Mathematics (General Prep) Exam Survival Guide

Window: All competitive & school exams (board + aptitude + olympiad foundations)
Components: Core skills · Problem-solving heuristics · Proof/solution writing · Practice cycles

Must-do topics (80/20 focus)

• Algebra Core: linear/quad eqns & inequalities (incl. absolute value), functions (domain/range/inverse), sequences (AP/GP), factorisation & identities.
• Number Sense & NT: primes/composites, gcd–lcm, divisibility, remainders/mod arithmetic (incl. negatives), parity, base/place value, last-digit cycles.
• Geometry/Mensuration: congruence/similarity, Pythagoras & triples, angle-chasing, circles (power of a point), area/CSA/TSA/volume; coordinate slope–distance–midpoint.
• Counting/Probability: permutations vs combinations, inclusion–exclusion, complements, independence vs mutual exclusivity, simple Bayes, expected value basics.
• Inequalities & Estimation: AM–GM, Cauchy–Schwarz (intuitive form), bounding & approximation; order-of-magnitude checks.
• Data/Graphs: TAUSL habit (Title–Axes–Units–Scale–Legend), mean/median choice with outliers.

Problem-solving heuristics (carry card)

• G–T–P–P–C: Given → To find → Plan → Proof/Proceed → Check.
• Draw / Label / Name: every geometry or word problem becomes a picture.
• Work backwards from desired form; try small cases to spot structure.
• Invariants & Monovariants (what never changes / always increases or decreases).
• Pigeonhole / Extremal (force repetition; push to the edge case).
• Symmetry & Parity (even/odd, reflections/rotations).
• Choose smart numbers (base 100 for %; LCM of denominators; unit side = 1 for geometry ratios).

Top traps (avoid)

• Treating percent change additively; wrong base.
• Inequality flip forgotten when ×/÷ by a negative.
• Cancelling in a+ba\frac{a+b}{a}aa+b​ → “1 + b” (illegal unless factoring).
• Mod with negatives mishandled; remainder vs last digit.
• Geometry answers from a not-to-scale diagram; mixing r,d,h,lr,d,h,lr,d,h,l.
• Probability: assuming independence; double-counting arrangements; ignoring complement.
• Proofs: “because it looks true”—no cited lemma, no general step, no edge-case check.

Time split (how to study each week)

• 40% Core skills (algebra/NT/geo drills).
• 30% Mixed problem-solving (timed sets across topics).
• 20% Review & error-log (re-solve with a different method).
• 10% Theory flash (identities, theorems, template proofs).

Last-48h checklist (universal)

• One mixed paper under exam timing (mark 50–50s; no rabbit holes).
• Formula/idea ring (1 page): quad formula + discriminant; identities; %–ratio/TSD/work; area/volume; PnC/Prob basics; mod rules; AM–GM/C-S statements; common triples (3-4-5, 5-12-13, 8-15-17).
• Review your error log: fix by type (algebra signs, mod negatives, diagram discipline, set-Venn counts).
• 10 mental-math sprints (squares to 30; reciprocals 1/2..1/12; %–fraction pairs).

Quick facts / frames

• Quadratic: x=−b±b2−4ac2ax=\frac{-b\pm\sqrt{b^2-4ac}}{2a}x=2a−b±b2−4ac​​; D<0⇒D<0\RightarrowD<0⇒ no real roots; vertex at (−b2a,−D4a)\left(-\frac{b}{2a},-\frac{D}{4a}\right)(−2ab​,−4aD​).
• Inequalities: multiply/divide by negative ⇒ flip sign.
• Remainders: a≡b(modn)a\equiv b\pmod na≡b(modn) ⇒ preserve +,−,×; for negatives, add nnn until 0≤r<n0\le r<n0≤r<n.
• Similar triangles: equal angles ⇒ sides in proportion; scale kkk: area k2k^2k2, volume k3k^3k3.
• PnC: order → nPrnP rnPr; no order → nCrnC rnCr. Use complement when “at least/at most.”
• Probability bounds: 0≤P≤10\le P\le10≤P≤1; sanity check with total outcomes.
• TAUSL for graphs: say scale aloud before answering “how many / how much change”.

Solution writing (for full/subjective papers)

• Skeleton first: State Given & To-Prove; list lemmas or identities you will use.
• Name the step: e.g., “by AM–GM”, “by Pythagoras”, “by inclusion–exclusion”.
• Edge cases: equality conditions; zero/negative/degenerate geometry cases.
• Finish with a sentence that answers exactly the question (with units where relevant).

Speed tactics (during test)

• Two-pass sweep: harvest singles → mark 50–50s → one revisit.
• Back-solve from options; plug numbers into variable-heavy statements.
• Bound answers: quick hi/lo estimates to kill implausible options.
• Unit & dimension audit (esp. mensuration, rates).
• If stuck @ 60–90 s: switch method (draw, choose numbers, symmetry, complement), else mark & move.

Exam-day mini-plan

• Warm up (3–4 min): 4 mental-math + 1 algebra easy.
• Order: single-step arithmetic/algebra → nice geometry/NT → DI/PnC → leftovers.
• Final 8–10 min: recheck bubbled items; units/signs; revisit one marked problem with a fresh method.

Practice cycle (repeat weekly)

1. Diagnose: 20–30 mixed Qs; tag misses (Sign, Mod, Diagram, Count, Time).
2. Deep fix: 5 problems per tag with written explanations.
3. Retest: new mixed set; aim for fewer repeats of the same tag.
4. Summarize: one-page insights, not just formulas (e.g., “Inequality flips: when & why”).