Sets, Relations & Functions
Functional Equation / Quadratic Functions
nta_pyq_2025_apr
Grade 11
Question:
Let $f : \mathbb{R} \to \mathbb{R}$ be a function defined by $f(x) = (2+3a)x^2 + \left(\frac{a+2}{a-1}\right)x + b$, $a \neq 1$. If $f(x+y) = f(x) + f(y) + 1 - \frac{2}{7}xy$, then the value of $28\displaystyle\sum_{i=1}^{5}|f(i)|$ is:
Step-by-Step Solution
Key Concept: Substitute $x=y=0$ to find $f(0)$, then $y=0$ to find the coefficient structure. Derive $b$ and $a$ from the functional equation.
$f(0)=-1\Rightarrow b=-1$. Comparing $f(x+y)$ with $f(x)+f(y)+1-\frac{2}{7}xy$: quadratic coeff $2+3a=0\Rightarrow$ doesn't work directly; detailed substitution gives $a=-5/7$, $f(x)=-\frac{x^2}{7}-\frac{3x}{4}-1$. $\sum_{i=1}^5 f(i) = -\frac{1}{7}\cdot\frac{55}{3}-\frac{3}{4}\cdot15-5 = \frac{-675}{28}$. $28|\cdot| = 675$.
Correct Answer: 675