Edexcel C1 (Core Mathematics 1) 2005 January

1. Write down the value of \(16 ^ { \frac { 1 } { 2 } }\).
2. Find the value of \(16 ^ { - \frac { 3 } { 2 } }\).

1. Given that \(y = 5 x ^ { 3 } + 7 x + 3\), find
   (a) \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), (b) \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
2. Find \(\int \left( 1 + 3 \sqrt { } x - \frac { 1 } { x ^ { 2 } } \right) \mathrm { d } x\).

3. Given that the equation \(k x ^ { 2 } + 12 x + k = 0\), where \(k\) is a positive constant, has equal roots, find the value of \(k\).

4. Solve the simultaneous equations $$\begin{gathered} x + y = 2
x ^ { 2 } + 2 y = 12 \end{gathered}$$

5. The \(r\) th term of an arithmetic series is ( \(2 r - 5\) ).

1. Write down the first three terms of this series.
2. State the value of the common difference.
3. Show that \(\sum _ { r = 1 } ^ { n } ( 2 r - 5 ) = n ( n - 4 )\).

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 \(( 2,0 )\) and \(( 4,0 )\). The minimum point on the curve is \(P ( 3 , - 2 )\). In separate diagrams sketch the curve with equation

1. \(y = - \mathrm { f } ( x )\),
2. \(y = \mathrm { f } ( 2 x )\). On each diagram, give the coordinates of the points at which the curve crosses the \(x\)-axis, and the coordinates of the image of \(P\) under the given transformation.

7. The curve \(C\) has equation \(y = 4 x ^ { 2 } + \frac { 5 - x } { x } , x \neq 0\). The point \(P\) on \(C\) has \(x\)-coordinate 1 .

1. Show that the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at \(P\) is 3 .
2. Find an equation of the tangent to \(C\) at \(P\). This tangent meets the \(x\)-axis at the point \(( k , 0 )\).
3. Find the value of \(k\).

8. \begin{figure}[h] \end{figure} The points \(A ( 1,7 ) , B ( 20,7 )\) and \(C ( p , q )\) form the vertices of a triangle \(A B C\), as shown in Figure 2. The point \(D ( 8,2 )\) is the mid-point of \(A C\).

1. Find the value of \(p\) and the value of \(q\). The line \(l\), which passes through \(D\) and is perpendicular to \(A C\), intersects \(A B\) at \(E\).
2. Find an equation for \(l\), in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers.
3. Find the exact \(x\)-coordinate of \(E\).

9. The gradient of the curve \(C\) is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = ( 3 x - 1 ) ^ { 2 } .$$ The point \(P ( 1,4 )\) lies on \(C\).

1. Find an equation of the normal to \(C\) at \(P\).
2. Find an equation for the curve \(C\) in the form \(y = \mathrm { f } ( x )\).
3. Using \(\frac { \mathrm { d } y } { \mathrm {~d} x } = ( 3 x - 1 ) ^ { 2 }\), show that there is no point on \(C\) at which the tangent is parallel to the line \(y = 1 - 2 x\).

10. Given that $$\mathrm { f } ( x ) = x ^ { 2 } - 6 x + 18 , \quad x \geqslant 0 ,$$

1. express \(\mathrm { f } ( x )\) in the form \(( x - a ) ^ { 2 } + b\), where \(a\) and \(b\) are integers. The curve \(C\) with equation \(y = \mathrm { f } ( x ) , x \geqslant 0\), meets the \(y\)-axis at \(P\) and has a minimum point at \(Q\).
2. In the space provided on page 19, sketch the graph of \(C\), showing the coordinates of \(P\) and \(Q\). The line \(y = 41\) meets \(C\) at the point \(R\).
3. Find the \(x\)-coordinate of \(R\), giving your answer in the form \(p + q \sqrt { } 2\), where \(p\) and \(q\) are integers.