TMUA · Paper 2: Mathematical Reasoning

Necessary and sufficient conditions on a parameter

5 practice questions, each with the answer and an explanation.

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All 5 questions

Let $P$ be the statement ' $k \geq 2$ ' and $Q$ be the statement ' $x^{2} - 2 x + k \geq 1$ for every real $x$ '. Which of the following is correct? (Here $k$ is a real number.)
Let $P$ be the statement ' $0 < k < 4$ ' and $Q$ be the statement ' $x^{2} + k x + k > 0$ for every real $x$ '. Which of the following is correct? (Here $k$ is a real number.)
Let $P$ be the statement ' $k > 0$ ' and $Q$ be the statement 'the function $f (x) = x^{3} + k x$ has no stationary point'. Which of the following is correct? (Here $k$ is a real number.)
Let $P$ be the statement ' $k > - \frac{1}{4}$ ' and $Q$ be the statement 'the line $y = x + k$ meets the parabola $y = x^{2}$ at two distinct points'. Which of the following is correct? (Here $k$ is a real n…
Let $P$ be the statement ' $∣ r ∣ < 1$ ' and $Q$ be the statement 'the sequence $u_{n} = r^{n}$ converges'. Which of the following is correct? (Here $r$ is a real number.)

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