Question

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

• A. $\alpha$
• B. $\gamma$
• C. $\beta$
• D. $\theta$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A $\alpha$
B $\gamma$
C $\beta$
D $\theta$

$\left|\begin{array}{ccc}f\left(\alpha \right)& g\left(\alpha \right)& f\left(\alpha \right)g\left(\theta \right)+g\left(\alpha \right)f\left(\theta \right)\\ f\left(\beta \right)& g\left(\beta \right)& f\left(\beta \right)g\left(\theta \right)+g\left(\beta \right)f\left(\theta \right)\\ f\left(\gamma \right)& g\left(\gamma \right)& f\left(\gamma \right)g\left(\theta \right)+g\left(\gamma \right)f\left(\theta \right)\end{array}\right|$
$C_3\to C_3-g\left(\theta \right)C_1-f\left(\theta \right)C_2$
$=\left|\begin{array}{ccc}f\left(\alpha \right)& g\left(\alpha \right)& 0\\ f\left(\beta \right)& g\left(\beta \right)& 0\\ f\left(\gamma \right)& g\left(\gamma \right)& 0\end{array}\right|=0$
Hence, a, b, c, d is answer.