Edexcel C1 (Core Mathematics 1) 2013 January

Factorise completely \(x - 4 x ^ { 3 }\)

Express \(8 ^ { 2 x + 3 }\) in the form \(2 ^ { y }\), stating \(y\) in terms of \(x\).

1. Express $$( 5 - \sqrt { } 8 ) ( 1 + \sqrt { } 2 )$$ in the form \(a + b \sqrt { } 2\), where \(a\) and \(b\) are integers.
2. Express $$\sqrt { } 80 + \frac { 30 } { \sqrt { } 5 }$$ in the form \(c \sqrt { } 5\), where \(c\) is an integer.

4. A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) satisfies $$u _ { n + 1 } = 2 u _ { n } - 1 , n \geqslant 1$$ Given that \(u _ { 2 } = 9\),

1. find the value of \(u _ { 3 }\) and the value of \(u _ { 4 }\),
2. evaluate \(\sum _ { r = 1 } ^ { 4 } u _ { r }\).

5. The line \(l _ { 1 }\) has equation \(y = - 2 x + 3\) The line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\) and passes through the point \(( 5,6 )\).

1. Find an equation for \(l _ { 2 }\) in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers. The line \(l _ { 2 }\) crosses the \(x\)-axis at the point \(A\) and the \(y\)-axis at the point \(B\).
2. Find the \(x\)-coordinate of \(A\) and the \(y\)-coordinate of \(B\). Given that \(O\) is the origin,
3. find the area of the triangle \(O A B\).

6. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \frac { 2 } { x } , x \neq 0\) The curve \(C\) has equation \(y = \frac { 2 } { x } - 5 , x \neq 0\), and the line \(l\) has equation \(y = 4 x + 2\)

1. Sketch and clearly label the graphs of \(C\) and \(l\) on a single diagram. On your diagram, show clearly the coordinates of the points where \(C\) and \(l\) cross the coordinate axes.
2. Write down the equations of the asymptotes of the curve \(C\).
3. Find the coordinates of the points of intersection of \(y = \frac { 2 } { x } - 5\) and \(y = 4 x + 2\)

1. Lewis played a game of space invaders. He scored points for each spaceship that he captured.

Lewis scored 140 points for capturing his first spaceship.
He scored 160 points for capturing his second spaceship, 180 points for capturing his third spaceship, and so on. The number of points scored for capturing each successive spaceship formed an arithmetic sequence.

1. Find the number of points that Lewis scored for capturing his 20th spaceship.
2. Find the total number of points Lewis scored for capturing his first 20 spaceships. Sian played an adventure game. She scored points for each dragon that she captured. The number of points that Sian scored for capturing each successive dragon formed an arithmetic sequence. Sian captured \(n\) dragons and the total number of points that she scored for capturing all \(n\) dragons was 8500 . Given that Sian scored 300 points for capturing her first dragon and then 700 points for capturing her \(n\)th dragon,
3. find the value of \(n\).

8. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = - x ^ { 3 } + \frac { 4 x - 5 } { 2 x ^ { 3 } } , \quad x \neq 0$$ Given that \(y = 7\) at \(x = 1\), find \(y\) in terms of \(x\), giving each term in its simplest form.

9. The equation $$( k + 3 ) x ^ { 2 } + 6 x + k = 5 , \text { where } k \text { is a constant, }$$ has two distinct real solutions for \(x\).

1. Show that \(k\) satisfies $$k ^ { 2 } - 2 k - 24 < 0$$
2. Hence find the set of possible values of \(k\).

10. $$4 x ^ { 2 } + 8 x + 3 \equiv a ( x + b ) ^ { 2 } + c$$

1. Find the values of the constants \(a , b\) and \(c\).
2. On the axes on page 27, sketch the curve with equation \(y = 4 x ^ { 2 } + 8 x + 3\), showing clearly the coordinates of any points where the curve crosses the coordinate axes. \includegraphics[max width=\textwidth, alt={}, center]{099016ad-e742-4679-9669-47dcd1d9cc5f-15_1283_1284_319_322}

11. The curve \(C\) has equation $$y = 2 x - 8 \sqrt { } x + 5 , \quad x \geqslant 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving each term in its simplest form. The point \(P\) on \(C\) has \(x\)-coordinate equal to \(\frac { 1 } { 4 }\)
2. Find the equation of the tangent to \(C\) at the point \(P\), giving your answer in the form \(y = a x + b\), where \(a\) and \(b\) are constants. The tangent to \(C\) at the point \(Q\) is parallel to the line with equation \(2 x - 3 y + 18 = 0\)
3. Find the coordinates of \(Q\).