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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M9 Three-Dimensional Geometry

All the topics in Mathematics: Three-Dimensional Geometry

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.

34 topics shown

#	Topic	Type	Marks	Times asked	Last asked	Status
1	Case study or linked parts on lines and planesModel the story (panel, tower, mic, bee path) as planes and lines, then answer each part in turn (equation, point on plane, parallel plane, distance, foot or image), carrying results forward and labelling every part.	Sum	3	5 papers	Specimen 2027	Asked
2	Foot of perpendicular and image of a point in planeWrite the line through the point along the normal, find where it meets the plane (the foot), then use the foot as the midpoint of the point and its image (or reflected point).	Sum	3	4 papers	2026	Asked
3	Convert a line between vector, Cartesian and non-standard formsRewrite each ratio so that x, y and z have coefficient +1 (dividing through, moving signs into the denominator), then read the point and direction ratios and write the other form.	Sum	1	3 papers	Specimen 2027	Asked
4	Distance of a point from a planeWrite the plane as ax + by + cz + d = 0 (or r.n = d), substitute the point, take the absolute value and divide by \|n\|.	Sum	1	3 papers	2026	Asked
5	Line through a point perpendicular to two given linesRead the direction vectors b1 and b2 of the given lines, find b1 x b2 as the direction, and write the line through the given point in vector and Cartesian form.	Sum	3	3 papers	2023	Asked
6	Show two lines intersect and find the pointWrite a general point of each line with parameters lambda and mu, equate two coordinates to solve for lambda and mu, check that the third coordinate agrees, and give the point.	Sum	3	2 papers	Specimen 2027	Asked
7	Plane equation basics: normal DRs, point on plane, form conversionRead the direction ratios of the normal from ax + by + cz = d or r.n = d, convert between vector and Cartesian forms, and substitute a point to check it lies on the plane or to find an unknown coordinate.	Sum	1	2 papers	Specimen 2027	Asked
8	Parallel planes: find a coefficient or plane through a pointUse proportional normal DRs to find an unknown, or keep the same a, b, c and find d by substituting the given point.	Sum	1	2 papers	Specimen 2027	Asked
9	Intercept form: plane with given or equal interceptsWrite x/a + y/b + z/c = 1 with the given intercepts (or with a = b = c and find a from the given point), and simplify.	Sum	1	2 papers	2026	Asked
10	Line through a point parallel to a given directionWrite the point a and direction b, give the vector form r = a + lambda b, and the Cartesian form (x - x1)/a = (y - y1)/b = (z - z1)/c.	Sum	1	2 papers	2025	Asked
11	Normal form: direction cosines of normal and distance from originDivide the equation by sqrt(a^2 + b^2 + c^2), make the right side positive, read l, m, n of the normal and the distance p from the origin.	Sum	1	2 papers	2025 improvement	Asked
12	Plane through three given pointsForm two vectors in the plane from the points, take their cross product as the normal, and use the one point form (or the 3x3 determinant form) to write the plane.	Sum	2	2 papers	2025	Asked
13	Angle between two lines from direction ratiosRead (or first rewrite into standard form) the direction ratios of each line, apply cos(theta) = \|a1 a2 + b1 b2 + c1 c2\| / (sqrt(a1^2 + b1^2 + c1^2) sqrt(a2^2 + b2^2 + c2^2)), and give theta.	Sum	2	2 papers	2020	Asked
14	One point form: plane through a point with given normalFind the normal n (given, or as the vector joining two given points), write a(x - x1) + b(y - y1) + c(z - z1) = 0 or (r - a).n = 0, and simplify.	Sum	2	2 papers	2020	Asked
15	Plane normal to a vector given by direction anglesFind the missing direction cosine from l^2 + m^2 + n^2 = 1 choosing the sign the angle condition allows, build n = (magnitude)(l i + m j + n k), and write (r - a).n = 0 in vector form, simplified.	Sum	3	1 paper	Specimen 2027	Asked
16	Point where a line meets a planeWrite a general point of the line in terms of lambda, substitute into the plane, solve for lambda, and give the point.	Sum	2	1 paper	Specimen 2027	Asked
17	Angle between a line and a plane, including show-typeRead the line's direction b and the plane's normal n, apply sin(phi) = \|b.n\| / (\|b\|\|n\|), simplify, and in a show-type item point out which parameters have cancelled.	Sum	2	1 paper	Specimen 2027	Asked
18	Direction cosines and ratios of a line through two pointsWrite the direction ratios as x2 - x1, y2 - y1, z2 - z1, find sqrt of the sum of their squares, divide each ratio by it, and give the direction cosines with consistent signs.	Sum	1	1 paper	2026	Asked
19	Word problem: unknown that makes two paths meetWrite general points of both paths, equate coordinates, solve two equations for the parameters, substitute in the third to find the value of the unknown that makes them meet, and state the value to avoid.	Sum	3	1 paper	2026	Asked
20	Line through two points, or parallel to the joinFind the direction b = (B - A), choose the point (A, or the given point for a parallel line), and write both the vector and Cartesian forms.	Sum	2	1 paper	2025 improvement	Asked
21	Show three points in space are collinearFind the direction ratios of PQ and QR, show they are proportional, and state that with the common point Q the three points lie on one line.	Proof	1	1 paper	2025	Asked
22	Shortest distance between two parallel linesCheck the directions are parallel, take the common direction b, compute (a2 - a1) x b and its magnitude, and divide by \|b\|.	Sum	2	1 paper	2025 improvement	Asked
23	Plane through two points, parallel to an axis or lineTake the normal as perpendicular to the join of the points and to the given axis or line (cross product, or two dot-product equations), then write the one point form.	Sum	2	1 paper	2025	Asked
24	Distance between two parallel planesMake the x, y, z coefficients identical in both equations, then divide the absolute difference of the constants by sqrt(a^2 + b^2 + c^2).	Sum	1	1 paper	2025 improvement	Asked
25	Angle between two planesRead the normals, apply cos(theta) = \|n1.n2\| / (\|n1\|\|n2\|), and give theta.	Sum	1	1 paper	2025	Asked
26	Equations of axes, coordinate planes and lines parallel to axesWrite the equations of the x, y and z axes, of lines parallel to them and of the xy, yz and zx planes, and read the distance of a point (a, b, c) from each coordinate plane.	Statement	1	1 paper	2024	Asked
27	Direction cosines: standard results and identitiesState l = cos(alpha), m = cos(beta), n = cos(gamma), l^2 + m^2 + n^2 = 1, and use it to evaluate expressions such as the sum of sin^2 of the direction angles or to find a missing angle.	Statement	1	1 paper	2024	Asked
28	Plane through a point perpendicular to two given planesRead the normals n1, n2 of the given planes, take n = n1 x n2 as the new normal, and write the one point form through the given point.	Sum	2	1 paper	2023	Asked
29	Condition for perpendicular or parallel lines: find the unknownRead both sets of direction ratios, write a1 a2 + b1 b2 + c1 c2 = 0 (perpendicular) or a1/a2 = b1/b2 = c1/c2 (parallel), and solve for the unknown.	Sum	1	never		Never asked
30	Test whether two lines are coplanar or skewWrite a1, a2, b1, b2, compute (a2 - a1).(b1 x b2) (or the 3x3 determinant of x2 - x1, y2 - y1, z2 - z1 and the two sets of DRs), and conclude coplanar if zero, skew if not zero and not parallel.	Sum	2	never		Never asked
31	Distance of a point from a line, via the footWrite a general point Q of the line, form PQ, set PQ.b = 0 to find the parameter, give the foot Q and the distance \|PQ\|.	Sum	3	never		Never asked
32	Shortest distance between two skew linesWrite a1, b1, a2, b2 from the equations, compute b1 x b2 and its magnitude, compute (a2 - a1).(b1 x b2), and give SD as the absolute value of the quotient.	Sum	3	never		Never asked
33	Two-statement or assertion-reason on lines and planesTest each statement with the matching condition (dot product zero, proportional DRs, angle formula), mark each true or false, and pick the option.	Give reason	1	never		Never asked
34	Plane through the intersection of two planesWrite the family (a1 x + b1 y + c1 z - d1) + k(a2 x + b2 y + c2 z - d2) = 0, use the extra condition (through a point, equal intercepts) to find k, and substitute back.	Sum	2	2 papers	2019	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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