75-Archetype SyllabusMathematics 2 • ARCH-MM-09 • Spec MM6

Full 6-Card Lesson 15-Question Timed Drill

UAT-UK SPEC CODE: MM6 (Differentiation)ASSUMED MATHS 1 PREREQUISITE: M5.11FREQUENCY: VERY HIGH

Stationary Point Classification, Monotonic Cubics & Constrained 2D/3D Optimisation

Deconstructed from 29 past UAT-UK questions (7.07% of all historical Mathematics 2 papers). Students using standard A-Level classroom methods spend an average of 175s on this archetype—blowing past the 90s ESAT budget. Using the 2-line Sub-90 heuristic below compresses solve time to 50s (saving 125s per question).

Historical Frequency

29 Past Qs

7.07% of Mathematics 2

Classroom Trap Time

175s

+85s over 90s budget

Sub-90 Target Time

50s

Banks +125s on the clock

Dominant UI Format

5-Option A–E

Maths 1 Skill: M5.11

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-09.

Launch All 6 Active-Recall Cards

CARD 1 OF 6 • THE CLASSROOM TIME TRAP (175s)

Why Students Bleed Time:

Differentiating gives

\frac{dy}{dx} = 3x^2 + 2px + q = 0

x = 1

and

x = 3

. Setting up simultaneous equations

3 + 2p + q = 0

and

27 + 6p + q = 0

and solving gives

p = -6

and

q = 9

, so

p + q = 3

(~140s).

In 29 past UAT-UK questions (7.07% of Mathematics 2), candidates lose 125s by forgetting that f'(a)=0 and f''(a)=0 with no sign change in f'(x) is a point of inflection. Instead, apply MM6 with Maths 1 skill M5.11.

CARD 2 OF 6 • THE 2-LINE SUB-90 HEURISTIC (<50s)

Line 1 (Factorised Derivative Identity):

y'(x) = 3x^2 + 2px + q \equiv 3(x - 1)(x - 3) = 3x^2 - 12x + 9

**Line 2 (Read Off

p, q

):**

2p = -12 \implies p = -6,\; q = 9 \implies p + q = 3

(25s).

Sanity & Limit Check: Verify units/signs against multi_statement_linchpin_foil (Assuming f'(a) = 0 guarantees a local maximum or minimum (ignoring stationary point of inflection)) and fence_post_or_domain_trap (Using b² - 3ac < 0 instead of b² - 3ac <= 0 for a cubic ax³ + bx² + cx + d to be strictly increasing).

Examiner Distractor Blueprint for ARCH-MM-09

FOIL #1multi_statement_linchpin_foil

Assuming f'(a) = 0 guarantees a local maximum or minimum (ignoring stationary point of inflection)

Forgetting that f'(a)=0 and f''(a)=0 with no sign change in f'(x) is a point of inflection.

Open Multi-Statement Linchpin Foil Clinic

FOIL #2fence_post_or_domain_trap

Using b² - 3ac < 0 instead of b² - 3ac <= 0 for a cubic ax³ + bx² + cx + d to be strictly increasing

Omitting the boundary case where a repeated root of f'(x) = 0 gives a single point of zero gradient.

Open Fence-Post / Domain Trap Clinic

FOIL #3intermediate_stop_trap

Outputting the optimal radius/length x_opt instead of the maximum volume/area V(x_opt)

Stopping after solving dV/dx = 0 for x without substituting back into V(x).

Open Intermediate Stop Trap Clinic

FOIL #4factor_of_two_or_stoichiometry_trap

Including a closed lid 2*pi*r² when the stem specifies an open-topped cylinder/box

Misreading open-top vs. closed-container surface area constraints.

Open Factor-of-2 / Stoichiometry 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-09)

ARCH-MM-09: Stationary Point Classification, Monotonic Cubics & Constrained 2D/3D Optimisation (Sample)Mathematics 2

00:00.0

SUB-90 ZONE (<50S)

0.0s50s Sub-9090s ESAT Budget150s Classroom

Question 1 of 1ARCH-MM-09 (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-09•Standard 5-Option

or press Enter to start