General Aptitude Test
Divisibility by 3 or 9 — Sum all digits; if divisible by 3 (or 9), the number is too. E.g., 372 → 3+7+2=12 → divisible by 3, not by 9.
Divisibility by 11 — Alternating digit sum: (odd-position digits) minus (even-position digits). If result is 0 or divisible by 11, number is divisible by 11.
Profit/Loss Percentage Base — Always on Cost Price unless stated "on SP." If marked price given, remember: Discount is always on marked price, profit/loss on CP.
Simple Interest Trick — SI = (P × R × T)/100. To find principal when SI, rate, time known: P = (SI × 100)/(R × T). Memorize the triangle: SI at top, P·R·T at bottom.
Compound Interest for 2 Years — CI = P[(1 + R/100)² – 1]. Quick shortcut: For 2 years, CI ≈ SI × (2 + R/100). Works well for small rates.
Ratio Mixing (Alligation) — Cheaper/Dearer items mixed at mean price: draw X-diagram. (Mean – Cheaper) : (Dearer – Mean) gives the ratio. Saves algebraic solving.
Percentage-Fraction Equivalents — Memorize: 1/8=12.5%, 1/6≈16.67%, 1/7≈14.29%, 1/9≈11.11%, 2/3≈66.67%, 3/8=37.5%. Speeds up option elimination and mental math.
Averages Group Shift — If n items average A, adding one item X makes new average [(nA + X)/(n+1)]. If X > A, average rises; if X < A, average drops.
Indian National Movement Timeline (1900–1947) — PANDA-CQR-JI: Partition-Bengal (1905), Arrival-Gandhi (1915), Non-Cooperation (1920), Dandi (1930), August-Offer (1940), Cabinet-Mission (1946), Quit-India (1942), Red-Fort-Trials (1945–46), Jan-1947-Partition-plan.
Polity: Fundamental Rights (Articles 12–35) — REEF-CP: Right to Equality (14–18), Exploitation (23–24), Freedom (19–22), Cultural/Educational (29–30), Constitutional-Remedies (32), Property (removed but remember 300A).
Environment: Biomes Mnemonic — TAMED-GS: Tundra, Alpine, Mediterranean, Equatorial-rainforest, Desert, Grassland, Savanna. Links climate zones to vegetation types quickly for ecology MCQs.
Data Interpretation Speed Tip — Round numbers first: 3789 ≈ 3800, 47% ≈ 50%. Eliminate obviously wrong options before exact calculation. Saves 30–40 seconds per DI set.
Mathematics / Applied Mathematics
Derivative Quick Formulas — d/dx(xⁿ)=nxⁿ⁻¹, d/dx(eˣ)=eˣ, d/dx(ln x)=1/x, d/dx(sin x)=cos x, d/dx(cos x)=–sin x. Chain rule: multiply by inner derivative. Practice "spot-the-chain" daily.
Integration Standard Forms — ∫xⁿdx = xⁿ⁺¹/(n+1); ∫(1/x)dx = ln|x|; ∫eˣdx = eˣ; ∫sin x dx = –cos x; ∫cos x dx = sin x. Memorize these six—90% of integral MCQs use them.
Inverse Trig Domains/Ranges — sin⁻¹, cos⁻¹: domain [–1,1]; sin⁻¹ range [–π/2,π/2], cos⁻¹ [0,π]. tan⁻¹: domain ℝ, range (–π/2,π/2). Check domain first to avoid invalid options.
Determinant Shortcut (2×2) — |a b; c d| = ad – bc. For 3×3, use Sarrus rule or cofactor. If two rows/columns identical or proportional, determinant = 0.
Matrix Multiplication Compatibility — (m×n) × (n×p) = (m×p). Number of columns in first must equal rows in second. If dimensions don't match, multiplication undefined.
Linear Programming Corner-Point Theorem — Optimal value (max/min) always at a vertex of feasible region. Plot constraints, find intersection points, test objective function Z at each corner.
Probability Complement Rule — P(A) + P(A') = 1. Faster to compute P(not-A) when "at least one" or "none" appears. E.g., P(at least 1 head in 3 tosses) = 1 – P(0 heads) = 1 – 1/8.
Vector Dot & Cross Product — Dot: a·b = |a||b|cos θ (scalar, gives angle). Cross: a×b = |a||b|sin θ n̂ (vector, perpendicular). Dot→projection, Cross→area of parallelogram.
3D Geometry Distance Formula — Distance between (x₁,y₁,z₁) and (x₂,y₂,z₂): √[(x₂–x₁)² + (y₂–y₁)² + (z₂–z₁)²]. Section formula: midpoint or m:n ratio division—memorize weighted average form.
Differential Equation Order/Degree — Order = highest derivative present; Degree = power of highest-order derivative (after clearing fractions/radicals). dy/dx + y² → order 1, degree 1.
Continuity Triple-Check — Function f continuous at x=a if: (i) f(a) exists, (ii) lim(x→a) f(x) exists, (iii) both are equal. If piecewise, always verify left-hand = right-hand limit at junction.
Applications of Derivatives: Maxima/Minima — f'(x)=0 gives critical points. f''(x)>0 → local minimum; f''(x)<0 → local maximum. For interval problems, also check boundary values.
Physics
Ohm's Law & Resistance Series/Parallel — V=IR. Series: R_total = R₁+R₂+...; Parallel: 1/R_total = 1/R₁+1/R₂+.... For two resistors in parallel, R = (R₁R₂)/(R₁+R₂).
Current Electricity: Kirchhoff's Laws — KCL: Sum of currents entering junction = sum leaving (conservation of charge). KVL: Sum of potential differences in closed loop = 0 (conservation of energy).
Electrostatics: Coulomb's Law & Gauss — F = k(q₁q₂/r²), k=9×10⁹. Gauss: Φ = q_enclosed/ε₀. For infinite sheet, E uniform; for sphere, treat as point charge outside, zero inside (conductor).
Magnetic Field Directions — Right-Hand Rule: Thumb = current direction, curled fingers = magnetic field loops (for straight wire). For solenoid, fingers = current, thumb = N-pole direction.
Electromagnetic Induction: Faraday & Lenz — EMF = –dΦ/dt. Lenz's law: induced current opposes flux change. Increasing flux → induced field opposes; decreasing flux → induced field aids.
AC Circuit Reactance — Inductive reactance X_L = ωL = 2πfL;