Functions
Functions
Allen Star Batch
Grade 12
Question:
Let $f(x) = 1 + 2\cos x + 3\sin x$. If real numbers $a, b, c$ are such that $a f(x) + b f(-x) = 1$ holds for any $x \in \mathbb{R}$ then $\frac{b\cos c}{a} =$
$1$
$-1$
$\frac{1}{2}$
$-\frac{1}{2}$
Step-by-Step Solution
Key Concept: Recognize that f(x) = 1 + 2cos(x) + 3sin(x) has a special symmetry property: f(x) + f(π - x) = 2. This allows you to express the condition af(x) + bf(-x) = 1 by substituting x with π - x and comparing coefficients to find a, b, and c.
Notice that $f(x) + f(\pi - x) = 2$. So $\frac{1}{2}f(x) + \frac{1}{2}f(\pi-x) = 1$. Hence $a = b = \frac{1}{2}$ and $c = \pi$.
Correct Answer: 2