Cameroon GCE A-Level Pure Math with Mechanics flashcards

33 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). For divisor (ax − b) the remainder is f(b/a).

Factor Theorem

(x − a) is a factor of polynomial f(x) if and only if f(a) = 0. Used to factorise cubics after finding one root by trial.

Discriminant of a quadratic

For ax² + bx + c, the value b² − 4ac. Positive gives two distinct real roots, zero gives one repeated root, negative gives no real roots.

Inverse function condition

A function has an inverse only if it is one-to-one. The graphs of f and its inverse are reflections in the line y = x, and domains and ranges swap.

Perpendicular line gradients

Two lines are perpendicular exactly when the product of their gradients equals −1, so the normal gradient is the negative reciprocal of the tangent gradient.

Circle equation in general form

x² + y² + 2gx + 2fy + c = 0 represents a circle with centre (−g, −f) and radius √(g² + f² − c), provided the expression under the root is positive.

Distance from point to line

For point (x₁, y₁) and line ax + by + c = 0, the perpendicular distance is |ax₁ + by₁ + c| divided by √(a² + b²).

Parametric curve gradient

For a curve given as x = f(t), y = g(t), the gradient dy/dx equals (dy/dt) divided by (dx/dt), provided dx/dt is not zero.

Sum of geometric progression

Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1. The infinite sum converges to a/(1 − r) only when the modulus of r is less than 1.

Arithmetic progression nth term

The nth term of an arithmetic progression with first term a and common difference d is a + (n − 1)d; the sum is n/2 times first plus last term.

Binomial expansion validity

For (1 + x)ⁿ with n negative or fractional, the expansion is infinite and valid only when the modulus of x is less than 1.

Method of differences

If each term can be written as f(r) − f(r+1), summing from r = 1 to n telescopes to f(1) − f(n+1), giving a closed form.

Sector arc length and area

With angle θ in radians and radius r: arc length s = rθ and sector area A = ½r²θ. Convert degrees to radians by multiplying by π/180.

Double angle formulae

sin 2A = 2 sin A cos A; cos 2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A; tan 2A = 2tan A/(1 − tan²A).

Harmonic form R sin(θ + α)

a sin θ + b cos θ equals R sin(θ + α) where R = √(a² + b²) and tan α = b/a. Useful for maxima and solving equations.

Change of base formula

log_a x equals log_b x divided by log_b a. In particular, log_a x = ln x / ln a, allowing any base to be converted to natural logarithms.

Exponential growth doubling time

For y = Ae^{kt} with positive k, the doubling time is ln 2 divided by k; for decay with negative k the half-life is ln 2 divided by |k|.

Linearising a power law

For y = axⁿ, taking natural logs gives ln y = ln a + n ln x. Plotting ln y against ln x gives gradient n and intercept ln a.

Quotient rule

For y = u/v, the derivative is (u′v − uv′)/v², where u′ and v′ are the derivatives of numerator and denominator with respect to x.

Second derivative test

At a stationary point, a negative second derivative indicates a local maximum; a positive second derivative indicates a local minimum; zero is inconclusive.

Implicit differentiation

Differentiate each term with respect to x, applying the chain rule to y terms, then rearrange to solve for dy/dx in terms of x and y.

Connected rates of change

When quantities are related, use the chain rule such as dV/dt = (dV/dr)(dr/dt) to convert a known rate into the required rate.

Volume of revolution

Rotating y = f(x) about the x-axis between a and b gives volume π∫y² dx. About the y-axis the formula is π∫x² dy.

Trapezium rule

With n equal strips of width h = (b − a)/n, the integral is approximately (h/2)[y₀ + yₙ + 2(sum of interior ordinates)].

Integration by parts

The rule ∫u dv = uv − ∫v du reverses the product rule. Choose u as the factor that simplifies when differentiated.

Scalar product and perpendicularity

a·b = |a||b|cos θ = a₁b₁ + a₂b₂ + a₃b₃. The vectors are perpendicular exactly when the scalar product equals zero.

Vector equation of a line

A line through point with position vector a and parallel to direction b is r = a + λb, where λ is a real parameter.

Projectile range formula

For launch speed u at angle θ on level ground, range R = u² sin 2θ / g and time of flight T = 2u sin θ / g.

Velocity-time graph interpretation

The gradient of a velocity-time graph gives acceleration, and the area between the graph and the time axis gives displacement over that interval.

Limiting friction

A body is on the point of sliding when friction reaches its maximum value F = μR, where μ is the coefficient of friction and R the normal reaction.

Weight components on incline

On a plane inclined at angle θ, weight mg resolves into mg sin θ along the plane and mg cos θ perpendicular to the plane.

Principle of moments

A rigid body in equilibrium has the total clockwise moment about any point equal to the total anticlockwise moment about that same point.

Moment of a couple

A couple is two equal, opposite, parallel forces separated by perpendicular distance d. Its moment is F × d and is the same about any point.