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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M8 Vector Algebra

All the topics in Mathematics: Vector Algebra

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.

26 topics shown

#	Topic	Type	Marks	Times asked	Last asked	Status
1	Scalar and vector projection of one vector on anotherCompute a.b and the magnitude of the vector projected ON, write scalar projection = a.b/\|b\|, and, if asked, vector projection = (a.b/\|b\|^2) b; in a find-lambda question set the projection equal to the given value and solve.	Sum	2	6 papers	Specimen 2027	Asked
2	Magnitudes from expanded dot products, e.g. \|a + b\|Square the given magnitude as a dot product, expand (a + b).(a + b) or (2x - 3a).(2x + 3a), use a.b = 0 or \|a^\| = 1 as given, and solve for the required magnitude.	Sum	2	4 papers	2026	Asked
3	Unit vector, or vector of given magnitude, along a directionFind the magnitude of the given vector (or of AB), divide to get the unit vector, and multiply by the required magnitude; write the final vector in i, j, k form.	Sum	1	3 papers	2024	Asked
4	Compute a x b by the determinant and use itWrite the 3x3 determinant with i, j, k in the first row, expand along it with the correct signs, and use the result (for example in a further dot product).	Sum	1	2 papers	Specimen 2027	Asked
5	Unit vector perpendicular to two vectors or to a planeFind the two vectors (for a triangle ABC use AB and AC), compute their cross product, find its magnitude, and divide; give both answers +/- (a x b)/\|a x b\|.	Sum	2	2 papers	Specimen 2027	Asked
6	Area of a parallelogram from adjacent sides or diagonalsFor adjacent sides a, b write area = \|a x b\|; for diagonals d1, d2 write area = (1/2)\|d1 x d2\|; compute the cross product and its magnitude and give square units.	Sum	1	2 papers	2026	Asked
7	Find the unknown that makes two vectors perpendicularSet a.b = 0, write it in components, and solve the linear equation for the unknown.	Sum	1	2 papers	2025 improvement	Asked
8	Area of a triangle by the cross productForm two side vectors from one common vertex (AB and AC), compute AB x AC, find its magnitude, and write area = (1/2)\|AB x AC\| in square units.	Sum	2	2 papers	2025	Asked
9	Structured case question: vectors, projection and area from three pointsFrom the given position vectors find the side vectors, then answer each linked part in turn (perpendicular check, projection, unit normal, area), carrying earlier results forward and labelling each part.	Sum	2	2 papers	2025 improvement	Asked
10	Angle or magnitude from \|a x b\| and a.bWrite \|a x b\| = \|a\|\|b\| sin(theta) and a.b = \|a\|\|b\| cos(theta), substitute the given conditions (unit vector, \|a x b\| = a.b), and solve for theta or the missing magnitude.	Sum	2	2 papers	2024	Asked
11	Proof from cross-product properties (zero product, perpendicularity)Rearrange to a single cross product (such as a x (b - c) = 0) or compute the needed products, use the zero-cross-product or perpendicularity condition, and state each case of the conclusion.	Proof	2	2 papers	2024	Asked
12	Word problem: position vectors on a map or roadFix the origin and the i, j directions from the story, write each place as a position vector, use the section formula or AB = OB - OA, and use a dot product of zero to justify a perpendicular or closest point.	Sum	2	1 paper	Specimen 2027	Asked
13	Two-statement or assertion-reason item on vector basicsJudge each statement (or the assertion and the reason) separately as true or false with a quick check or counter-example, then decide whether the reason explains the assertion and pick the option.	Give reason	1	1 paper	2026	Asked
14	Show three points are collinear by vectors (incl. word problem)Find AB and BC (or AC) from the position vectors, show one is a scalar multiple of the other (or AB x BC = 0), and state that with the common point B the three points lie on one line.	Sum	2	1 paper	2026	Asked
15	Vector joining two points from coordinates or position vectorsWrite AB = (position vector of B) - (position vector of A) component by component and give its magnitude if asked.	Sum	1	1 paper	2025	Asked
16	Position vector of the point dividing a joinWrite the internal (or external) section formula with the ratio m:n placed correctly, substitute the position vectors, and simplify to i, j, k form; for a midpoint use (a + b)/2.	Sum	1	1 paper	2025 improvement	Asked
17	Dot product: definition, standard results and propertiesState a.b = \|a\|\|b\| cos(theta), the results for i, j, k, a.a = \|a\|^2, commutative and distributive laws, and a.b = 0 for perpendicular vectors; use them to judge an assertion-reason item.	Statement	1	1 paper	2025	Asked
18	Components, sum, difference and scalar multiple; magnitude of resultWrite each vector in i, j, k form, combine like components for a + b, a - b or k a, then find the magnitude as sqrt of the sum of squares.	Sum	1	1 paper	2024	Asked
19	Find the unknown that makes two vectors collinear or equalWrite the condition alpha = k beta (or equate components for equal vectors), compare the coefficients of the non-collinear base vectors or of i, j, k, and solve for the unknown.	Sum	2	1 paper	2019	Asked
20	Prove a vector property using dot productsTranslate the claim into dot-product statements (equal angles, perpendicular, equal magnitudes), expand using the given conditions such as a.b = 0 and \|a\| = \|b\|, and conclude in a sentence.	Proof	2	1 paper	2017	Asked
21	Vectors and scalars, types of vectors, standard termsWrite the meaning of a vector as a directed line segment, its magnitude and direction, and state correctly whether given vectors are equal, unit, zero, negative, parallel or collinear, with a one-line reason.	Statement	1	never		Never asked
22	Direction cosines and direction ratios of a vectorRead the direction ratios as the components, divide each by the magnitude to get l, m, n, and check that l^2 + m^2 + n^2 = 1.	Sum	1	never		Never asked
23	Dot product and the angle between two vectorsCompute a.b from components, find \|a\| and \|b\|, write cos(theta) = a.b/(\|a\|\|b\|) and give theta, using cos^-1 if it is not a standard angle.	Sum	1	never		Never asked
24	Cross product: definition, standard results and propertiesState a x b = \|a\|\|b\| sin(theta) n^, the i, j, k cyclic results, a x b = -(b x a), a x a = 0, the distributive law, and that a x b is perpendicular to both a and b.	Statement	1	never		Never asked
25	Coplanarity of four points, or find lambda for itForm AB, AC and AD, show (AB x AC).AD = 0 (or find the plane through A, B, C and show D satisfies it), and for an unknown set the expression to zero and solve.	Sum	2	2 papers	2018	Not in syllabus
26	General distance formula proof using cross productsWrite the area of triangle ABC two ways, (1/2)\|AB x AC\| and (1/2)\|AB\| h, expand AB x AC in terms of a, b, c, and divide to get h.	Proof	2	1 paper	2018	Not in syllabus

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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