tmua 2021 Q6

tmua · Uk · paper2 1 marks Proof Proof of Equivalence or Logical Relationship Between Conditions

Consider the following two statements about the polynomial $\mathrm { f } ( x )$ : $P : \quad \mathrm { f } ( x ) = 0$ for exactly three real values of $x$ $Q : \quad \mathrm { f } ^ { \prime } ( x ) = 0$ for exactly two real values of $x$ Which one of the following is correct?
A $P$ is necessary but not sufficient for $Q$.
B $P$ is sufficient but not necessary for $Q$.
C $P$ is necessary and sufficient for $Q$.
D $P$ is not necessary and not sufficient for $Q$.

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Consider the following two statements about the polynomial $\mathrm { f } ( x )$ :\\
$P : \quad \mathrm { f } ( x ) = 0$ for exactly three real values of $x$\\
$Q : \quad \mathrm { f } ^ { \prime } ( x ) = 0$ for exactly two real values of $x$\\
Which one of the following is correct?\\
A $P$ is necessary but not sufficient for $Q$.\\
B $P$ is sufficient but not necessary for $Q$.\\
C $P$ is necessary and sufficient for $Q$.\\
D $P$ is not necessary and not sufficient for $Q$.

Paper Questions

Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Q13 Q14 Q15 Q16 Q17 Q18 Q19 Q20