\end{figure}
Figure 1 shows part of the graph of \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\). The graph consists of two line segments that meet at the point \(( 1 , a ) , a < 0\). One line meets the \(x\)-axis at \(( 3,0 )\). The other line meets the \(x\)-axis at \(( - 1,0 )\) and the \(y\)-axis at \(( 0 , b ) , b < 0\).
In separate diagrams, sketch the graph with equation
(a) \(y = \mathrm { f } ( x + 1 )\),
(b) \(y = \mathrm { f } ( | x | )\).
Indicate clearly on each sketch the coordinates of any points of intersection with the axes.
Given that \(\mathrm { f } ( x ) = | x - 1 | - 2\), find
(c) the value of \(a\) and the value of \(b\),
(d) the value of \(x\) for which \(\mathrm { f } ( x ) = 5 x\). [0pt]
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3. (a) Show on an Argand diagram the locus of points given by
$$| z - 10 - 12 i | = 8$$
Set \(A\) is defined by
$$A = \left\{ z : 0 \leqslant \arg ( z - 10 - 10 i ) \leqslant \frac { \pi } { 2 } \right\} \cap \{ z : | z - 10 - 12 i | \leqslant 8 \}$$
(b) Shade the region defined by \(A\) on your Argand diagram.
(c) Determine the area of the region defined by \(A\). [0pt]
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4. The curve with equation \(y = \mathrm { f } ( x )\) where
$$f ( x ) = x ^ { 2 } + \ln \left( 2 x ^ { 2 } - 4 x + 5 \right)$$
has a single turning point at \(x = \alpha\)
(a) Show that \(\alpha\) is a solution of the equation
$$2 x ^ { 3 } - 4 x ^ { 2 } + 7 x - 2 = 0$$
The iterative formula
$$x _ { n + 1 } = \frac { 1 } { 7 } \left( 2 + 4 x _ { n } ^ { 2 } - 2 x _ { n } ^ { 3 } \right)$$
is used to find an approximate value for \(\alpha\).
Starting with \(x _ { 1 } = 0.3\)
(b) calculate, giving each answer to 4 decimal places,
the value of \(x _ { 2 }\)
the value of \(x _ { 4 }\)
Using a suitable interval and a suitable function that should be stated,
(c) show that \(\alpha\) is 0.341 to 3 decimal places. [0pt]
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5. The triangle \(T\) has vertices at the points \(( 1 , k ) , ( 3,0 )\) and \(( 11,0 )\), where \(k\) is a positive constant. Triangle \(T\) is transformed onto the triangle \(T ^ { \prime }\) by the matrix
$$\left( \begin{array} { c c }
6 & - 2
1 & 2
\end{array} \right)$$
Given that the area of triangle \(T ^ { \prime }\) is 364 square units, find the value of \(k\). [0pt]
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6. The complex number \(w\) is given by
$$w = 10 - 5 \mathrm { i }$$
(a) Find \(| w |\).
(b) Find arg \(w\), giving your answer in radians to 2 decimal places.
The complex numbers \(z\) and \(w\) satisfy the equation
$$( 2 + \mathrm { i } ) ( z + 3 \mathrm { i } ) = w$$
(c) Use algebra to find \(z\), giving your answer in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real numbers.
Given that
$$\arg ( \lambda + 9 i + w ) = \frac { \pi } { 4 }$$
where \(\lambda\) is a real constant,
(d) find the value of \(\lambda\). [0pt]
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7.
\section*{Figure 1}
Figure 1 shows the finite region \(R\), which is bounded by the curve \(y = x \mathrm { e } ^ { x }\), the line \(x = 1\), the line \(x = 3\) and the \(x\)-axis.
The region \(R\) is rotated through 360 degrees about the \(x\)-axis.
Use integration by parts to find an exact value for the volume of the solid generated. [0pt]
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8. With respect to a fixed origin \(O\), the line \(l\) has equation
$$\mathbf { r } = \left( \begin{array} { c }
13
8
1
\end{array} \right) + \lambda \left( \begin{array} { r }