The value of $\left| {\begin{array}{*{20}{c}}
{\sin \alpha }&{\cos \alpha }&{\sin \left( {\alpha + \gamma } \right)}\\
{\sin \beta }&{\cos \beta }&{\sin \left( {\beta + \gamma } \right)}\\
{\sin \delta }&{\cos \delta }&{\sin \left( {\gamma + \delta } \right)}
\end{array}} \right|$ is
The value of $\left| {\begin{array}{*{20}{c}}
{\sin \alpha }&{\cos \alpha }&{\sin \left( {\alpha + \gamma } \right)}\\
{\sin \beta }&{\cos \beta }&{\sin \left( {\beta + \gamma } \right)}\\
{\sin \delta }&{\cos \delta }&{\sin \left( {\gamma + \delta } \right)}
\end{array}} \right|$ is
- A
$\sin \alpha \sin \beta \sin \delta $
- B
$\cos \alpha \cos \beta \cos \delta $
- C
$1$
- D
$0$
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$\left| {\,\begin{array}{*{20}{c}}{{{({a^x} + {a^{ - x}})}^2}}&{{{({a^x} - {a^{ - x}})}^2}}&1\\{{{({b^x} + {b^{ - x}})}^2}}&{{{({b^x} - {b^{ - x}})}^2}}&1\\{{{({c^x} + {c^{ - x}})}^2}}&{{{({c^x} - {c^{ - x}})}^2}}&1\end{array}\,} \right| = $
In a square matrix $A$ of order $3, a_{i i}'s$ are the sum of the roots of the equation $x^2 - (a + b)x + ab= 0$; $a_{i , i + 1}'s$ are the product of the roots, $a_{i , i - 1}'s$ are all unity and the rest of the elements are all zero. The value of the det. $(A)$ is equal to
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For some $a, b$, let $f(x)=\left|\begin{array}{ccc}a+\frac{\sin x}{x} & 1 & b \\ a & 1+\frac{\sin x}{x} & b \\ a & 1 & b+\frac{\sin x}{x}\end{array}\right|, \quad x \neq 0$, $\lim _{ x \rightarrow 0} f ( x )=\lambda+\mu a + vb$. Then $(\lambda+\mu+v)^2$ is equal to:
- [JEE MAIN 2025]