Ehime Entrance Mathematics study plan

Compute limits of rational, radical and trigonometric-type expressions by factoring, rationalising and applying sin x / x, reaching at least 18/20 on an untimed set.

Differentiate polynomials, rational functions, radicals and composite functions using the product, quotient and chain rules without consulting notes.

Find the equation of the tangent and the normal at a given point, including points on implicitly or parametrically defined curves.

State the domain and differentiability conditions for every function you differentiate so no mark is lost to a missing restriction.

Locate maxima, minima and inflection points from the sign charts of f′ and f″, then sketch the full curve with asymptotes, intercepts and symmetry labelled.

Set up and solve optimisation word problems (maximum area, minimum distance, minimum surface area) and justify the answer by checking endpoints and the sign of f′.

Evaluate definite integrals with the fundamental theorem, substitution and integration by parts, including limits that require splitting the interval.

Compute the area between two curves by first finding the intersection points and then assigning the correct upper and lower function on each sub-interval.

Calculate areas of regions bounded by curves, including regions that must be split at intersection points found algebraically.

Calculate volumes of solids of revolution about both the x-axis and the y-axis, choosing correctly between the disk/washer and shell methods.

Solve reverse problems in which an area or volume is given and an unknown constant in the curve equation must be determined.

Set up, without necessarily evaluating, the integral for a region described parametrically or implicitly, such as an ellipse or a rotated curve.

Derive closed forms for arithmetic and geometric sequences and use them to evaluate sums written in sigma notation.

Solve first-order recurrence relations of the form a(n+1) = p·a(n) + q by shifting to a geometric sequence, and handle variants where the added term depends on n.

Write complete induction proofs for sum formulas, divisibility statements and inequalities, stating the base case, the hypothesis and the algebraic step explicitly.

Classify series as convergent or divergent and evaluate infinite geometric series while stating the |r| < 1 condition each time.

Compute dot products in two and three dimensions and use them to find angles, perpendicularity conditions and scalar projections.

Write equations of lines and planes in vector, parametric and normal forms, and use them to find intersection points and distances.

Solve configuration problems involving position vectors: points dividing a segment in a given ratio, feet of perpendiculars, and areas or volumes obtained from vector products.

Solve trigonometric equations and inequalities on a stated interval by substitution and unit-circle reasoning, listing every solution in the required range.

Sketch, differentiate and integrate exponential and logarithmic functions while respecting the domain restriction of the logarithm.

Solve related-rates problems by writing the geometric constraint first and then differentiating both sides with respect to time, stating the chain-rule step.

Count outcomes with permutations, combinations and complementary counting, deciding explicitly whether order matters before choosing a formula.

Compute conditional probabilities, test for independence, and calculate the expected value of a random variable from its distribution table.

Handle 'exactly k' and 'at least one' probability questions using binomial reasoning and the complement rule.

Convert complex numbers between rectangular, polar and exponential forms, and multiply and divide them using modulus and argument.

Apply De Moivre's theorem to compute powers and to list all n-th roots of a complex number, placing each root correctly in the complex plane.

Translate equations and inequalities into loci (circle, perpendicular bisector, half-plane, ray) and sketch the resulting region.

Factor polynomials with the factor theorem, locate rational roots, and solve higher-degree equations whose remaining roots are complex.

Run the Euclidean algorithm to find a gcd and to solve a linear Diophantine equation, and convert integers between base 10 and another base in both directions.

Apply congruence and similarity to prove length and angle relations in triangles and circles, including cyclic quadrilaterals and the power of a point.

Compute lengths, areas and volumes for solids inscribed in or circumscribed about other solids by working through a cross-section.

Use coordinate geometry to derive circle equations, impose tangency conditions, and compute the distance from a point to a line.

Complete two full mock papers under timed conditions and classify every lost mark by domain to build a final revision list.