STPM Mathematics 954 cheat sheet

Quadratic ax²+bx+c=0: roots x=(−b±√(b²−4ac))/(2a); discriminant b²−4ac >0 two real roots, =0 repeated root, <0 no real roots. Sum of roots −b/a, product c/a.

Remainder Theorem: if polynomial P(x) is divided by (x−a), the remainder is P(a). Factor Theorem: (x−a) is a factor exactly when P(a)=0.

Partial fractions: for a proper rational function, use A/(x−a) for distinct linear factors, A/(x−a)+B/(x−a)² for a repeated factor, and (Ax+B)/(x²+bx+c) for an irreducible quadratic.

Modulus: |x|<a ⟺ −a<x<a; |x|>a ⟺ x<−a or x>a. For |f(x)|=|g(x)|, solve f(x)=g(x) and f(x)=−g(x).

Composite functions: (f∘g)(x)=f(g(x)) exists only where range of g lies inside domain of f. Inverse f⁻¹ exists only if f is one-to-one; graph of f⁻¹ is reflection of f in y=x.

Exponential and logarithm: log_a(xy)=log_a x+log_a y; log_a(x/y)=log_a x−log_a y; log_a(x^n)=n log_a x; log_a b=ln b/ln a.

Inequalities: multiplying or dividing by a negative quantity reverses the inequality sign; solve rational inequalities by sign analysis on a number line with critical values.

Binomial expansion: (a+b)^n=Σ C(n,r) a^(n−r) b^r for r=0..n; general term C(n,r) a^(n−r) b^r; valid for positive integer n only without convergence conditions.

Differentiation: d/dx(x^n)=nx^(n−1); product rule (uv)'=u'v+uv'; quotient rule (u/v)'=(u'v−uv')/v²; chain rule dy/dx=dy/du·du/dx.

Standard derivatives: d/dx(e^x)=e^x; d/dx(ln x)=1/x; d/dx(sin x)=cos x; d/dx(cos x)=−sin x; d/dx(tan x)=sec²x.

Implicit differentiation: differentiate both sides with respect to x, treating y as a function of x, then solve for dy/dx.

Stationary points: solve dy/dx=0; second derivative d²y/dx²>0 gives a minimum, <0 gives a maximum, =0 requires further investigation.

Integration: ∫x^n dx=x^(n+1)/(n+1)+C for n≠−1; ∫(1/x)dx=ln|x|+C; ∫e^x dx=e^x+C; ∫sin x dx=−cos x+C; ∫cos x dx=sin x+C.

Definite integrals: ∫_a^b f(x)dx=F(b)−F(a); area between curve and x-axis is ∫|f(x)|dx, splitting where f changes sign.

Volume of revolution about x-axis: V=π∫_a^b y² dx; about y-axis: V=π∫_c^d x² dy.

Integration by substitution: ∫f(g(x))g'(x)dx=∫f(u)du with u=g(x); change limits accordingly for definite integrals.

Trapezium rule: ∫a^b f(x)dx ≈ (h/2)[y₀+2(y₁+…+y(n−1))+y_n] with h=(b−a)/n.

Magnitude of vector a=(a₁,a₂,a₃) is |a|=√(a₁²+a₂²+a₃²); unit vector in direction of a is a/|a|.

Scalar (dot) product: a·b=|a||b|cos θ=a₁b₁+a₂b₂+a₃b₃; a·b=0 exactly when a and b are perpendicular.

Vector (cross) product: a×b has magnitude |a||b|sin θ, direction perpendicular to both by right-hand rule; a×b=0 when parallel.

Line equation: r=a+tb where a is a point on the line, b is the direction vector, t is a scalar parameter.

Plane equation: r·n=d where n is the normal vector; equivalently ax+by+cz=d for plane with normal (a,b,c).

Angle between two lines uses their direction vectors: cos θ=(b₁·b₂)/(|b₁||b₂|); angle between line and plane uses sin θ=(b·n)/(|b||n|).

Distance from point P to plane r·n=d: |(p−a)·n|/|n| where a is any point on the plane.

Shortest distance from point P to line r=a+tb: |(p−a)×b|/|b|.

Mean of data x₁,…,x_n: x̄=Σx/n; for frequency table x̄=Σfx/Σf; for grouped data use class midpoints.

Variance: s²=Σ(x−x̄)²/(n−1) for a sample; population variance σ²=Σ(x−μ)²/N; computational form s²=(Σx²−nx̄²)/(n−1).

Standard deviation is the positive square root of variance; adding a constant to every value leaves s unchanged, multiplying by k multiplies s by |k|.

Median is the middle value of ordered data (average of two middle values if n even); mode is the most frequent value.

Quartiles: Q₁, Q₂ (median), Q₃; interquartile range IQR=Q₃−Q₁; outliers often defined as values below Q₁−1.5·IQR or above Q₃+1.5·IQR.

Pearson correlation coefficient r=Σ(x−x̄)(y−ȳ)/√[Σ(x−x̄)²Σ(y−ȳ)²]; r lies in [−1,1]; r=±1 indicates perfect linear relationship.

Least squares regression line of y on x: y=a+bx with b=Σ(x−x̄)(y−ȳ)/Σ(x−x̄)² and a=ȳ−bx̄; line passes through (x̄,ȳ).

Spearman rank correlation: r_s=1−6Σd²/[n(n²−1)] where d is the difference in ranks of each pair.

Addition rule: P(A∪B)=P(A)+P(B)−P(A∩B); for mutually exclusive events P(A∩B)=0 so P(A∪B)=P(A)+P(B).

Multiplication rule: P(A∩B)=P(A)P(B|A); for independent events P(A∩B)=P(A)P(B).

Conditional probability: P(A|B)=P(A∩B)/P(B) for P(B)>0; rearranged to give P(A∩B)=P(B)P(A|B).

Total probability: P(B)=ΣP(A_i)P(B|A_i) over a partition {A_i}; Bayes' theorem: P(A_i|B)=P(A_i)P(B|A_i)/ΣP(A_j)P(B|A_j).

Binomial distribution: X~B(n,p), P(X=r)=C(n,r)p^r(1−p)^(n−r), mean=np, variance=np(1−p); conditions are fixed n, independent trials, constant p.

Poisson distribution: X~Po(λ), P(X=r)=e^(−λ)λ^r/r!, mean=variance=λ; approximates binomial when n large and p small with λ=np.

Normal distribution: X~N(μ,σ²); standardise with Z=(X−μ)/σ; use tables of Φ(z)=P(Z≤z); P(Z>z)=1−Φ(z).

Normal approximation to binomial: valid when np>5 and n(1−p)>5; use continuity correction ±0.5 when approximating discrete counts.

Permutations and combinations: number of arrangements of n distinct objects is n!; choosing r from n is C(n,r)=n!/[r!(n−r)!]; arrangements of r from n is P(n,r)=n!/(n−r)!.

Null hypothesis H₀ states no effect or no difference; alternative H₁ is one-tailed (>, <) or two-tailed (≠); test statistic is compared with critical value or p-value with significance level α.

Reject H₀ when the test statistic falls in the critical region; equivalently reject when p-value < α; otherwise fail to reject (never 'accept' H₀).

Type I error: rejecting H₀ when it is true (probability α); Type II error: failing to reject H₀ when it is false (probability β); power = 1−β.

One-sample z-test for mean with known σ: z=(x̄−μ₀)/(σ/√n); use when population is normal or n is large.

One-sample t-test for mean with unknown σ: t=(x̄−μ₀)/(s/√n) with n−1 degrees of freedom; use t tables.

Confidence interval for mean: x̄±z_(α/2)·σ/√n when σ known; x̄±t_(α/2)·s/√n when σ unknown with n−1 df.

Test for a proportion: z=(p̂−p₀)/√[p₀(1−p₀)/n] under H₀; valid when np₀ and n(1−p₀) are both at least 5.

Chi-squared goodness-of-fit: χ²=Σ(O−E)²/E with df=k−1−m where k is number of categories and m is number of estimated parameters; for contingency tables df=(r−1)(c−1).

Correlation test: under H₀ of zero correlation, t=r√[(n−2)/(1−r²)] follows a t distribution with n−2 degrees of freedom.

Quadratic equation ax²+bx+c=0 has roots x=(−b±√(b²−4ac))/(2a); discriminant b²−4ac determines number of real roots.

Sum of roots of ax²+bx+c=0 is −b/a and product is c/a.

Remainder Theorem: dividing polynomial P(x) by (x−a) gives remainder P(a); Factor Theorem: (x−a) is a factor iff P(a)=0.

Modulus inequality |x|<a is equivalent to −a<x<a, while |x|>a gives x<−a or x>a.

Logarithm laws: log_a(xy)=log_a x+log_a y, log_a(x/y)=log_a x−log_a y, log_a(x^n)=n log_a x.

Binomial expansion (a+b)^n=Σ C(n,r)a^(n−r)b^r for r=0 to n, where C(n,r)=n!/[r!(n−r)!].

Inverse function f⁻¹ exists only when f is one-to-one; its graph is the reflection of f in the line y=x.

Chain rule: if y=f(u) and u=g(x), then dy/dx=(dy/du)(du/dx).

Product rule (uv)'=u'v+uv'; quotient rule (u/v)'=(u'v−uv')/v².

Derivatives: d/dx(x^n)=nx^(n−1), d/dx(e^x)=e^x, d/dx(ln x)=1/x, d/dx(sin x)=cos x, d/dx(cos x)=−sin x.

Stationary point where dy/dx=0; d²y/dx²>0 gives minimum, d²y/dx²<0 gives maximum.

Integration: ∫x^n dx=x^(n+1)/(n+1)+C for n≠−1; ∫(1/x)dx=ln|x|+C; ∫e^x dx=e^x+C.

Definite integral ∫_a^b f(x)dx=F(b)−F(a) by the Fundamental Theorem of Calculus.

Volume of revolution about the x-axis: V=π∫_a^b y² dx.

Magnitude |a|=√(a₁²+a₂²+a₃²); unit vector is a/|a|.

Dot product a·b=|a||b|cos θ=a₁b₁+a₂b₂+a₃b₃; a·b=0 means perpendicular.

Cross product a×b has magnitude |a||b|sin θ and is perpendicular to both a and b.

Line: r=a+tb where a is a position vector of a point and b is the direction vector.

Plane: r·n=d where n is the normal vector; Cartesian form ax+by+cz=d.

Angle between line and plane: sin θ=(b·n)/(|b||n|) using direction b and normal n.

Distance from point P to plane r·n=d is |(p−a)·n|/|n| for any point a on the plane.

Mean x̄=Σx/n; grouped data x̄=Σfx/Σf using class midpoints.

Sample variance s²=Σ(x−x̄)²/(n−1); population variance σ²=Σ(x−μ)²/N.

Standard deviation is the square root of variance; adding a constant leaves s unchanged, multiplying by k scales s by |k|.

Median is the middle ordered value; IQR=Q₃−Q₁; outliers lie below Q₁−1.5·IQR or above Q₃+1.5·IQR.

Pearson r=Σ(x−x̄)(y−ȳ)/√[Σ(x−x̄)²Σ(y−ȳ)²], with −1≤r≤1.

Regression line y=a+bx with b=Σ(x−x̄)(y−ȳ)/Σ(x−x̄)² and a=ȳ−bx̄; it passes through (x̄,ȳ).

Spearman rank correlation r_s=1−6Σd²/[n(n²−1)] where d is difference in ranks.

P(A∪B)=P(A)+P(B)−P(A∩B); mutually exclusive events have P(A∩B)=0.

P(A∩B)=P(A)P(B|A); independent events satisfy P(A∩B)=P(A)P(B).

Conditional probability P(A|B)=P(A∩B)/P(B); Bayes' theorem reverses conditioning using a partition.

Binomial X~B(n,p): P(X=r)=C(n,r)p^r(1−p)^(n−r), mean np, variance np(1−p).

Poisson X~Po(λ): P(X=r)=e^(−λ)λ^r/r!, mean=variance=λ; approximates binomial when n large and p small.

Normal X~N(μ,σ²): standardise Z=(X−μ)/σ and use Φ(z)=P(Z≤z) tables.

Normal approximation to binomial valid when np>5 and n(1−p)>5, with continuity correction ±0.5.

Reject H₀ when test statistic lies in the critical region or p-value<α; otherwise fail to reject H₀.

Type I error rejects a true H₀ (probability α); Type II error fails to reject a false H₀ (probability β); power=1−β.

z-test for mean with known σ: z=(x̄−μ₀)/(σ/√n); t-test with unknown σ: t=(x̄−μ₀)/(s/√n) with n−1 df.

Confidence interval for mean: x̄±z_(α/2)σ/√n when σ known; x̄±t_(α/2)s/√n when σ unknown.

Proportion test: z=(p̂−p₀)/√[p₀(1−p₀)/n], valid when np₀≥5 and n(1−p₀)≥5.

Chi-squared goodness-of-fit χ²=Σ(O−E)²/E with df=k−1−m; contingency tables df=(r−1)(c−1).

Correlation test statistic t=r√[(n−2)/(1−r²)] follows t with n−2 df under H₀: ρ=0.