QBManager

Matrix and Determinant

79428 The equation \(\Delta=\left|\begin{array}{lll}\mathbf{x}-\mathbf{a} & \mathbf{x}-\mathbf{b} & \mathbf{x}-\mathbf{c} \\ \mathbf{x}-\mathbf{b} & \mathbf{x}-\mathbf{c} & \mathbf{x}-\mathbf{a} \\ \mathbf{x}-\mathbf{c} & \mathbf{x}-\mathbf{a} & \mathbf{x}-\mathbf{b}\end{array}\right|=\mathbf{0} \quad\) is satisfied when

1 \(x=\frac{1}{3(a+b+c)}\)

2 \(x=\frac{1}{2(a+b+c)}\)

3 \(\mathrm{x}=\mathrm{a}+\mathrm{b}+\mathrm{c}\)

4 \(\mathrm{x}=\frac{1}{3}(\mathrm{a}+\mathrm{b}+\mathrm{c})\)

SRM JEE-2010

Matrix and Determinant

79429 If \(A, B, C\) are the angles of a triangle, then
\(\Delta=\left|\begin{array}{ccc}-1 & \cos C & \cos B \\ \cos C & -1 & \cos A \\ \cos B & \cos A & -1\end{array}\right|\) equals

1 0

2 -1

3 \(2 \cos \mathrm{A} \cos \mathrm{B} \cos \mathrm{C}\)

4 none of these

SRM JEE-2014

Matrix and Determinant

79430 \(\left|\begin{array}{cccc} 21 & 17 & 7 & 10 \\ 24 & 22 & 6 & 10 \\ 6 & 8 & 2 & 3 \\ 6 & 7 & 1 & 2 \end{array}\right|=\)

1 2

2 -1

3 3

4 none of these

SRM JEE-2015

Matrix and Determinant

79431 If \(A+B+C=\pi\), then \(\left|\begin{array}{lll}\sin ^{2} A & \cot A & 1 \\ \sin ^{2} B & \cot B & 1\end{array}\right|=\) \(\sin ^{2} \mathrm{C} \cot C 1\)

1 1

2 -1

3 2

4 none of these

Explanation:

(D) : According to given summation,
Let, \(\quad \Delta=\left|\begin{array}{lll}\sin ^{2} \mathrm{~A} & \cot \mathrm{A} & 1 \\ \sin ^{2} \mathrm{~B} & \cot \mathrm{B} & 1 \\ \sin ^{2} \mathrm{C} & \cot \mathrm{C} & 1\end{array}\right|\)
Operating \(\mathrm{R}_{1} \rightarrow \mathrm{R}_{1}-\mathrm{R}_{2}\)
\(\mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-\mathrm{R}_{3}\)
\(\Delta=\left|\begin{array}{ccc} \sin ^{2} \mathrm{~A}-\sin ^{2} \mathrm{~B} & \cot \mathrm{A}-\cot \mathrm{B} & 0 \\ \sin ^{2} \mathrm{~B}-\sin ^{2} \mathrm{C} & \cot \mathrm{B}-\cot \mathrm{C} & 0 \\ \sin ^{2} \mathrm{C} & \cot \mathrm{C} & 1 \end{array}\right|\)
(\(\therefore\) Using: \(\quad \sin ^{2} \mathrm{~A}-\sin ^{2} \mathrm{~B}=\sin (\mathrm{A}+\mathrm{B}) \sin (\mathrm{A}-\mathrm{B})\)
\(\Delta=\left|\begin{array}{ccc}\sin (\mathrm{A}+\mathrm{B}) \sin (\mathrm{A}-\mathrm{B}) & \frac{\cos \mathrm{A} \cdot \sin \mathrm{B}-\sin \mathrm{A} \cos \mathrm{B}}{\sin \mathrm{A} \sin \mathrm{B}} & 0 \\ \sin (\mathrm{B}+\mathrm{C}) \sin (\mathrm{B}-\mathrm{C}) & \frac{\cos \mathrm{B} \cdot \sin \mathrm{C}-\sin \mathrm{B} \cos \mathrm{C}}{\sin \mathrm{B} \sin \mathrm{C}} & 0 \\ \sin ^{2} \mathrm{C} & \cot \mathrm{C} & 1\end{array}\right|\)
\((\therefore\) Using: A + B + C \(=\pi)\)
\(\Delta=\left|\begin{array}{ccc}\sin \mathrm{C} \sin (\mathrm{A}-\mathrm{B}) & \frac{-\sin (\mathrm{A}-\mathrm{B})}{\sin \mathrm{A} \sin \mathrm{B}} & 0 \\ \sin \mathrm{A} \sin (\mathrm{B}-\mathrm{C}) & \frac{-\sin (\mathrm{B}-\mathrm{C})}{\sin \mathrm{B} \sin \mathrm{C}} & 0 \\ \sin ^2 \mathrm{C} & \cot \mathrm{C} & 1\end{array}\right|\)
\(\Delta=-\frac{\sin (B-C) \sin C \sin (A-B)}{\sin B \sin C}+\frac{\sin A \sin (B-C) \sin (A-B)}{\sin A \sin B}\)
\(\Delta=-\frac{\sin (\mathrm{B}-\mathrm{C}) \sin (\mathrm{A}-\mathrm{B})}{\sin \mathrm{B}}+\frac{\sin (\mathrm{B}-\mathrm{C}) \sin (\mathrm{A}-\mathrm{B})}{\sin \mathrm{B}}\)
\(\Delta=0\)

SRM JEE-2016

Matrix and Determinant

79428 The equation \(\Delta=\left|\begin{array}{lll}\mathbf{x}-\mathbf{a} & \mathbf{x}-\mathbf{b} & \mathbf{x}-\mathbf{c} \\ \mathbf{x}-\mathbf{b} & \mathbf{x}-\mathbf{c} & \mathbf{x}-\mathbf{a} \\ \mathbf{x}-\mathbf{c} & \mathbf{x}-\mathbf{a} & \mathbf{x}-\mathbf{b}\end{array}\right|=\mathbf{0} \quad\) is satisfied when

1 \(x=\frac{1}{3(a+b+c)}\)

2 \(x=\frac{1}{2(a+b+c)}\)

3 \(\mathrm{x}=\mathrm{a}+\mathrm{b}+\mathrm{c}\)

4 \(\mathrm{x}=\frac{1}{3}(\mathrm{a}+\mathrm{b}+\mathrm{c})\)

SRM JEE-2010

Matrix and Determinant

79429 If \(A, B, C\) are the angles of a triangle, then
\(\Delta=\left|\begin{array}{ccc}-1 & \cos C & \cos B \\ \cos C & -1 & \cos A \\ \cos B & \cos A & -1\end{array}\right|\) equals

1 0

2 -1

3 \(2 \cos \mathrm{A} \cos \mathrm{B} \cos \mathrm{C}\)

4 none of these

SRM JEE-2014

Matrix and Determinant

79430 \(\left|\begin{array}{cccc} 21 & 17 & 7 & 10 \\ 24 & 22 & 6 & 10 \\ 6 & 8 & 2 & 3 \\ 6 & 7 & 1 & 2 \end{array}\right|=\)

1 2

2 -1

3 3

4 none of these

SRM JEE-2015

Matrix and Determinant

79431 If \(A+B+C=\pi\), then \(\left|\begin{array}{lll}\sin ^{2} A & \cot A & 1 \\ \sin ^{2} B & \cot B & 1\end{array}\right|=\) \(\sin ^{2} \mathrm{C} \cot C 1\)

1 1

2 -1

3 2

4 none of these

Explanation:

(D) : According to given summation,
Let, \(\quad \Delta=\left|\begin{array}{lll}\sin ^{2} \mathrm{~A} & \cot \mathrm{A} & 1 \\ \sin ^{2} \mathrm{~B} & \cot \mathrm{B} & 1 \\ \sin ^{2} \mathrm{C} & \cot \mathrm{C} & 1\end{array}\right|\)
Operating \(\mathrm{R}_{1} \rightarrow \mathrm{R}_{1}-\mathrm{R}_{2}\)
\(\mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-\mathrm{R}_{3}\)
\(\Delta=\left|\begin{array}{ccc} \sin ^{2} \mathrm{~A}-\sin ^{2} \mathrm{~B} & \cot \mathrm{A}-\cot \mathrm{B} & 0 \\ \sin ^{2} \mathrm{~B}-\sin ^{2} \mathrm{C} & \cot \mathrm{B}-\cot \mathrm{C} & 0 \\ \sin ^{2} \mathrm{C} & \cot \mathrm{C} & 1 \end{array}\right|\)
(\(\therefore\) Using: \(\quad \sin ^{2} \mathrm{~A}-\sin ^{2} \mathrm{~B}=\sin (\mathrm{A}+\mathrm{B}) \sin (\mathrm{A}-\mathrm{B})\)
\(\Delta=\left|\begin{array}{ccc}\sin (\mathrm{A}+\mathrm{B}) \sin (\mathrm{A}-\mathrm{B}) & \frac{\cos \mathrm{A} \cdot \sin \mathrm{B}-\sin \mathrm{A} \cos \mathrm{B}}{\sin \mathrm{A} \sin \mathrm{B}} & 0 \\ \sin (\mathrm{B}+\mathrm{C}) \sin (\mathrm{B}-\mathrm{C}) & \frac{\cos \mathrm{B} \cdot \sin \mathrm{C}-\sin \mathrm{B} \cos \mathrm{C}}{\sin \mathrm{B} \sin \mathrm{C}} & 0 \\ \sin ^{2} \mathrm{C} & \cot \mathrm{C} & 1\end{array}\right|\)
\((\therefore\) Using: A + B + C \(=\pi)\)
\(\Delta=\left|\begin{array}{ccc}\sin \mathrm{C} \sin (\mathrm{A}-\mathrm{B}) & \frac{-\sin (\mathrm{A}-\mathrm{B})}{\sin \mathrm{A} \sin \mathrm{B}} & 0 \\ \sin \mathrm{A} \sin (\mathrm{B}-\mathrm{C}) & \frac{-\sin (\mathrm{B}-\mathrm{C})}{\sin \mathrm{B} \sin \mathrm{C}} & 0 \\ \sin ^2 \mathrm{C} & \cot \mathrm{C} & 1\end{array}\right|\)
\(\Delta=-\frac{\sin (B-C) \sin C \sin (A-B)}{\sin B \sin C}+\frac{\sin A \sin (B-C) \sin (A-B)}{\sin A \sin B}\)
\(\Delta=-\frac{\sin (\mathrm{B}-\mathrm{C}) \sin (\mathrm{A}-\mathrm{B})}{\sin \mathrm{B}}+\frac{\sin (\mathrm{B}-\mathrm{C}) \sin (\mathrm{A}-\mathrm{B})}{\sin \mathrm{B}}\)
\(\Delta=0\)

SRM JEE-2016

Matrix and Determinant

79428 The equation \(\Delta=\left|\begin{array}{lll}\mathbf{x}-\mathbf{a} & \mathbf{x}-\mathbf{b} & \mathbf{x}-\mathbf{c} \\ \mathbf{x}-\mathbf{b} & \mathbf{x}-\mathbf{c} & \mathbf{x}-\mathbf{a} \\ \mathbf{x}-\mathbf{c} & \mathbf{x}-\mathbf{a} & \mathbf{x}-\mathbf{b}\end{array}\right|=\mathbf{0} \quad\) is satisfied when

1 \(x=\frac{1}{3(a+b+c)}\)

2 \(x=\frac{1}{2(a+b+c)}\)

3 \(\mathrm{x}=\mathrm{a}+\mathrm{b}+\mathrm{c}\)

4 \(\mathrm{x}=\frac{1}{3}(\mathrm{a}+\mathrm{b}+\mathrm{c})\)

SRM JEE-2010

Matrix and Determinant

79429 If \(A, B, C\) are the angles of a triangle, then
\(\Delta=\left|\begin{array}{ccc}-1 & \cos C & \cos B \\ \cos C & -1 & \cos A \\ \cos B & \cos A & -1\end{array}\right|\) equals

1 0

2 -1

3 \(2 \cos \mathrm{A} \cos \mathrm{B} \cos \mathrm{C}\)

4 none of these

SRM JEE-2014

Matrix and Determinant

79430 \(\left|\begin{array}{cccc} 21 & 17 & 7 & 10 \\ 24 & 22 & 6 & 10 \\ 6 & 8 & 2 & 3 \\ 6 & 7 & 1 & 2 \end{array}\right|=\)

1 2

2 -1

3 3

4 none of these

SRM JEE-2015

Matrix and Determinant

79431 If \(A+B+C=\pi\), then \(\left|\begin{array}{lll}\sin ^{2} A & \cot A & 1 \\ \sin ^{2} B & \cot B & 1\end{array}\right|=\) \(\sin ^{2} \mathrm{C} \cot C 1\)

1 1

2 -1

3 2

4 none of these

Explanation:

(D) : According to given summation,
Let, \(\quad \Delta=\left|\begin{array}{lll}\sin ^{2} \mathrm{~A} & \cot \mathrm{A} & 1 \\ \sin ^{2} \mathrm{~B} & \cot \mathrm{B} & 1 \\ \sin ^{2} \mathrm{C} & \cot \mathrm{C} & 1\end{array}\right|\)
Operating \(\mathrm{R}_{1} \rightarrow \mathrm{R}_{1}-\mathrm{R}_{2}\)
\(\mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-\mathrm{R}_{3}\)
\(\Delta=\left|\begin{array}{ccc} \sin ^{2} \mathrm{~A}-\sin ^{2} \mathrm{~B} & \cot \mathrm{A}-\cot \mathrm{B} & 0 \\ \sin ^{2} \mathrm{~B}-\sin ^{2} \mathrm{C} & \cot \mathrm{B}-\cot \mathrm{C} & 0 \\ \sin ^{2} \mathrm{C} & \cot \mathrm{C} & 1 \end{array}\right|\)
(\(\therefore\) Using: \(\quad \sin ^{2} \mathrm{~A}-\sin ^{2} \mathrm{~B}=\sin (\mathrm{A}+\mathrm{B}) \sin (\mathrm{A}-\mathrm{B})\)
\(\Delta=\left|\begin{array}{ccc}\sin (\mathrm{A}+\mathrm{B}) \sin (\mathrm{A}-\mathrm{B}) & \frac{\cos \mathrm{A} \cdot \sin \mathrm{B}-\sin \mathrm{A} \cos \mathrm{B}}{\sin \mathrm{A} \sin \mathrm{B}} & 0 \\ \sin (\mathrm{B}+\mathrm{C}) \sin (\mathrm{B}-\mathrm{C}) & \frac{\cos \mathrm{B} \cdot \sin \mathrm{C}-\sin \mathrm{B} \cos \mathrm{C}}{\sin \mathrm{B} \sin \mathrm{C}} & 0 \\ \sin ^{2} \mathrm{C} & \cot \mathrm{C} & 1\end{array}\right|\)
\((\therefore\) Using: A + B + C \(=\pi)\)
\(\Delta=\left|\begin{array}{ccc}\sin \mathrm{C} \sin (\mathrm{A}-\mathrm{B}) & \frac{-\sin (\mathrm{A}-\mathrm{B})}{\sin \mathrm{A} \sin \mathrm{B}} & 0 \\ \sin \mathrm{A} \sin (\mathrm{B}-\mathrm{C}) & \frac{-\sin (\mathrm{B}-\mathrm{C})}{\sin \mathrm{B} \sin \mathrm{C}} & 0 \\ \sin ^2 \mathrm{C} & \cot \mathrm{C} & 1\end{array}\right|\)
\(\Delta=-\frac{\sin (B-C) \sin C \sin (A-B)}{\sin B \sin C}+\frac{\sin A \sin (B-C) \sin (A-B)}{\sin A \sin B}\)
\(\Delta=-\frac{\sin (\mathrm{B}-\mathrm{C}) \sin (\mathrm{A}-\mathrm{B})}{\sin \mathrm{B}}+\frac{\sin (\mathrm{B}-\mathrm{C}) \sin (\mathrm{A}-\mathrm{B})}{\sin \mathrm{B}}\)
\(\Delta=0\)

SRM JEE-2016

Matrix and Determinant

79428 The equation \(\Delta=\left|\begin{array}{lll}\mathbf{x}-\mathbf{a} & \mathbf{x}-\mathbf{b} & \mathbf{x}-\mathbf{c} \\ \mathbf{x}-\mathbf{b} & \mathbf{x}-\mathbf{c} & \mathbf{x}-\mathbf{a} \\ \mathbf{x}-\mathbf{c} & \mathbf{x}-\mathbf{a} & \mathbf{x}-\mathbf{b}\end{array}\right|=\mathbf{0} \quad\) is satisfied when

1 \(x=\frac{1}{3(a+b+c)}\)

2 \(x=\frac{1}{2(a+b+c)}\)

3 \(\mathrm{x}=\mathrm{a}+\mathrm{b}+\mathrm{c}\)

4 \(\mathrm{x}=\frac{1}{3}(\mathrm{a}+\mathrm{b}+\mathrm{c})\)

SRM JEE-2010

Matrix and Determinant

79429 If \(A, B, C\) are the angles of a triangle, then
\(\Delta=\left|\begin{array}{ccc}-1 & \cos C & \cos B \\ \cos C & -1 & \cos A \\ \cos B & \cos A & -1\end{array}\right|\) equals

1 0

2 -1

3 \(2 \cos \mathrm{A} \cos \mathrm{B} \cos \mathrm{C}\)

4 none of these

SRM JEE-2014

Matrix and Determinant

79430 \(\left|\begin{array}{cccc} 21 & 17 & 7 & 10 \\ 24 & 22 & 6 & 10 \\ 6 & 8 & 2 & 3 \\ 6 & 7 & 1 & 2 \end{array}\right|=\)

1 2

2 -1

3 3

4 none of these

SRM JEE-2015

Matrix and Determinant

79431 If \(A+B+C=\pi\), then \(\left|\begin{array}{lll}\sin ^{2} A & \cot A & 1 \\ \sin ^{2} B & \cot B & 1\end{array}\right|=\) \(\sin ^{2} \mathrm{C} \cot C 1\)

1 1

2 -1

3 2

4 none of these

Explanation:

(D) : According to given summation,
Let, \(\quad \Delta=\left|\begin{array}{lll}\sin ^{2} \mathrm{~A} & \cot \mathrm{A} & 1 \\ \sin ^{2} \mathrm{~B} & \cot \mathrm{B} & 1 \\ \sin ^{2} \mathrm{C} & \cot \mathrm{C} & 1\end{array}\right|\)
Operating \(\mathrm{R}_{1} \rightarrow \mathrm{R}_{1}-\mathrm{R}_{2}\)
\(\mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-\mathrm{R}_{3}\)
\(\Delta=\left|\begin{array}{ccc} \sin ^{2} \mathrm{~A}-\sin ^{2} \mathrm{~B} & \cot \mathrm{A}-\cot \mathrm{B} & 0 \\ \sin ^{2} \mathrm{~B}-\sin ^{2} \mathrm{C} & \cot \mathrm{B}-\cot \mathrm{C} & 0 \\ \sin ^{2} \mathrm{C} & \cot \mathrm{C} & 1 \end{array}\right|\)
(\(\therefore\) Using: \(\quad \sin ^{2} \mathrm{~A}-\sin ^{2} \mathrm{~B}=\sin (\mathrm{A}+\mathrm{B}) \sin (\mathrm{A}-\mathrm{B})\)
\(\Delta=\left|\begin{array}{ccc}\sin (\mathrm{A}+\mathrm{B}) \sin (\mathrm{A}-\mathrm{B}) & \frac{\cos \mathrm{A} \cdot \sin \mathrm{B}-\sin \mathrm{A} \cos \mathrm{B}}{\sin \mathrm{A} \sin \mathrm{B}} & 0 \\ \sin (\mathrm{B}+\mathrm{C}) \sin (\mathrm{B}-\mathrm{C}) & \frac{\cos \mathrm{B} \cdot \sin \mathrm{C}-\sin \mathrm{B} \cos \mathrm{C}}{\sin \mathrm{B} \sin \mathrm{C}} & 0 \\ \sin ^{2} \mathrm{C} & \cot \mathrm{C} & 1\end{array}\right|\)
\((\therefore\) Using: A + B + C \(=\pi)\)
\(\Delta=\left|\begin{array}{ccc}\sin \mathrm{C} \sin (\mathrm{A}-\mathrm{B}) & \frac{-\sin (\mathrm{A}-\mathrm{B})}{\sin \mathrm{A} \sin \mathrm{B}} & 0 \\ \sin \mathrm{A} \sin (\mathrm{B}-\mathrm{C}) & \frac{-\sin (\mathrm{B}-\mathrm{C})}{\sin \mathrm{B} \sin \mathrm{C}} & 0 \\ \sin ^2 \mathrm{C} & \cot \mathrm{C} & 1\end{array}\right|\)
\(\Delta=-\frac{\sin (B-C) \sin C \sin (A-B)}{\sin B \sin C}+\frac{\sin A \sin (B-C) \sin (A-B)}{\sin A \sin B}\)
\(\Delta=-\frac{\sin (\mathrm{B}-\mathrm{C}) \sin (\mathrm{A}-\mathrm{B})}{\sin \mathrm{B}}+\frac{\sin (\mathrm{B}-\mathrm{C}) \sin (\mathrm{A}-\mathrm{B})}{\sin \mathrm{B}}\)
\(\Delta=0\)

SRM JEE-2016

Question Bank Series

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