# Cambridge AS Levels Pure Maths 1 (P1)

Knowledge of the content of O Level/IGCSE Mathematics is assumed. Candidates will be expected to be familiar with scientific notation for the expression of compound units, e.g. 5 m s–1 for 5 metres per second.
 Module 1 Quadratics - CIE P1 (105.56 mins) Carry out the process of completing the square for a quadratic polynomial ax2 + bx + c, and use this form, e.g. to locate the vertex of the graph of y = ax2 + bx + c or to sketch the graph. Find the discriminant of a quadratic polynomial ax2 + bx + c and use the discriminant, e.g. to determine the number of real roots of the equation ax2 + bx + c = 0 . Solve quadratic equations, and linear and quadratic inequalities, in one unknown. Solve by substitution a pair of simultaneous equations of which one is linear and one is quadratic. Recognise and solve equations in x which are quadratic in some function of x, e.g. x4 – 5x2 + 4 = 0. Unit 1 Quadratics – Pure Mathematics 1 (P1) PDF Unit 2 Quadratics – Pure Mathematics 1 (P1) Video Module 2 Functions - CIE P1 (82.11 mins) Understand the terms function, domain, range, one-one function, inverse function and composition of functions. Identify the range of a given function in simple cases, and find the composition of two given functions. Determine whether or not a given function is one-one, and find the inverse of a one-one function in simple cases. Illustrate in graphical terms the relation between a one-one function and its inverse. Unit 1 Functions – Pure Mathematics 1 (P1) PDF Unit 2 Functions – Pure Mathematics 1 (P1) Video Module 3 Coordinate geometry - CIE P1 (58.75 mins) Find the length, gradient and mid-point of a line segment, given the coordinates of the end-points. Find the equation of a straight line given sufficient information (e.g. the coordinates of two points on it, or one point on it and its gradient). Understand and use the relationships between the gradients of parallel and perpendicular lines. Interpret and use linear equations, particularly the forms y = mx + c  and  y – y1 = m(x – x1) Understand the relationship between a graph and its associated algebraic equation, and use the relationship between points of intersection of graphs and solutions of equations (including, in simple cases, the correspondence between a line being tangent to a curve and a repeated root of an equation). Unit 1 Coordinate geometry – Pure Mathematics 1 (P1) PDF Unit 2 Coordinate geometry – Pure Mathematics 1 (P1) Video Module 4 Circular measure - CIE P1 (28.36 mins) Understand the definition of a radian, and use the relationship between radians and degrees. Use the formulae  s = rθ  and  A = ½r2θ  in solving problems concerning the arc length and sector area of a circle. Unit 1 Circular measure - Pure Mathematics 1 (P1) PDF Unit 2 Circular measure - Pure Mathematics 1 (P1) Video Module 5 Trigonometry - CIE P1 (90.73 mins) Sketch and use graphs of the sine, cosine and tangent functions (for angles of any size, and using either degrees or radians). Use the exact values of the sine, cosine and tangent of 30°, 45°, 60°, and related angles, e.g. $$\cos$$ 150° = $$–\dfrac{1}{2} \sqrt{3}$$ Use the notations $$\sin^{−1}x$$, $$\cos^{−1}x$$, $$\tan^{−1}x$$ to denote the principal values of the inverse trigonometric relations. Use the identities $$\dfrac{\sin \theta}{\cos \theta} \equiv \tan \theta$$ and $$\sin^2 \theta + \cos^2 \theta \equiv 1$$ Find all the solutions of simple trigonometrical equations lying in a specified interval (general forms of solution are not included). Unit 1 Trigonometry – Pure Mathematics 1 (P1) PDF Unit 2 Trigonometry – Pure Mathematics 1 (P1) Video Module 6 Vectors - CIE P1 (97.15 mins) Use standard notations for vectors, i.e. $$\begin{pmatrix} x \\ y \end{pmatrix}, x\mathbf{i}+y\mathbf{j}, \begin{pmatrix} x \\ y \\ z \end{pmatrix}, x\mathbf{i}+y\mathbf{j}+z\mathbf{k}, \overrightarrow{AB}, \mathbf{a}$$ Carry out addition and subtraction of vectors and multiplication of a vector by a scalar, and interpret these operations in geometrical terms. Use unit vectors, displacement vectors and position vectors. Calculate the magnitude of a vector and the scalar product of two vectors. Use the scalar product to determine the angle between two directions and to solve problems concerning perpendicularity of vectors. Unit 1 Vectors - Pure Mathematics 1 (P1) PDF Unit 2 Vectors - Pure Mathematics 1 (P1) Video Module 7 Series - CIE P1 (120.91 mins) Use the expansion of $$(a + b)^n$$ , where n is a positive integer (knowledge of the greatest term and properties of the coefficients are not required, but the notations $$\begin{pmatrix} n \\ r \end{pmatrix}$$ and n! should be known) Recognise arithmetic and geometric progressions. Use the formulae for the nth term and for the sum of the first n terms to solve problems involving arithmetic or geometric progressions. Use the condition for the convergence of a geometric progression, and the formula for the sum to infinity of a convergent geometric progression. Unit 1 Series - Pure Mathematics 1 (P1) PDF Unit 2 Series - Pure Mathematics 1 (P1) Video Module 8 Differentiation - CIE P1 (132.55 mins) Understand the idea of the gradient of a curve, and use the notations $$f'(x), f"(x), \dfrac{dy}{dx}$$  and  $$\dfrac{d^2y}{dx^2}$$ (the technique of differentiation from first principles is not required). Use the derivative of $$x^n$$ (for any rational n), together with constant multiples, sums, differences of functions, and of composite functions using the chain rule. Apply differentiation to gradients, tangents and normals, increasing and decreasing functions and rates of change (including connected rates of change). Locate stationary points, and use information about stationary points in sketching graphs (the ability to distinguish between maximum points and minimum points is required, but identification of points of inflexion is not included). Unit 1 Differentiation - Pure Mathematics 1 (P1) PDF Unit 2 Differentiation - Pure Mathematics 1 (P1) Video Module 9 Integration - CIE P1 (114.05 mins) Understand integration as the reverse process of differentiation, and integrate $$(ax + b)^n$$ (for any rational n except -1), together with constant multiples, sums and differences. Solve problems involving the evaluation of a constant of integration, e.g. to find the equation of the curve through (1, -2) for which $$\dfrac{dy}{dx} = 2x + 1$$ evaluate definite integrals (including simple cases of ‘improper’ integrals, such as $$\int^{1}_{0}x^{-\frac{1}{2}}dx$$  and  $$\int^{\infty}_{1}x^{-2}dx$$ Use definite integration to find the area of a region bounded by a curve and lines parallel to the axes, or between two curves a volume of revolution about one of the axes. Unit 1 Integration – Pure Mathematics 1 (P1) PDF Unit 2 Integration – Pure Mathematics 1 (P1) Video