Edexcel C1 (Core Mathematics 1)

1. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 5 + \frac { 1 } { x ^ { 2 } } .$$

1. Use integration to find \(y\) in terms of \(x\).
2. Given that \(y = 7\) when \(x = 1\), find the value of \(y\) at \(x = 2\).

2. The sum of an arithmetic series is $$\sum _ { r = 1 } ^ { n } ( 80 - 3 r ) .$$

1. Write down the first two terms of the series.
2. Find the common difference of the series. Given that \(n = 50\),
3. find the sum of the series.

3. The points \(A\) and \(B\) have coordinates \(( 1,2 )\) and \(( 5,8 )\) respectively.

1. Find the coordinates of the mid-point of \(A B\).
2. Find, in the form \(y = m x + c\), an equation for the straight line through \(A\) and \(B\).

4. (a) Solve the equation \(4 x ^ { 2 } + 12 x = 0\). $$f ( x ) = 4 x ^ { 2 } + 12 x + c ,$$ where \(c\) is a constant.
(b) Given that \(\mathrm { f } ( x ) = 0\) has equal roots, find the value of \(c\) and hence solve \(\mathrm { f } ( x ) = 0\).

5. Find the set of values for \(x\) for which

1. \(6 x - 7 < 2 x + 3\),
2. \(2 x ^ { 2 } - 11 x + 5 < 0\),
3. both \(6 x - 7 < 2 x + 3\) and \(2 x ^ { 2 } - 11 x + 5 < 0\).

6. Given that \(\mathrm { f } ( x ) = 15 - 7 x - 2 x ^ { 2 }\),

1. find the coordinates of all points at which the graph of \(y = \mathrm { f } ( x )\) crosses the coordinate axes.
2. Sketch the graph of \(y = \mathrm { f } ( x )\).

7. Initially the number of fish in a lake is 500000 . The population is then modelled by the recurrence relation $$u _ { n + 1 } = 1.05 u _ { n } - d , \quad u _ { 0 } = 500000 .$$ In this relation \(u _ { n }\) is the number of fish in the lake after \(n\) years and \(d\) is the number of fish which are caught each year. Given that \(d = 15000\),

1. calculate \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\) and comment briefly on your results. Given that \(d = 100000\),
2. show that the population of fish dies out during the sixth year.
3. Find the value of \(d\) which would leave the population each year unchanged.

8. A curve \(C\) has equation \(y = x ^ { 3 } - 5 x ^ { 2 } + 5 x + 2\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\). The points \(P\) and \(Q\) lie on \(C\). The gradient of \(C\) at both \(P\) and \(Q\) is 2 . The \(x\)-coordinate of \(P\) is 3 .
2. Find the \(x\)-coordinate of \(Q\).
3. Find an equation for the tangent to \(C\) at \(P\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants. This tangent intersects the coordinate axes at the points \(R\) and \(S\).
4. Find the length of \(R S\), giving your answer as a surd.

9. The points \(A ( - 1 , - 2 ) , B ( 7,2 )\) and \(C ( k , 4 )\), where \(k\) is a constant, are the vertices of \(\triangle A B C\). Angle \(A B C\) is a right angle.

1. Find the gradient of \(A B\).
2. Calculate the value of \(k\).
3. Show that the length of \(A B\) may be written in the form \(p \sqrt { 5 }\), where \(p\) is an integer to be found.
4. Find the exact value of the area of \(\triangle A B C\).
5. Find an equation for the straight line \(l\) passing through \(B\) and \(C\). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.