OCR C1 (Core Mathematics 1) 2008 January

1 Express \(\frac { 4 } { 3 - \sqrt { 7 } }\) in the form \(a + b \sqrt { 7 }\), where \(a\) and \(b\) are integers.

2

1. Write down the equation of the circle with centre \(( 0,0 )\) and radius 7 .
2. A circle with centre \(( 3,5 )\) has equation \(x ^ { 2 } + y ^ { 2 } - 6 x - 10 y - 30 = 0\). Find the radius of the circle.

3 Given that \(3 x ^ { 2 } + b x + 10 = a ( x + 3 ) ^ { 2 } + c\) for all values of \(x\), find the values of the constants \(a , b\) and \(c\).

4 Solve the equations

1. \(10 ^ { p } = 0.1\),
2. \(\left( 25 k ^ { 2 } \right) ^ { \frac { 1 } { 2 } } = 15\),
3. \(t ^ { - \frac { 1 } { 3 } } = \frac { 1 } { 2 }\).

5

1. Sketch the curve \(y = x ^ { 3 } + 2\).
2. Sketch the curve \(y = 2 \sqrt { x }\).
3. Describe a transformation that transforms the curve \(y = 2 \sqrt { x }\) to the curve \(y = 3 \sqrt { x }\).

6

1. Solve the equation \(x ^ { 2 } + 8 x + 10 = 0\), giving your answers in simplified surd form.
2. Sketch the curve \(y = x ^ { 2 } + 8 x + 10\), giving the coordinates of the point where the curve crosses the \(y\)-axis.
3. Solve the inequality \(x ^ { 2 } + 8 x + 10 \geqslant 0\).

7

1. Find the gradient of the line \(l\) which has equation \(x + 2 y = 4\).
2. Find the equation of the line parallel to \(l\) which passes through the point ( 6,5 ), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
3. Solve the simultaneous equations $$y = x ^ { 2 } + x + 1 \quad \text { and } \quad x + 2 y = 4$$

8

1. Find the coordinates of the stationary points on the curve \(y = x ^ { 3 } + x ^ { 2 } - x + 3\).
2. Determine whether each stationary point is a maximum point or a minimum point.
3. For what values of \(x\) does \(x ^ { 3 } + x ^ { 2 } - x + 3\) decrease as \(x\) increases?

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

1. Find the equation of the line \(A B\), giving your answer in the form \(a x + b y + c = 0\).
2. Find the coordinates of the mid-point of \(A B\). The point \(C\) has coordinates (-3,4).
3. Calculate the length of \(A C\), giving your answer in simplified surd form.
4. Determine whether the line \(A C\) is perpendicular to the line \(B C\), showing all your working.

10 Given that \(\mathrm { f } ( x ) = 8 x ^ { 3 } + \frac { 1 } { x ^ { 3 } }\),

1. find \(\mathrm { f } ^ { \prime \prime } ( x )\),
2. solve the equation \(\mathrm { f } ( x ) = - 9\).