Further Mathematics

1. 1.

   Find the angle between the normals to planes x + 2y + 2z = 4 and 2x + y - 2z = 1.

   [3 marks] · no calculator

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. Read normal components from the coefficients, not the constants on the right.
   2. Both normals are nonzero; a zero dot product therefore proves perpendicularity.

   Marking points

   • Normals are (1,2,2) and (2,1,-2).
   • Their dot product is 0.
   • The angle is 90 degrees.

   Examiner tip: The plane constants affect position, not orientation.

2. 2.

   Find where r = (1,0,2) + t(2,1,-1) meets x + y + z = 7.

   [3 marks] · no calculator

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. A single parameter controls all three coordinates along the line.
   2. The coefficient of t is nonzero, so the line meets the plane at exactly one point.

   Marking points

   • Substitute x = 1 + 2t, y = t, z = 2 - t.
   • 3 + 2t = 7 gives t = 2.
   • Intersection: (5,2,0).

   Examiner tip: Report the point, not only the parameter value.

3. 3.

   Find a Cartesian equation of the plane through A = (1,0,0), B = (0,1,0), C = (0,0,2).

   [3 marks] · no calculator

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. The cross product is perpendicular to two non-parallel vectors within the plane.
   2. Insert A to find the constant; B and C also satisfy the resulting equation.

   Marking points

   • AB = (-1,1,0) and AC = (-1,0,2).
   • AB cross AC = (2,2,1).
   • The plane is 2x + 2y + z = 2.

   Examiner tip: A reversed cross product gives an equivalent plane after changing every sign.

4. 4.

   Find the perpendicular foot from P = (3,2,1) to plane x + 2y + 2z = 0, and the distance.

   [3 marks] · no calculator

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. Move from P in the normal direction until the plane equation is satisfied.
   2. The displacement is (-1,-2,-2), of length three; the foot satisfies 2 + 0 - 2 = 0.

   Marking points

   • The normal is n = (1,2,2), with P dot n = 9 and n dot n = 9.
   • Foot P - [(P dot n)/(n dot n)]n = (2,0,-1).
   • Distance is 3 units.

   Examiner tip: Use n dot n in the projection coefficient, not |n|.

5. 5.

   Lines L1: r = (0,0,0) + s(1,0,0) and L2: r = (0,1,1) + t(0,1,0). Show they are skew and find their shortest distance and closest points.

   [3 marks] · no calculator

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. A shortest connector must have zero x and y components because either line permits adjustment in one of those directions.
   2. The remaining vertical separation is fixed at one, giving a lower bound attained at the stated points.

   Marking points

   • Directions are not parallel and the z coordinates 0 and 1 rule out intersection.
   • The connector (-s,1 + t,1) is perpendicular to both directions only when s = 0 and t = -1.
   • Closest points are (0,0,0) and (0,0,1); distance is 1 unit.

   Examiner tip: Non-parallel lines in space can be skew; a two-dimensional intersection assumption is invalid.

6. 6.

   Find a vector equation of the intersection of x + y + z = 3 and x - y + z = 1. Find the point on that line nearest the origin.

   [3 marks] · no calculator

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. Choose equal x and z as a convenient point, then vary them oppositely to preserve their sum.
   2. The chosen point vector has zero dot product with the line direction, independently confirming perpendicularity to the origin connector.

   Marking points

   • Subtracting gives y = 1 and then x + z = 2.
   • r = (1,1,1) + t(1,0,-1).
   • Squared distance is 3 + 2t^2, minimised at t = 0; nearest point is (1,1,1).

   Examiner tip: Minimise distance squared; it has the same minimiser and simpler algebra.

Original, unofficial practice questions from SubjectScout — not official past-paper questions.