Cambridge International AS and A Level Further Mathematics 1: Further Pure Mathematics 1
Cambridge International AS and A Level Further Mathematics 1: Further Pure Mathematics 1
Study guide//Updated 15 Oct 2026
Cambridge International Further Mathematics 9231 Paper 1 notes on polynomial roots, rational graphs, series, matrices, polar coordinates, vectors and induction.
Further Pure Mathematics 1 is the compulsory Cambridge International AS and A Level Further Mathematics 9231 Paper 1. Its seven official sections cover roots of polynomial equations, rational functions and graphs, summation of series, matrices, polar coordinates, vectors and proof by induction.
1. Roots of polynomial equations
For a monic polynomial, relations between roots and coefficients provide symmetric sums without solving for every root. For a quadratic with roots alpha and beta, their sum and product come from the x coefficient and constant term. Cubic and quartic relations alternate signs through sums of single roots, pairwise products, triple products and the product of all roots. Paper 1 restricts this work to degrees 2, 3 and 4.
Use these relations to evaluate symmetric expressions such as sums of reciprocal roots or squared roots. Rewrite the target in terms of known elementary symmetric sums. For example, the sum of squares equals the square of the sum minus twice the sum of pairwise products.
To construct an equation whose roots are related to old roots, introduce the transformation, express the old variable in terms of the new one where possible, substitute into the original equation and clear denominators. For simple reciprocal, square or linear transformations, Cambridge may not supply the substitution. Check whether the transformation combines distinct old roots into the same new root.
2. Rational functions and graphs
Sketch simple rational functions whose numerator and denominator degrees are at most 2. Find domain exclusions, intercepts and vertical asymptotes first. Polynomial division exposes any horizontal or oblique asymptote. Turning points may require differentiation.
To find the range, set y equal to the rational function, rearrange as a polynomial equation in x and require a real solution. A discriminant condition often helps find the permitted y-values. Retain the original domain exclusions and check any y-value for which the rearranged equation drops in degree; a real root of the cleared equation is valid only if it belongs to the original domain. A sketch should show significant features, not a dense plot.
Relate y = f(x) to y2
Check this topic from memory
Attempt the matching topic bank before reopening the notes. Use each missed idea to decide what to review next.
= f(x), y = 1/f(x), y = f(sqrt x) and the other syllabus-listed graph relations. Apply domain and sign restrictions before drawing branches. Use the sketches to solve equations or inequalities by identifying intersections or signed regions.
3. Summation of series
Recall the standard finite sums of r, r2 and r3. Shift indices or expand algebraically to convert a related polynomial sum into these forms. State the limits after any index change.
The method of differences writes a general term as a difference such as f(r + 1) - f(r), so most intermediate terms cancel. Partial fractions may reveal the telescoping structure. Write several initial and final terms to prove which survive instead of claiming cancellation from pattern alone.
To decide convergence, derive the sum to n terms and take its limit as n grows. This is direct consideration of partial sums. A term tending to zero is necessary but not sufficient for convergence.
4. Matrices
Add and subtract matrices of equal order. Multiply AB only when the column count of A equals the row count of B. Matrices may be non-square but have at most 3 rows and columns. Matrix multiplication is generally not commutative.
A square matrix is singular when its determinant is zero. Evaluate determinants and inverses for non-singular 2 by 2 and 3 by 3 matrices. For non-singular A and B, (AB)−1 = B−1A−1; the order reverses and the result extends to longer products.
A 2 by 2 matrix represents a linear transformation in the coordinate plane. The columns are the images of the coordinate basis vectors. Know rotations, reflections, enlargements, stretches and shears. In AB, transformation B happens first and A second. The absolute determinant is the area scale factor. A positive determinant preserves orientation and a negative determinant reverses it; a zero determinant collapses area to zero, so no orientation-preserving or reversing inverse is available.
A point with position vector x is invariant under A when Ax = x. For an invertible transformation, an invariant line through the origin maps onto itself, with its direction scaled by a nonzero factor. Distinguish it from a line of fixed points, where every point is unchanged. A singular transformation can instead collapse a line to the origin; inspect the image rather than calling every point fixed.
5. Polar coordinates
Polar coordinates use x = r cos theta and y = r sin theta, with r non-negative in this syllabus. Also r2 = x2 + y2 and tan theta = y/x where defined. Convert equations carefully and retain the angular interval.
Sketch simple polar curves over 0 to 2pi, -pi to pi or a stated subinterval. Test symmetry, intersections with the initial line, behaviour at the pole and least or greatest radius. A repeated pole value can correspond to different tangential directions, so inspect nearby angles.
The area swept from angle a to b is one half of the integral of r2 with respect to theta. For an area between polar curves, identify which radius is outer on each interval and split where the curves intersect.
6. Vectors
A plane can be written in Cartesian form ax + by + cz = d, normal form r dot n = p, or parametric form r = a + lambda b + mu c. The Cartesian coefficients a, b and c must not all be zero, the normal vector must be nonzero, and the two parametric direction vectors must be linearly independent. Convert between valid forms using a nonzero normal vector perpendicular to both direction vectors.
The vector product a cross b is perpendicular to a and b, has magnitude |a||b| sin theta and follows its orientation convention. Compute it from components and use it to find plane normals and areas.
Combine line and plane equations with scalar and vector products. Determine whether a line lies in, is parallel to or intersects a plane; find line-plane intersections; drop a perpendicular from a point to a plane; and find angles between a line and plane or two planes. Two distinct nonparallel planes intersect in a line whose direction is parallel to the nonzero cross product of their normals. Parallel or coincident planes do not determine such an intersection line.
For skew lines, find the common perpendicular direction using a cross product of their direction vectors. The shortest distance is the magnitude of the connecting displacement projected onto the corresponding unit direction. To find the common perpendicular line itself, locate compatible points on both skew lines and verify perpendicularity to each direction.
7. Proof by induction
An induction proof has a base case, an induction hypothesis and an inductive step. Verify the statement at its first valid integer. Assume it holds for n = k, then use that assumption to prove the n = k + 1 case. Conclude explicitly that the result holds for all integers in the stated domain.
Induction can prove sums, recurrences, matrix powers and divisibility. For a recurrence, substitute the assumed closed form into the recurrence and simplify to the next case. For divisibility, expose the required factor plus a term already divisible by the target integer.
When the formula is unknown, calculate several cases to form a conjecture, then prove it by induction. Trial supports discovery but does not replace proof.
Worked application: roots transformed by reciprocals
Let alpha, beta and gamma be the roots of x3 - 4x2 + 2x + 3 = 0, and construct the equation with roots 1/alpha, 1/beta and 1/gamma. Put the new variable y = 1/x, so x = 1/y. Substitution gives 1/y3 - 4/y2 + 2/y + 3 = 0. Multiplying by y3 gives 1 - 4y + 2y2 + 3y3 = 0, or 3y3 + 2y2 - 4y + 1 = 0. The constant term of the original is non-zero, so none of the original roots is zero and every reciprocal is defined. The transformed equation needs no numerical root solving.
Common misconceptions and corrections
Applying root-coefficient relations beyond degree 4. This Paper 1 section is restricted to degrees 2, 3 and 4.
Forgetting alternating signs in cubic or quartic relations. Read them from the expanded factorisation.
Using a reciprocal transformation when an old root is zero. Check the original constant term.
Sketching a rational graph before division. Expose the oblique or horizontal asymptote first.
Giving a range without a real-solution condition. Use a discriminant or equivalent argument.
Ignoring domain restrictions in related graph transformations. Determine where every expression is real and defined.
Assuming terms tending to zero proves convergence. Study the partial-sum limit.
Claiming a series telescopes without showing survivors. Write boundary terms.
Multiplying matrices in the written transformation order. The rightmost transformation acts first.
Inverting a singular matrix. Check the determinant.
Using determinant rather than its magnitude for area scale. Area scale is the absolute value.
Calling every invariant-line point fixed. The line may map onto itself while points move along it.
Allowing negative r despite the syllabus convention. Use r non-negative.
Using sector area without squaring r. Integrate one half of r2.
Using a dot product to create a plane normal from two directions. Use the cross product.
Calling non-intersecting non-parallel lines parallel. In three dimensions they may be skew.
Projecting onto a non-unit normal and forgetting its magnitude. Normalise or divide appropriately.
Assuming the induction statement inside the base case. Verify the base directly.
Proving only the k + 1 algebra without using the k case. The inductive link must be explicit.
Treating several successful trials as proof. Use them only to conjecture.
Assessment guidance
State the controlling restriction before each method: polynomial degree, rational-function degree, matrix order, polar interval or induction domain. In transformed-root problems, define the new variable and clear denominators visibly. Rational sketches need intercepts, asymptotes, turning points and branch placement, while range arguments need a real-root condition. Show index changes and surviving terms in sums. Matrix work must preserve multiplication order and determinant checks. Polar areas need intersection angles and outer-radius decisions. Vector solutions should define every line parameter, normal and projection. An induction proof earns its logic from the labelled base, hypothesis, step and final universal conclusion, not from algebra alone.
Retrieval practice
Evaluate symmetric root functions and construct equations for reciprocal, squared and shifted roots. Sketch a quadratic-over-linear and quadratic-over-quadratic function, including range. Derive a telescoping sum and test its infinite limit. Invert and compose 3 by 3 matrices, then find invariant points and lines. Convert and sketch polar curves and calculate an enclosed area. Solve line-plane, plane-plane and skew-line problems using both scalar and vector products. Finish by proving one sum, recurrence, matrix identity and divisibility statement by induction.
