A-LevelFurther MathematicsVectors9231/12 · May/June 2023 · Q714 Marks
The plane Π₁ has equation r = -4j −3k+ λ(i- j+k)+μ(i+j-k).
- (a)Obtain an equation of Π₁ in the form px+qy+rz = d.[4]
- (b)The plane Π₂ has equation r.(-5i+3j+5k) = 4. Find a vector equation of the line of intersection of Π₁ and Π₂.[4]
- (c)The line l passes through the point A with position vector ai+aj+(a-7)k and is parallel to (1-b)i+bj+bk, where a and b are positive constants. Given that the perpendicular distance from A to Π₁ is √2, find the value of a.[2]
- (d)Given that the obtuse angle between l and Π₁ is π/3, find the exact value of b.[4]
Mark scheme
Marking points from the official mark scheme (May/June 2023). The mark or mark type is shown in brackets.
- M1 A1 Finds common perpendicular.
- M1 A1 Substitutes point.
- B1 FT States point common to both planes e.g. (-7, -2, -5)
- M1 A1FT Finds direction of line.
- A1 OE.
- M1 Uses correct formula for distance from A to Π₁.
- A1
- M1 A1 Uses correct formula.
- M1 Solves for b.
- A1 CAO
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