17
\end{array} \right) + \lambda \left( \begin{array} { c } 
- 2 \\
1 \\
- 4
\end{array} \right) \quad l _ { 2 } : \quad \mathbf { r } = \left( \begin{array} { c } 
- 5 \\
11 \\
p
\end{array} \right) + \mu \left( \begin{array} { l } 
q \\
2 \\
2
\end{array} \right)$$

where $\lambda$ and $\mu$ are parameters and $p$ and $q$ are constants. Given that $l _ { 1 }$ and $l _ { 2 }$ are perpendicular,\\
(a) show that $q = - 3$.

Given further that $l _ { 1 }$ and $l _ { 2 }$ intersect, find\\
(b) the value of $p$,\\
(c) the coordinates of the point of intersection.

The point $A$ lies on $l _ { 1 }$ and has position vector $\left( \begin{array} { c } 9 \\ 3 \\ 13 \end{array} \right)$. The point $C$ lies on $l _ { 2 }$.\\
Given that a circle, with centre $C$, cuts the line $l _ { 1 }$ at the points $A$ and $B$,\\
(d) find the position vector of $B$.\\

5.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{a5579938-e202-4543-8513-6483ede49850-09_696_686_196_626}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}

A container is made in the shape of a hollow inverted right circular cone. The height of the container is 24 cm and the radius is 16 cm , as shown in Figure 2. Water is flowing into the container. When the height of water is $h \mathrm {~cm}$, the surface of the water has radius $r \mathrm {~cm}$ and the volume of water is $V \mathrm {~cm} ^ { 3 }$.\\
(a) Show that $V = \frac { 4 \pi h ^ { 3 } } { 27 }$.\\[0pt]
[The volume $V$ of a right circular cone with vertical height $h$ and base radius $r$ is given by the formula $V = \frac { 1 } { 3 } \pi r ^ { 2 } h$.]

Water flows into the container at a rate of $8 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }$.\\
(b) Find, in terms of $\pi$, the rate of change of $h$ when $h = 12$.

6. (a) Find $\int \tan ^ { 2 } x \mathrm {~d} x$.\\
(b) Use integration by parts to find $\int \frac { 1 } { x ^ { 3 } } \ln x \mathrm {~d} x$.\\
(c) Use the substitution $u = 1 + e ^ { x }$ to show that

$$\int \frac { \mathrm { e } ^ { 3 x } } { 1 + \mathrm { e } ^ { x } } \mathrm {~d} x = \frac { 1 } { 2 } \mathrm { e } ^ { 2 x } - \mathrm { e } ^ { x } + \ln \left( 1 + \mathrm { e } ^ { x } \right) + k$$

where $k$ is a constant.\\

7.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{a5579938-e202-4543-8513-6483ede49850-13_511_714_237_612}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}

The curve $C$ shown in Figure 3 has parametric equations

$$x = t ^ { 3 } - 8 t , \quad y = t ^ { 2 }$$

where $t$ is a parameter. Given that the point $A$ has parameter $t = - 1$,\\
(a) find the coordinates of $A$.

The line $l$ is the tangent to $C$ at $A$.\\
(b) Show that an equation for $l$ is $2 x - 5 y - 9 = 0$.

The line $l$ also intersects the curve at the point $B$.\\
(c) Find the coordinates of $B$.\\

\hfill \mbox{\textit{Edexcel C4 2009 Q17}}