Edexcel C1 (Core Mathematics 1)

1. Find the value of \(y\) such that

$$4 ^ { y + 3 } = 8 .$$

1. Find

$$\int \left( 3 x ^ { 2 } + \frac { 1 } { 2 x ^ { 2 } } \right) \mathrm { d } x$$

3. \begin{figure}[h] \end{figure} Figure 1 shows the rectangles \(A B C D\) and \(E F G H\) which are similar.
Given that \(A B = ( 3 - \sqrt { 5 } ) \mathrm { cm } , A D = \sqrt { 5 } \mathrm {~cm}\) and \(E F = ( 1 + \sqrt { 5 } ) \mathrm { cm }\), find the length \(E H\) in cm, giving your answer in the form \(a + b \sqrt { 5 }\) where \(a\) and \(b\) are integers.

4. (a) Sketch on the same diagram the curves \(y = x ^ { 2 } - 4 x\) and \(y = - \frac { 1 } { x }\).
(b) State, with a reason, the number of real solutions to the equation $$x ^ { 2 } - 4 x + \frac { 1 } { x } = 0 .$$

1. (a) By completing the square, find in terms of the constant \(k\) the roots of the equation

$$x ^ { 2 } + 2 k x + 4 = 0 .$$ (b) Hence find the exact roots of the equation $$x ^ { 2 } + 6 x + 4 = 0 .$$

1. (a) Evaluate

$$\sum _ { r = 1 } ^ { 50 } ( 80 - 3 r )$$ (b) Show that $$\sum _ { r = 1 } ^ { n } \frac { r + 3 } { 2 } = k n ( n + 7 )$$ where \(k\) is a rational constant to be found.

1. Given that

$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x ^ { 3 } - 4 } { x ^ { 3 } } , \quad x \neq 0$$

1. find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\). Given also that \(y = 0\) when \(x = - 1\),
2. find the value of \(y\) when \(x = 2\).

9. A curve has the equation \(y = ( \sqrt { x } - 3 ) ^ { 2 } , x \geq 0\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1 - \frac { 3 } { \sqrt { x } }\). The point \(P\) on the curve has \(x\)-coordinate 4 .
2. Find an equation for the normal to the curve at \(P\) in the form \(y = m x + c\).
3. Show that the normal to the curve at \(P\) does not intersect the curve again.

10. The straight line \(l\) has gradient 3 and passes through the point \(A ( - 6,4 )\).

1. Find an equation for \(l\) in the form \(y = m x + c\). The straight line \(m\) has the equation \(x - 7 y + 14 = 0\).
   Given that \(m\) crosses the \(y\)-axis at the point \(B\) and intersects \(l\) at the point \(C\),
2. find the coordinates of \(B\) and \(C\),
3. show that \(\angle B A C = 90 ^ { \circ }\),
4. find the area of triangle \(A B C\).