Why isn't the answer to TMUA-P2-PRF5-002 shown upfront?

The options are lettered without marking the correct one so you attempt TMUA-P2-PRF5-002 first — TMUA questions reward spotting the trap before you check the worked solution below.

Do I need a calculator for TMUA-P2-PRF5-002?

No — calculators are banned in the TMUA. TMUA-P2-PRF5-002 is set for a no-calculator target of about 190 seconds; the worked solution shows the fastest by-hand route.

Which TMUA specification point is TMUA-P2-PRF5-002?

TMUA-P2-PRF5-002 is Paper P2 · Following proofs (specification code PRF5), level 2. Drill more of the same topic from this page.

TMUA-P2-PRF5-002 — Paper P2 · Following proofs · Level 2

P2PRF5Level 2~190s target

Spec: PRF5 · No calculator · ~190s

A proof that the square of an odd integer is odd reads: (1) Write

n = 2k + 1

. (2) Then

n^2 = 4k^2 + 4k + 1

. (3) ??? (4) Hence

n^2

is odd. Which line (3) correctly bridges (2) to (4)?

Options

So $n^2 = 4k^2 + 4k$ , which is even.
So $n^2 = 2(2k^2 + 2k) + 1$ with $2k^2 + 2k$ an integer.
So $n^2 = 2k + 1$ again.
So $n^2$ is divisible by $4$ .
So $k$ must be even.

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: line-number-fault, even-odd-split-missing. Name the trap first, work the stem by hand, then open the solution below.

Show solution

Odd means

2\times(\text{integer}) + 1

; factoring gives

2(2k^2+2k)+1

with integer bracket. Trap: stopping at the expanded polynomial without matching the definition, or “simplifying” back to

2k+1

which confuses the variable roles.

Keep practising

Same topic · same level
P2PRF5Level 2
TMUA-P2-PRF5-003
A question asks: “Prove that for an integer n, n is even if and only if n^2 is even.” Which proof structure is required?
Following proofs
Open TMUA-P2-PRF5-003 →
Same trap: line-number-fault
P2ERR1Level 1
TMUA-P2-ERR1-001
A “proof” that 1 = 2 reads: (1) Let a = b. (2) Then a^2 = ab. (3) So a^2 - b^2 = ab - b^2. (4) Factorising gives (a-b)(a…
Identifying errors in proofs
Open TMUA-P2-ERR1-001 →
Step up a level
P2PRF5Level 3
TMUA-P2-PRF5-005
A revision guide prints a short proof without a theorem statement: suppose x and y are rational, write x = a/b and y = c…
Following proofs
Open TMUA-P2-PRF5-005 →

Browse more

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Options

n^2 = 4k^2 + 4k

, which is even.

n^2 = 2(2k^2 + 2k) + 1

with

2k^2 + 2k

an integer.

n^2 = 2k + 1

again.

n^2

is divisible by

4

k

must be even.

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

Keep practising

Same topic · same level

P2PRF5Level 2

TMUA-P2-PRF5-003

A question asks: “Prove that for an integer n, n is even if and only if n^2 is even.” Which proof structure is required?

Following proofs

Open TMUA-P2-PRF5-003 →

Same trap: line-number-fault

P2ERR1Level 1

TMUA-P2-ERR1-001

A “proof” that 1 = 2 reads: (1) Let a = b. (2) Then a^2 = ab. (3) So a^2 - b^2 = ab - b^2. (4) Factorising gives (a-b)(a…

Identifying errors in proofs

Open TMUA-P2-ERR1-001 →

Step up a level

P2PRF5Level 3

TMUA-P2-PRF5-005

A revision guide prints a short proof without a theorem statement: suppose x and y are rational, write x = a/b and y = c…

Following proofs

Open TMUA-P2-PRF5-005 →

TMUA-P2-PRF5-002 — Paper P2 · Following proofs · Level 2

Options

Keep practising

Master Following proofs

TMUA-P2-PRF5-002 — Paper P2 · Following proofs · Level 2

Options

Keep practising

Master Following proofs