tldr; i put every insane ext1/ext2 trick i learned into one place (aka The Vault), check it out here
thevaultexclusive.framer.website

hello on this fine (rainy asf) evening yall

so i graduated from one of those schools where everyone breathes maths and i had like 3 diff extension teachers all telling me random hacks over the years, plus all my mates and mates of mates (and mates of mates of mates…) were state rankers or close to it

and i thought, why not just dump every single obscure little thing into one resource
so yeah, i made something called the vault and i promise this isnt just a glaze but it’s literally almost like i gathered together 5 state rankers in one room and just took notes on how they think

i’m not gonna do the whole fake hype thing, i’ll just show you the type of stuff i mean
1. ext2 sneaky integral
Evaluate ∫ from 0 to π/2 of ln(sin x) dx
everyone panics and starts expanding with power series. the Vault way: use the identity ∫0→a f(x) dx = ∫0→a f(a - x) dx, pair it up with ln(cos x) and boom, you turn it into ∫ ln(sin x) + ln(cos x) dx = ∫ ln(sin 2x) dx - π ln(2). that reduces into a constant so clean its kinda chefs kiss
2. ext1 MCQ elimination trick
question says: A parabola with vertex (0,0) and focus (a,0) has equation y² = 4ax. Find the tangent with slope m.
Vault shortcut —> don’t mess with focus-directrix definitions
straight formula: y = m x + (a / m). works for ALL parabolas in that form and shows up like every second year
3. harder ext2 trick
Show that if |z - 1| = |z + 1| then arg(z) = π/2 or -π/2.
Vault approach is don’t draw, don’t overthink
condition screams “perpendicular bisector” but algebra route is faster: |z - 1|² = |z + 1|² → expand → cancel → Re(z) = 0. one line instead of some long ahhh geometric explanation
4. proof hack (the double swap)
when they throw Σ(k=1 to n) [k * r^k] at you in a series proof, everyone tries induction. but here’s the double swap
u write S = r + 2r² + 3r³ + … + n rⁿ, then compute rS, subtract: S - rS = r + r² + … + rⁿ - n rⁿ⁺¹, then simplify. you’ve got the closed form without induction (but dw tho Vault teaches the “pattern catch” so you do this in 20 seconds flat)
5. projectile motion boss trick
instead of splitting into two trig equations and getting lost, check if they gave you max height or range. if they did, you can set u² sin(2θ) = R g and u² sin²(θ) = 2 H g, combine them straight away → boom, both unknowns dead in like two lines

and like… there’s full sections for
• parametric sketching cheats (when x² + y² pops up → polar switch instantly)
• vector hidden identities (dot product is your friend in circle proofs)
• complex shortcuts (don’t expand cis form unless you want pain)
• the half-angle substitution hack in integrals that kills every sec² question

if ur grinding for high E4 im talking like easy 95+ and you want every little sneaky technique distilled, that’s basically what The Vault is legit the 20% of effort that gets you the 80% of results (in this case its 95% but yeah ykwim)

if ur keen its js gonna be at this link so feel free to type it into your search bar (bos dont allow clickable links for some scuffed reason but oh well)

thevaultexclusive.framer.website