Edexcel C1 (Core Mathematics 1)

1. Solve the inequality
   \(10 + x ^ { 2 } > x ( x - 2 )\).
   (3)
   

\begin{center} \begin{tabular}{|l|l|} \hline

2. Find \(\int \left( x ^ { 2 } - \frac { 1 } { x ^ { 2 } } + \sqrt [ 3 ] { x } \right) \mathrm { d } x\) & Leave blank
\hline \end{tabular} \end{center}

Find the value of

1. \(81 ^ { \frac { 1 } { 2 } }\),
2. \(81 ^ { \frac { 3 } { 4 } }\),
3. \(81 ^ { - \frac { 3 } { 4 } }\).

A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$a _ { 1 } = k , \quad a _ { n + 1 } = 4 a _ { n } - 7 ,$$ where \(k\) is a constant.

1. Write down an expression for \(a _ { 2 }\) in terms of \(k\).
2. Find \(a _ { 3 }\) in terms of \(k\), simplifying your answer. Given that \(a _ { 3 } = 13\),
3. find the value of \(k\).

5. (a) Show that eliminating \(y\) from the equations $$\begin{gathered} 2 x + y = 8
3 x ^ { 2 } + x y = 1 \end{gathered}$$ produces the equation $$x ^ { 2 } + 8 x - 1 = 0$$ (b) Hence solve the simultaneous equations $$\begin{gathered} 2 x + y = 8
3 x ^ { 2 } + x y = 1 \end{gathered}$$ giving your answers in the form \(a + b \sqrt { } 17\), where \(a\) and \(b\) are integers.

6. $$f ( x ) = \frac { ( 2 x + 1 ) ( x + 4 ) } { \sqrt { } x } , \quad x > 0$$

1. Show that \(\mathrm { f } ( x )\) can be written in the form \(P x ^ { \frac { 3 } { 2 } } + Q x ^ { \frac { 1 } { 2 } } + R x ^ { - \frac { 1 } { 2 } }\), stating the values of the constants \(P , Q\) and \(R\).
2. Find f \({ } ^ { \prime } ( x )\).
3. Show that the tangent to the curve with equation \(y = \mathrm { f } ( x )\) at the point where \(x = 1\) is parallel to the line with equation \(2 y = 11 x + 3\).
   (3)
   

   6. continued

   Leave blank

   \begin{center} \begin{tabular}{|l|l|} \hline \begin{tabular}{l}

7. (a) Factorise completely \(x ^ { 3 } - 4 x\).
(3)
(b) Sketch the curve with equation \(y = x ^ { 3 } - 4 x\), showing the coordinates of the points where the curve crosses the \(x\)-axis.
(3)
(c) On a separate diagram, sketch the curve with equation \(y = ( x - 1 ) ^ { 3 } - 4 ( x - 1 ) ,\)
showing the coordinates of the points where the curve crosses the \(x\)-axis.
(3)
\end{tabular} & Leave blank
\hline \end{tabular} \end{center} \begin{center} \begin{tabular}{|l|l|} \hline \begin{tabular}{l}

8. The straight line \(l _ { 1 }\) has equation \(y = 3 x - 6\).
The straight line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\) and passes through the point (6, 2).

1. Find an equation for \(l _ { 2 }\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
   (3)
   The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(C\).
2. Use algebra to find the coordinates of \(C\).
   (3)
   The lines \(l _ { 1 }\) and \(l _ { 2 }\) cross the \(x\)-axis at the points \(A\) and \(B\) respectively.
3. Calculate the exact area of triangle \(A B C\).
   (4) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
   \end{tabular} & Leave blank
   \hline \end{tabular} \end{center}

   8. continued	Leave blank

   \begin{center} \begin{tabular}{|l|l|} \hline \begin{tabular}{l}

9. An arithmetic series has first term \(a\) and common difference \(d\).

1. Prove that the sum of the first \(n\) terms of the series is \(\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ] .\)
   (4)
   A polygon has 16 sides. The lengths of the sides of the polygon, starting with the shortest side, form an arithmetic sequence with common difference \(d \mathrm {~cm}\).
   The longest side of the polygon has length 6 cm and the perimeter of the polygon is 72 cm .
   Find
2. the length of the shortest side of the polygon,
   (5)
3. the value of \(d\).
   (2) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
   \end{tabular} & Leave blank
   \hline \end{tabular} \end{center}

   Leave blank

   \begin{center} \begin{tabular}{|l|l|} \hline \begin{tabular}{l}

10. For the curve \(C\) with equation \(y = \mathrm { f } ( x )\), \(\frac { d y } { d x } = x ^ { 3 } + 2 x - 7 .\)

1. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
   (2)
2. Show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } \geq 2\) for all values of \(x\).
   (1)
   Given that the point \(P ( 2,4 )\) lies on \(C\),
3. find \(y\) in terms of \(x\),
   (5)
4. find an equation for the normal to \(C\) at \(P\) in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers.
   (5)
   \end{tabular} & Leave blank
   \hline \end{tabular} \end{center}
   1. continued