Edexcel C1 (Core Mathematics 1)

1. A sequence is defined by the recurrence relation

$$u _ { n + 1 } = \sqrt { \left( \frac { u _ { n } } { 2 } + \frac { a } { u _ { n } } \right) } , \quad n = 1,2,3 , \ldots ,$$ where \(a\) is a constant.

1. Given that \(a = 20\) and \(u _ { 1 } = 3\), find the values of \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\), giving your answers to 2 decimal places.
2. Given instead that \(u _ { 1 } = u _ { 2 } = 3\),
   1. calculate the value of \(a\),
   2. write down the value of \(u _ { 5 }\).

2. The equation \(x ^ { 2 } + 5 k x + 2 k = 0\), where \(k\) is a constant, has real roots.

1. Prove that \(k ( 25 k - 8 ) \geq 0\).
2. Hence find the set of possible values of \(k\).
3. Write down the values of \(k\) for which the equation \(x ^ { 2 } + 5 k x + 2 k = 0\) has equal roots.

3. (a) Given that \(3 ^ { x } = 9 ^ { y - 1 }\), show that \(x = 2 y - 2\).
(b) Solve the simultaneous equations $$\begin{aligned} & x = 2 y - 2
& x ^ { 2 } = y ^ { 2 } + 7 \end{aligned}$$

1. The curve \(C\) with equation \(y = \mathrm { f } ( x )\) is such that

$$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 \sqrt { } x + \frac { 12 } { \sqrt { } x } , \quad x > 0$$

1. Show that, when \(x = 8\), the exact value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) is \(9 \sqrt { } 2\). The curve \(C\) passes through the point \(( 4,30 )\).
2. Using integration, find \(\mathrm { f } ( x )\).

5. The points \(A\) and \(B\) have coordinates \(( 4,6 )\) and \(( 12,2 )\) respectively. The straight line \(l _ { 1 }\) passes through \(A\) and \(B\).

1. Find an equation for \(l _ { 1 }\) in the form \(a x + b y = c\), where \(a\), b and \(c\) are integers. The straight line \(l _ { 2 }\) passes through the origin and has gradient - 4 .
2. Write down an equation for \(l _ { 2 }\). The lines \(l _ { 1 }\) and \(l _ { 2 }\) intercept at the point \(C\).
3. Find the exact coordinates of the mid-point of \(A C\).

6. $$f ( x ) = 9 - ( x - 2 ) ^ { 2 }$$

1. Write down the maximum value of \(\mathrm { f } ( x )\).
2. Sketch the graph of \(y = \mathrm { f } ( x )\), showing the coordinates of the points at which the graph meets the coordinate axes. The points \(A\) and \(B\) on the graph of \(y = \mathrm { f } ( x )\) have coordinates \(( - 2 , - 7 )\) and \(( 3,8 )\) respectively.
3. Find, in the form \(y = m x + c\), an equation of the straight line through \(A\) and \(B\).
4. Find the coordinates of the point at which the line \(A B\) crosses the \(x\)-axis. The mid-point of \(A B\) lies on the line with equation \(y = k x\), where \(k\) is a constant.
5. Find the value of \(k\).

7. For the curve \(C\) with equation \(y = x ^ { 4 } - 8 x ^ { 2 } + 3\),

1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), The point \(A\), on the curve \(C\), has \(x\)-coordinate 1 .
2. Find an equation for the normal to \(C\) at \(A\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

8. $$f ( x ) = \frac { \left( x ^ { 2 } - 3 \right) ^ { 2 } } { x ^ { 3 } } , x \neq 0$$

1. Show that \(\mathrm { f } ( x ) \equiv x - 6 x ^ { - 1 } + 9 x ^ { - 3 }\).
2. Hence, or otherwise, differentiate \(\mathrm { f } ( x )\) with respect to \(x\). END