Cambridge: Pure Mathematics 1

Pure Mathematics 1 is the first major step into advanced mathematics, building upon IGCSE or O-Level skills. It covers a wide range of foundational topics required for further study in science and engineering. The main areas are algebra, including quadratics and functions; coordinate geometry of lines and circles; trigonometry using radians; and an introduction to series, such as arithmetic and geometric progressions and the binomial expansion.

By the end of this course, you are expected to solve quadratic equations and inequalities, sketch and transform graphs of functions, and work with equations of straight lines and circles. You will also learn to solve problems involving arcs and sectors of circles, prove and apply trigonometric identities, sum series, and use the foundational principles of calculus: differentiation to find gradients and integration to find areas under curves.

The most significant challenge for many students is the introduction of calculus. The concepts of differentiation and integration can be abstract at first. The course heavily relies on strong algebraic manipulation skills from previous studies, which are essential for nearly every topic, especially coordinate geometry and calculus. The shift from degrees to radians in trigonometry also requires careful adjustment.

Formulas for Cambridge: Pure Mathematics 1

Solution of a quadratic equation: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ For an equation of the form ax^2 + bx + c = 0.
Equation of a circle: $(x - a)^2 + (y - b)^2 = r^2$ For a circle with centre (a, b) and radius r.
Arc length: $s = r\theta$ Where θ is the angle in radians.
Area of a sector: $A = \frac{1}{2}r^2\theta$ Where θ is the angle in radians.
Pythagorean identity: $\sin^2{\theta} + \cos^2{\theta} \equiv 1$ This identity is true for all values of θ.
Tangent identity: $\tan{\theta} \equiv \frac{\sin{\theta}}{\cos{\theta}}$ This identity is true for all values of θ where cos(θ) ≠ 0.
Binomial expansion: $(a+b)^n = a^n + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^2 + \dots + b^n$ For a positive integer n, where \binom{n}{r} = \frac{n!}{r!(n-r)!}.
Sum of an arithmetic progression: $S_n = \frac{n}{2}(2a + (n-1)d)$ a is the first term, d is the common difference.
Sum of a geometric progression: $S_n = \frac{a(1-r^n)}{1-r}$ a is the first term, r is the common ratio (r ≠ 1).
Sum to infinity of a geometric progression: $S_\infty = \frac{a}{1-r}$ This converges only when |r| < 1.
Power rule for differentiation: $\frac{d}{dx}(x^n) = nx^{n-1}$ The fundamental rule for differentiating polynomials.