Edexcel C1 (Core Mathematics 1)

1. (a) Given that \(8 = 2 ^ { k }\), write down the value of \(k\).
   (b) Given that \(4 ^ { x } = 8 ^ { 2 - x }\), find the value of \(x\).
2. Given that \(( 2 + \sqrt { 7 } ) ( 4 - \sqrt { 7 } ) = a + b \sqrt { 7 }\), where a and \(b\) are integers,
   (a) find the value of a and the value of \(b\).

Given that \(\frac { 2 + \sqrt { 7 } } { 4 + \sqrt { 7 } } = c + d \sqrt { 7 }\) where \(c\) and \(d\) are rational numbers,
(b) find the value of \(c\) and the value of \(d\).

3. (a) Solve the inequality \(3 x - 8 > x + 13\).
(b) Solve the inequality \(x ^ { 2 } - 5 x - 14 > 0\).

4. (a) Prove, by completing the square, that the roots of the equation \(x ^ { 2 } + 2 k x + c = 0\), where \(k\) and \(c\) are constants, are \(- k \pm \sqrt { } \left( k ^ { 2 } - c \right)\). The equation \(x ^ { 2 } + 2 k x \pm 81 = 0\) has equal roots.
(b) Find the possible values of \(k\).

5. Solve the simultaneous equations \(x - 3 y + 1 = 0 , \quad x ^ { 2 } - 3 x y + y ^ { 2 } = 11\).

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

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

7. Each year, for 40 years, Anne will pay money into a savings scheme. In the first year she pays \(\pounds 500\). Her payments then increase by \(\pounds 50\) each year, so that she pays \(\pounds 550\) in the second year, \(\pounds 600\) in the third year, and so on.

1. Find the amount that Anne will pay in the 40th year.
2. Find the total amount that Anne will pay in over the 40 years. Over the same 40 years, Brian will also pay money into the savings scheme. In the first year he pays in \(\pounds 890\) and his payments then increase by \(\pounds d\) each year. Given that Brian and Anne will pay in exactly the same amount over the 40 years,
3. find the value of \(d\).

8. The points \(A\) and \(B\) have coordinates \(( 4,6 )\) and \(( 12,2 )\) respectively. The straight line \(l _ { 1 }\) passes through \(A\) and \(B\).

1. Find an equation for \(l _ { 1 }\) in the form \(a x + b y = c\), where \(a\), b and \(c\) are integers. The straight line \(l _ { 2 }\) passes through the origin and has gradient - 4 .
2. Write down an equation for \(l _ { 2 }\). The lines \(l _ { 1 }\) and \(l _ { 2 }\) intercept at the point \(C\).
3. Find the exact coordinates of the mid-point of \(A C\).

9. 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.

10. \begin{figure}[h] \end{figure} The points \(A ( 3,0 )\) and \(B ( 0,4 )\) are two vertices of the rectangle \(A B C D\), as shown in Fig. 1.

1. Write down the gradient of \(A B\) and hence the gradient of \(B C\). The point \(C\) has coordinates \(( 8 , k )\), where \(k\) is a positive constant.
2. Find the length of \(B C\) in terms of \(k\). Given that the length of \(B C\) is 10 and using your answer to part (b),
3. find the value of \(k\),
4. find the coordinates of \(D\).