Edexcel C1 (Core Mathematics 1) 2011 June

Find the value of

1. \(25 ^ { \frac { 1 } { 2 } }\)
2. \(25 ^ { - \frac { 3 } { 2 } }\)

Given that \(y = 2 x ^ { 5 } + 7 + \frac { 1 } { x ^ { 3 } } , x \neq 0\), find, in their simplest form, (a) \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
(b) \(\int y \mathrm {~d} x\).

The points \(P\) and \(Q\) have coordinates \(( - 1,6 )\) and \(( 9,0 )\) respectively. The line \(l\) is perpendicular to \(P Q\) and passes through the mid-point of \(P Q\).
Find an equation for \(l\), giving your answer in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers.

4. Solve the simultaneous equations $$\begin{aligned} x + y & = 2
4 y ^ { 2 } - x ^ { 2 } & = 11 \end{aligned}$$

5. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$\begin{aligned} a _ { 1 } & = k
a _ { n + 1 } & = 5 a _ { n } + 3 , \quad n \geqslant 1 , \end{aligned}$$ where \(k\) is a positive integer.

1. Write down an expression for \(a _ { 2 }\) in terms of \(k\).
2. Show that \(a _ { 3 } = 25 k + 18\).
3. 1. Find \(\sum _ { r = 1 } ^ { 4 } a _ { r }\) in terms of \(k\), in its simplest form.
   2. Show that \(\sum _ { r = 1 } ^ { 4 } a _ { r }\) is divisible by 6 .

6. Given that \(\frac { 6 x + 3 x ^ { \frac { 5 } { 2 } } } { \sqrt { } x }\) can be written in the form \(6 x ^ { p } + 3 x ^ { q }\),

1. write down the value of \(p\) and the value of \(q\). Given that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 x + 3 x ^ { \frac { 5 } { 2 } } } { \sqrt { } x }\), and that \(y = 90\) when \(x = 4\),
2. find \(y\) in terms of \(x\), simplifying the coefficient of each term.

7. $$\mathrm { f } ( x ) = x ^ { 2 } + ( k + 3 ) x + k$$ where \(k\) is a real constant.

1. Find the discriminant of \(\mathrm { f } ( x )\) in terms of \(k\).
2. Show that the discriminant of \(\mathrm { f } ( x )\) can be expressed in the form \(( k + a ) ^ { 2 } + b\), where \(a\) and \(b\) are integers to be found.
3. Show that, for all values of \(k\), the equation \(\mathrm { f } ( x ) = 0\) has real roots.

8. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\).
The curve \(C\) passes through the origin and through \(( 6,0 )\).
The curve \(C\) has a minimum at the point \(( 3 , - 1 )\). On separate diagrams, sketch the curve with equation

1. \(y = \mathrm { f } ( 2 x )\),
2. \(y = - \mathrm { f } ( x )\),
3. \(y = \mathrm { f } ( x + p )\), where \(p\) is a constant and \(0 < p < 3\). On each diagram show the coordinates of any points where the curve intersects the \(x\)-axis and of any minimum or maximum points.

1. (a) Calculate the sum of all the even numbers from 2 to 100 inclusive,

$$2 + 4 + 6 + \ldots \ldots + 100$$ (b) In the arithmetic series $$k + 2 k + 3 k + \ldots \ldots + 100$$ \(k\) is a positive integer and \(k\) is a factor of 100 .

1. Find, in terms of \(k\), an expression for the number of terms in this series.
2. Show that the sum of this series is $$50 + \frac { 5000 } { k }$$ (c) Find, in terms of \(k\), the 50th term of the arithmetic sequence $$( 2 k + 1 ) , ( 4 k + 4 ) , ( 6 k + 7 ) , \ldots \ldots ,$$ giving your answer in its simplest form.

10. The curve \(C\) has equation $$y = ( x + 1 ) ( x + 3 ) ^ { 2 }$$

1. Sketch \(C\), showing the coordinates of the points at which \(C\) meets the axes.
2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } + 14 x + 15\). The point \(A\), with \(x\)-coordinate - 5 , lies on \(C\).
3. Find the equation of the tangent to \(C\) at \(A\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants. Another point \(B\) also lies on \(C\). The tangents to \(C\) at \(A\) and \(B\) are parallel.
4. Find the \(x\)-coordinate of \(B\).