Why isn't the answer to TMUA-P2-ERR1-006 shown upfront?

The options are lettered without marking the correct one so you attempt TMUA-P2-ERR1-006 first — TMUA questions reward spotting the trap before you check the worked solution below.

Do I need a calculator for TMUA-P2-ERR1-006?

No — calculators are banned in the TMUA. TMUA-P2-ERR1-006 is set for a no-calculator target of about 210 seconds; the worked solution shows the fastest by-hand route.

Which TMUA specification point is TMUA-P2-ERR1-006?

TMUA-P2-ERR1-006 is Paper P2 · Identifying errors in proofs (specification code ERR1), level 2. Drill more of the same topic from this page.

TMUA-P2-ERR1-006 — Paper P2 · Identifying errors in proofs · Level 2

P2ERR1Level 2~210s target

Spec: ERR1 · No calculator · ~210s

A viral video “proves” that

-3 = 3

with the following line-numbered argument: line one notes that

(-3)^2 = 9

and

3^2 = 9

, so the squares are equal; line two takes the square root of both sides to obtain

\sqrt{(-3)^2} = \sqrt{3^2}

; line three simplifies each side by cancelling the square against the root to leave

-3 = 3

. The presenter insists every line is reversible. Which response correctly identifies the faulty line and explains the underlying illegal move?

Options

The argument fails only because $-3$ is negative.
All lines are valid, so $-3 = 3$ .
Line two is the first fault: square roots of equal numbers need not be equal.
Line three is the first fault: $\sqrt{x^2} = |x|$ , not $x$ , so the left side is $3$ , and squaring destroys sign information so it cannot be reversed.
Line one is false: $(-3)^2 \ne 3^2$ .

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. Name the trap first, work the stem by hand, then open the solution below.

Show solution

Squaring is not injective:

(-3)^2 = 3^2 = 9

is true, and roots of equal inputs are equal, but

\sqrt{x^2} = |x|

gives

3 = 3

— no contradiction and no proof. Trap: assuming inverse operations simply “undo” each other;

x \mapsto x^2

loses the sign, so the step is irreversible.

Keep practising

Same topic · same level
P2ERR1Level 2
TMUA-P2-ERR1-004
A “proof” for acute A, B reads: (1) Suppose A = B. (2) “Cancel” to get A = B. (3) Hence equal sines imply equal angles.…
Identifying errors in proofs
Open TMUA-P2-ERR1-004 →
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
P2ERR1Level 3
TMUA-P2-ERR1-009
A six-line algebra proof submitted for a scholarship problem claims to show that x = 5 follows from the equation (x - 2)…
Identifying errors in proofs
Open TMUA-P2-ERR1-009 →

Browse more

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Options

The argument fails only because

-3

is negative.

All lines are valid, so

-3 = 3

Line two is the first fault: square roots of equal numbers need not be equal.

Line three is the first fault:

\sqrt{x^2} = |x|

, not

x

, so the left side is

3

, and squaring destroys sign information so it cannot be reversed.

Line one is false:

(-3)^2 \ne 3^2

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

Keep practising

Same topic · same level

P2ERR1Level 2

TMUA-P2-ERR1-004

A “proof” for acute A, B reads: (1) Suppose A = B. (2) “Cancel” to get A = B. (3) Hence equal sines imply equal angles.…

Identifying errors in proofs

Open TMUA-P2-ERR1-004 →

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

P2ERR1Level 3

TMUA-P2-ERR1-009

A six-line algebra proof submitted for a scholarship problem claims to show that x = 5 follows from the equation (x - 2)…

Identifying errors in proofs

Open TMUA-P2-ERR1-009 →

TMUA-P2-ERR1-006 — Paper P2 · Identifying errors in proofs · Level 2

Options

Keep practising

Master Identifying errors in proofs

TMUA-P2-ERR1-006 — Paper P2 · Identifying errors in proofs · Level 2

Options

Keep practising

Master Identifying errors in proofs