tmua.me Extreme Set C Paper 2 · Question 19

Let $n$ be a positive integer. Consider the following statement.

There exists a polynomial $P$ with real coefficients and degree exactly $3$ such that, for every integer $k$ with $1\leq k\leq n$ ,

P(k)=k\qquad\text{or}\qquad P(k)=k^2.

The choice between the two equalities may be different for different values of $k$ .

Which of the following correctly describes the condition $n\leq4$ in relation to this statement?

AThe condition is necessary but not sufficient for the statement.

BThe condition is sufficient but not necessary for the statement.

CThe condition is both necessary and sufficient for the statement.

DThe condition is neither necessary nor sufficient for the statement.

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