Edexcel C1 (Core Mathematics 1)

1. Find in exact form the real solutions of the equation

$$x ^ { 4 } = 5 x ^ { 2 } + 14 .$$

3. (a) Solve the equation $$x ^ { \frac { 3 } { 2 } } = 27 .$$ (b) Express \(\left( 2 \frac { 1 } { 4 } \right) ^ { - \frac { 1 } { 2 } }\) as an exact fraction in its simplest form.

4. \begin{figure}[h] \end{figure} Figure 1 shows the curve with equation \(y = x ^ { 3 } + a x ^ { 2 } + b x + c\), where \(a , b\) and \(c\) are constants. The curve crosses the \(x\)-axis at the point \(( - 1,0 )\) and touches the \(x\)-axis at the point \(( 3,0 )\). Show that \(a = - 5\) and find the values of \(b\) and \(c\).

5. Given that $$y = \frac { x ^ { 4 } - 3 } { 2 x ^ { 2 } }$$

1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
2. show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { x ^ { 4 } - 9 } { x ^ { 4 } }\).

6. (a) Sketch on the same diagram the curve with equation \(y = ( x - 2 ) ^ { 2 }\) and the straight line with equation \(y = 2 x - 1\). Label on your sketch the coordinates of any points where each graph meets the coordinate axes.
(b) Find the set of values of \(x\) for which $$( x - 2 ) ^ { 2 } > 2 x - 1$$

1. A curve has the equation \(y = \frac { x } { 2 } + 3 - \frac { 1 } { x } , x \neq 0\).

The point \(A\) on the curve has \(x\)-coordinate 2 .

1. Find the gradient of the curve at \(A\).
2. Show that the tangent to the curve at \(A\) has equation $$3 x - 4 y + 8 = 0$$ The tangent to the curve at the point \(B\) is parallel to the tangent at \(A\).
3. Find the coordinates of \(B\).

8. The straight line \(l _ { 1 }\) has gradient \(\frac { 3 } { 2 }\) and passes through the point \(A ( 5,3 )\).

1. Find an equation for \(l _ { 1 }\) in the form \(y = m x + c\). The straight line \(l _ { 2 }\) has the equation \(3 x - 4 y + 3 = 0\) and intersects \(l _ { 1 }\) at the point \(B\).
2. Find the coordinates of \(B\).
3. Find the coordinates of the mid-point of \(A B\).
4. Show that the straight line parallel to \(l _ { 2 }\) which passes through the mid-point of \(A B\) also passes through the origin.

9. The third term of an arithmetic series is \(5 \frac { 1 } { 2 }\). The sum of the first four terms of the series is \(22 \frac { 3 } { 4 }\).

1. Show that the first term of the series is \(6 \frac { 1 } { 4 }\) and find the common difference.
2. Find the number of positive terms in the series.
3. Hence, find the greatest value of the sum of the first \(n\) terms of the series.

10. The curve \(C\) has the equation \(y = \mathrm { f } ( x )\). Given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 8 x - \frac { 2 } { x ^ { 3 } } , \quad x \neq 0$$ and that the point \(P ( 1,1 )\) lies on \(C\),

1. find an equation for the tangent to \(C\) at \(P\) in the form \(y = m x + c\),
2. find an equation for \(C\),
3. find the \(x\)-coordinates of the points where \(C\) meets the \(x\)-axis, giving your answers in the form \(k \sqrt { 2 }\).