MATSEC IM Pure Mathematics study plan

Factorise cubic and quartic polynomials using the factor theorem combined with long division or synthetic division.

Apply the remainder theorem to determine unknown coefficients in a polynomial.

Decompose rational expressions into partial fractions, including cases with repeated linear factors.

Solve polynomial, rational and modulus inequalities, and express each solution set on a number line.

Simplify expressions involving indices and surds, including rationalising denominators with conjugate pairs.

Differentiate polynomials, products, quotients and composite functions using the product, quotient and chain rules.

Differentiate trigonometric, exponential and logarithmic functions from standard results.

Calculate gradients and construct equations of tangents and normals at specified points on a curve.

Locate stationary points and classify them using the first- or second-derivative test.

Solve optimisation and connected-rates-of-change problems by forming and differentiating a suitable model.

Convert fluently between degrees and radians, and compute arc lengths and sector areas.

Prove trigonometric identities using Pythagorean, compound-angle and double-angle formulae.

Solve trigonometric equations over a stated interval and list every solution within it.

Sketch graphs of the form y = a sin(bx + c) + d and state amplitude, period and phase shift.

Apply the sine and cosine rules to determine unknown sides, angles and areas in triangles.

Integrate standard functions, including polynomials, exponential, reciprocal, sine and cosine forms.

Evaluate definite integrals and interpret them as signed areas, handling regions below the x-axis correctly.

Apply integration by substitution to composite integrands, changing limits appropriately for definite integrals.

Apply integration by parts to products, choosing u and dv deliberately.

Approximate a definite integral using the trapezium rule and compare the estimate with the exact value.

Find equations of straight lines, perpendicular bisectors and circles from given geometric data.

Solve intersection problems between lines and circles, and use distance, midpoint and gradient formulae.

Sketch graphs of standard functions and apply transformations of the form y = af(bx + c) + d.

Determine domain and range, and construct composite and inverse functions where they exist.

Solve equations graphically by identifying intersections of a line with a curve.

Apply the laws of logarithms to solve exponential and logarithmic equations exactly.

Sketch exponential and logarithmic graphs, stating intercepts and asymptotes.

Use the nth-term and sum formulae for arithmetic and geometric sequences and series.

Determine the sum to infinity and state the condition for a geometric series to converge.

Model growth, decay and savings problems using exponential and series formulae.

Perform vector addition, scalar multiplication and the scalar product, and calculate the angle between two vectors.

Use position vectors to prove geometric results such as collinearity, parallelism and perpendicularity.

Express complex numbers in Cartesian, modulus-argument and polar forms, and convert between them.

Carry out arithmetic on complex numbers and solve quadratic equations with complex roots, using conjugate pairs.

Plot complex numbers on an Argand diagram and interpret multiplication and division geometrically.

Sit two full timed papers under exam conditions without notes or formula aids.

Record every lost mark in a topic-error table and rework each incorrect question to a fully correct solution.

Reproduce the complete formula sheet from memory and check it against the syllabus.

Re-drill the two weakest domains identified by the error table until accuracy exceeds 85%.