A Level Mathematics Glossary

Rate of change of velocity with respect to time. a = dv/dt = d²s/dt². Measured in m/s². Positive acceleration means speeding up (in chosen positive direction); negative means decelerating. For constant acceleration, use SUVAT equations.

Identities for trig functions of sums/differences: sin(A±B)=sinAcosB±cosAsinB, cos(A±B)=cosAcosB∓sinAsinB, tan(A±B)=(tanA±tanB)/(1∓tanAtanB). Used to find exact values and prove identities.

The maximum displacement from the equilibrium position of a periodic function. For y = a sin(bx+c)+d, the amplitude is |a|. The range is [d−|a|, d+|a|].

A 2D diagram representing complex numbers geometrically. The horizontal axis represents the real part Re(z) and the vertical axis the imaginary part Im(z). The complex number z = a+bi is plotted as the point (a, b). Used to visualise loci, modulus, and argument.

The angle θ that a complex number z makes with the positive real axis in the Argand diagram. Principal argument: θ ∈ (−π, π]. Found using arctan(b/a) adjusted for the correct quadrant. z = r(cosθ + i sinθ) where r = |z|.

A sequence in which consecutive terms differ by a fixed constant d (common difference). uₙ = a + (n−1)d. Sum: Sₙ = n/2[2a+(n−1)d]. The sequence diverges — no sum to infinity.

Inverse trigonometric functions with restricted domains to make them one-to-one. arcsin: domain [−1,1], range [−π/2, π/2]. arccos: domain [−1,1], range [0, π]. arctan: domain ℝ, range (−π/2, π/2). Their derivatives: d/dx(arcsin x)=1/√(1−x²), d/dx(arctan x)=1/(1+x²).

A line that a curve approaches but never reaches. Vertical asymptotes occur where the function is undefined (e.g. x=0 for y=1/x). Horizontal asymptotes describe the long-run behaviour as x→±∞ (e.g. y=0 for y=1/x). The Normal distribution curve is asymptotic to the x-axis.

Two masses connected by a string over a smooth pulley. The heavier mass descends, the lighter ascends. System acceleration: a=(m₁−m₂)g/(m₁+m₂). Tension: T=2m₁m₂g/(m₁+m₂). A standard connected-particle problem in Mechanics 1.

A first-order ODE of the form dy/dx + P(x)y = Q(x)yⁿ. Solved by substituting v = y^(1−n) to convert to a linear ODE. Special cases: n=0 (linear), n=1 (separable).

X~B(n,p): probability distribution for the number of successes in n independent trials each with probability p of success. P(X=r)=ⁿCᵣpʳ(1−p)ⁿ⁻ʳ. Mean E(X)=np, Var(X)=np(1−p). Normal approximation valid when np>5 and nq>5.

Expansion of (a+b)ⁿ for positive integer n: T_{r+1}=ⁿCᵣaⁿ⁻ʳbʳ. For non-integer n (P2/P3): (1+x)ⁿ=1+nx+n(n−1)x²/2!+... valid for |x|<1. Used to find specific terms, approximate functions, and expand compound expressions.

Given values of y and/or y' at specific x values, used to determine the constants of integration A and B in the general solution of a differential equation. Without boundary conditions, only the general solution can be stated.

States that for large n (n≥30), the sample mean X̄ from any population with mean μ and variance σ² is approximately normally distributed: X̄~N(μ, σ²/n). Fundamental to hypothesis testing — allows use of the Normal distribution regardless of the population's shape.

Differentiation rule for composite functions: d/dx[f(g(x))] = f'(g(x))·g'(x). Equivalently: dy/dx = (dy/du)(du/dx). Essential for differentiating expressions like (3x²+1)⁵, sin(eˣ), or ln(cos x).

Measure of elasticity in a collision: e = speed of separation / speed of approach = (v_B−v_A)/(u_A−u_B). Range: 0 ≤ e ≤ 1. e=0: perfectly inelastic (coalesce). e=1: perfectly elastic (no KE lost). For bouncing: e = √(h_after/h_before).

The general solution of the homogeneous ODE aẏ'+by'+cy=0, found from the characteristic equation aλ²+bλ+c=0. Three cases: distinct real roots (Ae^λ₁x+Be^λ₂x), repeated root ((A+Bx)eλx), complex roots α±βi (e^αx(Acosβx+Bsinβx)).

Rewriting ax²+bx+c in the form a(x+p)²+q. Used to find the vertex of a parabola, solve quadratic equations, find the range of a quadratic function, and convert a circle's general equation to standard form. ax²+bx+c = a(x+b/2a)²+(c−b²/4a).

The conjugate of z=a+bi is z̄=a−bi. Key properties: z·z̄=|z|² (real), z+z̄=2a (real), z−z̄=2bi (pure imaginary). Conjugate root theorem: if z is a root of a polynomial with real coefficients, then z̄ is also a root.

A number of the form z=a+bi where a=Re(z) is the real part, b=Im(z) is the imaginary part, and i=√(−1). The set ℂ contains all complex numbers. Every real number is a complex number with b=0. Complex numbers cannot be ordered (no "greater than").

The probability of event A given that event B has occurred: P(A|B) = P(A∩B)/P(B). Only defined when P(B)>0. Events are independent if P(A|B)=P(A), equivalently P(A∩B)=P(A)P(B).

In a closed system with no friction or air resistance, total mechanical energy (KE+GPE) is constant. With friction: initial ME = final ME + energy dissipated by friction. ½mu²+mgh₁ = ½mv²+mgh₂+W_friction.

In the absence of external forces, total momentum of a system is constant: m₁u₁+m₂u₂=m₁v₁+m₂v₂. Holds for all collisions. Combined with Newton's law of restitution for two equations in two unknowns in collision problems.

Adjustment made when using the Normal distribution to approximate a discrete distribution (Binomial or Poisson). A half-unit is added/subtracted at the boundary: P(X≤k)→P(Y<k+0.5), P(X≥k)→P(Y>k−0.5), where Y is the Normal approximation.

A series converges if the sum of its terms approaches a finite limit as the number of terms increases. A geometric series converges if and only if |r|<1, giving sum to infinity S∞=a/(1−r). A divergent series has no finite sum.

For vectors a=(a₁,a₂,a₃) and b=(b₁,b₂,b₃): a×b=(a₂b₃−a₃b₂, a₃b₁−a₁b₃, a₁b₂−a₂b₁). Result is a vector perpendicular to both a and b. |a×b|=|a||b|sinθ equals the area of the parallelogram with sides a and b. Used to find the normal vector to a plane.

A running total of frequencies up to and including each class. Plotted against the upper class boundary to give an ogive (S-shaped curve). Used to estimate the median (at n/2), quartiles Q₁ (at n/4), Q₃ (at 3n/4), and any percentile.

(cosθ+i sinθ)ⁿ = cos(nθ)+i sin(nθ) for any integer n. Allows computation of powers of complex numbers in polar form. Used to derive multiple-angle formulae (e.g. cos3θ in terms of cosθ) and find nth roots of complex numbers.

The integral ∫ₐᵇ f(x) dx evaluated between limits a and b, equal to [F(x)]ₐᵇ=F(b)−F(a) where F is an antiderivative of f. Represents the signed area between the curve y=f(x) and the x-axis from x=a to x=b. No constant of integration is needed.

An equation relating a function to its derivatives. First-order ODEs involve dy/dx; second-order involve d²y/dx². Solved by: separation of variables (if separable), integrating factor method (if linear first-order), or CF+PI method (if linear second-order with constant coefficients).

For ax²+bx+c: Δ=b²−4ac. Δ>0: two distinct real roots. Δ=0: one repeated real root (tangent to x-axis). Δ<0: no real roots (graph entirely above or below x-axis). Used in tangency conditions — setting Δ=0 to find values of parameters for which a line is tangent to a curve.

The vector change in position — distance moved in a specified direction, measured in metres. Displacement s = ∫v dt. Different from distance (which is always positive and counts total path length). A body that moves 5m forward then 5m back has displacement 0 but distance 10m.

The set of valid input values (x-values) for a function. Must be specified alongside the function rule. Restricting the domain to make a function one-to-one allows an inverse to exist. The domain of f⁻¹ equals the range of f.

For vectors a=(a₁,a₂,a₃) and b=(b₁,b₂,b₃): a·b=a₁b₁+a₂b₂+a₃b₃=|a||b|cosθ. Result is a scalar. Key applications: finding the angle between vectors, checking perpendicularity (a·b=0), and finding the foot of a perpendicular. The dot product is commutative: a·b=b·a.

sin2A=2sinAcosA; cos2A=cos²A−sin²A=2cos²A−1=1−2sin²A; tan2A=2tanA/(1−tan²A). The three forms of cos2A are all equivalent — use whichever simplifies the problem. Half-angle: sin²A=½(1−cos2A), cos²A=½(1+cos2A).

Energy stored in an elastic string or spring when extended or compressed: EPE = λe²/(2l₀) where λ=modulus of elasticity, e=extension, l₀=natural length. Equivalently: EPE=½ke² where k=λ/l₀ is the spring constant. Released as kinetic energy when the string/spring returns to natural length.

A body is in equilibrium when the net force and net moment about any point are both zero: ΣF=0 and ΣM=0. For a particle: ΣFₓ=0 and ΣFᵧ=0. For a rigid body: additionally ΣM=0 about any point. Negative reaction forces indicate tilting/instability.

Numerical method for approximating ODE solutions: y_{n+1}=yₙ+h·f(xₙ,yₙ) where h is the step size. Advances the solution using the tangent at each point. Smaller h gives greater accuracy but more computation. Subject to accumulation of error.

The mean (long-run average) of a random variable. For DRV: E(X)=ΣxP(X=x). Key results: E(aX+b)=aE(X)+b; E(X+Y)=E(X)+E(Y) always. For Binomial: E=np. For Poisson: E=λ. For Normal: E=μ. Expectation is a linear operator.

If f(a)=0, then (x−a) is a factor of the polynomial f(x), and vice versa. Extension: (ax−b) is a factor iff f(b/a)=0. Used to find linear factors of polynomials before performing long division to obtain the remaining quadratic or cubic factor.

Numerical method: rearrange f(x)=0 to x=g(x), then iterate x_{n+1}=g(xₙ). Converges when |g'(x)|<1 near the root. Diverges when |g'(x)|>1. Different rearrangements of the same equation may converge or diverge — must check the derivative condition.

Used as the y-axis in a histogram for grouped continuous data: FD = frequency ÷ class width. Ensures area of each bar equals the frequency. Using raw frequency on the y-axis is incorrect when class widths differ — a common error that loses marks.

A contact force opposing relative motion (or tendency of motion) between surfaces. F≤μR where μ=coefficient of friction, R=normal reaction. F=μR at the point of sliding (limiting equilibrium) or during sliding. Acts along the surface, always opposing the direction of motion.

A rule that maps each element of the domain to exactly one element of the range — every input gives a unique output. Notation: f: x↦f(x). A many-to-one function (like x²) can have an inverse only if the domain is restricted to make it one-to-one.

The complete solution of a differential equation containing arbitrary constants (A, B, c). For a first-order ODE: one constant. For second-order: two constants A and B. The particular solution is obtained by applying initial/boundary conditions to find specific values of the constants.

X~Geo(p): models the number of trials until the first success. P(X=r)=(1−p)^(r−1)p for r=1,2,3,... E(X)=1/p, Var(X)=(1−p)/p². P(X>r)=(1−p)^r — useful shortcut for cumulative probabilities.

A sequence in which each term is obtained by multiplying the previous term by a fixed ratio r (common ratio). uₙ=arⁿ⁻¹. Sum of n terms: Sₙ=a(1−rⁿ)/(1−r). Sum to infinity (|r|<1): S∞=a/(1−r). Must state |r|<1 condition when using S∞.

The slope of a curve at a point, equal to dy/dx at that point. For a straight line: m=(y₂−y₁)/(x₂−x₁). Two lines are perpendicular when the product of their gradients is −1: m₁×m₂=−1. The gradient of a tangent equals the derivative of the function at that point.

Energy stored by virtue of position in a gravitational field: GPE=mgh where h is height above a chosen reference level. Taking h=0 at the lowest point is usually most convenient. GPE is gained when moving upward and lost when moving downward.

Rearrangements of double angle: sin²A=½(1−cos2A), cos²A=½(1+cos2A). Essential for integrating even powers of sin and cos: ∫sin²x dx=x/2−sin2x/4+c, ∫cos²x dx=x/2+sin2x/4+c. Also used to prove further identities.

Tension in an elastic string/spring is proportional to extension: T=λe/l₀=ke where λ=modulus of elasticity (N), l₀=natural length, e=extension. Valid only within the elastic limit. Elastic strings can only pull (T≥0); springs can push (thrust) when compressed.

A formal procedure to assess whether sample evidence supports or contradicts a claim about a population parameter. Involves: H₀ (null hypothesis — no change), H₁ (alternative — claimed direction), test statistic, critical region or p-value, and a conclusion stated in context. The conclusion never proves H₀ — only provides evidence for or against it.

A multiple of i=√(−1). Powers cycle: i¹=i, i²=−1, i³=−i, i⁴=1. A purely imaginary number has Re(z)=0. The imaginary axis is the vertical axis of the Argand diagram. Complex numbers extend the real number system to allow square roots of negative numbers.

Differentiating an equation in x and y without making y the explicit subject. Differentiate both sides with respect to x, applying the chain rule to every y term: d/dx[f(y)]=f'(y)·dy/dx. Collect dy/dx terms and solve. Used for circles, ellipses, and other implicit curves.

Force × time = change in momentum: J=Ft=m(v−u). Measured in N s or kg m/s. For a variable force: J=∫F dt. The impulse-momentum theorem connects force, time, mass, and velocity change — particularly useful when the exact forces and times are not separately known.

An integral with infinite limits or an unbounded integrand. Evaluated by replacing the problematic limit with a parameter t and taking the limit: ∫_a^∞f(x)dx=lim_{t→∞}∫_a^tf(x)dx. Converges if the limit exists and is finite; diverges otherwise. ∫₁^∞1/xᵖ dx converges iff p>1.

A function μ(x)=e^(∫P(x)dx) that, when multiplied through a linear first-order ODE dy/dx+P(x)y=Q(x), converts the left side to d(μy)/dx. The ODE then integrates directly to μy=∫μQ(x)dx+c. No constant is needed when computing the integrating factor itself.

Formula for integrating products: ∫u dv=uv−∫v du. Choose u using LIATE (Logarithm>Inverse trig>Algebraic>Trig>Exponential). Cyclic IBP: when ∫eˣsinx dx, applying IBP twice returns the original integral — set equal to I and solve 2I=... to find the answer.

IQR=Q₃−Q₁. Measures the spread of the middle 50% of the data — unaffected by outliers. Preferred over standard deviation for skewed distributions or when outliers are present. Outlier rule: a value is an outlier if it lies below Q₁−1.5×IQR or above Q₃+1.5×IQR.

f⁻¹(x) undoes f: ff⁻¹(x)=x and f⁻¹f(x)=x. Found by: write y=f(x), swap x and y, solve for y. Domain of f⁻¹=Range of f. Graph of f⁻¹ is the reflection of f in y=x. Only exists when f is one-to-one — restrict the domain if necessary.

Energy of a moving body: KE=½mv². Measured in Joules (J). Always non-negative. The work-energy theorem states: net work done on a body equals its change in KE: W_net=½mv²−½mu². Kinetic energy is transferred to other forms (heat, sound, deformation) during inelastic collisions.

For limits of the indeterminate form 0/0 or ∞/∞: lim f(x)/g(x) = lim f'(x)/g'(x). In Cambridge A Level, the preferred approach is to use Maclaurin series to evaluate such limits — series expansion avoids the need for L'Hôpital's rule explicitly.

Mnemonic for choosing u in integration by parts: Logarithm > Inverse trig > Algebraic > Trigonometric > Exponential. The function type appearing earlier in LIATE should be chosen as u (to be differentiated). Example: in ∫x²ln x dx, u=ln x (L before A).

Sets of points in the Argand diagram satisfying conditions on z: |z−z₁|=r (circle, centre z₁, radius r), |z−z₁|=|z−z₂| (perpendicular bisector of segment z₁z₂), arg(z−z₁)=θ (half-line from z₁ at angle θ). All loci must be sketched and described geometrically in exam answers.

log_a(b) is the power to which a must be raised to give b. Laws: log(xy)=logx+logy; log(x/y)=logx−logy; log(xⁿ)=nlogx. Change of base: log_ab=lnb/lna. Natural logarithm: ln=log_e. Used to solve exponential equations and in modelling y=Aeᵇˣ via linearisation ln y=bx+ln A.

Power series expansion of f(x) about x=0: f(x)=f(0)+f'(0)x+f''(0)x²/2!+f'''(0)x³/3!+.... Standard series: eˣ=1+x+x²/2!+...; sinx=x−x³/3!+...; cosx=1−x²/2!+...; ln(1+x)=x−x²/2+... (|x|≤1). Used to approximate functions, evaluate limits, and find series compositions.

The distance of z=a+bi from the origin in the Argand diagram: |z|=√(a²+b²). Properties: |z₁z₂|=|z₁||z₂|; |z₁/z₂|=|z₁|/|z₂|; |z|²=zz̄. The modulus is always non-negative. In polar form: z=r(cosθ+i sinθ) where r=|z|.

|f(x)| gives the absolute (non-negative) value. Graph: reflect negative portions upward. To solve |f(x)|=g(x): consider f(x)=g(x) and f(x)=−g(x). To solve |f(x)|<|g(x)|: square both sides (valid since both sides are non-negative) to get [f(x)]²<[g(x)]².

The turning effect of a force about a pivot: Moment=Force×perpendicular distance from pivot to line of action. Measured in Nm. For equilibrium: net clockwise moment = net anticlockwise moment about any point. Taking moments about a support eliminates its reaction from the equation.

A vector quantity: p=mv. Measured in kg m/s or N s. Newton's 2nd Law in momentum form: F=dp/dt. The impulse-momentum theorem: Ft=Δp=m(v−u). Conservation of momentum holds in the absence of external forces — the total momentum of a closed system is constant.

Iterative root-finding method: x_{n+1}=xₙ−f(xₙ)/f'(xₙ). Converges quadratically near a simple root — much faster than fixed-point iteration. May fail if f'(xₙ)=0 or if the starting point is far from the root. Cambridge questions typically require 2–3 iterations.

1st: A body stays at rest or constant velocity unless acted on by a resultant force. 2nd: F=ma — resultant force equals mass × acceleration. 3rd: Every action has an equal and opposite reaction (acting on different bodies). The second law is the cornerstone of all Mechanics problems.

X~N(μ,σ²): a continuous, symmetric, bell-shaped distribution characterised by mean μ and variance σ². Standardise: Z=(X−μ)/σ~N(0,1). 68% of data within μ±σ; 95% within μ±2σ; 99.7% within μ±3σ. The most important distribution in statistics — used directly and as an approximation.

The contact force from a surface on a body, always perpendicular to the surface. On a horizontal surface: R=mg (if no other vertical forces). On an inclined plane at angle θ: R=mgcosθ. The normal reaction is NOT equal to the weight when other forces have vertical components.

Finding the maximum or minimum value of a function subject to a constraint. Method: express the objective function in one variable using the constraint, differentiate and set equal to zero, classify the stationary point using d²y/dx². Always verify it is a maximum or minimum (not just a stationary point) and check boundary values if the domain is restricted.

An observation that lies unusually far from the rest of the data. Formal rule: an outlier lies below Q₁−1.5×IQR or above Q₃+1.5×IQR. Outliers are plotted individually on a box-and-whisker plot. They may indicate errors in measurement or genuinely unusual events.

A curve expressed as x=f(t), y=g(t) where t is a parameter. Differentiate using: dy/dx=(dy/dt)/(dx/dt). For the second derivative: d²y/dx²=[d(dy/dx)/dt]/(dx/dt). To find Cartesian equation: eliminate the parameter t between the two equations.

Decomposing a proper rational function into simpler fractions for integration or expansion. Four forms: distinct linear factors (A/(ax+b)+B/(cx+d)), repeated factor (A/(ax+b)+B/(ax+b)²), irreducible quadratic ((Ax+B)/(ax²+b)+C/(cx+d)), improper (divide first, then decompose remainder). Always check the fraction is proper before decomposing.

Any specific solution to the non-homogeneous ODE aẏ''+by'+cy=f(x). Found by substituting a trial function based on the form of f(x). If the trial PI coincides with a term in the CF (modification rule), multiply by x (or x² for a double root). The general solution is y=CF+PI.

The smallest positive value T for which f(x+T)=f(x) for all x — the length of one complete cycle. For y=a sin(bx+c)+d, the period is 2π/b. Increasing b compresses the graph horizontally (reduces period). sin x and cos x have period 2π; tan x has period π.

Permutations (order matters): ⁿPᵣ=n!/(n−r)!. Combinations (order doesn't matter): ⁿCᵣ=n!/[r!(n−r)!]. Arrangements with repeated items: n!/p!q!... Circular arrangements: (n−1)!. Always identify whether order matters — this determines which formula to use.

Vector form: r·n=d where n is the normal to the plane. Cartesian form: ax+by+cz=d. To find the plane through three points: find two vectors in the plane, compute their cross product to get n, then use d=a·n for any point a on the plane. Distance from point P to plane: |p·n−d|/|n|.

X~Po(λ): models the number of random independent events in a fixed interval at constant average rate λ. P(X=r)=e^(−λ)λʳ/r!. Key property: E(X)=Var(X)=λ. Additive: if X~Po(λ₁) and Y~Po(λ₂) independently, then X+Y~Po(λ₁+λ₂). Normal approx valid when λ>15.

Representing a complex number as z=r(cosθ+i sinθ)=reⁱᶿ where r=|z| and θ=arg(z). Multiplication: multiply moduli, add arguments. Division: divide moduli, subtract arguments. Power: zⁿ=rⁿ(cos nθ+i sin nθ) by De Moivre's theorem. Preferred form for multiplication and powers.

Rate of doing work: P=W/t. For constant force: P=Fv. Measured in Watts (W). Maximum speed occurs when driving force equals total resistance (acceleration=0): v_max=P/F_resistance. At speeds below v_max, the driving force exceeds resistance giving positive acceleration.

For differentiating the product of two functions: d/dx(uv)=u'v+uv'. Remember: differentiate the first, keep the second, PLUS keep the first, differentiate the second. For three functions: d/dx(uvw)=u'vw+uv'w+uvw'. Essential for expressions like x²eˣ, x sin x, or x³ ln x.

A body moving freely under gravity after an initial projection — no air resistance. Horizontal and vertical motions are independent. Horizontal: constant velocity uₓ=u cosα. Vertical: SUVAT with a=−g. Range: R=u²sin2α/g. Maximum range when α=45°. Maximum height: H=u²sin²α/(2g).

For differentiating the quotient of two functions: d/dx(u/v)=(u'v−uv')/v². Mnemonic: "vdu minus udv over v squared." The denominator is always v² (not (v')²). Used for expressions like (x²+1)/(2x−3), sin x/eˣ, or tan x=sin x/cos x (recovers sec²x).

Expressing a sinx+b cosx as a single sinusoidal function: R sin(x+α) where R=√(a²+b²) and tanα=b/a. Useful for finding the maximum/minimum of expressions and solving equations. The maximum value of the expression is R; minimum is −R.

Unit of angle measurement: 1 radian is the angle subtended at the centre by an arc equal to the radius. 2π radians = 360°. Key conversions: π/6=30°, π/4=45°, π/3=60°, π/2=90°. All A Level trigonometric calculus requires angles in radians — degrees give incorrect derivatives and integrals.

The set of actual output values (y-values) of a function. Found by determining all possible values f(x) can take for x in the domain. For a quadratic with vertex (p,q): range is f≥q (if a>0) or f≤q (if a<0). The range of f becomes the domain of f⁻¹.

A recurrence relation expressing Iₙ (an integral with parameter n) in terms of Iₙ₋₁ or Iₙ₋₂. Derived using integration by parts. Classic example: Iₙ=∫₀^(π/2) sinⁿx dx = ((n−1)/n)Iₙ₋₂ with I₀=π/2 and I₁=1. Allows systematic evaluation of high-power trig integrals.

When polynomial f(x) is divided by (x−a), the remainder equals f(a). For (ax−b): substitute x=b/a. This allows the remainder to be found without performing the full long division. The Factor Theorem is a special case: if f(a)=0, the remainder is zero and (x−a) is a factor.

The mean of a random sample. As a random variable: E(X̄)=μ and Var(X̄)=σ²/n. The standard deviation of X̄ is SE=σ/√n (standard error). If the population is Normal, X̄ is exactly Normal. By the CLT, X̄ is approximately Normal for large n regardless of population shape.

A first-order ODE that can be written as g(y)dy=f(x)dx — all y terms on the left, all x terms on the right. Integrate both sides and apply initial conditions to find the constant. Common error: forgetting the constant of integration or failing to apply the initial condition.

The probability threshold for rejecting H₀ — typically 5% or 1%. Represents the maximum acceptable probability of a Type I error (rejecting a true H₀). A result is statistically significant if the p-value ≤ α. Lower α means stronger evidence required to reject H₀.

Two lines in 3D that are neither parallel nor intersecting — they pass each other at different heights without meeting. To identify: show direction vectors are not parallel, then demonstrate the system of simultaneous equations for intersection has no solution. Shortest distance: d=|(a₂−a₁)·(d₁×d₂)|/|d₁×d₂|.

The square root of the variance — measures the typical spread of data about the mean. Population: σ=√(Σx²/n−x̄²). A larger standard deviation means data is more spread out. For the sample mean: SE=σ/√n (standard error — the standard deviation of X̄).

The standard deviation of the sampling distribution of the sample mean: SE=σ/√n. As n increases, SE decreases — larger samples give more precise estimates of the population mean. Used in the denominator of the Z-test statistic: Z=(X̄−μ₀)/SE.

A point where dy/dx=0. Three types: local maximum (d²y/dx²<0), local minimum (d²y/dx²>0), and point of inflection (d²y/dx²=0 — use sign test on dy/dx). Global maximum/minimum may be at endpoints of a restricted domain, not at a stationary point — always check both.

Five equations for constant acceleration: v=u+at; s=ut+½at²; v²=u²+2as; s=½(u+v)t; s=vt−½at². Each equation omits one of the five variables — choose the one whose omitted variable is not needed. Critical: these equations are only valid for constant acceleration.

Substitution t=tan(x/2) converting trig integrals/equations to rational form: sinx=2t/(1+t²), cosx=(1−t²)/(1+t²), dx=2dt/(1+t²). Used to solve equations like a cosx+b sinx=c or integrate 1/(a+b cosx). Note: t=±1 corresponds to x=±π/2.

The line that touches a curve at a point and has the same gradient as the curve there. At point (x₁,y₁): gradient m=dy/dx|_{x=x₁}, equation y−y₁=m(x−x₁). A line is tangent to a curve if substituting it into the curve equation gives a repeated root (Δ=0). The normal is perpendicular to the tangent.

The pulling force transmitted through a string, rope, or cable — always acts along the string, away from the object. For a light inextensible string over a smooth pulley, the tension is the same on both sides. For an elastic string: T=λe/l₀ (Hooke's Law). Tension is zero when the string is slack.

Changes to a function's graph: y=f(x)+a (translate a up); y=f(x+a) (translate a left); y=af(x) (vertical stretch ×a); y=f(ax) (horizontal stretch ×1/a); y=−f(x) (reflect in x-axis); y=f(−x) (reflect in y-axis); y=|f(x)| (reflect negative portions up). The horizontal transformations are counterintuitive — +a means left, not right.

Type I error: rejecting H₀ when it is true (false positive). P(Type I error) = significance level α. Type II error: failing to reject H₀ when it is false (false negative). P(Type II) = P(not in critical region | true parameter value). Reducing α decreases Type I errors but increases Type II errors (and vice versa).

A measure of spread: the mean of squared deviations from the mean. Population: σ²=Σx²/n−x̄²=E(X²)−[E(X)]². Key properties: Var(aX+b)=a²Var(X); Var(X+Y)=Var(X)+Var(Y) only if X and Y are independent. Variance is always non-negative; standard deviation is its square root.

A quantity with magnitude and direction in 3D space, written as a column vector or aî+bĵ+cк̂. Magnitude: |a|=√(a₁²+a₂²+a₃²). Unit vector: â=a/|a|. Addition and scalar multiplication follow the parallelogram law. Vectors can represent positions, displacements, velocities, or forces.

Graph of velocity against time. Gradient = acceleration (positive: speeding up; negative: slowing down; zero: constant velocity). Area under graph = displacement (signed). Area below x-axis = negative displacement. Total distance = sum of absolute areas. A horizontal line indicates zero acceleration (constant velocity).

Volume formed by rotating a region 360° about an axis. About x-axis: V=π∫ₐᵇy²dx. About y-axis: V=π∫_c^dx²dy. Square the function FIRST, then integrate. The π comes from the circular cross-section at each point. Most common error: forgetting to square y before integrating.

Energy transferred by a force: W=Fdcosθ where θ is the angle between force and displacement. Measured in Joules. Work done against gravity = mgh. Work done by friction = −μRd (always negative — energy removed from system). The work-energy theorem: net work done = change in kinetic energy.

The net work done on a body equals its change in kinetic energy: W_net = ½mv² − ½mu². For constant force: (F−R)d = ½mv² − ½mu² on a horizontal surface. This theorem connects forces, distances, and speeds without requiring knowledge of time — often faster than Newton's Law + SUVAT.

The number of standard deviations a value lies from the mean: z=(x−μ)/σ. Used to convert any Normal distribution problem to the Standard Normal N(0,1). Key z-values: z=1.645 (upper 5%), z=1.960 (upper 2.5%), z=2.326 (upper 1%), z=2.576 (upper 0.5%). Φ(z) gives P(Z<z) from tables.

Hypothesis test for a population mean when σ is known: test statistic Z=(X̄−μ₀)/(σ/√n). Compare to critical values: ±1.645 (5% two-tailed or 1% one-tailed? — be careful: 5% one-tailed=1.645; 5% two-tailed=1.960). Always state the distribution of the test statistic under H₀ and write the conclusion in context.