Edexcel C1 (Core Mathematics 1) 2008 January

Find \(\int \left( 3 x ^ { 2 } + 4 x ^ { 5 } - 7 \right) d x\).

1. Write down the value of \(16 ^ { \frac { 1 } { 4 } }\).
2. Simplify \(\left( 16 x ^ { 12 } \right) ^ { \frac { 3 } { 4 } }\).

Simplify $$\frac { 5 - \sqrt { 3 } } { 2 + \sqrt { 3 } } ,$$ giving your answer in the form \(a + b \sqrt { } 3\), where \(a\) and \(b\) are integers.

4. The point \(A ( - 6,4 )\) and the point \(B ( 8 , - 3 )\) lie on the line \(L\).

1. Find an equation for \(L\) in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers.
2. Find the distance \(A B\), giving your answer in the form \(k \sqrt { 5 }\), where \(k\) is an integer.

5. (a) Write \(\frac { 2 \sqrt { } x + 3 } { x }\) in the form \(2 x ^ { p } + 3 x ^ { q }\) where \(p\) and \(q\) are constants. Given that \(y = 5 x - 7 + \frac { 2 \sqrt { } x + 3 } { x } , \quad x > 0\),
(b) find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), simplifying the coefficient of each term.

6. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\). The curve crosses the \(x\)-axis at the points \(( 1,0 )\) and \(( 4,0 )\). The maximum point on the curve is \(( 2,5 )\).
In separate diagrams sketch the curves with the following equations.
On each diagram show clearly the coordinates of the maximum point and of each point at which the curve crosses the \(x\)-axis.

1. \(y = 2 \mathrm { f } ( x )\),
2. \(y = \mathrm { f } ( - x )\). The maximum point on the curve with equation \(y = \mathrm { f } ( x + a )\) is on the \(y\)-axis.
3. Write down the value of the constant \(a\).

1. A sequence is given by:

$$\begin{aligned} & x _ { 1 } = 1 \\ & x _ { n + 1 } = x _ { n } \left( p + x _ { n } \right) \end{aligned}$$ where \(p\) is a constant ( \(p \neq 0\) ) .

1. Find \(x _ { 2 }\) in terms of \(p\).
2. Show that \(x _ { 3 } = 1 + 3 p + 2 p ^ { 2 }\). Given that \(x _ { 3 } = 1\),
3. find the value of \(p\),
4. write down the value of \(x _ { 2008 }\).

8. The equation $$x ^ { 2 } + k x + 8 = k$$ has no real solutions for \(x\).

1. Show that \(k\) satisfies \(k ^ { 2 } + 4 k - 32 < 0\).
2. Hence find the set of possible values of \(k\).

9. The curve \(C\) has equation \(y = \mathrm { f } ( x ) , x > 0\), and \(\mathrm { f } ^ { \prime } ( x ) = 4 x - 6 \sqrt { } x + \frac { 8 } { x ^ { 2 } }\). Given that the point \(P ( 4,1 )\) lies on \(C\),

1. find \(\mathrm { f } ( x )\) and simplify your answer.
2. Find an equation of the normal to \(C\) at the point \(P ( 4,1 )\).

1. The curve \(C\) has equation

$$y = ( x + 3 ) ( x - 1 ) ^ { 2 }$$

1. Sketch \(C\) showing clearly the coordinates of the points where the curve meets the coordinate axes.
2. Show that the equation of \(C\) can be written in the form $$y = x ^ { 3 } + x ^ { 2 } - 5 x + k ,$$ where \(k\) is a positive integer, and state the value of \(k\). There are two points on \(C\) where the gradient of the tangent to \(C\) is equal to 3 .
3. Find the \(x\)-coordinates of these two points.

11. The first term of an arithmetic sequence is 30 and the common difference is - 1.5

1. Find the value of the 25th term. The \(r\) th term of the sequence is 0 .
2. Find the value of \(r\). The sum of the first \(n\) terms of the sequence is \(S _ { n }\).
3. Find the largest positive value of \(S _ { n }\).