Edexcel C1 (Core Mathematics 1) 2014 January

1. Simplify fully
   1. \(( 2 \sqrt { } x ) ^ { 2 }\)
   2. \(\frac { 5 + \sqrt { 7 } } { 2 + \sqrt { 7 } }\)
      

2. $$y = 2 x ^ { 2 } - \frac { 4 } { \sqrt { } x } + 1 , \quad x > 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving each term in its simplest form.
2. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\), giving each term in its simplest form.
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3. Solve the simultaneous equations $$\begin{gathered} x - 2 y - 1 = 0
x ^ { 2 } + 4 y ^ { 2 } - 10 x + 9 = 0 \end{gathered}$$

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of a curve with equation \(y = \mathrm { f } ( x )\). The curve crosses the \(y\)-axis at \(( 0,3 )\) and has a minimum at \(P ( 4,2 )\). On separate diagrams, sketch the curve with equation

1. \(y = \mathrm { f } ( x + 4 )\),
2. \(y = 2 \mathrm { f } ( x )\). On each diagram, show clearly the coordinates of the minimum point and any point of intersection with the \(y\)-axis.

5. Given that for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } a _ { r } = 12 + 4 n ^ { 2 }$$

1. find the value of \(\sum _ { r = 1 } ^ { 5 } a _ { r }\)
2. Find the value of \(a _ { 6 }\)

6. \begin{figure}[h] \end{figure} The straight line \(l _ { 1 }\) has equation \(2 y = 3 x + 7\)
The line \(l _ { 1 }\) crosses the \(y\)-axis at the point \(A\) as shown in Figure 2.

1. 1. State the gradient of \(l _ { 1 }\)
   2. Write down the coordinates of the point \(A\). Another straight line \(l _ { 2 }\) intersects \(l _ { 1 }\) at the point \(B ( 1,5 )\) and crosses the \(x\)-axis at the point \(C\), as shown in Figure 2. Given that \(\angle A B C = 90 ^ { \circ }\),
2. find an equation of \(l _ { 2 }\) in the form \(a x + b y + c = 0\), where \(a\), b and \(c\) are integers. The rectangle \(A B C D\), shown shaded in Figure 2, has vertices at the points \(A , B , C\) and \(D\).
3. Find the exact area of rectangle \(A B C D\).

1. Shelim starts his new job on a salary of \(\pounds 14000\). He will receive a rise of \(\pounds 1500\) a year for each full year that he works, so that he will have a salary of \(\pounds 15500\) in year 2 , a salary of \(\pounds 17000\) in year 3 and so on. When Shelim’s salary reaches \(\pounds 26000\), he will receive no more rises. His salary will remain at \(\pounds 26000\).
   1. Show that Shelim will have a salary of \(\pounds 26000\) in year 9 .
   2. Find the total amount that Shelim will earn in his job in the first 9 years.
   Anna starts her new job at the same time as Shelim on a salary of \(\pounds A\). She receives a rise of \(\pounds 1000\) a year for each full year that she works, so that she has a salary of \(\pounds ( A + 1000 )\) in year \(2 , \pounds ( A + 2000 )\) in year 3 and so on. The maximum salary for her job, which is reached in year 10 , is also \(\pounds 26000\).
2. Find the difference in the total amount earned by Shelim and Anna in the first 10 years.

1. The equation \(2 x ^ { 2 } + 2 k x + ( k + 2 ) = 0\), where \(k\) is a constant, has two distinct real roots.
   1. Show that \(k\) satisfies
   $$k ^ { 2 } - 2 k - 4 > 0$$
2. Find the set of possible values of \(k\).

9. A curve with equation \(y = \mathrm { f } ( x )\) passes through the point ( 3,6 ). Given that $$f ^ { \prime } ( x ) = ( x - 2 ) ( 3 x + 4 )$$

1. use integration to find \(\mathrm { f } ( x )\). Give your answer as a polynomial in its simplest form.
2. Show that \(\mathrm { f } ( x ) \equiv ( x - 2 ) ^ { 2 } ( x + p )\), where \(p\) is a positive constant. State the value of \(p\).
3. Sketch the graph of \(y = \mathrm { f } ( x )\), showing the coordinates of any points where the curve touches or crosses the coordinate axes.

10. The curve \(C\) has equation \(y = x ^ { 3 } - 2 x ^ { 2 } - x + 3\) The point \(P\), which lies on \(C\), has coordinates \(( 2,1 )\).

1. Show that an equation of the tangent to \(C\) at the point \(P\) is \(y = 3 x - 5\) The point \(Q\) also lies on \(C\).
   Given that the tangent to \(C\) at \(Q\) is parallel to the tangent to \(C\) at \(P\),
2. find the coordinates of the point \(Q\).