Fukushima Entrance Mathematics study plan

Expand and simplify expressions containing brackets and nested brackets without errors, completing 30 items in 40 minutes.

Factorise quadratic expressions of the form x² + bx + c and ax² + bx + c, including difference of two squares, at 90% accuracy.

Rationalise denominators containing surds and simplify expressions such as 6/(√5−√2) correctly.

State and apply the quadratic formula and the relationship between roots and coefficients in at least 10 practice items.

Solve linear equations and simultaneous equations in two and three unknowns, checking each answer by substitution.

Solve quadratic equations by factorisation, completing the square and the formula, choosing the fastest method for each.

Solve linear inequalities and represent the solution set on a number line, including compound inequalities.

Solve quadratic inequalities by sketching the parabola and reading off the sign of the expression.

Sketch quadratic functions in general form by completing the square to locate the vertex, axis of symmetry and intercepts.

Determine the maximum or minimum of a quadratic function on a restricted interval, distinguishing interior and endpoint cases.

Solve problems relating the number of intersections of a line and a parabola to the discriminant of the resulting quadratic.

Convert between standard and vertex forms and read off transformations of the graph.

Apply the Pythagorean theorem, similarity and the angle bisector theorem to compute unknown lengths in triangles.

Use the sine and cosine rules and the area formula (1/2)ab sin C to solve non-right triangles.

Prove congruence and similarity of triangles using the standard conditions, writing each proof in the accepted format.

Compute lengths, angles and areas involving circles, including the inscribed angle theorem and tangents.

Calculate means, medians, modes and ranges from raw and grouped data, and interpret what each measure reveals.

Compute variance and standard deviation and explain how outliers affect each measure of spread.

Read and construct histograms, box plots and scatter plots, and describe correlation and trend.

Count outcomes using permutations and combinations and apply them to selection problems.

Compute probabilities of independent and dependent events and of complementary events, showing the sample space explicitly.

Distinguish conditional probability from joint probability and solve at least 8 conditional problems.

Identify the hypothesis and conclusion in a statement and state its converse, contrapositive and negation.

Construct direct and contradiction proofs, and explain why a single counterexample disproves a universal statement.

Translate word problems into equations or functions, defining variables explicitly before solving.

Solve applied problems combining geometry and algebra, such as optimising the area of an enclosed region under a fixed perimeter.

Break multi-step problems into sub-goals and record the reasoning chain for each step.

Manage time by allocating a target number of minutes per problem type and reviewing errors afterwards.

Complete two full-length timed practice papers under exam conditions and record the score and time per section.

Revisit every item from the error analysis and re-solve it correctly without notes.

Consolidate the formula sheet into a single page you can reproduce from memory.

Set a per-section time budget based on your weakest domains and rehearse it once.