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1.Relation and Function
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If $f(x)$ satisfies $f(7 -x) = f(7 + x)\ \forall \,x\, \in \,R$ such that $f(x)$ has exactly $5$ real roots which are all distinct such that sum of the real roots is $S$ then $S/7$ is equal to
A
$1$
B
$3$
C
$5$
D
$7$
Solution
$f(x)$ is symmetrical about the line $x=7.$
Let $\mathrm{x}_{1}, \mathrm{x}_{2}, \mathrm{x}_{3}, \mathrm{x}_{4}$ and $\mathrm{x}_{5}$ are the real and distinct
roots of $f(x)=0 .$ Then $x_{3}=7, \frac{x_{1}+x_{5}}{2}=7,$
$\frac{\mathrm{x}_{2}+\mathrm{x}_{4}}{2}=7.$
$\mathrm{S}=\mathrm{x}_{1}+\mathrm{x}_{2}+\mathrm{x}_{3}+\mathrm{x}_{4}+\mathrm{x}_{5}=35$
$\mathrm{S} / 7=5$
Standard 12
Mathematics
