OCR C1 (Core Mathematics 1) 2008 June

1 Express each of the following in the form \(4 ^ { n }\) :

1. \(\frac { 1 } { 16 }\),
2. 64 ,
3. 8 .

2

1. The curve \(y = x ^ { 2 }\) is translated 2 units in the positive \(x\)-direction. Find the equation of the curve after it has been translated.
2. The curve \(y = x ^ { 3 } - 4\) is reflected in the \(x\)-axis. Find the equation of the curve after it has been reflected.

3 Express each of the following in the form \(k \sqrt { 2 }\), where \(k\) is an integer:

1. \(\sqrt { 200 }\),
2. \(\frac { 12 } { \sqrt { 2 } }\),
3. \(5 \sqrt { 8 } - 3 \sqrt { 2 }\).

4 Solve the equation \(2 x - 7 x ^ { \frac { 1 } { 2 } } + 3 = 0\).

5 Find the gradient of the curve \(y = 8 \sqrt { x } + x\) at the point whose \(x\)-coordinate is 9 .

6

1. Expand and simplify \(( x - 5 ) ( x + 2 ) ( x + 5 )\).
2. Sketch the curve \(y = ( x - 5 ) ( x + 2 ) ( x + 5 )\), giving the coordinates of the points where the curve crosses the axes.

7 Solve the inequalities

1. \(8 < 3 x - 2 < 11\),
2. \(y ^ { 2 } + 2 y \geqslant 0\).

8 The curve \(y = x ^ { 3 } - k x ^ { 2 } + x - 3\) has two stationary points.

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Given that there is a stationary point when \(x = 1\), find the value of \(k\).
3. Determine whether this stationary point is a minimum or maximum point.
4. Find the \(x\)-coordinate of the other stationary point.

9

1. Find the equation of the circle with radius 10 and centre ( 2,1 ), giving your answer in the form \(x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0\).
2. The circle passes through the point \(( 5 , k )\) where \(k > 0\). Find the value of \(k\) in the form \(p + \sqrt { q }\).
3. Determine, showing all working, whether the point \(( - 3,9 )\) lies inside or outside the circle.
4. Find an equation of the tangent to the circle at the point ( 8,9 ).

10

1. Express \(2 x ^ { 2 } - 6 x + 11\) in the form \(p ( x + q ) ^ { 2 } + r\).
2. State the coordinates of the vertex of the curve \(y = 2 x ^ { 2 } - 6 x + 11\).
3. Calculate the discriminant of \(2 x ^ { 2 } - 6 x + 11\).
4. State the number of real roots of the equation \(2 x ^ { 2 } - 6 x + 11 = 0\).
5. Find the coordinates of the points of intersection of the curve \(y = 2 x ^ { 2 } - 6 x + 11\) and the line \(7 x + y = 14\).