Question 1

Paper 2, calculatorHard6 marks

For x,y,z,k∈Rx,y,z,k\in\mathbb{R} consider the planes

P1: x−z=1,P2: y+z=2,Qk: x+ky+(1−k)z=3.P_1:\ x-z=1,\qquad P_2:\ y+z=2,\qquad Q_k:\ x+ky+(1-k)z=3.
(a)

Determine a direction vector of the line of intersection of P1P_1 and P2P_2

[1]
(b)

The acute angle between this line and QkQ_k is 30∘30^\circ Determine all possible values of kk and, for each value, the common solution of the three plane equations.

[5]

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