75-Archetype SyllabusMathematics 1 • ARCH-M1-02 • Spec M2

Full 6-Card Lesson 15-Question Timed Drill

UAT-UK SPEC CODE: M2 (Number)ASSUMED MATHS 1 PREREQUISITE: M2.5FREQUENCY: HIGH

Surd Rationalisation, Difference-of-Squares & Laws of Indices

Deconstructed from 17 past UAT-UK questions (7.56% of all historical Mathematics 1 papers). Students using standard A-Level classroom methods spend an average of 125s on this archetype—blowing past the 90s ESAT budget. Using the 2-line Sub-90 heuristic below compresses solve time to 35s (saving 90s per question).

Historical Frequency

17 Past Qs

7.56% of Mathematics 1

Classroom Trap Time

125s

+35s over 90s budget

Sub-90 Target Time

35s

Banks +90s on the clock

Dominant UI Format

5-Option A–E

Maths 1 Skill: M2.5

Card 1 & Card 2 Preview: The Classroom Time Trap vs. Sub-90 2-Line Rule

Excerpted from our interactive 6-Step Topic Lesson for ARCH-M1-02.

Launch All 6 Active-Recall Cards

CARD 1 OF 6 • THE CLASSROOM TIME TRAP (125s)

Why Students Bleed Time:

Expand

(2\sqrt{3} - \sqrt{2})(2\sqrt{3} - \sqrt{2}) = 4(3) - 2\sqrt{6} - 2\sqrt{6} + 2 = 14 - 4\sqrt{6}

. Comparing with

a - b\sqrt{6}

gives

a = 14

and

b = 4

, so

a + b = 18

(~105s).

In 17 past UAT-UK questions (7.56% of Mathematics 1), candidates lose 90s by subtracting the squared surd term in expansion or multiplying by (a - sqrt(b)) instead of conjugate (a + sqrt(b)). Instead, apply M2 with Maths 1 skill M2.5.

CARD 2 OF 6 • THE 2-LINE SUB-90 HEURISTIC (<35s)

Line 1 (Binomial Square):

(2\sqrt{3})^2 + (\sqrt{2})^2 = 12 + 2 = 14 = a

, and cross-term coefficient

b = 2 \times 2 = 4

Line 2 (Sum):

a + b = 14 + 4 = 18

(25s).

Sanity & Limit Check: Verify units/signs against sign_or_direction_flip ((a - b*sqrt(c))² = a² - b²*c or failing to flip conjugate sign) and factor_of_two_or_stoichiometry_trap (Omitting the cross-term 2ab*sqrt(c) when squaring (a + b*sqrt(c))²).

Examiner Distractor Blueprint for ARCH-M1-02

FOIL #1sign_or_direction_flip

(a - b*sqrt(c))² = a² - b²*c or failing to flip conjugate sign

Subtracting the squared surd term in expansion or multiplying by (a - sqrt(b)) instead of conjugate (a + sqrt(b)).

Open Sign / Direction Flip Clinic

FOIL #2factor_of_two_or_stoichiometry_trap

Omitting the cross-term 2ab*sqrt(c) when squaring (a + b*sqrt(c))²

Distributing the exponent directly across a sum: (a + b)² -> a² + b².

Open Factor-of-2 / Stoichiometry Clinic

FOIL #3fence_post_or_domain_trap

Writing sqrt(a) * sqrt(a) = a² instead of a

Squaring the radicand when multiplying identical square roots.

Open Fence-Post / Domain Trap Clinic

FOIL #4intermediate_stop_trap

Leaving unsimplified radical factor (e.g. sqrt(12) instead of 2*sqrt(3)) before cancelling denominator

Failing to extract square factors before rationalising.

Open Intermediate Stop Trap Clinic

Try 1 Live Sample Question Under the 90-Second Stopwatch

Press W to open the 2-line dry-wipe scratchpad, select A–E, and lock in to reveal the Dual-View solution.

Full 6-Card Lesson Launch All 15 Questions (ARCH-M1-02)

ARCH-M1-02: Surd Rationalisation, Difference-of-Squares & Laws of Indices (Sample)Mathematics 1

00:00.0

SUB-90 ZONE (<35S)

0.0s35s Sub-9090s ESAT Budget150s Classroom

Question 1 of 1ARCH-M1-02 (6-Card Lesson)Format: Standard 5-Option

Right-click any option to cross out • Keys A–E + Enter

Question 1 of 1 • 90s Target

Ready to Start Question 1?

The question stem, diagram, and answer options are hidden behind this pre-start gate. When you click below, the question will reveal and your 90-second split stopwatch will immediately begin ticking.

Mathematics 1•ARCH-M1-02•Standard 5-Option

or press Enter to start