Playbook
Line and Plane
205 q - 4.96/paper - 42% HARD. The single heaviest chapter on recent papers. Seven subtopics, and the HARD is spread rather than concentrated (top two carry only 47%), so there is no cherry-pick here: you own the whole chapter or you lose ten marks.
- Questions in the bank
- 205
- q/paper (2024–25 shifts)
- 4.96
- Tagged HARD
- 42%
- Subtopics
- 7
Strand: Cornerstone
When you’ll see it
Anything stated in three coordinates: a line through two points, a plane through a normal, a distance, an angle, a foot of perpendicular, or two lines you are asked to test for intersection.
How this chapter is tested
205 q at 4.96 per paper makes this the single heaviest chapter on recent MHT-CET papers — roughly ten marks, every time. It is also 42% HARD, and the important structural fact is that the HARD does not concentrate. Seven subtopics, and the top two carry only 47% of the chapter's HARD between them. Compare that with Vectors, where two subtopics carry 74%. There is no cherry-pick available here: you own the whole chapter or you lose ten marks.
The cheapest corner is Line — Equation, Direction Cosines, and Vector Form: 29 q at 21% HARD, the lowest rate in the chapter and the foundation everything else stands on. The most expensive is Intersection, Coplanarity, and Skew Lines at 62% HARD, which is exactly where a student who learned formulas without learning the geometry falls over. Learn in that order, not in book order.
Almost every question in this chapter is one of two moves wearing different clothes: an ANGLE between two objects, or a DISTANCE from a point to an object. Perpendicularity is just angle 90 and parallelism is just angle 0 — the survey found the angle idea running through 87 q across 7 chapters and perpendicularity through 83 q across 7. Once you see that, the seven subtopics stop being seven separate syllabi.
At 1.8 minutes a question you cannot afford to derive a standard result on the paper. The direction-ratio, normal-form, distance and coplanarity formulas have to be automatic, and the 3-D picture has to be sketchable in ten seconds.
The sub-skills
The distinct skills inside the chapter, in the order to learn them.
Line — equation, direction cosines, and vector form
Write a line three ways (vector, symmetric, parametric) and move between them without thinking. Direction ratios are any proportional triple; direction cosines are the normalised ones, and their squares sum to 1. 29 q at 21% HARD — the cheapest block in the chapter and the one everything else assumes.
Plane — equation, normal, and construction
Build a plane from a normal and a point, from three points, or from a line plus a point. The normal vector IS the plane for exam purposes. 47 q at 38% HARD, the largest subtopic here.
Angles — line, plane, and direction conditions
Line-to-line uses the two direction vectors; plane-to-plane uses the two normals; line-to-plane uses direction against normal and then takes the complement — this is where the sine-versus-cosine slip lives. 29 q at 45% HARD.
Distances in 3-D
Point to plane, point to line, and the shortest distance between two skew lines. Three different formulas with one shared shape: project a connecting vector onto the perpendicular direction, then take the modulus. 33 q at 42% HARD.
Foot of perpendicular, image, and projection
Drop a perpendicular by parametrising the line, imposing the perpendicularity condition, and solving for the parameter. The image is the foot doubled, not the foot. 19 q at 53% HARD.
Intersection, coplanarity, and skew lines
Two lines in space usually miss each other. Coplanarity is a vanishing 3x3 determinant built from the two directions and the joining vector — the same degeneracy test that appears as concurrency in 2-D and as scalar triple product zero in Vectors. 37 q at 62% HARD, the hardest cell in the chapter.
Tetrahedron geometry — centroid, volume, and vertices
A small, closed block: centroid as the average of four vertices, volume as one sixth of a scalar triple product. 11 q at 27% HARD, so it is cheap once the triple product is already yours.
Traps to expect
Distractor shapes this chapter reuses. The Traps page covers the patterns that cut across chapters.
Direction ratios used as direction cosines
A distractor built by plugging unnormalised ratios straight into a cosine formula. If the triple's squares do not sum to 1, it is not a set of direction cosines. Check before you use it.
Line-to-plane angle given as the direction-normal angle
The angle between a line and a plane is the COMPLEMENT of the angle between the line's direction and the plane's normal. The wrong option is the uncomplemented value, and it is always present.
Foot of perpendicular offered where the image was asked
The image sits twice as far along the same perpendicular. Both points appear as options, and the foot is the more tempting-looking of the two because it comes out of the algebra first.
Skew treated as intersecting
Solving two of the three parametric equations always produces a parameter pair. The third equation is what tells you whether the lines actually meet, and skipping it turns a skew pair into a fabricated intersection point.
Learn it before you drill it
This chapter has full teaching notes — foundations, worked examples, self-checks and a per-subtopic mastery checkpoint. Read the notes once, then drill subtopic by subtopic below.
Line and Plane notesDrill every line and plane question
205 questions from the bank, scoped to 7 bundled subtopics.
Drill one subtopic at a time
The 7 subtopics this playbook covers, in catalog order.
- Plane — Equation, Normal, and ConstructionDrill
- Intersection, Coplanarity, and Skew LinesDrill
- Distances in 3-DDrill
- Angles — Line, Plane, and Direction ConditionsDrill
- Line — Equation, Direction Cosines, and Vector FormDrill
- Foot of Perpendicular, Image, and ProjectionDrill
- Tetrahedron Geometry — Centroid, Volume, and VerticesDrill
Related playbooks
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