Algebra: quadratics, sequences and binomial
- ✓Quadratic roots: x = (−b ± √(b² − 4ac)) / 2a, with discriminant D = b² − 4ac. Real and distinct if D > 0, equal if D = 0, complex if D < 0.
- ✓Sum and product of roots: α + β = −b/a, αβ = c/a. Quadratic from roots: x² − (α+β)x + αβ = 0.
- ✓AP: aₙ = a + (n−1)d; Sₙ = (n/2)[2a + (n−1)d] = (n/2)(a + l).
- ✓GP: aₙ = arⁿ⁻¹; Sₙ = a(rⁿ − 1)/(r − 1) for r ≠ 1; S∞ = a/(1 − r) for |r| < 1.
- ✓Means: AM ≥ GM ≥ HM, with equality only when all terms are equal. GM² = AM × HM for two numbers.
- ✓Standard sums: Σn = n(n+1)/2; Σn² = n(n+1)(2n+1)/6; Σn³ = [n(n+1)/2]².
- ✓Binomial theorem: (a + b)ⁿ = Σ ⁿCᵣ aⁿ⁻ʳ bʳ, with general term Tᵣ₊₁ = ⁿCᵣ aⁿ⁻ʳ bʳ.
- ✓Logarithms: log(mn) = log m + log n; log(m/n) = log m − log n; log mⁿ = n log m; log_b a = log a / log b.
- ✓Modulus: |x| < a means −a < x < a; |x| > a means x < −a or x > a.
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Trigonometry
- ✓Pythagorean identities: sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ.
- ✓Compound angles: sin(A ± B) = sinA cosB ± cosA sinB; cos(A ± B) = cosA cosB ∓ sinA sinB; tan(A ± B) = (tanA ± tanB)/(1 ∓ tanA tanB).
- ✓Double angle: sin2θ = 2 sinθ cosθ = 2tanθ/(1 + tan²θ); cos2θ = cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1 = (1 − tan²θ)/(1 + tan²θ); tan2θ = 2tanθ/(1 − tan²θ).
- ✓Triple angle: sin3θ = 3sinθ − 4sin³θ; cos3θ = 4cos³θ − 3cosθ; tan3θ = (3tanθ − tan³θ)/(1 − 3tan²θ).
- ✓Sum to product: sinA + sinB = 2 sin((A+B)/2) cos((A−B)/2); cosA + cosB = 2 cos((A+B)/2) cos((A−B)/2).
- ✓Triangle: a/sinA = b/sinB = c/sinC = 2R (sine rule); a² = b² + c² − 2bc cosA (cosine rule); area = ½ab sinC = rs = abc/4R.
- ✓General solutions: sinθ = 0 gives θ = nπ; cosθ = 0 gives θ = (2n+1)π/2; tanθ = 0 gives θ = nπ; sinθ = sinα gives θ = nπ + (−1)ⁿα; cosθ = cosα gives θ = 2nπ ± α; tanθ = tanα gives θ = nπ + α.
- ✓Inverse trig: sin⁻¹x + cos⁻¹x = π/2; tan⁻¹x + cot⁻¹x = π/2; tan⁻¹x + tan⁻¹y = tan⁻¹((x+y)/(1−xy)) when xy < 1.
- ✓Ranges matter: sin⁻¹x ∈ [−π/2, π/2], cos⁻¹x ∈ [0, π], tan⁻¹x ∈ (−π/2, π/2). Most inverse-trig errors are range errors.
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Coordinate geometry: lines and circles
- ✓Distance: √((x₂−x₁)² + (y₂−y₁)²). Section formula (internal, ratio m:n): ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)).
- ✓Line forms: slope-intercept y = mx + c; point-slope y − y₁ = m(x − x₁); two-point; intercept x/a + y/b = 1; normal x cosα + y sinα = p.
- ✓Distance from a point to a line: |ax₁ + by₁ + c| / √(a² + b²). Distance between parallel lines: |c₁ − c₂| / √(a² + b²).
- ✓Angle between two lines: tanθ = |(m₁ − m₂)/(1 + m₁m₂)|. Parallel if m₁ = m₂; perpendicular if m₁m₂ = −1.
- ✓Circle: (x − h)² + (y − k)² = r². General form x² + y² + 2gx + 2fy + c = 0 has centre (−g, −f) and radius √(g² + f² − c).
- ✓Tangent to a circle at (x₁, y₁) on x² + y² = a²: xx₁ + yy₁ = a². Condition for y = mx + c to touch it: c² = a²(1 + m²).
- ✓Length of the tangent from an external point: √(x₁² + y₁² + 2gx₁ + 2fy₁ + c).
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Conic sections at a glance
Conics reward a table over prose, because almost every question needs one of these four parameters and the differences between the curves are systematic.
| Conic | Standard form | Eccentricity | Foci | Latus rectum |
|---|---|---|---|---|
| Parabola | y² = 4ax | e = 1 | (a, 0) | 4a |
| Ellipse (a > b) | x²/a² + y²/b² = 1 | e = √(1 − b²/a²) | (±ae, 0) | 2b²/a |
| Hyperbola | x²/a² − y²/b² = 1 | e = √(1 + b²/a²) | (±ae, 0) | 2b²/a |
| Rectangular hyperbola | xy = c² | e = √2 | (±c√2, ±c√2) | 2√2 c |
- ✓Parabola y² = 4ax: vertex (0,0), directrix x = −a, axis along x. Parametric point (at², 2at).
- ✓Ellipse: directrices x = ±a/e, sum of focal distances = 2a. Parametric point (a cosθ, b sinθ).
- ✓Hyperbola: asymptotes y = ±(b/a)x, difference of focal distances = 2a. Parametric point (a secθ, b tanθ).
- ✓Tangent at (x₁, y₁) is obtained by the T = 0 rule: replace x² with xx₁, y² with yy₁, x with (x + x₁)/2 and y with (y + y₁)/2.
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Calculus: limits and derivatives
- ✓Standard limits: lim(x→0) sinx/x = 1; lim(x→0) tanx/x = 1; lim(x→0) (1 − cosx)/x² = ½; lim(x→0) (eˣ − 1)/x = 1; lim(x→0) ln(1+x)/x = 1; lim(x→0) (1 + x)^(1/x) = e; lim(x→a) (xⁿ − aⁿ)/(x − a) = naⁿ⁻¹.
- ✓L'Hôpital applies only to 0/0 and ∞/∞ forms - check the form before differentiating.
- ✓Derivatives: d/dx(xⁿ) = nxⁿ⁻¹; sinx → cosx; cosx → −sinx; tanx → sec²x; secx → secx tanx; cosecx → −cosecx cotx; cotx → −cosec²x.
- ✓Exponentials and logs: d/dx(eˣ) = eˣ; d/dx(aˣ) = aˣ ln a; d/dx(ln x) = 1/x.
- ✓Inverse trig: d/dx(sin⁻¹x) = 1/√(1 − x²); d/dx(tan⁻¹x) = 1/(1 + x²); d/dx(sec⁻¹x) = 1/(|x|√(x² − 1)).
- ✓Rules: (uv)′ = u′v + uv′; (u/v)′ = (u′v − uv′)/v²; chain rule dy/dx = (dy/du)(du/dx).
- ✓Applications: tangent slope = dy/dx at the point; normal slope = −1/(dy/dx); maxima and minima where f′(x) = 0, classified by f″(x); increasing if f′ > 0, decreasing if f′ < 0.
- ✓Rolle's theorem needs continuity on [a,b], differentiability on (a,b) and f(a) = f(b). The Mean Value Theorem drops the last condition.
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Calculus: integration
- ✓∫xⁿ dx = xⁿ⁺¹/(n+1) + C for n ≠ −1; ∫dx/x = ln|x| + C; ∫eˣ dx = eˣ + C; ∫aˣ dx = aˣ/ln a + C.
- ✓∫sinx dx = −cosx; ∫cosx dx = sinx; ∫sec²x dx = tanx; ∫cosec²x dx = −cotx; ∫secx tanx dx = secx.
- ✓∫tanx dx = ln|secx|; ∫cotx dx = ln|sinx|; ∫secx dx = ln|secx + tanx|; ∫cosecx dx = ln|cosecx − cotx|.
- ✓∫dx/(x² + a²) = (1/a) tan⁻¹(x/a); ∫dx/(x² − a²) = (1/2a) ln|(x − a)/(x + a)|; ∫dx/(a² − x²) = (1/2a) ln|(a + x)/(a − x)|.
- ✓∫dx/√(a² − x²) = sin⁻¹(x/a); ∫dx/√(x² + a²) = ln|x + √(x² + a²)|; ∫dx/√(x² − a²) = ln|x + √(x² − a²)|.
- ✓Integration by parts: ∫u dv = uv − ∫v du, choosing u by the ILATE order (inverse, log, algebraic, trigonometric, exponential).
- ✓Definite integral properties: ∫₀ᵃ f(x)dx = ∫₀ᵃ f(a − x)dx; ∫₋ₐᵃ f(x)dx = 2∫₀ᵃ f(x)dx if f is even and 0 if f is odd.
- ✓Area between curves: ∫|f(x) − g(x)|dx over the interval where they bound the region.
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Vectors and three-dimensional geometry
- ✓Dot product: a·b = |a||b| cosθ = a₁b₁ + a₂b₂ + a₃b₃. Perpendicular when a·b = 0.
- ✓Cross product: |a × b| = |a||b| sinθ, direction by the right-hand rule. Parallel when a × b = 0.
- ✓Projection of a on b = (a·b)/|b|. Area of a triangle = ½|a × b|; area of a parallelogram = |a × b|.
- ✓Scalar triple product [a b c] = a·(b × c) = the volume of the parallelepiped. Coplanar when [a b c] = 0.
- ✓Line through a with direction b: r = a + λb. Cartesian form (x − x₁)/a = (y − y₁)/b = (z − z₁)/c.
- ✓Plane: r·n̂ = d, or ax + by + cz + d = 0 with normal (a, b, c). Distance from a point: |ax₁ + by₁ + cz₁ + d|/√(a² + b² + c²).
- ✓Angle between planes: cosθ = |n₁·n₂|/(|n₁||n₂|). Angle between a line and a plane: sinθ = |b·n|/(|b||n|).
- ✓Shortest distance between skew lines: |(a₂ − a₁)·(b₁ × b₂)| / |b₁ × b₂|.
- ✓Direction cosines satisfy l² + m² + n² = 1.
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Complex numbers, matrices and determinants
- ✓For z = x + iy: |z| = √(x² + y²), arg z = tan⁻¹(y/x) adjusted for quadrant, and z z̄ = |z|².
- ✓Polar form z = r(cosθ + i sinθ). De Moivre: (cosθ + i sinθ)ⁿ = cos nθ + i sin nθ.
- ✓Cube roots of unity: 1, ω, ω² with ω³ = 1 and 1 + ω + ω² = 0. These two facts solve most ω questions.
- ✓|z₁ + z₂| ≤ |z₁| + |z₂| (triangle inequality); |z₁z₂| = |z₁||z₂|; arg(z₁z₂) = arg z₁ + arg z₂.
- ✓Matrices: (AB)ᵀ = BᵀAᵀ; (AB)⁻¹ = B⁻¹A⁻¹; A⁻¹ = adj(A)/|A| provided |A| ≠ 0.
- ✓Determinants: |AB| = |A||B|; |Aᵀ| = |A|; |kA| = kⁿ|A| for an n × n matrix; |A⁻¹| = 1/|A|.
- ✓A determinant is zero if two rows or columns are identical or proportional. Row operations that add a multiple of one row to another leave it unchanged.
- ✓System of equations: unique solution if |A| ≠ 0; for |A| = 0 the system is either inconsistent or has infinitely many solutions.
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Permutations, probability and differential equations
- ✓ⁿPᵣ = n!/(n − r)!; ⁿCᵣ = n!/(r!(n − r)!); ⁿCᵣ = ⁿCₙ₋ᵣ; ⁿCᵣ + ⁿCᵣ₋₁ = ⁿ⁺¹Cᵣ.
- ✓Arrangements of n items with repeats: n!/(p! q! ...). Circular arrangements of n items: (n − 1)!.
- ✓P(A ∪ B) = P(A) + P(B) − P(A ∩ B). Independent events: P(A ∩ B) = P(A)P(B).
- ✓Conditional probability: P(A|B) = P(A ∩ B)/P(B). Bayes: P(Eᵢ|A) = P(Eᵢ)P(A|Eᵢ) / Σ P(Eⱼ)P(A|Eⱼ).
- ✓Binomial distribution: P(X = r) = ⁿCᵣ pʳ q^(n−r), with mean np and variance npq.
- ✓Variance: σ² = (Σxᵢ²)/n − ((Σxᵢ)/n)². Standard deviation is its square root.
- ✓Variable-separable differential equation: write it as f(y)dy = g(x)dx and integrate both sides.
- ✓Linear differential equation dy/dx + Py = Q: the integrating factor is IF = e^(∫P dx), and the solution is y·IF = ∫Q·IF dx.
- ✓Homogeneous equation dy/dx = f(y/x): substitute y = vx and it becomes separable.
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How to use a formula sheet properly
A formula sheet used as a reference makes you dependent on it; used as a test it makes you fast. The difference is direction: read the chapter name, recite the formulas from memory, then check.
- ✓Daily, in the final month: cover this page, name the chapter, write every formula you can, then check and mark the gaps.
- ✓Mark each formula with a dot every time you hesitate. After two weeks the dots tell you exactly what to drill.
- ✓Learn conditions alongside formulas - r ≠ 1 for a GP sum, |r| < 1 for the infinite sum, |A| ≠ 0 for an inverse, xy < 1 for the tan⁻¹ addition rule. Most careless losses are a right formula applied outside its domain.
- ✓Derive the ones that derive quickly (double-angle from compound-angle, cos2θ variants from the Pythagorean identity) so a blank under pressure is recoverable.
- ✓Never learn a formula you have not used in a problem. Pair every new formula with two questions the same day.
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