UPMSP Intermediate Mathematics study plan

Build a one-page antiderivative reference sheet covering standard algebraic, trigonometric, exponential and logarithmic forms, then reproduce it from memory by day 4.

Evaluate 40 indefinite integrals by substitution, checking each answer by differentiating it back.

Classify and solve 25 definite integrals using the fundamental theorem of calculus, including problems requiring a change of limits after substitution.

Refresh the algebraic manipulation skills integrals depend on: partial fractions, completing the square, and trigonometric identities, by working 20 mixed simplification drills.

Compute areas under curves and between two curves for at least 15 standard region types, sketching each region before integrating.

Set up and evaluate integrals for area of circles, parabolas and ellipses to derive known area formulas rather than memorising them.

Apply the first derivative test and second derivative test to classify critical points in 25 functions.

Solve 15 applied optimisation word problems (maxima and minima) by translating the constraint and objective into equations before differentiating.

Compute equations of tangents and normals at given points for 20 curves.

Perform matrix addition, scalar multiplication and multiplication on 30 problems, verifying non-commutativity with counterexamples.

Compute determinants of order 2 and 3 by cofactor expansion, and of order 3 by the shortcut rule, across 25 problems.

Find inverses of 2x2 and 3x3 matrices using the adjoint method and check each by multiplying back to the identity.

Solve systems of linear equations by matrix inversion and by Cramer's rule for 12 systems, including one inconsistent and one with infinitely many solutions.

Determine whether given relations are reflexive, symmetric and transitive, and identify equivalence classes in 15 examples.

Test continuity of piecewise functions at junction points for 20 examples using left and right limits.

Differentiate 40 functions using chain rule, product rule, quotient rule, and implicit differentiation.

Differentiate logarithmic, exponential and inverse trigonometric composites in 20 problems.

State and apply Rolle's theorem and the mean value theorem to verify hypotheses and find the guaranteed point in 10 functions.

Compute dot and cross products for 25 vector pairs and use them to find angles, projections and areas of triangles.

Find direction cosines, direction ratios and equations of lines in both vector and Cartesian form for 20 line problems.

Determine shortest distances between skew lines and between a point and a plane in 12 problems.

Derive and apply the standard forms of a straight line (slope-intercept, two-point, normal) to 25 problems.

Identify focus, directrix, vertex and eccentricity of parabolas, ellipses and hyperbolas from their equations in 20 cases.

Solve 20 conic-section problems that require finding the equation from given geometric conditions.

Compute distances, midpoints, slopes and angle bisectors in coordinate geometry problems involving triangles and quadrilaterals.

Memorise and apply the principal-value ranges of all six inverse trigonometric functions to evaluate 30 expressions.

Simplify composite inverse trigonometric expressions using identities such as the sum and double-angle forms in 20 problems.

Solve 10 inverse trigonometric equations, checking that each solution lies in the principal range.

Classify differential equations by order, degree and linearity across 20 examples.

Solve variable-separable and homogeneous differential equations in 20 problems, applying initial conditions to find particular solutions.

Solve first-order linear equations using the integrating factor in 10 problems.

Apply the addition and multiplication theorems of probability, including independent events, to 20 problems.

Compute conditional probability and apply Bayes' theorem to 10 realistic scenario problems.

Formulate and graphically solve 8 linear programming problems, correctly identifying feasible regions, corner points and optimal values.

Compose and invert functions in 15 problems, including finding the inverse of a given function and verifying composition.

Revise the entire syllabus by reworking every item in your error logs from weeks 1 to 7.

Complete two full-length timed mock papers of 3 hours each and mark them strictly.