Find parameter value for geometric condition

10 The points \(A\) and \(B\) have position vectors, relative to the origin \(O\), given by $$\overrightarrow { O A } = \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } \quad \text { and } \quad \overrightarrow { O B } = 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } .$$ The line \(l\) has vector equation $$\mathbf { r } = ( 1 - 2 t ) \mathbf { i } + ( 5 + t ) \mathbf { j } + ( 2 - t ) \mathbf { k }$$

1. Show that \(l\) does not intersect the line passing through \(A\) and \(B\).
2. The point \(P\) lies on \(l\) and is such that angle \(P A B\) is equal to \(60 ^ { \circ }\). Given that the position vector of \(P\) is \(( 1 - 2 t ) \mathbf { i } + ( 5 + t ) \mathbf { j } + ( 2 - t ) \mathbf { k }\), show that \(3 t ^ { 2 } + 7 t + 2 = 0\). Hence find the only possible position vector of \(P\).

1. With respect to a fixed origin \(O\), the line \(l _ { 1 }\) is given by the equation

$$\mathbf { r } = \left( \begin{array} { r } 8 \\ 1 \\ - 3 \end{array} \right) + \mu \left( \begin{array} { r } - 5 \\ 4 \\ 3 \end{array} \right)$$ where \(\mu\) is a scalar parameter.
The point \(A\) lies on \(l _ { 1 }\) where \(\mu = 1\)

1. Find the coordinates of \(A\). The point \(P\) has position vector \(\left( \begin{array} { l } 1 \\ 5 \\ 2 \end{array} \right)\) The line \(l _ { 2 }\) passes through the point \(P\) and is parallel to the line \(l _ { 1 }\)
2. Write down a vector equation for the line \(l _ { 2 }\)
3. Find the exact value of the distance \(A P\). Give your answer in the form \(k \sqrt { 2 }\), where \(k\) is a constant to be found. The acute angle between \(A P\) and \(l _ { 2 }\) is \(\theta\)
4. Find the value of \(\cos \theta\) A point \(E\) lies on the line \(l _ { 2 }\) Given that \(A P = P E\),
5. find the area of triangle \(A P E\),
6. find the coordinates of the two possible positions of \(E\).

1. With respect to a fixed origin \(O\), the points \(A\) and \(B\) have coordinates \(( 2,2 , - 1 )\) and ( \(4,2 p , 1\) ) respectively, where \(p\) is a constant.

For each of the following, determine the possible values of \(p\) for which,

1. \(O B\) makes an angle of \(45 ^ { \circ }\) with the positive \(x\)-axis
2. \(\overrightarrow { O A } \times \overrightarrow { O B }\) is parallel to \(\left( \begin{array} { r } 4 \\ - p \\ 2 \end{array} \right)\)
3. the area of triangle \(O A B\) is \(3 \sqrt { 2 }\)

8 In this question you must show detailed reasoning. A sequence of vectors \(\mathbf { a } _ { 1 } , \mathbf { a } _ { 2 } , \mathbf { a } _ { 3 } , \ldots\) is defined by

• \(\mathbf { a } _ { 1 } = \left( \begin{array} { l } 1 \\ 1 \\ 1 \end{array} \right)\)
• \(\quad \mathbf { a } _ { n + 1 } = \left( \mathbf { a } _ { n } \times \mathbf { b } \right) \times \mathbf { b }\), for integers \(n \geqslant 1\), where \(\mathbf { b }\) is the vector \(\frac { 1 } { 4 } \left( \begin{array} { c } - 3 \\ 1 \\ 2 \end{array} \right)\).
  1. Prove by induction that \(\mathbf { a } _ { n } = \left( - \frac { 7 } { 8 } \right) ^ { n - 1 } \left( \begin{array} { l } 1 \\ 1 \\ 1 \end{array} \right)\). for all integers \(n \geqslant 1\).
  2. Use an algebraic method to find the smallest value of \(n\) such that \(\left| \mathbf { a } _ { n } \right| < 0.001\).

\section*{END OF QUESTION PAPER}

3 The equations of two intersecting lines are \(\mathbf { r } = \left( \begin{array} { c } - 12 \\ a \\ - 1 \end{array} \right) + \lambda \left( \begin{array} { l } 2 \\ 2 \\ 1 \end{array} \right) \quad \mathbf { r } = \left( \begin{array} { l } 2 \\ 0 \\ 5 \end{array} \right) + \mu \left( \begin{array} { c } - 3 \\ 1 \\ - 1 \end{array} \right)\) where \(a\) is a constant.

1. Find a vector, \(\mathbf { b }\), which is perpendicular to both lines.
2. Show that b. \(\left( \begin{array} { c } - 12 \\ a \\ - 1 \end{array} \right) =\) b. \(\left( \begin{array} { l } 2 \\ 0 \\ 5 \end{array} \right)\).
3. Hence, or otherwise, find the value of \(a\).

With respect to a fixed origin \(O\), the line \(l_1\) is given by the equation $$\mathbf{r} = \mathbf{i} + 2\mathbf{j} + 5\mathbf{k} + \lambda(8\mathbf{i} - \mathbf{j} + 4\mathbf{k})$$ where \(\lambda\) is a scalar parameter. The point \(A\) lies on \(l_1\) Given that \(|\overrightarrow{OA}| = 5\sqrt{10}\)

1. show that at \(A\) the parameter \(\lambda\) satisfies $$81\lambda^2 + 52\lambda - 220 = 0$$ [3]

Hence

1. 1. show that one possible position vector for \(A\) is \(-15\mathbf{i} + 4\mathbf{j} - 3\mathbf{k}\)
   2. find the other possible position vector for \(A\). [3]

The line \(l_2\) is parallel to \(l_1\) and passes through \(O\). Given that • \(\overrightarrow{OA} = -15\mathbf{i} + 4\mathbf{j} - 3\mathbf{k}\) • point \(B\) lies on \(l_2\) where \(|\overrightarrow{OB}| = 4\sqrt{10}\)

1. find the area of triangle \(OAB\), giving your answer to one decimal place. [4]

Vectors \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\), are given by \(\mathbf{a} = \mathbf{i} + (1-p)\mathbf{j} + (p+2)\mathbf{k}\), \(\mathbf{b} = 2\mathbf{i} + \mathbf{j} + \mathbf{k}\) and \(\mathbf{c} = \mathbf{i} + 14\mathbf{j} + (p-3)\mathbf{k}\) where \(p\) is a constant. You are given that \(\mathbf{a} \times \mathbf{b}\) is perpendicular to \(\mathbf{c}\). Determine the possible values of \(p\). [6]

In this question, you must show detailed reasoning. A curve is defined parametrically by \(x = t^3 - 3t + 1\), \(y = 3t^2 - 1\), for \(0 \leq t \leq 5\). Find, in exact form,

1. the length of the curve, [6]
2. the area of the surface generated when the curve is rotated completely about the \(x\)-axis. [4]