Edexcel C1 (Core Mathematics 1)

1. Given that \(2 ^ { x } = \frac { 1 } { \sqrt { 2 } }\) and \(2 ^ { y } = 4 \sqrt { } 2\),
   1. find the exact value of \(x\) and the exact value of \(y\),
   2. calculate the exact value of \(2 ^ { y - x }\).
   3. \(f ( x ) = \frac { \left( x ^ { 2 } - 3 \right) ^ { 2 } } { x ^ { 3 } } , x \neq 0\).
   4. Show that \(\mathrm { f } ( x ) \equiv x - 6 x ^ { - 1 } + 9 x ^ { - 3 }\).
   5. Hence, or otherwise, differentiate \(\mathrm { f } ( x )\) with respect to \(x\).
   6. The sum of an arithmetic series is \(\sum _ { r = 1 } ^ { n } ( 80 - 3 r )\).
   7. Write down the first two terms of the series.
   8. Find the common difference of the series.
   Given that \(n = 50\),
2. find the sum of the series.

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

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

5. The equation \(x ^ { 2 } + 5 k x + 2 k = 0\), where \(k\) is a constant, has real roots.

1. Prove that \(k ( 25 k - 8 ) \geq 0\).
2. Hence find the set of possible values of \(k\).
3. Write down the values of \(k\) for which the equation \(x ^ { 2 } + 5 k x + 2 k = 0\) has equal roots.

6. Initially the number of fish in a lake is 500000 . The population is then modelled by the recurrence relation \(\quad u _ { n + 1 } = 1.05 u _ { n } - d , \quad u _ { 0 } = 500000\).
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.

7. \begin{figure}[h] \end{figure} Fig. 1 shows the curve with equation \(y ^ { 2 } = 4 ( x - 2 )\) and the line with equation \(2 x - 3 y = 12\).
The curve crosses the \(x\)-axis at the point \(A\), and the line intersects the curve at the points \(P\) and \(Q\).

1. Write down the coordinates of \(A\).
2. Find, using algebra, the coordinates of \(P\) and \(Q\).
3. Show that \(\angle P A Q\) is a right angle.

8. 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.

9. The curve \(C\) has equation \(y = \mathrm { f } ( x )\). Given that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - 20 x + 29\) and that \(C\) passes through the point \(P ( 2,6 )\),

1. find \(y\) in terms of \(x\).
2. Verify that \(C\) passes through the point \(( 4,0 )\).
3. Find an equation of the tangent to \(C\) at \(P\). The tangent to \(C\) at the point \(Q\) is parallel to the tangent at \(P\).
4. Calculate the exact \(x\)-coordinate of \(Q\).