UPMSP High School Mathematics study plan
UPMSP High School Mathematics · Give each week a clear focus.
8 weeks · About 6 hours per week
Triangles and Similarity & Trigonometry and Its Applications
State and apply the Basic Proportionality Theorem (Thales) and its converse to set up correct ratios in triangle problems
Prove the three similarity criteria (AA, SSS, SAS) on paper from memory and label corresponding parts correctly
Derive the exact values of sin, cos and tan for 0°, 30°, 45°, 60° and 90° without a table
Solve 15 two-triangle similarity problems that require finding an unknown side or proving a ratio
Your checkpoint
A handwritten one-page similarity proof sheet plus 15 solved triangle problems with reasoning shown
Trigonometry and Its Applications & Mensuration (Areas, Surface Areas, Volumes)
Convert between trigonometric ratios and use identities such as sin²θ + cos²θ = 1 to simplify expressions
Solve 12 height-and-distance word problems by drawing the correct diagram and choosing the right ratio
Compute surface area and volume of spheres, hemispheres and spherical shells using π = 22/7
Work out 10 mensuration problems combining a cone, cylinder or hemisphere into one solid
Your checkpoint
A log of 22 solved problems (12 trigonometry applications, 10 combined-solids mensuration) with diagrams and units stated
Mensuration (Areas, Surface Areas, Volumes) & Quadratic Equations
Find surface area and volume of frustums of cones and solve bucket and glass-shaped problems
Solve quadratic equations by factorisation, completing the square and the quadratic formula
Discriminate between real, equal and no real roots using the discriminant and justify each case
Set up and solve 8 word problems that reduce to quadratic equations (age, speed, area, number problems)
Your checkpoint
A completed 20-question mixed worksheet on frustum mensuration and quadratic equations, scored and corrected
Quadratic Equations & Linear Equations in Two Variables
Solve pairs of linear equations in two variables by substitution, elimination and cross-multiplication
Identify consistent, inconsistent and dependent systems using the ratio condition on coefficients
Translate 8 real-life situations into a pair of linear equations and solve them
Solve 10 quadratic word problems under timed conditions (under 8 minutes each)
Your checkpoint
A timed 90-minute mock test on Domains 3 and 4, with every error reworked and annotated
Arithmetic Progressions & Number Systems and Real Numbers
Use the nth term formula aₙ = a + (n−1)d and the sum formula Sₙ = n/2[2a + (n−1)d] fluently
Solve problems finding the number of terms, a missing term or a common difference from given conditions
Prove that √2, √3 and √5 are irrational by contradiction
Apply Euclid's division lemma and the Fundamental Theorem of Arithmetic to find HCF and LCM
Your checkpoint
A formula card for AP and real numbers plus 20 solved problems covering nth term, sum, HCF/LCM and irrationality proofs
Polynomials and Rational Expressions & Coordinate Geometry
Divide polynomials and verify the division algorithm for polynomials
Find the zeros of a quadratic polynomial and relate them to the sum and product of zeros
Use the distance formula and section formula to solve 10 coordinate geometry problems
Find the area of a triangle from its vertices and check collinearity using the area condition
Your checkpoint
A 15-problem coordinate geometry set and a 10-problem polynomial set, fully solved with steps
Circles and Tangents & Constructions
State and apply the tangent–radius perpendicularity theorem and the equal-tangent theorem in numerical problems
Prove that the lengths of two tangents from an external point are equal
Construct a triangle similar to a given triangle with a given scale factor and justify the construction
Construct tangents to a circle from an external point and from a point on the circle
Your checkpoint
A construction portfolio of 6 neat, labelled constructions with justification steps written beside each
Statistics and Probability & Number Systems and Real Numbers & Arithmetic Progressions
Compute mean, median and mode for grouped data using the direct, assumed-mean and step-deviation methods
Draw an ogive and read the median from it
Solve probability problems using the classical definition and the complement rule
Complete 3 full-length timed papers and review every wrong answer
Your checkpoint
Three completed full-length papers with a written error log listing each mistake and the rule to avoid it
Milestones
Week 2: You can draw a height-and-distance diagram from a written description and solve it without hints, and you can compute combined-solid volumes with correct units in under 5 minutes per problem.
Week 4: You score at least 70% on the timed Domains 3 and 4 test and can classify any linear system as consistent or inconsistent from its coefficients alone.
Week 6: You solve coordinate geometry and polynomial questions correctly with all intermediate steps shown, and you can state every formula used from memory.
Week 8: You complete a full-length paper within the time limit with at least 75% accuracy, and your error log shows no repeated mistake across the last two papers.