ilovepdf_merged (6).pdf). Content covers sections 35.1–35.x.A plane in 3D can be described by a point and a normal vector, or by intercepts, or by three non-collinear points.
General plane: ax + by + cz + d = 0 with normal vector n⃗ = (a,b,c). Plane through (x₀,y₀,z₀) with normal (a,b,c): a(x−x₀)+b(y−y₀)+c(z−z₀)=0.
Distance from (x₁,y₁,z₁) to ax+by+cz+d=0 is |ax₁+by₁+cz₁+d|/√(a²+b²+c²).
Most exam-important points from this chapter:
Start every answer with the key definition or standard form from Plane.
Copy the formula, then substitute values; never jump to the number alone.
Check domain, quadrant, non-zero denominators, and applicability of the theorem.
Memorise and apply: n =
Memorise and apply: is the equation of the plane then d is not the distance of
English questions from NIOS Mathematics (311) public / sample papers (Sample 2024, Apr 2024, Oct 2024, Apr 2025; board-style where scanned papers had no extractable text). Mapped exclusively to this chapter — no cross-chapter duplicates. Use Model Answer for the marking key and Explanation for working.
4 question(s) · Sources: Board-style, Board-style distance, Board-style intercept, Board-style parallel
PYQ1. Find the equation of the plane through (1,0,0) with normal 2î+3ĵ−k̂.
Model Answer
1) 2x + 3y − z − 2 = 0
Explanation
1) 2(x−1)+3y−z=0 ⇒ 2x+3y−z−2=0.
PYQ2. Distance from (1,1,1) to the plane x+y+z−3=0.
Model Answer
1) 0 (point lies on the plane)
Explanation
1) |1+1+1−3|/√3=0.
PYQ3. Are the planes 2x−y+z=1 and 4x−2y+2z=7 parallel?
Model Answer
1) Yes (normals proportional; distinct planes)
Explanation
1) Normals (2,−1,1) and (4,−2,2)=2(2,−1,1). Constants not in same ratio
2) parallel distinct.
PYQ4. Write x/2 + y/3 + z/6 = 1 in the form ax+by+cz+d=0.
Model Answer
1) 3x + 2y + z − 6 = 0
Explanation
1) Multiply through by 6.
10 English exam-style problems for this chapter (NIOS Mathematics 311). Each question states the formula, gives full step-by-step working with the final answer, plus formal/easy explanations. English only.
Find a Cartesian equation of the plane through (1,0,0) with normal 2i+3j−k.
1) 2(x−1)+3(y−0)−1(z−0)=0 ⇒ 2x+3y−z−2=0.
Answer: 2x + 3y − z − 2 = 0
Point-normal form of a plane.
Normal components times (r−r₀).
Can multiply by any non-zero constant.
Include the constant term −2.
Distance from (1,1,1) to x+y+z−3=0.
1) |1+1+1−3|/√3=0.
Answer: 0 (point lies on the plane)
Zero distance means the point satisfies the plane equation.
1+1+1−3=0.
Quick check: plug into plane.
Non-zero would use absolute value over √3.
Are planes 2x−y+z=1 and 4x−2y+2z=7 parallel?
1) Normals (2,−1,1) and (4,−2,2)=2(2,−1,1) proportional ⇒ parallel.
2) Constants not proportional in the same ratio for identity ⇒ distinct parallel planes.
Answer: Yes, parallel (distinct)
Parallel planes have proportional normals.
Second normal is double the first.
If also d scales the same, planes coincide.
4≠2·1 so not the same plane.
Write x/2 + y/3 + z/6 = 1 in general form ax+by+cz+d=0.
1) Multiply by 6: 3x+2y+z−6=0.
Answer: 3x + 2y + z − 6 = 0
Clear denominators of intercept form.
LCM of 2,3,6 is 6.
Intercepts 2,3,6 on the axes.
Do not forget −6.
Does the plane 2x−y+3z=0 pass through the origin? Find a normal vector.
1) 0=0 true ⇒ passes through origin.
2) Normal n⃗=2i−j+3k.
Answer: Yes; normal (2, −1, 3)
No constant term implies origin satisfies the equation.
Plug (0,0,0).
Normal from coefficients.
Any multiple is also a normal.
Find the acute angle between planes x+y+z=1 and x−y=2.
1) n₁=(1,1,1), n₂=(1,−1,0).
2) n₁·n₂=1−1+0=0 ⇒ θ=90°.
Answer: 90° (planes perpendicular)
Normals are perpendicular ⇒ planes are perpendicular.
Dot product of normals is zero.
Acute angle between planes equals angle between normals (taken acute).
Do not use direction of a line in the plane without care.
Write a Cartesian equation of the plane through (0,0,0) with normal <1,1,1>.
1) 1(x−0)+1(y−0)+1(z−0)=0 ⇒ x+y+z=0.
Answer: x + y + z = 0
Normal form through a point.
Plane of points with sum of coordinates zero.
d=0 because it passes through origin.
Check each algebraic step carefully.
Distance from (0,0,0) to x+2y+2z−6=0.
1) |−6|/√(1+4+4)=6/3=2.
Answer: 2
Point-to-plane distance formula.
Normal magnitude 3.
Absolute value on numerator.
Check each algebraic step carefully.
Are x+y+z=1 and 2x+2y+2z=5 parallel?
1) Normals <1,1,1> and <2,2,2> are proportional.
2) Yes — parallel (distinct because 1≠5/2).
Answer: Yes — parallel distinct planes
Parallel iff normals are scalar multiples.
Second is double the first’s left side.
Constants not proportional the same way ⇒ not the same plane.
Check each algebraic step carefully.
Show that x=0 and y=0 are perpendicular planes.
1) Normals i and j; i·j=0 ⇒ planes perpendicular.
Answer: Perpendicular (normals · = 0)
Angle between planes equals angle between normals.
Coordinate planes x=0 and y=0 meet at 90°.
z=0 is also ⊥ to each.
Check each algebraic step carefully.