What the board asks

We went through the ISC board papers, question by question, and matched each question to a topic. Here is what was asked, how often, and what was never asked.

Mathematics: 309 topics in the 2027 syllabus, across 11 units. The board has asked 255 of them at least once. 54 were never asked in the papers we checked.

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M1 Relations and Functions

All the topics in Mathematics: Relations and Functions

Asked topics first, the most asked at the top. Tap a topic to see what to write, the result, the years, and the real board questions.

31 topics shown

#	Topic	Type	Marks	Times asked	Last asked	Status
1	Find the inverse of an invertible algebraic function and verifyWrite y = f(x), solve for x in terms of y, swap to write f^-1(x) with its domain, and when asked verify (f o f^-1)(x) = x by substitution.	Sum	2	5 papers	2026	Asked
2	Principal value of expressions and nested inverse trigEvaluate the innermost inverse function on its principal branch, work outward using exact trig values, and give the final angle in its principal range.	Sum	2	5 papers	2026	Asked
3	Apply the double-angle formulae to evaluate or solveRewrite 2tan^-1 x (or 2sin^-1, 2cos^-1) using the right formula or let tan^-1 x = theta, evaluate the trig of the multiple angle, and solve or simplify to the final value.	Sum	2	4 papers	2026	Asked
4	Equations solved with the complementary identitiesReplace cot^-1 x by pi/2 - tan^-1 x (or cos^-1 x by pi/2 - sin^-1 x), solve for the single inverse function, take the trig function of both sides, and check the answer is in the domain.	Sum	2	4 papers	2025 improvement	Asked
5	Domain and range of a real-valued functionFind the natural domain by excluding values that give division by zero or roots of negatives, then find the range by bounding the expression or solving y = f(x) for x, and write both in interval or set notation.	Sum	2	3 papers	Specimen 2027	Asked
6	Test a listed relation for reflexive, symmetric, transitiveFor each property, check every required pair in the list and either confirm it or name one missing pair as the counter-example, then state the conclusion with a reason.	Give reason	1	3 papers	2026	Asked
7	Convert one inverse trig function into anotherLet the inverse function equal theta, draw the right triangle or use identities to express the same angle through another inverse function, then evaluate or simplify.	Sum	2	3 papers	2026	Asked
8	tan^-1 sum and difference formula: evaluate or proveCombine two tan^-1 terms with the sum or difference formula, state the condition on xy, simplify the fraction, and match the required value or relation.	Proof	3	3 papers	2026	Asked
9	Simplify or prove by trig substitution for xChoose the substitution (x = tan theta, cos theta, cos 2theta, or x^2 = cos 2theta) that turns the argument into one trig ratio, simplify with half- or double-angle identities, and convert back to x.	Proof	3	3 papers	2026	Asked
10	Write relations on a set: identity, empty, universal, inverseList the ordered pairs of the required relation (identity, empty, universal, smallest equivalence, or inverse R^-1) in set notation, and state its domain or range when asked.	Statement	1	3 papers	2025 improvement	Asked
11	sin^-1 and cos^-1 sum formulas: evaluate or prove a relationCombine the sin^-1 or cos^-1 terms with the sum formula (or rewrite one term and take cos or sin of both sides), square to remove roots where needed, and simplify to the required relation.	Proof	3	3 papers	2025	Asked
12	Show a function is one-one and ontoAssume f(x1) = f(x2) and deduce x1 = x2 for one-one; then take any y in the codomain, solve y = f(x) for x, check x lies in the domain and f(x) = y for onto; conclude f is a bijection.	Proof	2	3 papers	2024	Asked
13	Test a rule-defined relation on Z or linesTake a general element and test each property from the rule; give a specific counter-example (numbers or a picture of lines) where a property fails, then choose the option.	Give reason	1	2 papers	Specimen 2027	Asked
14	Complementary identities: sin^-1 x + cos^-1 x = pi/2 typeState the complementary identities with their conditions and use them to rewrite or evaluate an expression in one inverse function.	Statement	1	2 papers	Specimen 2027	Asked
15	Count functions, one-one functions and bijections between finite setsUse the counting rule for the number of functions, one-one functions or bijections from a set of m elements to a set of n elements and state when the count is zero.	Sum	1	2 papers	2026	Asked
16	Inverse trig functions: domain, range, principal value branchesWrite the domain and the principal value range of each of the six inverse trigonometric functions, and the principal value of a standard argument within that range.	Statement	1	2 papers	2026	Asked
17	Formulae for 2sin^-1 x, 2cos^-1 x, 2tan^-1 x, 3tan^-1 xState each formula with its range of validity, and prove one by putting x = sin theta, cos theta or tan theta when asked.	Statement	1	2 papers	2026	Asked
18	Function or not, and its type, from a graphApply the vertical line test to decide if a graph is a function of x, and the horizontal line test and the range to decide one-one or onto, then choose the option with a reason.	Give reason	1	2 papers	2025 improvement	Asked
19	Case study: linked parts on one inverse trig functionAnswer the linked parts in order: identify the graph, state domain and range, prove or apply the double-angle formula, and evaluate the given expressions, carrying earlier results forward.	Sum	5	2 papers	2025 improvement	Asked
20	Solve sin^-1 and cos^-1 equations by taking sine or cosineMove terms so one side is a single angle, take sin or cos of both sides, expand with compound-angle formulas, solve for x, and check each root in the original equation.	Sum	3	2 papers	2024	Asked
21	Solve equations using the tan^-1 sum formulaCombine the tan^-1 terms with the sum formula, take tan of both sides, solve the resulting algebraic equation, and reject roots that break the formula's condition or the principal range.	Sum	3	2 papers	2023	Asked
22	Composite functions fog and gofWrite (fog)(x) = f(g(x)) and (gof)(x) = g(f(x)), substitute carefully, simplify, and state any value of x excluded from the domain.	Sum	2	2 papers	2020	Asked
23	Decide whether a function is invertible (horizontal line test)State that f is invertible exactly when it is one-one and onto, test this from the graph or the formula, and if not, name a restriction of the domain that makes it invertible.	Give reason	1	1 paper	Specimen 2027	Asked
24	Case study: inverse on a restricted domain from a graphRead the domain and turning point from the graph, explain why f is not invertible on its whole domain, find the inverse on each restricted piece by choosing the correct sign of the root, verify g(g^-1(a)) = a, and solve equations between the two inverses.	Sum	5	1 paper	Specimen 2027	Asked
25	Sketch a function and its inverse by reflecting in y=xDraw y = f(x), the line y = x and the mirror image of the curve in that line, marking the swapped points (a,b) and (b,a); in assertion-reason items, judge both statements and the link between them.	Graph	1	1 paper	Specimen 2027	Asked
26	Use inverse trig domains to restrict xWrite the condition that each argument lies in the domain (for example -1 <= x^2 - 2x + 2 <= 1), find the only possible x or the interval for the expression, then finish the question.	Give reason	2	1 paper	Specimen 2027	Asked
27	Word problem: tan^-1 sum in a right triangleTranslate the story into sides, combine the two tan^-1 terms with the sum formula, simplify using Pythagoras (r^2 = p^2 + q^2), and give the angle.	Sum	2	1 paper	Specimen 2027	Asked
28	Validity of sin^-1(sin x), tan^-1(cot x) type identitiesCheck whether the angle lies in the principal range; if not, rewrite it as an equivalent angle inside the range, then judge each statement true or false with a reason.	Give reason	1	1 paper	2026	Asked
29	Graphs of inverse trig functions: sketch or identifySketch y = sin^-1 x, cos^-1 x or tan^-1 x on the principal branch with end points and asymptotes marked, or pick the correct graph and explain from domain and range.	Graph	1	1 paper	2025	Asked
30	Prove a relation is an equivalence relationProve reflexive, symmetric and transitive in turn for general elements, each with its own heading and a one-line reason, and conclude that R is an equivalence relation.	Proof	3	never		Never asked
31	Show a function is not one-one or not ontoGive two different inputs with the same output to disprove one-one, or one codomain value with no pre-image to disprove onto, and state the conclusion.	Give reason	2	never		Never asked

This list was made from the 2027 syllabus and checked with 10 board papers, including the 2027 specimen paper. It is not yet checked with the textbook.

The 10 topics Mathematics papers ask most

Counted on the board papers from 2018 to the 2027 specimen paper, which every unit has been checked against.

1.Formulate an LPP from a word problemM10 Linear Programming · asked 7 times since 2018
2.Scalar and vector projection of one vector on anotherM8 Vector Algebra · asked 6 times since 2018
3.Mean (expected value) of a random variableM11 Probability · asked 6 times since 2018
4.Find the inverse of an invertible algebraic function and verifyM1 Relations and Functions · asked 5 times since 2018
5.Principal value of expressions and nested inverse trigM1 Relations and Functions · asked 5 times since 2018
6.Evaluate or show zero without expanding, using propertiesM2 Algebra: Matrices and Determinants · asked 5 times since 2018
7.Prove a determinant equals a factorised expression using propertiesM2 Algebra: Matrices and Determinants · asked 5 times since 2018
8.Definite integral of a modulus functionM5 Calculus: Integrals · asked 5 times since 2018
9.Case study or linked parts on lines and planesM9 Three-Dimensional Geometry · asked 5 times since 2018
10.Equations solved with the complementary identitiesM1 Relations and Functions · asked 4 times since 2018

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