SQA Higher Mathematics Area 2 Relationships and Calculus: quadratics, trig equations, differentiation, integration and the addition formulae

What Area 2 actually demands

Relationships and Calculus is the core of Higher Mathematics, where the algebra of Area 1 meets the new ideas of differentiation and integration. The examiners test secure algebra, clear method-led working, and the ability to choose the right tool for an unfamiliar problem. This guide walks through all five topics of the area, then sets out the patterns the SQA repeats. Each topic has a matching dot-point page with practice questions; this overview ties them together.

Polynomials and quadratics

The area opens with polynomials and quadratics. Completing the square rewrites $ax^2 + bx + c$ as $a(x + p)^2 + q$, revealing the turning point. The discriminant $b^2 - 4ac$ fixes the nature of the roots, and a quadratic with $a > 0$ is always positive when the discriminant is negative. The factor theorem says that if $f(h) = 0$ then $(x - h)$ is a factor, which factorises and solves cubics.

Trigonometric equations

Trigonometric equations are solved over a stated interval using the CAST diagram and graph symmetry, with the identities $\sin^2\theta + \cos^2\theta = 1$ and $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$. Many reduce to a quadratic in one ratio, which you factorise and solve across the interval, taking care not to lose solutions.

Differentiation

Differentiation gives the gradient function. The power rule $\dfrac{d}{dx}(x^n) = nx^{n-1}$ handles polynomials, roots and reciprocals once written as powers, and in radians $\dfrac{d}{dx}(\sin x) = \cos x$ and $\dfrac{d}{dx}(\cos x) = -\sin x$. It gives tangents, decides where a function is increasing or decreasing, and finds and classifies stationary points.

Integration

Integration reverses differentiation: $\displaystyle\int x^n\,dx = \dfrac{x^{n+1}}{n+1} + c$ for $n \ne -1$, with the constant of integration on every indefinite integral. The definite integral $\displaystyle\int_a^b f(x)\,dx$ substitutes the limits and subtracts, giving the area under a curve and, by integrating a difference, the area between two curves.

The addition formulae

The addition formulae $\sin(A \pm B)$ and $\cos(A \pm B)$ expand compound angles, and setting $B = A$ gives the double angle formulae such as $\cos 2A = 2\cos^2 A - 1$. The wave function writes $a\sin x + b\cos x$ as $k\sin(x + \alpha)$ with $k = \sqrt{a^2 + b^2}$, making the maximum value easy to read.

How Area 2 is examined

A typical SQA profile for Relationships and Calculus:

• Secure algebra. Completing the square, the discriminant and the factor theorem.
• Calculus technique. Tangents, stationary points and area, with full method shown.
• Trigonometric method. Solving equations over an interval and using identities.
• Compound angles. The addition and double angle formulae and the wave function.

Check your knowledge

A mix of recall and method questions covering Area 2. Attempt them, then check against the solutions.

1. Find the discriminant of $x^2 - 4x + 4$ and state the nature of the roots. (2 marks)
2. Solve $\sin x = \dfrac{1}{2}$ for $0 \le x \le 360^\circ$. (2 marks)
3. Differentiate $y = x^4 - 3x$. (2 marks)
4. Find $\displaystyle\int (3x^2 + 2)\,dx$. (2 marks)
5. State the amplitude $k$ when $5\sin x + 12\cos x$ is written as $k\sin(x + \alpha)$. (1 mark)