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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M2 Algebra: Matrices and Determinants

All the topics in Mathematics: Algebra: Matrices and Determinants

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.

33 topics shown

#	Topic	Type	Marks	Times asked	Last asked	Status
1	Prove a determinant equals a factorised expression using propertiesUse operations such as C1 -> C1 + C2 + C3 or R1 -> R1 - R2 to create common factors, take them out, create zeros, and expand only at the end to reach the given product; for two-determinant identities, multiply rows by variables and divide out.	Proof	3	6 papers	Specimen 2027	Asked
2	Evaluate or show zero without expanding, using propertiesApply named row or column operations (write each, e.g. C1 -> C1 + C2), take out common factors, and reach two identical or proportional rows or a simple form, giving the value.	Sum	2	5 papers	2026	Asked
3	Symmetric or skew-symmetric matrix: find unknown entriesWrite A' (or the condition a_ij = a_ji or a_ij = -a_ji), equate the matching entries, solve for the unknown, and check the diagonal of a skew-symmetric matrix is zero.	Sum	1	4 papers	2025	Asked
4	Word problem: put data in a matrix and multiplyArrange the data from the story in a matrix with labelled rows and columns, set up a product of compatible orders (quantity times price etc.), multiply, and state each total in words with units.	Sum	3	3 papers	Specimen 2027	Asked
5	Word problem or case study: form 3 equations, matrix methodName the unknowns with units, turn each condition of the story into a linear equation, write AX = B, solve by X = A^-1 B, and answer the question asked in words (e.g. the height of the cliff, number of items).	Sum	5	3 papers	Specimen 2027	Asked
6	Standard properties of determinants (\|kA\|, \|A'\|, \|AB\|, row operations)Recall and apply the standard results: value unchanged by transpose and by R_i -> R_i + kR_j, sign changes on swapping rows, scalar factor out of a row, splitting a row that is a sum, \|kA\| = k^n \|A\|, \|AB\| = \|A\|\|B\|.	Statement	1	3 papers	2026	Asked
7	Non-commutativity: AB vs BA, (A+B)(A-B), commuting matricesExpand the matrix product keeping the order of factors, show by an example or by algebra why AB != BA in general, and decide when an identity from numbers (like (a+b)(a-b) = a^2 - b^2) does or does not carry over.	Give reason	1	2 papers	Specimen 2027	Asked
8	Area of a triangle by determinant; find unknown vertexWrite area = (1/2)\|det [[x1,y1,1],[x2,y2,1],[x3,y3,1]]\|, expand, set equal to +- the given area (both signs), and solve for the unknown.	Sum	2	2 papers	Specimen 2027	Asked
9	Inverse of a 3x3 matrix by the adjoint methodFind \|A\| and check it is not zero, write the matrix of cofactors with signs, transpose it to get adj A, and write A^-1 = (1/\|A\|) adj A; check one entry of A A^-1.	Sum	3	2 papers	Specimen 2027	Asked
10	Types and order of matrices; counting possible matricesName the type of a given matrix (row, column, square, zero, diagonal, scalar, identity, triangular), state its order m x n, and count matrices of a given order when each entry has a fixed number of choices.	Statement	1	2 papers	2026	Asked
11	Consistency and number of solutions: conditions (MCQ, assertion-reason)State when AX = B has a unique solution (\|A\| != 0), no solution or infinitely many (\|A\| = 0 with (adj A)B != O or = O), and use this to choose and justify an option.	Give reason	1	2 papers	2026	Asked
12	Properties of symmetric and skew-symmetric matrices (MCQ, assertion-reason)Use transpose rules to decide whether combinations such as AB - BA, AB + BA, A'A or a 3x3 skew-symmetric determinant are symmetric, skew-symmetric or zero, and choose with a one-line justification.	Give reason	1	2 papers	2025 improvement	Asked
13	Matrix method for equations in 1/x, 1/y, 1/zSubstitute u = 1/x, v = 1/y, w = 1/z, solve the linear system in u, v, w by X = A^-1 B, then convert back to x, y, z.	Sum	5	2 papers	2023	Asked
14	Solve 3 linear equations in 3 unknowns by matrix methodWrite the equations as AX = B, find \|A\| != 0, find adj A and A^-1, compute X = A^-1 B, and state x, y, z; check in one equation.	Sum	5	2 papers	2020	Asked
15	Expand a 3x3 determinant; solve a determinant equation for xExpand along the row or column with most zeros using cofactors, simplify to a polynomial, equate to the given value and solve for x.	Sum	1	1 paper	2026	Asked
16	Powers of a square matrix and matrix polynomialsCompute A^2 (and A^3 if needed), substitute into an expression such as A^2 - 5A + 7I, simplify to the required matrix, or spot a pattern (nilpotent, idempotent) for high powers.	Sum	2	1 paper	2025	Asked
17	Singular or non-singular: find value making matrix non-invertibleFind \|A\| in terms of the unknown, set \|A\| = 0 for singular (or != 0 for invertible), solve, and state which values make A singular.	Sum	1	1 paper	2025 improvement	Asked
18	Properties of adjoint and inverse (MCQ, two-statement)Recall and apply the standard results on adj A and A^-1 to decide true/false statements or compute values such as \|adj A\| or \|A^-1\| from \|A\|.	Statement	1	1 paper	2025	Asked
19	Test a system for consistency; find parameter for consistencyWrite A and B, find \|A\|; if zero compute (adj A)B to decide; for three equations in two unknowns set the 3x3 determinant of coefficients and constants to zero and solve for the parameter.	Sum	2	1 paper	2025 improvement	Asked
20	Multiply two matrices; check compatibility; find unknownCheck that columns of the first equal rows of the second, write the order of the product, compute each entry as row times column, and equate entries to find any unknown.	Sum	1	1 paper	2020	Asked
21	Show A+A' symmetric and A-A' skew; split AFind A', form A + A' and A - A', show their transposes equal themselves or their negatives, and if asked write A = (1/2)(A + A') + (1/2)(A - A') with each part checked.	Proof	2	1 paper	2019	Asked
22	Solve a determinant equation using propertiesAdd all rows or columns to bring out a common factor in x, take it out, simplify the rest by operations, set the product to zero and give all roots allowed by the conditions.	Sum	3	1 paper	2018	Asked
23	Given AB = kI, find A^-1 and solve the systemMultiply the given matrices to get kI, deduce A^-1 = (1/k)B (watch whether the system's matrix is A or A'), write X = A^-1 B, and solve.	Sum	5	1 paper	2017	Asked
24	Equality of matrices: find unknown entriesEquate corresponding entries of two equal matrices of the same order, solve the resulting equations, and check every equation is satisfied by the values found.	Sum	1	never		Never asked
25	Addition, subtraction and scalar multiple; solve matrix equationsCheck the orders match, add or subtract entry by entry, multiply every entry by the scalar, and solve equations like 2X + A = B for the unknown matrix X.	Sum	2	never		Never asked
26	Non-zero matrices whose product is the zero matrixGive or verify an example of two non-zero 2x2 (or 3x3) matrices with AB = O, multiply them out, and state why this means AB = O does not force A = O or B = O.	Sum	1	never		Never asked
27	Proof that the inverse of a matrix is uniqueAssume B and C are both inverses of A, write B = BI = B(AC) = (BA)C = IC = C, naming the property used at each step.	Proof	2	never		Never asked
28	Minors and cofactors of elements of a determinantDelete the row and column of the element to get its minor, attach the sign (-1)^(i+j) to get the cofactor, and list them in the required order.	Sum	1	never		Never asked
29	Collinearity of three points; line through two pointsWrite the determinant of the three points with a column of 1s, show it is zero (or set it to zero to find an unknown or the equation of the line through two points), and conclude.	Sum	2	never		Never asked
30	Adjoint and inverse of a 2x2 matrixCheck \|A\| != 0, write adj A by swapping the diagonal entries and changing the sign of the others, and write A^-1 = (1/\|A\|) adj A.	Sum	1	never		Never asked
31	Verify A(adj A) = \|A\|I or (AB)^-1 = B^-1A^-1Compute adj A (or A^-1, B^-1, AB) for the given matrices, multiply out both sides, and show they are equal entry by entry.	Proof	3	never		Never asked
32	Inverse from a matrix equation like A^2-kA+I=OShow the given matrix satisfies the polynomial equation, multiply through by A^-1, and rearrange to express A^-1 in terms of A and I, then compute it.	Sum	3	never		Never asked
33	Solve 2 linear equations in 2 unknowns by matrix methodWrite AX = B, find \|A\| and A^-1 by the 2x2 rule, compute X = A^-1 B, and state x and y.	Sum	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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