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Determinants — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDeterminants1 Mark
Question
State True or False for the statements of the following Exercise:
The determinant $\begin{vmatrix}\sin\text{A}&\cos\text{A}&\sin\text{A}+\cos\text{B}\\\sin\text{B}&\cos\text{A}&\sin\text{B}+\cos\text{B}\\\sin\text{C}&\cos\text{A}&\sin\text{C}+\cos\text{B}\end{vmatrix}$ is equal to zero.
✓

Answer

True.
Solution:
Since, $\begin{vmatrix}\sin\text{A}&\cos\text{A}&\sin\text{A}+\cos\text{B}\\\sin\text{B}&\cos\text{A}&\sin\text{B}+\cos\text{B}\\\sin\text{C}&\cos\text{A}&\sin\text{C}+\cos\text{B}\end{vmatrix}$
$=\begin{vmatrix}\sin\text{A}&\cos\text{A}&\sin\text{A}\\\sin\text{B}&\cos\text{A}&\sin\text{B}\\\sin\text{C}&\cos\text{A}&\sin\text{C}\end{vmatrix}+\begin{vmatrix}\sin\text{A}&\cos\text{A}&\cos\text{B}\\\sin\text{B}&\cos\text{A}&\cos\text{B}\\\sin\text{C}&\cos\text{A}&\cos\text{B}\end{vmatrix}$
$=0+\begin{vmatrix}\sin\text{A}&\cos\text{A}&\cos\text{B}\\\sin\text{B}&\cos\text{A}&\cos\text{B}\\\sin\text{C}&\cos\text{A}&\cos\text{B}\end{vmatrix}$
[Since, in first determinant $C_1$ and $C_3$ are identicals]
$=\cos\text{A}.\cos\text{B}\begin{vmatrix}\sin\text{A}&1&1\\\sin\text{B}&1&1\\\sin\text{C}&1&1\end{vmatrix}$
[Taking cos A common from $C_2$ and cos B common from $C_3$]
= 0 [since, $C_2$ and $C_3$ are identicals]

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