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CAT 2020 Slot 1 — QA Question 24

Even / Odd FunctionsMedium

If f(5 + x) = f(5 - x) for every real x, and f(x) = 0 has four distinct real roots, then the sum of these roots is

Answer & solution

  • 20

  • B

    40

  • C

    0

  • D

    10

Solution

Medium

The condition f(5+x)=f(5−x)f(5+x)=f(5-x) says the graph of ff is symmetric about the vertical line x=5x=5. Roots therefore come in mirror pairs around 55: if 5+α5+\alpha is a root, so is 5−α5-\alpha. Pair up the four roots and add.

1

Read off the symmetry. Putting any root 5+α5+\alpha into the identity shows its mirror image is also a root.

f(5+α)=f(5−α)(given identity)⇒ f(5+α)=0 ⟹ f(5−α)=0\begin{aligned} &f(5+\alpha)=f(5-\alpha) \quad\text{(given identity)}\\ &\Rightarrow\ f(5+\alpha)=0\ \Longrightarrow\ f(5-\alpha)=0 \end{aligned}
2

Form the two mirror pairs. Let the four distinct roots be 5±α5\pm\alpha and 5±β5\pm\beta.

roots={ 5+α, 5−α, 5+β, 5−β }\begin{aligned} &\text{roots}=\{\,5+\alpha,\ 5-\alpha,\ 5+\beta,\ 5-\beta\,\} \end{aligned}
3

Add the roots. The ±α\pm\alpha and ±β\pm\beta terms cancel in pairs.

(5+α)+(5−α)+(5+β)+(5−β)⇒ 4×5=20(α,β cancel)\begin{aligned} &(5+\alpha)+(5-\alpha)+(5+\beta)+(5-\beta)\\ &\Rightarrow\ 4\times 5=20 \quad\text{($\alpha,\beta$ cancel)} \end{aligned}

A quartic symmetric about x=5x=5 pairs its roots so each pair averages 55. With 44 roots that is 4×5=204\times 5=20 directly.

Sum of the roots=20\text{Sum of the roots}=20
CAT 2020 Slot 1 QA Q24: If f(5 + x) = f(5 - x) for every real x, and f(x) = 0 has four distinct real roots, then the sum of these root — Solution | TheCATExam