Why isn't the answer to TMUA-P2-PRF5-005 shown upfront?

The options are lettered without marking the correct one so you attempt TMUA-P2-PRF5-005 first — TMUA questions reward spotting the trap before you check the worked solution below.

Do I need a calculator for TMUA-P2-PRF5-005?

No — calculators are banned in the TMUA. TMUA-P2-PRF5-005 is set for a no-calculator target of about 250 seconds; the worked solution shows the fastest by-hand route.

Which TMUA specification point is TMUA-P2-PRF5-005?

TMUA-P2-PRF5-005 is Paper P2 · Following proofs (specification code PRF5), level 3. Drill more of the same topic from this page.

TMUA-P2-PRF5-005 — Paper P2 · Following proofs · Level 3

P2PRF5Level 3~250s target

Spec: PRF5 · No calculator · ~250s

A revision guide prints a short proof without a theorem statement: suppose

x

and

y

are rational, write

x = \frac{a}{b}

and

y = \frac{c}{d}

with integers

a, c

and non-zero integers

b, d

; then

x + y = \frac{ad+bc}{bd}

where the numerator is an integer and the denominator is a non-zero integer; hence the sum is rational. The guide then offers five candidate theorem statements and asks which one this argument proves, warning that stronger and converse versions are listed as traps. Which statement is proved?

Options

The sum of two rational numbers is rational.
The sum of two irrational numbers is irrational.
Every rational number is an integer.
If $x + y$ is rational then $x$ and $y$ are rational.
The product of two rationals is irrational.

Letters only — the answer key is hidden. Work it out, then check the solution below.

Hint (check trope)

No written hint yet — check the trope: proof-direction-reversed, converse-vs-contrapositive. Name the trap first, work the stem by hand, then open the solution below.

Show solution

Assumption to conclusion: rational inputs

\Rightarrow

rational sum — exactly the first statement. Trap: the fourth option is the converse (rational sum

\Rightarrow

rational summands), refuted by

x = \sqrt{2}

y = -\sqrt{2}

; the proof never argues backwards.

Keep practising

Same topic · same level
P2PRF5Level 3
TMUA-P2-PRF5-011
Three statements P, Q, R satisfy P ⇒ Q, Q ⇒ R and R ⇒ P. A study group offers five verdicts about the trio. Which is cor…
Following proofs
Open TMUA-P2-PRF5-011 →
Same trap: converse-vs-contrapositive
P2ARG1Level 1
TMUA-P2-ARG1-001
Let P and Q be mathematical statements. The converse of the statement “If P then Q ” is which of the following?
Statements and connectives
Open TMUA-P2-ARG1-001 →
More in this topic
P2PRF5Level 3
TMUA-P2-PRF5-012
A student establishes “ n is odd if and only if n^3 is odd” by passing through squares: odd n gives odd n^2 (direct), od…
Following proofs
Open TMUA-P2-PRF5-012 →

Browse more

Loading…

Options

The sum of two rational numbers is rational.

The sum of two irrational numbers is irrational.

Every rational number is an integer.

x + y

is rational then

x

and

y

are rational.

The product of two rationals is irrational.

Letters only — the answer key is hidden. Work it out, then check the solution below.

Keep practising

Same topic · same level

P2PRF5Level 3

TMUA-P2-PRF5-011

Three statements P, Q, R satisfy P ⇒ Q, Q ⇒ R and R ⇒ P. A study group offers five verdicts about the trio. Which is cor…

Following proofs

Open TMUA-P2-PRF5-011 →

Same trap: converse-vs-contrapositive

P2ARG1Level 1

TMUA-P2-ARG1-001

Let P and Q be mathematical statements. The converse of the statement “If P then Q ” is which of the following?

Statements and connectives

Open TMUA-P2-ARG1-001 →

TMUA-P2-PRF5-005 — Paper P2 · Following proofs · Level 3

Options

Keep practising

Master Following proofs

TMUA-P2-PRF5-005 — Paper P2 · Following proofs · Level 3

Options

Keep practising

Master Following proofs