Question 8 15 marks Paper 2 Section A Hard Calculator Free
Maximum mark: 15
The position vector at time \( t \) of a point \( P \) is given by
\[ \overrightarrow{OP} = (1+t)\mathbf{i} + (2-2t)\mathbf{j} + (3t-1)\mathbf{k}, \quad t \geq 0. \]
(a) Find the coordinates of \( P \) when \( t = 0 \).[1]
(b) Show that \( P \) moves along the line \( L \) with Cartesian equations
\[ x - 1 = \frac{y-2}{-2} = \frac{z+1}{3}. \]
[2]
(c) (i) Find the value of \( t \) when \( P \) lies on the plane with equation \( 2x + y + z = 6 \).[2]
(ii) State the coordinates of \( P \) at this time.[1]
(iii) Hence find the total distance travelled by \( P \) before it meets the plane.[2]
The position vector at time \( t \) of another point, \( Q \), is given by
\[ \overrightarrow{OQ} = \begin{pmatrix} t^{2} \\ 1-t \\ 1-t^{2} \end{pmatrix}, \quad t \geq 0. \]
(d) (i) Find the value of \( t \) for which the distance from \( Q \) to the origin is minimum.[3]
(ii) Find the coordinates of \( Q \) at this time.[1]
(e) Let \( \mathbf{a} \), \( \mathbf{b} \) and \( \mathbf{c} \) be the position vectors of \( Q \) at times \( t = 0 \), \( t = 1 \) and \( t = 2 \) respectively.
(i) Show that the equation \( \mathbf{a} - \mathbf{b} = k(\mathbf{b} - \mathbf{c}) \) has no solution for \( k \).[2]
(ii) Hence show that the path of \( Q \) is not a straight line.[1]
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