Question

If $f\left(x\right)$ and $g\left(x\right)$ are functions such that $f(x+)=f\left(x\right)g\left(y\right)+g\left(x\right)f\left(y\right)$, then the value of $\left|\begin{array}{ccc}f\left(\alpha \right)& g\left(\alpha \right)& f(\alpha +\theta )\\ f\left(\beta \right)& g\left(\beta \right)& f(\beta +\theta )\\ f\left(\gamma \right)& g\left(\gamma \right)& f(\gamma +\theta )\end{array}\right|$ is

• A. $f\left(\alpha \right).f\left(\beta \right).f\left(\gamma \right)$
• B. $0$
• C. $g\left(\alpha \right).g\left(\beta \right).g\left(\gamma \right)$
• D. $Noneofthese$

Answer 1

The correct option is B $0$

$\begin{array}{c}f\left(x+y\right)=f\left(x\right)g\left(y\right)+g\left(x\right)f\left(y\right)\\ \left|\begin{array}{ccc}f\left(\alpha \right)& g\left(\alpha \right)& f\left(\alpha +\theta \right)\\ f\left(\beta \right)& f\left(\beta \right)& f\left(\beta +\theta \right)\\ f\left(\lambda \right)& f\left(\lambda \right)& f\left(\lambda +\theta \right)\end{array}\right|\\ =\left|\begin{array}{ccc}f\left(\alpha \right)& g\left(\alpha \right)& f\left(\alpha \right)g\left(\theta \right)+g\left(\alpha \right)f\left(\theta \right)\\ f\left(\beta \right)& f\left(\beta \right)& f\left(\beta \right)g\left(\theta \right)+g\left(\beta \right)f\left(\theta \right)\\ f\left(\lambda \right)& f\left(\lambda \right)& f\left(\lambda \right)g\left(\theta \right)+g\left(\lambda \right)f\left(\theta \right)\end{array}\right|\\ =\left|\begin{array}{ccc}f\left(\alpha \right)& g\left(\alpha \right)& f\left(\alpha \right)g\left(\theta \right)\\ f\left(\beta \right)& f\left(\beta \right)& f\left(\beta \right)g\left(\theta \right)\\ f\left(\lambda \right)& f\left(\lambda \right)& f\left(\lambda \right)g\left(\theta \right)\end{array}\right|+\left|\begin{array}{ccc}f\left(\alpha \right)& g\left(\alpha \right)& g\left(\alpha \right)f\left(\theta \right)\\ f\left(\beta \right)& f\left(\beta \right)& g\left(\beta \right)f\left(\theta \right)\\ f\left(\lambda \right)& f\left(\lambda \right)& g\left(\lambda \right)f\left(\theta \right)\end{array}\right|\end{array}$

$=0$

Option $B$ is correct.