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CSCA Mathematics: formula sheet, syllabus map and exam-day facts

Reference · Mathematics

Exam guide

95 formulas · 4 modules · 88 syllabus topics · 48 questions in 60 minutes
Mathematics

Formula sheet

Every formula the questions lean on, grouped by topic.

95 formulas · 10 topics

CSCA Mathematics · Formula sheet

  1. Sets & inequalities

    • UnionA ∪ B = { x : x ∈ A or x ∈ B }
    • IntersectionA ∩ B = { x : x ∈ A and x ∈ B }
    • Complement (relative to universe U)Ac = { x ∈ U : x ∉ A }
    • DifferenceA − B = { x : x ∈ A and x ∉ B }
    • Inclusion-exclusionn(A ∪ B) = n(A) + n(B) − n(A ∩ B)
    • Subset count of an n-element set2n subsets2n − 1 proper subsets2n − 1 nonempty subsets
    • Empty set ∅ is a subset of every set
    • Quadratic inequality ax2+bx+c>0solution keyed to root(s) and sign of a / discriminant Δ=b2−4ac (Δ>0 two roots, Δ=0 one, Δ<0 none)
    • Roots straddle k (one >k, one <k) for a>0f(k) < 0
    • Rational inequality f(x)/g(x) ≤ 0sign chart on numerator/denominator zeros (g(x) ≠ 0)
    • Absolute value|x − a| < b ⇔ a − b < x < a + b (b>0)
    • Inequality multiplicative propertymultiplying/dividing by a negative reverses the inequality

  2. Functions (general, power/exp/log)

    • Functiony = f(x), domain D, range Requal functions require same domain AND same rule
    • Domain conditionsdenominator ≠ 0even-root radicand ≥ 0log argument > 0base of non-integer power > 0intersect all conditions
    • Even: f(−x) = f(x) (symmetric about y-axis)Odd: f(−x) = −f(x) (symmetric about origin)parity requires a domain symmetric about 0
    • Power functiony = xa (coefficient must be 1)
    • Exponentialy = ax (a>0, a≠1)natural base e
    • Logarithmy = loga x, inverse of axln = loge, lg = log10
    • Log rulesloga(MN)=loga M+loga Nloga(M/N)=loga M−loga Nloga(Mk)=k·loga M
    • Change of baseloga b = logc b / logc aidentities aloga N=N, loga(an)=n
    • Periodicityf(x + T) = f(x)

  3. Sequences and series

    • Arithmetic nth terman = a1 + (n − 1)d
    • Arithmetic sumSn = n(a1 + an)/2 = n·a1 + n(n−1)d/2 (quadratic in n)
    • Geometric nth terman = a1·rn−1
    • Geometric sum (r ≠ 1): Sn = a1(1 − rn)/(1 − r)
    • Geometric mean of a,bG = ±√ab
    • Term from partial suman = Sn − Sn−1 for n ≥ 2, a1 = S1
    • Geometric-series partial-sum form Sn = A·rn + C requires A + C = 0

  4. Trigonometry

    • Degree-radianradians = degrees × π/180
    • Pythagorean identitysin2 x + cos2 x = 1tan x = sin x / cos x
    • Periods2π for sin and cos, π for tan
    • Sum/differencesin(A±B)=sinA cosB ± cosA sinBcos(A±B)=cosA cosB ∓ sinA sinBtan(A±B)=(tanA ± tanB)/(1 ∓ tanA tanB)
    • Double-anglesin 2x = 2 sin x cos xcos 2x = cos2 x − sin2 x = 2cos2 x − 1 = 1 − 2sin2 x
    • Half-angle and product-to-sum / sum-to-product formulas (standard set)
    • Triangle angle sumA + B + C = π (so C = π − (A+B))
    • Sine Rulea/sin A = b/sin B = c/sin C = 2R
    • Law of Cosinesc2 = a2 + b2 − 2ab cos C
    • Triangle areaS = (1/2)ab sin C
    • y = A sin(ωx + φ): amplitude A, period 2π/ωhorizontal stretch by factor k uses ω → ω/k

  5. Calculus (derivatives)

    • Definitionf'(x) = limh→0 [f(x+h) − f(x)]/h
    • Tangent-line slope at x0k = f'(x0)tangent line y − f(x0) = f'(x0)(x − x0)
    • Power ruled/dx xn = n xn−1d/dx (1/x) = −1/x2d/dx √x = 1/(2√x)
    • d/dx ex = exd/dx ax = ax ln a
    • d/dx ln x = 1/xd/dx loga x = 1/(x ln a)
    • d/dx sin x = cos xd/dx cos x = −sin xd/dx tan x = sec2 x
    • Rules(u±v)'=u'±v'(uv)'=u'v+uv'(u/v)'=(u'v−uv')/v2(cu)'=cu'
    • Chain rule[f(g(x))]' = f'(g(x))·g'(x)
    • Monotonicityf'(x) > 0 increasing, f'(x) < 0 decreasing
    • Extremaf'(x0)=0 and f' changes + to − ⇒ local max− to + ⇒ local mincheck endpoints for closed-interval max/min
    • Convexity (per CSCA convention)f''(x) < 0 on an interval defines a 'convex' function there

  6. Analytic geometry (lines, circles, conics)

    • Slope from two pointsk = (y2 − y1)/(x2 − x1)vertical line slope undefinedslope from inclination θ: k = tan θ
    • Line formspoint-slope y−y1=k(x−x1)slope-intercept y=kx+bintercept x/a+y/b=1general Ax+By+C=0
    • Parallel: k1=k2 (or A1B2=A2B1)Perpendicular: k1·k2=−1 (or A1A2+B1B2=0)
    • Point-to-line distanced = |Ax0+By0+C| / √A2+B2
    • Circle standard(x−a)2+(y−b)2=r2general x2+y2+Dx+Ey+F=0, center (−D/2,−E/2), r=√D2+E2−4F/2 (need D2+E2−4F>0)
    • Ellipsex2/a2 + y2/b2 = 1 (a>b>0)a2 = b2 + c2|PF1|+|PF2| = 2ae = c/a < 1
    • Hyperbolax2/a2 − y2/b2 = 1c2 = a2 + b2asymptotes y = ±(b/a)x||PF1|−|PF2|| = 2ae = c/a > 1
    • Parabolay2 = 2px (focus (p/2,0), directrix x=−p/2)e = 1distance to focus = distance to directrix
    • Common chord of two circlessubtract the two general equations to get the chord (radical) line

  7. Vectors

    • Coordinates of ABB − A (component-wise)magnitude |a| = √ax2 + ay2 (+ az2)
    • Dot producta·b = |a||b| cos θ = ax bx + ay by (+ az bz)
    • Angle between vectorscos θ = (a·b)/(|a||b|)
    • Perpendicular: a·b = 0Parallel/collinear (2D): ax by − ay bx = 0 (components proportional)
    • Magnitude of a sum|a+b|2 = |a|2 + 2(a·b) + |b|2
    • Projection vector of a onto b((a·b)/|b|2) bscalar projection (a·b)/|b|

  8. Complex numbers

    • z = a + bi, i2 = −1, a,b ∈ RRe(z)=a, Im(z)=b
    • Modulus: |z| = √a2 + b2Conjugate: z̄ = a − bi
    • Multiplication(a1+b1 i)(a2+b2 i) = (a1a2 − b1b2) + (a1b2 + a2b1)i
    • Divisionmultiply numerator and denominator by the conjugate of the denominator
    • Polar form: z = r(cos θ + i sin θ)De Moivre: zn = rn(cos nθ + i sin nθ)
    • Rotation by ±π/2 about origin corresponds to multiplication by ±i

  9. Solid geometry

    • 3D point P(x,y,z)3D distance AB = √(x2−x1)2 + (y2−y1)2 + (z2−z1)2
    • Cuboid volume = lwhsurface = 2(lw+lh+wh)cube volume = s3, surface = 6s2
    • CylinderV = πr2 h, S = 2πr2 + 2πrh
    • ConeV = (1/3)πr2 h, lateral S = πrl, total S = πrl + πr2, slant l = √r2+h2
    • SphereV = (4/3)πR3, S = 4πR2
    • Cone lateral unrolls to a sectorarc length 2πr = l·(central angle)base radius r = l·(angle)/(2π)
    • Tetrahedron from coordinatesV = (1/6)|scalar triple product|pyramid V = (1/3)·base·height
    • Oblique (cavalier) projection area ratioS' = (√2/4) S
    • Equilateral triangle of side sarea = (√3/4)s2

  10. Probability & statistics

    • Classical probabilityP(A) = n(A)/n(S), 0 ≤ P(A) ≤ 1P(certain)=1, P(impossible)=0
    • Combinations C(n,k)=n!/(k!(n−k)!)permutations A(n,k)=n!/(n−k)!
    • Addition ruleP(A∪B) = P(A) + P(B) − P(A∩B)
    • Complement: P(Ac) = 1 − P(A)at-least-one: P(≥1) = 1 − P(none)
    • Conditional: P(A|B) = P(A∩B)/P(B)Multiplication: P(A∩B)=P(A)P(B|A)Independence: P(A∩B)=P(A)P(B)
    • Meanx̄ = (Σ xi)/nweighted/combined mean weights by group sizes
    • Variances2 = (1/n) Σ (xi − x̄)2standard deviation s = √s2range = xmax − xmin
    • Linear transformE(aX+b)=aE(X)+bD(aX+b)=a2 D(X)
    • Pooled variance of two groupsweight within-group variances by size AND add between-group (mean-shift) term
    • Normal PDFf(x) = 1/(σ√2π) · e−(x−μ2/(2σ2))curve symmetric about μtotal area = 1
    • Empirical rule≈68% within μ±σ, ≈95% within μ±2σ, ≈99.7% within μ±3σ
    • Standard normalZ = (x−μ)/σΦ(x)+Φ(−x)=1
    • Estimated count in an interval = P(interval) × population size
    • Least-squares slopeb̂ = (Σ xi yi − n x̄ ȳ)/(Σ xi2 − n x̄2)intercept â = ȳ − b̂ x̄
    • Normal approximation to binomial (De Moivre-Laplace)X~B(n,p) ≈ N(np, np(1−p))