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75-Archetype SyllabusMathematics 2 • ARCH-MM-02 • Spec MM1
Full 6-Card Lesson 15-Question Timed Drill
UAT-UK SPEC CODE: MM1 (Algebra and functions)ASSUMED MATHS 1 PREREQUISITE: M2.5FREQUENCY: HIGH

Disguised Radical Equations & Principal Square Root Domain Filtering (√x ≥ 0)

Deconstructed from 15 past UAT-UK questions (3.66% of all historical Mathematics 2 papers). Students using standard A-Level classroom methods spend an average of 150s on this archetype—blowing past the 90s ESAT budget. Using the 2-line Sub-90 heuristic below compresses solve time to 40s (saving 110s per question).

Historical Frequency
15 Past Qs
3.66% of Mathematics 2
Classroom Trap Time
150s
+60s over 90s budget
Sub-90 Target Time
40s
Banks +110s 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-MM-02.

Launch All 6 Active-Recall Cards
CARD 1 OF 6 • THE CLASSROOM TIME TRAP (150s)
Why Students Bleed Time:
Rearranging to x=…\sqrt{x} = \dots and squaring both sides gives a quadratic in xx with roots x=9x = 9 and x=4x = 4. Checking both roots back in the original equation takes ~135s and shows x=4x = 4 fails.
In 15 past UAT-UK questions (3.66% of Mathematics 2), candidates lose 110s by violating the uat-uk convention that sqrt(x) denotes strictly the non-negative root (sqrt(x) >= 0). Instead, apply MM1 with Maths 1 skill M2.5.
CARD 2 OF 6 • THE 2-LINE SUB-90 HEURISTIC (<40s)
**Line 1 (Domain Filter x≥0\sqrt{x} \ge 0):** Let u=x≥0u = \sqrt{x} \ge 0; factor u2−1u−6=(u−3)(u+2)=0u^2 - 1u - 6 = (u - 3)(u + 2) = 0.
Line 2 (Prune & Square): Reject u=−2<0u = -2 < 0 immediately; only u=3  ⟹  x=32=9u = 3 \implies x = 3^2 = 9 (25s).
Sanity & Limit Check: Verify units/signs against fence_post_or_domain_trap (Counting extraneous roots where sqrt(f(x)) = g(x) < 0 created by squaring both sides) and sign_or_direction_flip (Writing sqrt((a - b*sqrt(c))²) = a - b*sqrt(c) when a < b*sqrt(c)).

Examiner Distractor Blueprint for ARCH-MM-02

FOIL #1fence_post_or_domain_trap
Counting extraneous roots where sqrt(f(x)) = g(x) < 0 created by squaring both sides

Violating the UAT-UK convention that sqrt(x) denotes strictly the non-negative root (sqrt(x) >= 0).

Open Fence-Post / Domain Trap Clinic
FOIL #2sign_or_direction_flip
Writing sqrt((a - b*sqrt(c))²) = a - b*sqrt(c) when a < b*sqrt(c)

Failing to take the positive magnitude |a - b*sqrt(c)| = b*sqrt(c) - a.

Open Sign / Direction Flip Clinic
FOIL #3factor_of_two_or_stoichiometry_trap
Counting a repeated root (u - r)² = 0 as 2 distinct x-solutions

Failing to check discriminant b² - 4ac = 0 multiplicity.

Open Factor-of-2 / Stoichiometry Clinic
FOIL #4intermediate_stop_trap
Sum of roots of u = sqrt(x) reported instead of sum of roots x = u²

Stopping at the substitution variable u instead of squaring back to x.

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-MM-02)
sub90 ESAT Stopwatch Emblemsub90 ESAT
ARCH-MM-02: Disguised Radical Equations & Principal Square Root Domain Filtering (√x ≥ 0) (Sample)Mathematics 2
00:00.0
SUB-90 ZONE (<40S)
0.0s40s Sub-9090s ESAT Budget150s Classroom
Question 1 of 1ARCH-MM-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 2•ARCH-MM-02•Standard 5-Option
or press Enter to start