NSSCAS Mathematics (8227) study plan

Simplify and rationalise surd expressions, including denominators of the form a+√b, and convert confidently between surd, index and exponential forms without a calculator.

Apply the laws of indices to expressions with negative and fractional exponents, and solve equations such as 2^(3x−1)=5 by taking logarithms of both sides.

Use the laws of logarithms to combine and split expressions, solve equations such as log_a(x)+log_a(x−3)=1, and state the domain restriction that validates each solution.

Sketch y=a^x and y=log_a(x) for a>1 and for 0<a<1, labelling the asymptote, the axis intercept and two key points on each curve.

Complete the square on quadratic expressions to read off the vertex and range, and use the discriminant to classify the number of real roots of a given equation.

Divide polynomials by linear and quadratic divisors, and apply the factor and remainder theorems to determine unknown coefficients in a given polynomial.

Solve linear and quadratic inequalities, including cases with the unknown in a denominator, and mark solution sets on a number line with correct open or closed endpoints.

Expand (a+b)^n using the binomial theorem and extract a named term or a specific coefficient without writing out the whole expansion.

Determine domain and range of given functions, build composite functions, and find inverse functions algebraically while stating the domain restriction that makes each inverse valid.

Sketch curves after translations, stretches, reflections and modulus operations, and describe a sequence of transformations that maps one given curve onto another.

Differentiate polynomials, sine and cosine, exponential and logarithmic functions, and apply the chain rule to composite expressions such as (3x+1)^5 and sin(2x).

Find equations of tangents and normals at a stated point, including points whose coordinates must first be obtained from the curve equation.

Locate stationary points, classify each with the second derivative or a sign table, and state the intervals on which a function increases or decreases.

Set up and solve connected rates of change and simple optimisation problems, defining each variable and quoting units at every stage of the solution.

Integrate powers, exponentials, 1/x and trigonometric functions, and evaluate definite integrals with the limits substituted in the correct order.

Use integration to find areas between a curve and the x-axis and between two curves, splitting the integral wherever the curve crosses the axis.

Set up and evaluate a volume of revolution about the x-axis for a simple curve, writing the integral, limits and expansion explicitly before evaluating.

Re-work every differentiation question missed in Week 3 and write one sentence per error naming the specific slip and the correct technique.

Convert fluently between degrees and radians and apply arc length, sector area and segment area formulae inside multi-step problems involving compound shapes.

Prove and apply the identities sin²θ+cos²θ=1 and tanθ=sinθ/cosθ, and use the double-angle forms for sin2θ and cos2θ to simplify given expressions.

Solve trigonometric equations over a stated interval, listing every solution in the required range and rejecting any produced by squaring or by division by zero.

Sketch y=sin x, y=cos x, y=tan x and transformed variants, stating period, amplitude, asymptotes and range for each.

Find the centre and radius of a circle from its expanded equation, and solve line–circle intersection problems using the discriminant to decide the number of intersections.

Represent data in histograms, cumulative frequency curves and box plots, then read off median, quartiles and interquartile range and identify outliers by a stated rule.

Compute mean and standard deviation for raw and grouped data and explain in writing how adding or multiplying every value changes each summary statistic.

Count arrangements and selections with permutations and combinations, including repeated items and restrictions, and apply the addition and multiplication rules to conditional probability problems.

Standardise values to find normal distribution probabilities, work backwards from a probability to an unknown value, and use the binomial distribution for counted outcomes with fixed trials.

Use nth term and sum formulae for arithmetic and geometric progressions and find the sum to infinity when |r|<1, justifying whether convergence applies in each case.

Add, subtract and scale vectors in two and three dimensions, compute magnitudes and unit vectors, and interpret position vectors on a simple geometric diagram.

Use the scalar product to find the angle between two vectors, test perpendicularity and solve for an unknown component given a stated angle.

Apply vectors to prove three points are collinear, find the position vector of a point dividing a segment in a given ratio, and solve a simple geometric configuration problem.

Re-attempt every entry in the error log from Weeks 1 to 6 and tag each remaining weakness to its domain so revision time can be allocated deliberately.

Sit two full-length papers under strict examination conditions, with no notes, a single sitting per paper and the timing enforced by a clock.

Re-derive from memory the chain rule, the standard integrals and the area and volume formulae, then check each against the formula sheet and repair gaps immediately.

Re-solve every calculus question that lost more than two marks in the mocks, writing the corrected solution in full rather than annotating the original attempt.

Compress all nine domains onto a single revision card of formulae, conditions and standard traps, then test recall by reproducing it from blank paper twice.