Edexcel C1 (Core Mathematics 1) 2011 January

1. Find the value of \(16 ^ { - \frac { 1 } { 4 } }\)
2. Simplify \(x \left( 2 x ^ { - \frac { 1 } { 4 } } \right) ^ { 4 }\)

Find $$\int \left( 12 x ^ { 5 } - 3 x ^ { 2 } + 4 x ^ { \frac { 1 } { 3 } } \right) \mathrm { d } x$$ giving each term in its simplest form.

3. Simplify $$\frac { 5 - 2 \sqrt { 3 } } { \sqrt { 3 } - 1 }$$ giving your answer in the form \(p + q \sqrt { } 3\), where \(p\) and \(q\) are rational numbers.

4. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$\begin{aligned} a _ { 1 } & = 2
a _ { n + 1 } & = 3 a _ { n } - c \end{aligned}$$ where \(c\) is a constant.

1. Find an expression for \(a _ { 2 }\) in terms of \(c\). Given that \(\sum _ { i = 1 } ^ { 3 } a _ { i } = 0\)
2. find the value of \(c\).

5. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \frac { x } { x - 2 } , \quad x \neq 2$$ The curve passes through the origin and has two asymptotes, with equations \(y = 1\) and \(x = 2\), as shown in Figure 1.

1. In the space below, sketch the curve with equation \(y = \mathrm { f } ( x - 1 )\) and state the equations of the asymptotes of this curve.
2. Find the coordinates of the points where the curve with equation \(y = \mathrm { f } ( x - 1 )\) crosses the coordinate axes.

6. An arithmetic sequence has first term \(a\) and common difference \(d\). The sum of the first 10 terms of the sequence is 162 .

1. Show that \(10 a + 45 d = 162\) Given also that the sixth term of the sequence is 17 ,
2. write down a second equation in \(a\) and \(d\),
3. find the value of \(a\) and the value of \(d\).

7. The curve with equation \(y = \mathrm { f } ( x )\) passes through the point \(( - 1,0 )\). Given that $$\mathrm { f } ^ { \prime } ( x ) = 12 x ^ { 2 } - 8 x + 1$$ find \(\mathrm { f } ( x )\).

8. The equation \(x ^ { 2 } + ( k - 3 ) x + ( 3 - 2 k ) = 0\), where \(k\) is a constant, has two distinct real roots.

1. Show that \(k\) satisfies $$k ^ { 2 } + 2 k - 3 > 0$$
2. Find the set of possible values of \(k\).

9. The line \(L _ { 1 }\) has equation \(2 y - 3 x - k = 0\), where \(k\) is a constant. Given that the point \(A ( 1,4 )\) lies on \(L _ { 1 }\), find

1. the value of \(k\),
2. the gradient of \(L _ { 1 }\). The line \(L _ { 2 }\) passes through \(A\) and is perpendicular to \(L _ { 1 }\).
3. Find an equation of \(L _ { 2 }\) giving your answer 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 \(B\).
4. Find the coordinates of \(B\).
5. Find the exact length of \(A B\).

10. (a) On the axes below, sketch the graphs of

1. \(y = x ( x + 2 ) ( 3 - x )\)
2. \(y = - \frac { 2 } { x }\)
   showing clearly the coordinates of all the points where the curves cross the coordinate axes.
   (b) Using your sketch state, giving a reason, the number of real solutions to the equation $$x ( x + 2 ) ( 3 - x ) + \frac { 2 } { x } = 0$$ \includegraphics[max width=\textwidth, alt={}, center]{95e11fd7-765c-477d-800b-7574bc1af81f-13_994_997_1270_479}

11. The curve \(C\) has equation $$y = \frac { 1 } { 2 } x ^ { 3 } - 9 x ^ { \frac { 3 } { 2 } } + \frac { 8 } { x } + 30 , \quad x > 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Show that the point \(P ( 4 , - 8 )\) lies on \(C\).
3. Find an equation of the normal to \(C\) at the point \(P\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
   \includegraphics[max width=\textwidth, alt={}, center]{95e11fd7-765c-477d-800b-7574bc1af81f-15_113_129_2405_1816}