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75-Archetype SyllabusMathematics 2 • ARCH-MM-07 • Spec MM5
Full 6-Card Lesson 15-Question Timed Drill
UAT-UK SPEC CODE: MM5 (Exponentials and logarithms)ASSUMED MATHS 1 PREREQUISITE: M2.6FREQUENCY: VERY HIGH

Laws of Logarithms, Disguised Exponential Quadratics (a^{2x} + b a^x + c = 0) & Log Inequalities

Deconstructed from 47 past UAT-UK questions (11.46% 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
47 Past Qs
11.46% 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: M2.6

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-07.

Launch All 6 Active-Recall Cards
CARD 1 OF 6 • THE CLASSROOM TIME TRAP (150s)
Why Students Bleed Time:
Using log⁡2(A)+log⁡2(B)=log⁡2(AB)\log_{2}(A) + \log_{2}(B) = \log_{2}(AB), the sum becomes log⁡2 ⁣(43×54×⋯×4847)=log⁡2 ⁣(483)=log⁡2(16)=4\log_{2}\!\left(\frac{4}{3} \times \frac{5}{4} \times \dots \times \frac{48}{47}\right) = \log_{2}\!\left(\frac{48}{3}\right) = \log_{2}(16) = 4 (~90s).
In 47 past UAT-UK questions (11.46% of Mathematics 2), candidates lose 110s by failing to reject non-positive roots when solving quadratic in u = a^x or combining log(x-a) + log(x-b). Instead, apply MM5 with Maths 1 skill M2.6.
CARD 2 OF 6 • THE 2-LINE SUB-90 HEURISTIC (<40s)
Line 1 (Telescoping Log Product): All interior numerators and denominators cancel, leaving log⁡2 ⁣(last topfirst bottom)\log_{2}\!\left(\frac{\text{last top}}{\text{first bottom}}\right).
Line 2 (Evaluate Power): log⁡2 ⁣(483)=log⁡2(16)=4\log_{2}\!\left(\frac{48}{3}\right) = \log_{2}(16) = 4 (20s).
Sanity & Limit Check: Verify units/signs against fence_post_or_domain_trap (Accepting roots where u = a^x <= 0 or log argument x <= 0) and sign_or_direction_flip (Failing to flip inequality sign when dividing by log_a(b) where 0 < b < 1 (since log_a(b) < 0)).

Examiner Distractor Blueprint for ARCH-MM-07

FOIL #1fence_post_or_domain_trap
Accepting roots where u = a^x <= 0 or log argument x <= 0

Failing to reject non-positive roots when solving quadratic in u = a^x or combining log(x-a) + log(x-b).

Open Fence-Post / Domain Trap Clinic
FOIL #2sign_or_direction_flip
Failing to flip inequality sign when dividing by log_a(b) where 0 < b < 1 (since log_a(b) < 0)

Treating log(0.5) as a positive number during inequality division.

Open Sign / Direction Flip Clinic
FOIL #3factor_of_two_or_stoichiometry_trap
Confusing log(A) + log(B) = log(A+B) instead of log(AB), or (log x)² with 2*log(x)

Applying linear distribution to logarithmic laws.

Open Factor-of-2 / Stoichiometry Clinic
FOIL #4intermediate_stop_trap
Reporting u = a^x instead of x = log_a(u)

Stopping at the auxiliary quadratic root u.

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-07)
sub90 ESAT Stopwatch Emblemsub90 ESAT
ARCH-MM-07: Laws of Logarithms, Disguised Exponential Quadratics (a^{2x} + b a^x + c = 0) & Log Inequalities (Sample)Mathematics 2
00:00.0
SUB-90 ZONE (<40S)
0.0s40s Sub-9090s ESAT Budget150s Classroom
Question 1 of 1ARCH-MM-07 (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-07•Standard 5-Option
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