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Mark the tick against the correct answer in the following: \left|\begin{array}{ccc} \sin \alpha & \cos \alpha & \sin (\alpha+\delta) \\ \sin \beta & \cos \beta & \sin (\beta+\delta) \\ \sin \gamma & \cos \gamma & \sin (\gamma+\delta) \end{array}\right|=?
A. 0
B. 1
C. \sin (\alpha+\delta)+\sin (\beta+\delta)+\sin (\gamma+\delta)
D. none of these
CBSE | Class 12 | Determinants | Maths | RS Aggarwal
Mark the tick against the correct answer in the following: \left|\begin{array}{ccc} \sin \alpha & \cos \alpha & \sin (\alpha+\delta) \\ \sin \beta & \cos \beta & \sin (\beta+\delta) \\ \sin \gamma & \cos \gamma & \sin (\gamma+\delta) \end{array}\right|=?
A. 0
B. 1
C. \sin (\alpha+\delta)+\sin (\beta+\delta)+\sin (\gamma+\delta)
D. none of these

Solution:

Option(A)
To find: Value of \left|\begin{array}{lll}\operatorname{sind} & \cos \alpha & \sin (\alpha+\bar{\delta}) \\ \sin \beta & \cos \beta & \sin (\beta+\bar{\delta}) \\ \sin \gamma & \cos \gamma & \sin (\gamma+\delta)\end{array}\right|
Formula Used: \sin (A+B)=\sin A \cos B+\cos A \sin B
We have, \left|\begin{array}{lll}\operatorname{sind} & \cos \alpha & \sin (\alpha+\bar{\delta}) \\ \sin \beta & \cos \beta & \sin (\beta+\delta) \\ \sin \gamma & \cos \gamma & \sin (\gamma+\delta)\end{array}\right|
Applying C_{1} \rightarrow \cos (\bar{\delta}) C_{1}
\Rightarrow\left|\begin{array}{lll} \sin a \cos \delta & \cos a & \sin (\alpha+\delta) \\ \sin \beta \cos \delta & \cos \beta & \sin (\beta+\delta) \\ \sin \gamma \cos \bar{\delta} & \cos \gamma & \sin (\gamma+\delta) \end{array}\right|
Applying C_{2} \rightarrow \sin (\delta) C_{2}

*** QuickLaTeX cannot compile formula:
\Rightarrow\left|\begin{array}{lll}
\sin a \cos \delta \overline & \cos a \sin \overline{ } & \sin (a+\bar{\delta}) \\
\sin \beta \cos \bar{\sigma} & \cos \beta \sin \bar{\sigma} & \sin (\beta+\bar{\delta}) \\
\sin \gamma \cos \delta & \cos \gamma \sin \bar{\delta} & \sin (\gamma+\bar{\delta})
\end{array}\right|

*** Error message:
Missing } inserted.
leading text: \sin a \cos \delta \overline &
Missing \cr inserted.
leading text: \sin a \cos \delta \overline &
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leading text: \sin a \cos \delta \overline &
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leading text: \sin a \cos \delta \overline &
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leading text: \sin a \cos \delta \overline &
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leading text: \sin a \cos \delta \overline &
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leading text: \sin a \cos \delta \overline &
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leading text: \sin a \cos \delta \overline &
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leading text: \sin a \cos \delta \overline &
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leading text: \sin a \cos \delta \overline &
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leading text: \sin a \cos \delta \overline &
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leading text: \sin a \cos \delta \overline &

We know, \sin (A+B)=\sin A \cos B+\cos A \sin B
Applying \mathrm{C}_{3} \rightarrow \mathrm{C}_{3}-\mathrm{C}_{1}
=0
When two columns are identical then the value of determinant is 0

i

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