Edexcel C1 (Core Mathematics 1) 2015 June

Simplify

1. \(( 2 \sqrt { } 5 ) ^ { 2 }\)
2. \(\frac { \sqrt { } 2 } { 2 \sqrt { } 5 - 3 \sqrt { } 2 }\) giving your answer in the form \(a + \sqrt { } b\), where \(a\) and \(b\) are integers.

Solve the simultaneous equations $$\begin{gathered} y - 2 x - 4 = 0 \\ 4 x ^ { 2 } + y ^ { 2 } + 20 x = 0 \end{gathered}$$

Given that \(y = 4 x ^ { 3 } - \frac { 5 } { x ^ { 2 } } , x \neq 0\), find in their simplest form

1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
2. \(\int y \mathrm {~d} x\)

1. A sequence \(U _ { 1 } , U _ { 2 } , U _ { 3 } , \ldots\) is defined by $$\begin{gathered} U _ { n + 2 } = 2 U _ { n + 1 } - U _ { n } , \quad n \geqslant 1 \\ U _ { 1 } = 4 \text { and } U _ { 2 } = 4 \end{gathered}$$ Find the value of
   1. \(U _ { 3 }\)
   2. \(\sum _ { n = 1 } ^ { 20 } U _ { n }\)
   3. Another sequence \(V _ { 1 } , V _ { 2 } , V _ { 3 } , \ldots\) is defined by
      (a) Find \(V _ { 3 }\) and \(V _ { 4 }\) in terms of \(k\). $$\begin{gathered} V _ { n + 2 } = 2 V _ { n + 1 } - V _ { n } , \quad n \geqslant 1 \\ V _ { 1 } = k \text { and } V _ { 2 } = 2 k , \text { where } k \text { is a constant } \end{gathered}$$ a) Find \(V _ { 3 }\)

1. The equation

$$( p - 1 ) x ^ { 2 } + 4 x + ( p - 5 ) = 0 , \text { where } p \text { is a constant }$$ has no real roots.

1. Show that \(p\) satisfies \(p ^ { 2 } - 6 p + 1 > 0\)
2. Hence find the set of possible values of \(p\).

1. The curve \(C\) has equation

$$y = \frac { \left( x ^ { 2 } + 4 \right) ( x - 3 ) } { 2 x } , \quad x \neq 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in its simplest form.
2. Find an equation of the tangent to \(C\) at the point where \(x = - 1\) Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

1. Given that \(y = 2 ^ { x }\),
   1. express \(4 ^ { x }\) in terms of \(y\).
   2. Hence, or otherwise, solve
   $$8 \left( 4 ^ { x } \right) - 9 \left( 2 ^ { x } \right) + 1 = 0$$

1. (a) Factorise completely \(9 x - 4 x ^ { 3 }\) (b) Sketch the curve \(C\) with equation

$$y = 9 x - 4 x ^ { 3 }$$ Show on your sketch the coordinates at which the curve meets the \(x\)-axis. The points \(A\) and \(B\) lie on \(C\) and have \(x\) coordinates of - 2 and 1 respectively.
(c) Show that the length of \(A B\) is \(k \sqrt { } 10\) where \(k\) is a constant to be found.

Jess started work 20 years ago. In year 1 her annual salary was \(\pounds 17000\). Her annual salary increased by \(\pounds 1500\) each year, so that her annual salary in year 2 was \(\pounds 18500\), in year 3 it was \(\pounds 20000\) and so on, forming an arithmetic sequence. This continued until she reached her maximum annual salary of \(\pounds 32000\) in year \(k\). Her annual salary then remained at \(\pounds 32000\).

1. Find the value of the constant \(k\).
2. Calculate the total amount that Jess has earned in the 20 years.

A curve with equation \(y = \mathrm { f } ( x )\) passes through the point \(( 4,9 )\). Given that $$f ^ { \prime } ( x ) = \frac { 3 \sqrt { } x } { 2 } - \frac { 9 } { 4 \sqrt { } x } + 2 , \quad x > 0$$

1. find \(\mathrm { f } ( x )\), giving each term in its simplest form. Point \(P\) lies on the curve. The normal to the curve at \(P\) is parallel to the line \(2 y + x = 0\)
2. Find the \(x\) coordinate of \(P\).