CAPE Pure Maths Unit 1 flashcards

34 cards. Read the term and the definition here. The deck at the top of this page is optional.

Remainder Theorem

When a polynomial f(x) is divided by (x − a), the remainder equals f(a). This allows quick evaluation of remainders without long division.

Factor Theorem

(x − a) is a factor of polynomial f(x) if and only if f(a) = 0. Used to factorise polynomials and find roots.

Discriminant

For quadratic ax² + bx + c = 0, Δ = b² − 4ac. If Δ > 0, two distinct real roots; Δ = 0, one repeated real root; Δ < 0, two complex conjugate roots.

Sum and product of roots

For quadratic ax² + bx + c = 0 with roots α and β: α + β = −b/a and αβ = c/a. Useful for forming equations from root properties.

Binomial expansion general term

In (a + b)^n, the term containing b^r is C(n, r) a^{n−r} b^r, where C(n, r) = n!/(r!(n−r)!). Used to find specific coefficients.

Logarithm laws

log_a(xy) = log_a x + log_a y; log_a(x/y) = log_a x − log_a y; log_a(x^k) = k log_a x. Also log_a b = (log_c b)/(log_c a).

Partial fractions decomposition

A proper rational function is split into simpler fractions: each linear factor (ax+b) gives A/(ax+b); repeated factors give multiple terms; irreducible quadratics give (Bx+C)/(ax²+bx+c).

Inverse function condition

A function f has an inverse f⁻¹ if and only if it is one-to-one and onto (a bijection). The domain of f⁻¹ is the range of f.

Composite function domain

The composite function f∘g is defined only for x in the domain of g such that g(x) lies in the domain of f. Its domain is the set of such x.

Even and odd functions

f is even if f(−x) = f(x) for all x (graph symmetric about y-axis). f is odd if f(−x) = −f(x) for all x (graph symmetric about origin).

Graph transformation rules

y = f(x) + a shifts up by a; y = f(x + a) shifts left by a; y = a f(x) stretches vertically by factor a; y = f(ax) stretches horizontally by factor 1/a.

Rational function asymptotes

For y = (ax + b)/(cx + d) with c ≠ 0, the vertical asymptote is x = −d/c and the horizontal asymptote is y = a/c. The domain excludes x = −d/c.

Exponential and logarithmic graphs

y = e^x has domain ℝ, range (0, ∞), asymptote y = 0. y = ln x has domain (0, ∞), range ℝ, asymptote x = 0. They are inverse functions.

Modulus graph transformations

y = |f(x)| reflects the part of y = f(x) below the x-axis above it. y = f(|x|) discards x < 0 and reflects the x ≥ 0 part in the y-axis.

Distance and midpoint formulas

Distance between (x₁, y₁) and (x₂, y₂) is √((x₂−x₁)² + (y₂−y₁)²). Midpoint is ((x₁+x₂)/2, (y₁+y₂)/2).

Perpendicular lines condition

Two lines with gradients m₁ and m₂ are perpendicular if and only if m₁m₂ = −1 (provided neither is vertical). Parallel lines have equal gradients.

Circle equation

Circle with centre (h, k) and radius r: (x − h)² + (y − k)² = r². Expanded form x² + y² + 2gx + 2fy + c = 0 has centre (−g, −f) and radius √(g² + f² − c).

Arc length and sector area

For a sector of radius r and angle θ in radians: arc length s = rθ and area A = ½ r² θ. These formulas require θ in radians.

Pythagorean identities

sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ. These are fundamental for simplifying trigonometric expressions.

Double angle formulas

sin 2θ = 2 sin θ cos θ; cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ; tan 2θ = 2tan θ/(1 − tan²θ).

Harmonic form R cos(θ − α)

a cos θ + b sin θ = R cos(θ − α), where R = √(a² + b²) and tan α = b/a. The maximum value is R and minimum is −R.

General solution for sine

If sin θ = sin α, then θ = nπ + (−1)^n α for integer n. For cosine: θ = 2nπ ± α. For tangent: θ = nπ + α.

Derivative from first principles

f′(x) = lim_{h→0} [f(x + h) − f(x)]/h, provided the limit exists. This defines the gradient of the tangent at x.

Chain rule

If y = f(u) and u = g(x), then dy/dx = (dy/du)(du/dx). Used to differentiate composite functions.

Product and quotient rules

Product: (uv)′ = u′v + uv′. Quotient: (u/v)′ = (u′v − uv′)/v², where v ≠ 0. Essential for differentiating products and ratios.

Implicit differentiation

To differentiate an equation in x and y, differentiate both sides with respect to x, treating y as a function of x, then solve for dy/dx.

Parametric differentiation

If x = f(t) and y = g(t), then dy/dx = (dy/dt)/(dx/dt), provided dx/dt ≠ 0. Used to find gradients of parametric curves.

Integration by parts

∫ u dv = uv − ∫ v du. Choose u using LIATE: Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential. Used for products.

Standard integrals

∫ x^n dx = x^{n+1}/(n+1) + C (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.

Volume of revolution

About the x-axis: V = π ∫_a^b y² dx. About the y-axis: V = π ∫_c^d x² dy. The curve is rotated through 2π radians about the axis.

Second derivative test

At a stationary point where f′(x) = 0: if f″(x) > 0, local minimum; if f″(x) < 0, local maximum; if f″(x) = 0, test is inconclusive.

Tangent and normal equations

At x = a, tangent gradient is f′(a), so tangent: y − f(a) = f′(a)(x − a). Normal gradient is −1/f′(a) (if f′(a) ≠ 0).

Area between two curves

If f(x) ≥ g(x) on [a, b], the area between y = f(x) and y = g(x) from x = a to x = b is ∫_a^b [f(x) − g(x)] dx.

Kinematics with calculus

If displacement s(t), then velocity v = ds/dt and acceleration a = dv/dt = d²s/dt². Conversely, displacement is the integral of velocity.