{y}) \cdot(\log \mathrm{z})\}\)
WB JEE-2016
Matrix and Determinant

79187 Let \(A=\left(\begin{array}{lll}1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1\end{array}\right)\) then for positive integer \(\mathrm{n}, \mathrm{A}^{\mathrm{n}}\) is

1 \(\left(\begin{array}{ccc}1 & n & n^{2} \\ 0 & n^{2} & n \\ 0 & 0 & n\end{array}\right) \quad\)
2 \(\left(\begin{array}{ccc}1 & n & n\left(\frac{n+1}{2}\right) \\ 0 & 1 & n \\ 0 & 0 & 1\end{array}\right)\)
3 \(\left(\begin{array}{ccc}1 & \mathrm{n}^{2} & \mathrm{n} \\ 0 & \mathrm{n} & \mathrm{n}^{2} \\ 0 & 0 & \mathrm{n}^{2}\end{array}\right)\)
4 \(\left(\begin{array}{ccc}1 & \mathrm{n} & 2 \mathrm{n}-1 \\ 0 & \frac{\mathrm{n}+1}{2} & \mathrm{n} \\ 0 & 0 & \frac{\mathrm{n}+1}{2}\end{array}\right)\)
WB JEE-2017
Matrix and Determinant

79188 The value of det \(A\), where
\(A=\left(\begin{array}{ccc}1 & \cos \theta & 0 \\ -\cos \theta & 1 & \cos \theta \\ -1 & -\cos \theta & 1\end{array}\right)\),lies

1 in the closed interval \([1,2]\)
2 in the closed interval \([0,1]\)
3 in the open interval \((0,1)\)
4 in the open interval \((1,2)\)
WB JEE-2017
Matrix and Determinant

79189 If the polynomial
\(f(x)=\left|\begin{array}{ccc}(1+x)^{a} & (2+x)^{b} & 1 \\ 1 & (1+x)^{a} & (2+x)^{b} \\ (2+x)^{b} & 1 & (1+x)^{a}\end{array}\right|\), then the

1 \(2-3 \cdot 2^{\mathrm{b}}+2^{3 \mathrm{~b}}\)
2 \(2+3 \cdot 2^{\mathrm{b}}+2^{3 \mathrm{~b}}\)
3 \(2+3 \cdot 2^{\mathrm{b}}-2^{3 \mathrm{~b}}\)
4 \(2-3 \cdot 2^{\mathrm{b}}-2^{3 \mathrm{~b}}\)
[ \(a\) and \(b\) are positive integers]
WB JEE-2018
NEET DIGITAL LIBRARY
Matrix and Determinant

79186 If \(x, y\) and \(z\) are greater than 1 , then the value of
\(\left|\begin{array}{ccc}1 & \log _x y & \log _x z \\ \log _y x & 1 & \log _y z \\ \log _z x & \log _z y & 1\end{array}\right|\) is

1 \(\log x \cdot \log y \cdot \log z\)
2 \(\log x+\log y+\log z\)
3 0
4 \(1-\{(\log \mathrm{x}) \cdot(\log \mathrm{y}) \cdot(\log \mathrm{z})\}\)
WB JEE-2016
Matrix and Determinant

79187 Let \(A=\left(\begin{array}{lll}1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1\end{array}\right)\) then for positive integer \(\mathrm{n}, \mathrm{A}^{\mathrm{n}}\) is

1 \(\left(\begin{array}{ccc}1 & n & n^{2} \\ 0 & n^{2} & n \\ 0 & 0 & n\end{array}\right) \quad\)
2 \(\left(\begin{array}{ccc}1 & n & n\left(\frac{n+1}{2}\right) \\ 0 & 1 & n \\ 0 & 0 & 1\end{array}\right)\)
3 \(\left(\begin{array}{ccc}1 & \mathrm{n}^{2} & \mathrm{n} \\ 0 & \mathrm{n} & \mathrm{n}^{2} \\ 0 & 0 & \mathrm{n}^{2}\end{array}\right)\)
4 \(\left(\begin{array}{ccc}1 & \mathrm{n} & 2 \mathrm{n}-1 \\ 0 & \frac{\mathrm{n}+1}{2} & \mathrm{n} \\ 0 & 0 & \frac{\mathrm{n}+1}{2}\end{array}\right)\)
WB JEE-2017
Matrix and Determinant

79188 The value of det \(A\), where
\(A=\left(\begin{array}{ccc}1 & \cos \theta & 0 \\ -\cos \theta & 1 & \cos \theta \\ -1 & -\cos \theta & 1\end{array}\right)\),lies

1 in the closed interval \([1,2]\)
2 in the closed interval \([0,1]\)
3 in the open interval \((0,1)\)
4 in the open interval \((1,2)\)
WB JEE-2017
Matrix and Determinant

79189 If the polynomial
\(f(x)=\left|\begin{array}{ccc}(1+x)^{a} & (2+x)^{b} & 1 \\ 1 & (1+x)^{a} & (2+x)^{b} \\ (2+x)^{b} & 1 & (1+x)^{a}\end{array}\right|\), then the

1 \(2-3 \cdot 2^{\mathrm{b}}+2^{3 \mathrm{~b}}\)
2 \(2+3 \cdot 2^{\mathrm{b}}+2^{3 \mathrm{~b}}\)
3 \(2+3 \cdot 2^{\mathrm{b}}-2^{3 \mathrm{~b}}\)
4 \(2-3 \cdot 2^{\mathrm{b}}-2^{3 \mathrm{~b}}\)
[ \(a\) and \(b\) are positive integers]
WB JEE-2018
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