Determinants and their Properties
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Matrix and Determinant

79013 If \(\Delta(r)=\left|\begin{array}{cc}r & r^{3} \\ 1 & n(n+1)\end{array}\right|\), then \(\sum_{r=1}^{n} \Delta(r)\) is equal to

1 \(\sum_{\mathrm{r}=1}^{\mathrm{n}} \mathrm{r}^{2}\)
2 \(\sum_{\mathrm{r}=1}^{\mathrm{n}} \mathrm{r}^{3}\)
3 \(\sum_{\mathrm{r}=1}^{\mathrm{n}} \mathrm{r}\)
4 \(\sum_{\mathrm{r}=1}^{\mathrm{n}} \mathrm{r}^{4}\)
VITEEE-2015
Matrix and Determinant

79014 The value of the determinant
\(\left|\begin{array}{ccc} 1 & \cos (\alpha-\beta) & \cos \alpha \\ \cos (\alpha-\beta) & 1 & \cos \beta \\ \cos \alpha & \cos \beta & 1
\end{array}\right|\) is

1 \(\alpha^{2}+\beta^{2}\)
2 \(\alpha^{2}-\beta^{2}\)
3 1
4 0
VITEEE-2013
Matrix and Determinant

79015 If \(a, b\) and \(c\) are in \(A P\), then determinant
\(\left|\begin{array}{lll}
x+2 & x+3 & x+2 a \\ x+3 & x+4 & x+2 b \\ x+4 & x+5 & x+2 c \end{array}\right| \text { is }\)

1 0
2 1
3 \(\mathrm{x}\)
4 \(2 \mathrm{x}\)
VITEEE-2013
Matrix and Determinant

79016 The value of the determinant
\(\left|\begin{array}{ccc}\cos \alpha & -\sin \alpha & 1 \\ \sin \alpha & \cos \alpha & 1 \\ \cos (\alpha+\beta) & -\sin (\alpha+\beta) & 1\end{array}\right|\) is

1 independent of \(\alpha\)
2 independent of \(\beta\)
3 independent of \(\alpha\) and \(\beta\)
4 None of the above
VITEEE-2012
Matrix and Determinant

79019 If in a \(\triangle \mathrm{ABC}\),
\(\sin ^{3} \mathrm{~A}+\sin ^{3} \mathrm{~B}+\sin ^{3} \mathrm{C}=3 \sin \mathrm{A} \sin \mathrm{B} \sin \mathrm{C}\), then
the value of determinant \(\left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|\) is equal to

1 0
2 1
3 2
4 3
UPSEE-2014
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Matrix and Determinant

79013 If \(\Delta(r)=\left|\begin{array}{cc}r & r^{3} \\ 1 & n(n+1)\end{array}\right|\), then \(\sum_{r=1}^{n} \Delta(r)\) is equal to

1 \(\sum_{\mathrm{r}=1}^{\mathrm{n}} \mathrm{r}^{2}\)
2 \(\sum_{\mathrm{r}=1}^{\mathrm{n}} \mathrm{r}^{3}\)
3 \(\sum_{\mathrm{r}=1}^{\mathrm{n}} \mathrm{r}\)
4 \(\sum_{\mathrm{r}=1}^{\mathrm{n}} \mathrm{r}^{4}\)
VITEEE-2015
Matrix and Determinant

79014 The value of the determinant
\(\left|\begin{array}{ccc} 1 & \cos (\alpha-\beta) & \cos \alpha \\ \cos (\alpha-\beta) & 1 & \cos \beta \\ \cos \alpha & \cos \beta & 1
\end{array}\right|\) is

1 \(\alpha^{2}+\beta^{2}\)
2 \(\alpha^{2}-\beta^{2}\)
3 1
4 0
VITEEE-2013
Matrix and Determinant

79015 If \(a, b\) and \(c\) are in \(A P\), then determinant
\(\left|\begin{array}{lll}
x+2 & x+3 & x+2 a \\ x+3 & x+4 & x+2 b \\ x+4 & x+5 & x+2 c \end{array}\right| \text { is }\)

1 0
2 1
3 \(\mathrm{x}\)
4 \(2 \mathrm{x}\)
VITEEE-2013
Matrix and Determinant

79016 The value of the determinant
\(\left|\begin{array}{ccc}\cos \alpha & -\sin \alpha & 1 \\ \sin \alpha & \cos \alpha & 1 \\ \cos (\alpha+\beta) & -\sin (\alpha+\beta) & 1\end{array}\right|\) is

1 independent of \(\alpha\)
2 independent of \(\beta\)
3 independent of \(\alpha\) and \(\beta\)
4 None of the above
VITEEE-2012
Matrix and Determinant

79019 If in a \(\triangle \mathrm{ABC}\),
\(\sin ^{3} \mathrm{~A}+\sin ^{3} \mathrm{~B}+\sin ^{3} \mathrm{C}=3 \sin \mathrm{A} \sin \mathrm{B} \sin \mathrm{C}\), then
the value of determinant \(\left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|\) is equal to

1 0
2 1
3 2
4 3
UPSEE-2014
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Matrix and Determinant

79013 If \(\Delta(r)=\left|\begin{array}{cc}r & r^{3} \\ 1 & n(n+1)\end{array}\right|\), then \(\sum_{r=1}^{n} \Delta(r)\) is equal to

1 \(\sum_{\mathrm{r}=1}^{\mathrm{n}} \mathrm{r}^{2}\)
2 \(\sum_{\mathrm{r}=1}^{\mathrm{n}} \mathrm{r}^{3}\)
3 \(\sum_{\mathrm{r}=1}^{\mathrm{n}} \mathrm{r}\)
4 \(\sum_{\mathrm{r}=1}^{\mathrm{n}} \mathrm{r}^{4}\)
VITEEE-2015
Matrix and Determinant

79014 The value of the determinant
\(\left|\begin{array}{ccc} 1 & \cos (\alpha-\beta) & \cos \alpha \\ \cos (\alpha-\beta) & 1 & \cos \beta \\ \cos \alpha & \cos \beta & 1
\end{array}\right|\) is

1 \(\alpha^{2}+\beta^{2}\)
2 \(\alpha^{2}-\beta^{2}\)
3 1
4 0
VITEEE-2013
Matrix and Determinant

79015 If \(a, b\) and \(c\) are in \(A P\), then determinant
\(\left|\begin{array}{lll}
x+2 & x+3 & x+2 a \\ x+3 & x+4 & x+2 b \\ x+4 & x+5 & x+2 c \end{array}\right| \text { is }\)

1 0
2 1
3 \(\mathrm{x}\)
4 \(2 \mathrm{x}\)
VITEEE-2013
Matrix and Determinant

79016 The value of the determinant
\(\left|\begin{array}{ccc}\cos \alpha & -\sin \alpha & 1 \\ \sin \alpha & \cos \alpha & 1 \\ \cos (\alpha+\beta) & -\sin (\alpha+\beta) & 1\end{array}\right|\) is

1 independent of \(\alpha\)
2 independent of \(\beta\)
3 independent of \(\alpha\) and \(\beta\)
4 None of the above
VITEEE-2012
Matrix and Determinant

79019 If in a \(\triangle \mathrm{ABC}\),
\(\sin ^{3} \mathrm{~A}+\sin ^{3} \mathrm{~B}+\sin ^{3} \mathrm{C}=3 \sin \mathrm{A} \sin \mathrm{B} \sin \mathrm{C}\), then
the value of determinant \(\left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|\) is equal to

1 0
2 1
3 2
4 3
UPSEE-2014
NEET DIGITAL LIBRARY
Matrix and Determinant

79013 If \(\Delta(r)=\left|\begin{array}{cc}r & r^{3} \\ 1 & n(n+1)\end{array}\right|\), then \(\sum_{r=1}^{n} \Delta(r)\) is equal to

1 \(\sum_{\mathrm{r}=1}^{\mathrm{n}} \mathrm{r}^{2}\)
2 \(\sum_{\mathrm{r}=1}^{\mathrm{n}} \mathrm{r}^{3}\)
3 \(\sum_{\mathrm{r}=1}^{\mathrm{n}} \mathrm{r}\)
4 \(\sum_{\mathrm{r}=1}^{\mathrm{n}} \mathrm{r}^{4}\)
VITEEE-2015
Matrix and Determinant

79014 The value of the determinant
\(\left|\begin{array}{ccc} 1 & \cos (\alpha-\beta) & \cos \alpha \\ \cos (\alpha-\beta) & 1 & \cos \beta \\ \cos \alpha & \cos \beta & 1
\end{array}\right|\) is

1 \(\alpha^{2}+\beta^{2}\)
2 \(\alpha^{2}-\beta^{2}\)
3 1
4 0
VITEEE-2013
Matrix and Determinant

79015 If \(a, b\) and \(c\) are in \(A P\), then determinant
\(\left|\begin{array}{lll}
x+2 & x+3 & x+2 a \\ x+3 & x+4 & x+2 b \\ x+4 & x+5 & x+2 c \end{array}\right| \text { is }\)

1 0
2 1
3 \(\mathrm{x}\)
4 \(2 \mathrm{x}\)
VITEEE-2013
Matrix and Determinant

79016 The value of the determinant
\(\left|\begin{array}{ccc}\cos \alpha & -\sin \alpha & 1 \\ \sin \alpha & \cos \alpha & 1 \\ \cos (\alpha+\beta) & -\sin (\alpha+\beta) & 1\end{array}\right|\) is

1 independent of \(\alpha\)
2 independent of \(\beta\)
3 independent of \(\alpha\) and \(\beta\)
4 None of the above
VITEEE-2012
Matrix and Determinant

79019 If in a \(\triangle \mathrm{ABC}\),
\(\sin ^{3} \mathrm{~A}+\sin ^{3} \mathrm{~B}+\sin ^{3} \mathrm{C}=3 \sin \mathrm{A} \sin \mathrm{B} \sin \mathrm{C}\), then
the value of determinant \(\left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|\) is equal to

1 0
2 1
3 2
4 3
UPSEE-2014
Join With Us for More Updates
Matrix and Determinant

79013 If \(\Delta(r)=\left|\begin{array}{cc}r & r^{3} \\ 1 & n(n+1)\end{array}\right|\), then \(\sum_{r=1}^{n} \Delta(r)\) is equal to

1 \(\sum_{\mathrm{r}=1}^{\mathrm{n}} \mathrm{r}^{2}\)
2 \(\sum_{\mathrm{r}=1}^{\mathrm{n}} \mathrm{r}^{3}\)
3 \(\sum_{\mathrm{r}=1}^{\mathrm{n}} \mathrm{r}\)
4 \(\sum_{\mathrm{r}=1}^{\mathrm{n}} \mathrm{r}^{4}\)
VITEEE-2015
Matrix and Determinant

79014 The value of the determinant
\(\left|\begin{array}{ccc} 1 & \cos (\alpha-\beta) & \cos \alpha \\ \cos (\alpha-\beta) & 1 & \cos \beta \\ \cos \alpha & \cos \beta & 1
\end{array}\right|\) is

1 \(\alpha^{2}+\beta^{2}\)
2 \(\alpha^{2}-\beta^{2}\)
3 1
4 0
VITEEE-2013
Matrix and Determinant

79015 If \(a, b\) and \(c\) are in \(A P\), then determinant
\(\left|\begin{array}{lll}
x+2 & x+3 & x+2 a \\ x+3 & x+4 & x+2 b \\ x+4 & x+5 & x+2 c \end{array}\right| \text { is }\)

1 0
2 1
3 \(\mathrm{x}\)
4 \(2 \mathrm{x}\)
VITEEE-2013
Matrix and Determinant

79016 The value of the determinant
\(\left|\begin{array}{ccc}\cos \alpha & -\sin \alpha & 1 \\ \sin \alpha & \cos \alpha & 1 \\ \cos (\alpha+\beta) & -\sin (\alpha+\beta) & 1\end{array}\right|\) is

1 independent of \(\alpha\)
2 independent of \(\beta\)
3 independent of \(\alpha\) and \(\beta\)
4 None of the above
VITEEE-2012
Matrix and Determinant

79019 If in a \(\triangle \mathrm{ABC}\),
\(\sin ^{3} \mathrm{~A}+\sin ^{3} \mathrm{~B}+\sin ^{3} \mathrm{C}=3 \sin \mathrm{A} \sin \mathrm{B} \sin \mathrm{C}\), then
the value of determinant \(\left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|\) is equal to

1 0
2 1
3 2
4 3
UPSEE-2014