O Level A Math Full Syllabus Cheatsheet Notes — Upper Secondary G3

Discriminant and nature of roots

$b^2 - 4ac > 0 \Rightarrow 2 \text{ roots};\; b^2 - 4ac = 0 \Rightarrow 1 \text{ root};\; b^2 - 4ac < 0 \Rightarrow 0 \text{ roots}$

a, b and c are the coefficients of ax² + bx + c. For curve and line questions, equate the curve and line and bring everything to the left first.

Division of polynomials

$\frac{\text{Dividend}}{\text{Divisor}} = \text{Quotient} + \frac{\text{Remainder}}{\text{Divisor}}$

Used in long division and for improper fractions in partial fractions. Fill any missing terms with 0 before dividing.

Linear law

y = mx + c

Replace x and y with the new axes to linearise the equation. The gradient m and the y-intercept c must be constants with no variables.

Gradient of normal

$m_{\text{normal}} = -\frac{1}{m_{\text{tangent}}}$

Use with the equation of a line to find tangents and normals to a curve.

Angle conversion

180° = π rad

Converts between degrees and radians. Check the calculator mode before solving.

Ratios of n(90°) ± θ

$\sin[n(90^\circ) \pm \theta],\ \cos[n(90^\circ) \pm \theta],\ \tan[n(90^\circ) \pm \theta]$

If n is even, the ratio stays the same. If n is odd, change to the co-ratio (e.g. sin to cos). The sign depends on the quadrant.