TMUA · Paper 2: Mathematical Reasoning

Necessary and sufficient conditions

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

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Let $P$ be the statement ' $x > 3$ ' and $Q$ be the statement ' $x^{2} > 9$ '. Which of the following is correct? (Here $x$ is a real number.)
Let $P$ be the statement ' $n$ is divisible by 3' and $Q$ be the statement ' $n$ is divisible by 6'. Which of the following is correct? (Here $n$ is an integer.)
Let $P$ be the statement ' $x^{2} = 4$ ' and $Q$ be the statement ' $x = 2$ '. Which of the following is correct? (Here $x$ is a real number.)
Let $P$ be the statement ' $ab = 0$ ' and $Q$ be the statement ' $a = 0$ or $b = 0$ '. Which of the following is correct? (Here $a$ and $b$ are real numbers.)
Let $P$ be the statement ' $x^{2} - 3 x + 2 < 0$ ' and $Q$ be the statement ' $1 < x < 2$ '. Which of the following is correct? (Here $x$ is a real number.)
Let $P$ be the statement ' $x$ is an integer' and $Q$ be the statement ' $x^{2}$ is an integer'. Which of the following is correct? (Here $x$ is a real number.)
Let $P$ be the statement ' $x > 0$ and $y > 0$ ' and $Q$ be the statement ' $x + y > 0$ and $x y > 0$ '. Which of the following is correct? (Here $x$ and $y$ are real numbers.)
Let $P$ be the statement ' $x + y$ is even' and $Q$ be the statement ' $x$ and $y$ are both even'. Which of the following is correct? (Here $x$ and $y$ are integers.)
Let $P$ be the statement ' $n^{2}$ is divisible by 4' and $Q$ be the statement ' $n$ is even'. Which of the following is correct? (Here $n$ is an integer.)
Let $P$ be the statement ' $∣ x ∣ < 1$ ' and $Q$ be the statement ' $x^{2} < x$ '. Which of the following is correct? (Here $x$ is a real number.)
Let $P$ be the statement ' $x^{2} + y^{2} = 0$ ' and $Q$ be the statement ' $x = 0$ and $y = 0$ '. Which of the following is correct? (Here $x$ and $y$ are real numbers.)
Let $P$ be the statement ' $n$ is a multiple of 4' and $Q$ be the statement ' $n^{2}$ is a multiple of 8'. Which of the following is correct? (Here $n$ is a positive integer.)
Let $P$ be the statement ' $x$ is irrational' and $Q$ be the statement ' $x^{2}$ is irrational'. Which of the following is correct? (Here $x$ is a real number.)
Let $P$ be the statement ' $a^{2} > b^{2}$ ' and $Q$ be the statement ' $a > b$ '. Which of the following is correct? (Here $a$ and $b$ are real numbers.)
Let $P$ be the statement ' $ab > 0$ ' and $Q$ be the statement ' $a > 0$ and $b > 0$ '. Which of the following is correct? (Here $a$ and $b$ are real numbers.)
Let $P$ be the statement 'the quadratic $x^{2} + b x + c$ has real roots' and $Q$ be the statement ' $c \leq \frac{b ^{2}}{4}$ '. Which of the following is correct? (Here $b$ and $c$ are real numbers.)
Let $P$ be the statement ' $n$ is divisible by 9' and $Q$ be the statement 'the sum of the digits of $n$ is divisible by 9'. Which of the following is correct? (Here $n$ is a positive integer.)
Let $P$ be the statement ' $a, b, c$ is a geometric sequence' and $Q$ be the statement ' $b^{2} = a c$ '. Which of the following is correct? (Here $a$ , $b$ and $c$ are real numbers, and a geometric sequence…
Let $P$ be the statement ' $sin x = 0$ ' and $Q$ be the statement ' $cos x = 1$ '. Which of the following is correct? (Here $x$ is a real number.)
Let $P$ be the statement ' $x^{4} = 1$ ' and $Q$ be the statement ' $x^{2} = 1$ '. Which of the following is correct? (Here $x$ is a real number.)
Let $P$ be the statement ' $n > 1$ ' and $Q$ be the statement ' $n^{2} > n$ '. Which of the following is correct? (Here $n$ is an integer.)
Let $P$ be the statement ' $x$ is a root of $x^{2} - 5 x + 6 = 0$ ' and $Q$ be the statement ' $x = 2$ or $x = 3$ '. Which of the following is correct? (Here $x$ is a real number.)
Let $P$ be the statement ' $n$ is a prime number' and $Q$ be the statement ' $n$ has exactly two positive divisors'. Which of the following is correct? (Here $n$ is a positive integer.)
Let $P$ be the statement ' $n$ is divisible by 8' and $Q$ be the statement 'the number formed by the last three digits of $n$ is divisible by 8'. Which of the following is correct? (Here $n$ is a posit…
Let $P$ be the statement ' $x^{2}$ is divisible by 18' and $Q$ be the statement ' $x$ is divisible by 6'. Which of the following is correct? (Here $x$ is a positive integer.)
Let $P$ be the statement ' $n^{2} + n + 1$ is divisible by 3' and $Q$ be the statement ' $n$ leaves remainder 1 when divided by 3'. Which of the following is correct? (Here $n$ is a positive integer.)

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