Course Description

Course Content

The subject content for the Cambridge O Level Mathematics (Additional) is organized into the following fourteen main topics:

1. Functions

• Understand the terms: function, domain, range (image set), one–one function, many–one function, inverse function and composition of functions.
• Find the domain and range of functions.
• Recognize and use function notation.
• Understand the relationship between y=f(x) and y=∣f(x)∣, where f(x) may be linear, quadratic, cubic or trigonometric.
• Explain in words why a given function does not have an inverse.
• Find the inverse of a one–one function.
• Form and use composite functions.
• Use sketch graphs to show the relationship between a function and its inverse.

2. Quadratic functions

• Find the maximum or minimum value of the quadratic function f:x↦ax2+bx+c by completing the square or by differentiation.
• Use the maximum or minimum value of f(x) to sketch the graph of y=f(x) or determine the range for a given domain.
• Know the conditions for f(x)=0 to have: (i) two real roots (ii) two equal roots (iii) no real roots and the related conditions for a given line to: (i) intersect a given curve (ii) be a tangent to a given curve (iii) not intersect a given curve.
• Solve quadratic equations for real roots.
• 5 Find the solution set for quadratic inequalities either graphically or algebraically.

3. Factors of polynomials 

• Know and use the remainder and factor theorems.
• Find factors of polynomials.
• Solve cubic equations.

4. Equations, inequalities and graphs

• Solve equations of the type ∣ax+b∣=c, ∣ax+b∣=cx+d, ∣ax+b∣=∣cx+d∣ and ∣ax2+bx+c∣=d using algebraic or graphical methods.
• Solve graphically or algebraically inequalities of the type k∣ax+b∣>c, k∣ax+b∣≤c, k∣ax+b∣≤∣cx+d∣, ∣ax+b∣≤cx+d, ∣ax2+bx+c∣>d and ∣ax2+bx+c∣≤d.
• Use substitution to form and solve a quadratic equation in order to solve a related equation.
• Sketch the graphs of cubic polynomials and their moduli, when given as a product of three linear factors.
• Solve graphically cubic inequalities of the form f(x)≥d, f(x)>d, f(x)≤d or f(x)<d where f(x) is a product of three linear factors and d is a constant.

5. Simultaneous equations 

• Solve simultaneous equations in two unknowns by elimination or substitution.

6. Logarithmic and exponential functions

• Know and use simple properties and graphs of the logarithmic and exponential functions, including lnx and ex.
• Know and use the laws of logarithms, including change of base of logarithms.
• Solve equations of the form ax=b.

7. Straight-line graphs

• Use the equation of a straight line.
• Know and use the condition for two lines to be parallel or perpendicular.
• Solve problems involving midpoint and length of a line, including finding and using the equation of a perpendicular bisector.
• Transform given relationships to and from straight-line form, including determining unknown constants by calculating the gradient or intercept of the transformed graph.

8. Coordinate geometry of the circle

• Know and use the equation of a circle.
• Solve problems involving the intersection of a line and a circle.
• Solve problems involving tangents to a circle.

9. Circular measure

• Solve problems involving arc length and sector area, including composite shapes.
• Convert between degrees and radians.

10. Trigonometry

• Understand and use the relationship tanx=cosxsinx and sin2x+cos2x=1.
• Use the identities secx=cosx1, cosecx=sinx1 and cotx=tanx1.
• Know and use the relationships sec2x=1+tan2x and cosec2x=1+cot2x.
• Solve simple trigonometric equations for values of the angle between 0∘ and 360∘ (or 0 and 2π radians).
• Sketch and use the graphs of y=asinbx+c, y=acosbx+c and y=atanbx+c, where a is a positive integer, b is a simple fraction or integer, and c is an integer.
• Solve simple trigonometric equations for values of the angle between 0∘ and 360∘ (or 0 and 2π radians) by considering the shape of a graph.

11. Permutations and combinations 

• Understand the terms permutation and combination.
• Calculate the number of permutations of n items from n and r items from n.
• Calculate the number of combinations of r items from n items.

12. Series

• Use the notation n! and (rn).
• Use the binomial theorem for the expansion of (a+b)n for positive integer n.
• Recognize arithmetic and geometric progressions.
• Use the formulae for the nth term and the sum of the first n terms of an arithmetic progression.
• Use the formulae for the nth term and the sum of the first n terms of a geometric progression.
• Use the condition for and the formula for the sum to infinity of a geometric progression.

13. Vectors in two dimensions

• Use displacement vectors and position vectors.
• Understand and use the terms scalar and vector.
• Represent a vector by (xy), AB, a or xi+yj.
• Calculate the magnitude of a vector x2+y2.
• Use the magnitude of a vector to calculate a unit vector.
• Add and subtract vectors, and multiply a vector by a scalar.
• Use the notion of two vectors being parallel.
• Use the position vector xi+yj to find the coordinates of a point.
• Calculate the scalar product of two vectors and use the property a⋅b=∣a∣∣b∣cosθ.
• Use the scalar product to determine if two vectors are perpendicular.

14. Calculus

• Understand that differentiation is the limit of δxδy as δx→0.
• Differentiate products and quotients of functions.
• Differentiate eax+b, ln(ax+b), sin(ax+b), cos(ax+b) and tan(ax+b).
• Apply differentiation to gradients, stationary points (maxima and minima), connected rates of change, and small increments and approximations.
• Understand integration as the reverse process of differentiation.
• Integrate powers of x (excluding x1) and expressions of the form (ax+b)n.
• Integrate eax+b, ax+b1, sin(ax+b), cos(ax+b) and sec2(ax+b).
• Use definite integrals to find areas under curves, including those involving the modulus function.
• Apply differentiation and integration to kinematics problems.
• Use integration to find the value of a constant of integration for an indefinite integral.
• Understand that area under velocity-time graph represents displacement.

Learning Objectives

By the end of the Cambridge O Level Additional Mathematics (4037) course with Eduva, students will be able to:

1. Advanced Algebraic and Functional Fluency (AO1)

• Master advanced algebraic techniques including the Factor and Remainder Theorems to solve cubic equations and factor polynomials.
• Understand and use the concept of functions, including domain, range, inverse functions, and composite functions.
• Apply logarithmic and exponential laws to solve equations of the form ax=b.
• Use the Binomial Theorem for expansion and apply formulae for arithmetic and geometric progressions, including the sum to infinity.

2. Calculus and Trigonometry (AO2)

• Perform differentiation of complex functions, including products, quotients, and trigonometric/logarithmic/exponential functions.
• Apply differentiation to solve problems involving gradients, stationary points (maxima and minima), and connected rates of change.
• Perform integration as the reverse of differentiation to find indefinite and definite integrals (including finding areas under curves).
• Use and prove advanced trigonometric identities and solve trigonometric equations for a given range of angles.

3. Application and Problem-Solving (AO3)

• Solve complex problems involving simultaneous equations where one or both equations are non-linear (e.g., intersection of a line and a circle).
• Apply vectors in two dimensions to solve problems involving position, magnitude, and the scalar product (dot product) to determine perpendicularity.
• Apply circular measure to solve problems involving arc length and sector area in both degrees and radians.
• Use permutations and combinations to count the number of arrangements and selections in complex scenarios.
• Translate complex real-world problems (such as those in kinematics) into mathematical models using calculus techniques and solve them effectively.