TMUA Challenge Paper 2 Tmua-p2-challenge-1

• A.necessary and sufficient for $x^2 > 4$
• B.necessary but not sufficient for $x^2 > 4$
• C.sufficient but not necessary for $x^2 > 4$
• D.neither necessary nor sufficient for $x^2 > 4$
• E.equivalent to $x < -2$

Answer: C

• A.No child in the class owns a bicycle
• B.Every child in the class does not own a bicycle
• C.At least one child in the class does not own a bicycle
• D.At least one child in the class owns a bicycle
• E.Not every child in the class owns a car

Answer: C

• A.If the ground is wet then it is raining
• B.If the ground is not wet then it is not raining
• C.If it is not raining then the ground is not wet
• D.The ground is wet only if it is not raining
• E.It is raining and the ground is dry

Answer: B

• A.$N$ is itself prime
• B.$N$ is even
• C.$N$ leaves remainder $1$ on division by every prime up to $p$
• D.$N$ is a perfect square
• E.$N$ is divisible by $p$

Answer: C

• A.$n$ is a multiple of $12$
• B.$n$ is odd
• C.$n$ is not a multiple of $12$
• D.$n$ is a multiple of $3$
• E.nothing can be deduced

Answer: C

• A.$2$
• B.$3$
• C.$4$
• D.$5$
• E.$1$

Answer: C

• A.valid
• B.invalid
• C.valid only if at least one square exists
• D.valid because every square is a rhombus
• E.equivalent to its own converse

Answer: B

• A.$n$ is divisible by $2$ or by $3$
• B.$n$ is divisible by $2$ and by $3$
• C.$n$ is divisible by $12$
• D.$n$ is even
• E.the digits of $n$ sum to a multiple of $6$

Answer: B

• A.$1$
• B.$2$
• C.$3$
• D.$4$
• E.$0$

Answer: B

• A.For every real $x$ there is no real $y$ with $y^2 = x$
• B.There is a real $x$ such that, for every real $y$, $y^2 \ne x$
• C.There is a real $y$ with $y^2 \ne x$ for every real $x$
• D.For every real $y$ there is a real $x$ with $y^2 = x$
• E.No real number has a real square root

Answer: B

• A.$P$ true and $Q$ true
• B.$P$ true and $Q$ false
• C.$P$ false and $Q$ true
• D.$P$ false and $Q$ false
• E.whenever $P$ is false

Answer: B

• A.writing $a^2 = ab$
• B.writing $a^2 - b^2 = ab - b^2$
• C.factorising both sides
• D.cancelling the factor $(a-b)$
• E.concluding $2b = b$

Answer: D

• A.its diagonals are perpendicular
• B.its diagonals are equal in length
• C.all four sides are equal
• D.it has a pair of parallel sides
• E.its diagonals bisect each other

Answer: B

• A.$B \subseteq A$
• B.$A \cap B = B$
• C.$A \cup B = A$
• D.$A \cap B = A$
• E.$A$ and $B$ are disjoint

Answer: D

• A.true
• B.false, and $a = 4$, $b = 2$, $c = 2$ is a counterexample
• C.false, and $a = 5$, $b = 2$, $c = 3$ is a counterexample
• D.false, and $a = 3$, $b = 2$, $c = 6$ is a counterexample
• E.true, because it holds whenever $a$ is prime

Answer: B

• A.only for $x > 0$
• B.for all real $x$
• C.for all $x \ne \tfrac{1}{2}$
• D.for no real $x$
• E.only for $x > 1$

Answer: B

• A.If a function is continuous at a point then it is differentiable there
• B.If a function is not differentiable at a point then it is not continuous there
• C.If a function is not continuous at a point then it is not differentiable there
• D.A function is differentiable at a point if and only if it is continuous there
• E.If a function is continuous at a point then it is not differentiable there

Answer: A

• A.the box labelled "Apples"
• B.the box labelled "Oranges"
• C.the box labelled "Mixed"
• D.any single box will work
• E.it cannot be done with a single draw

Answer: C

• A.$3$
• B.$6$
• C.$9$
• D.$0$
• E.there is no minimum

Answer: B

• A.I only
• B.I and II only
• C.I, II and III
• D.II only
• E.III only

Answer: B