OCR C1 (Core Mathematics 1) 2006 June

1 The points \(A ( 1,3 )\) and \(B ( 4,21 )\) lie on the curve \(y = x ^ { 2 } + x + 1\).

1. Find the gradient of the line \(A B\).
2. Find the gradient of the curve \(y = x ^ { 2 } + x + 1\) at the point where \(x = 3\).

2

1. Evaluate \(27 ^ { - \frac { 2 } { 3 } }\).
2. Express \(5 \sqrt { 5 }\) in the form \(5 ^ { n }\).
3. Express \(\frac { 1 - \sqrt { 5 } } { 3 + \sqrt { 5 } }\) in the form \(a + b \sqrt { 5 }\).

3

1. Express \(2 x ^ { 2 } + 12 x + 13\) in the form \(a ( x + b ) ^ { 2 } + c\).
2. Solve \(2 x ^ { 2 } + 12 x + 13 = 0\), giving your answers in simplified surd form.

4

1. By expanding the brackets, show that $$( x - 4 ) ( x - 3 ) ( x + 1 ) = x ^ { 3 } - 6 x ^ { 2 } + 5 x + 12 .$$
2. Sketch the curve $$y = x ^ { 3 } - 6 x ^ { 2 } + 5 x + 12 ,$$ giving the coordinates of the points where the curve meets the axes. Label the curve \(C _ { 1 }\).
3. On the same diagram as in part (ii), sketch the curve $$y = - x ^ { 3 } + 6 x ^ { 2 } - 5 x - 12$$ Label this curve \(C _ { 2 }\).

5 Solve the inequalities

1. \(1 < 4 x - 9 < 5\),
2. \(y ^ { 2 } \geqslant 4 y + 5\).

6

1. Solve the equation \(x ^ { 4 } - 10 x ^ { 2 } + 25 = 0\).
2. Given that \(y = \frac { 2 } { 5 } x ^ { 5 } - \frac { 20 } { 3 } x ^ { 3 } + 50 x + 3\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
3. Hence find the number of stationary points on the curve \(y = \frac { 2 } { 5 } x ^ { 5 } - \frac { 20 } { 3 } x ^ { 3 } + 50 x + 3\).
4. Solve the simultaneous equations $$y = x ^ { 2 } - 5 x + 4 , \quad y = x - 1$$
5. State the number of points of intersection of the curve \(y = x ^ { 2 } - 5 x + 4\) and the line \(y = x - 1\).
6. Find the value of \(c\) for which the line \(y = x + c\) is a tangent to the curve \(y = x ^ { 2 } - 5 x + 4\).

8 A cuboid has a volume of \(8 \mathrm {~m} ^ { 3 }\). The base of the cuboid is square with sides of length \(x\) metres. The surface area of the cuboid is \(A \mathrm {~m} ^ { 2 }\).

1. Show that \(A = 2 x ^ { 2 } + \frac { 32 } { x }\).
2. Find \(\frac { \mathrm { d } A } { \mathrm {~d} x }\).
3. Find the value of \(x\) which gives the smallest surface area of the cuboid, justifying your answer.

9 The points \(A\) and \(B\) have coordinates \(( 4 , - 2 )\) and \(( 10,6 )\) respectively. \(C\) is the mid-point of \(A B\). Find

1. the coordinates of \(C\),
2. the length of \(A C\),
3. the equation of the circle that has \(A B\) as a diameter,
4. the equation of the tangent to the circle in part (iii) at the point \(A\), giving your answer in the form \(a x + b y = c\).