Check the assumed knowledge

Use this checklist to identify what to revise. The source groups the skills as A1–A4, B5–B6 and C7.

A1: Quadratics, equations and inequalities

• Complete the square to find a quadratic’s maximum or minimum.
• Distinguish two real roots, two equal real roots and no real roots.
• Determine when $ax^2+bx+c$ is always positive or always negative.
• Solve two simultaneous equations by substitution when one equation is linear.

A2: Surds

• Perform all four operations on surds, including rationalising denominators.
• Solve equations involving surds.

A3: Polynomials and partial fractions

• Multiply and divide polynomials; apply the remainder and factor theorems.
• Resolve partial fractions with denominators no more complicated than $(ax+b)(cx+d)$, $(ax+b)(cx+d)^2$ and $(ax+b)(x^2+c^2)$.

A4: Exponentials and logarithms

• Work with $a^x$, $e^x$, $\log_a x$, $\ln x$ and their graphs.
• Apply logarithm laws, the equivalence $y=a^x\iff x=\log_a y$, and change of base.
• Simplify expressions and solve simple exponential and logarithmic equations.

For real logarithms, the base must satisfy $a>0$, $a\ne1$ and the argument must be positive; these domains make the source equivalence meaningful.

B5: Coordinate geometry

Recognise the circle in either form $(x-a)^2+(y-b)^2=r^2$ or $x^2+y^2+2gx+2fy+c=0$, with the appropriate radius condition for a real circle.

B6: Trigonometry

• Use all six trigonometric functions and the principal values of inverse sine, cosine and tangent.
• Solve trigonometric equations and use identities from the source’s List of Formulae reference.
• Express $a\cos\theta+b\sin\theta$ as $r\sin(\theta\pm\alpha)$ or $r\cos(\theta\pm\alpha)$.

C7: Differentiation and integration

• Interpret a derivative as a tangent gradient and as a rate of change.
• Differentiate $x^n$ for rational $n$, $\sin x$, $\cos x$, $\tan x$, $e^x$, $\ln x$, and constant multiples, sums and differences.
• Apply chain, product and quotient rules.
• Study increasing/decreasing functions and stationary points, including maxima, minima and stationary points of inflexion.
• Use the second derivative test, connected rates and optimisation.
• Treat integration as reverse differentiation.
• Integrate $x^n$ for rational $n$, $e^x$, $\sin x$, $\cos x$, $\sec^2x$, and their constant multiples, sums and differences.
• Integrate $(ax+b)^n$, $\sin(ax+b)$, $\cos(ax+b)$ and $e^{ax+b}$.

Work on real intervals where the functions and rules are defined. In particular, $n=-1$ uses a logarithmic antiderivative rather than the ordinary power rule; linear-inner substitutions need $a\ne0$, or separate treatment of a constant integrand. A zero second derivative alone does not establish a turning point or inflexion.

Revise these methods in the readiness chapter.