GATE 2018 EE GA Q.3: solution and answer

GATE 2018 · EEGA Q.3 · 1 mark · MCQ

The three roots of the equation $f (x) = 0$ are $x = {- 2, 0, 3}$ . What are the three values of $x$ for which $f (x - 3) = 0$ ?

(A)
$- 5, - 3, 0$
(B)
$- 2, 0, 3$
(C)
$0, 6, 8$
(D)
$1, 3, 6$

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Answer and solution

ANSWER

The values of $x$ for which $f (x - 3) = 0$ are $1, 3, 6$ . The answer is (D).

Official answer: DSource: Official answer key, GATE 2018 EE (IIT Guwahati)

$f (u) = 0$ exactly when $u \in {- 2, 0, 3}$ , because these are the three roots of $f$ .
Put $u = x - 3$ . Then $f (x - 3) = 0$ exactly when $x - 3 \in {- 2, 0, 3}$ .
Solve each case: $x - 3 = - 2 \Rightarrow x = 1$ ; $x - 3 = 0 \Rightarrow x = 3$ ; $x - 3 = 3 \Rightarrow x = 6$ .
Another way to see it: the graph of $f (x - 3)$ is the graph of $f (x)$ moved 3 units to the right, so every root goes up by 3: ${- 2 + 3, 0 + 3, 3 + 3} = {1, 3, 6}$ .
Check: $f (1 - 3) = f (- 2) = 0$ , $f (3 - 3) = f (0) = 0$ , $f (6 - 3) = f (3) = 0$ . The three values are $1, 3, 6$ , which is option (D).

KEY IDEA

If $r$ is a root of $f$ , then $f (x - 3) = 0$ when $x - 3 = r$ , that is, when $x = r + 3$ . Replacing $x$ by $x - 3$ moves the graph 3 units to the right, so every root goes up by 3.

WATCH OUT FOR

Seeing the minus sign in $x - 3$ and subtracting 3 from each root. That gives $- 5, - 3, 0$ , which is option (A). Another slip is to think the shift leaves the roots alone and pick option (B), $- 2, 0, 3$ . In $f (x - h)$ the graph moves right by $h$ , so the roots go up by $h$ . A quick check settles it: substitute the candidate $x$ into $x - 3$ and see whether you get an original root.

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GATE 2018 EE GA Q.3: solution and answer

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