OCR C1 (Core Mathematics 1) 2013 June

1 Express each of the following in the form \(a \sqrt { 5 }\), where \(a\) is an integer.

1. \(4 \sqrt { 15 } \times \sqrt { 3 }\)
2. \(\frac { 20 } { \sqrt { 5 } }\)
3. \(5 ^ { \frac { 3 } { 2 } }\)

2 Solve the equation \(8 x ^ { 6 } + 7 x ^ { 3 } - 1 = 0\).

3 It is given that \(\mathrm { f } ( x ) = \frac { 6 } { x ^ { 2 } } + 2 x\).

1. Find \(\mathrm { f } ^ { \prime } ( x )\).
2. Find \(\mathrm { f } ^ { \prime \prime } ( x )\).

4

1. Express \(3 x ^ { 2 } + 9 x + 10\) in the form \(3 ( x + p ) ^ { 2 } + q\).
2. State the coordinates of the minimum point of the curve \(y = 3 x ^ { 2 } + 9 x + 10\).
3. Calculate the discriminant of \(3 x ^ { 2 } + 9 x + 10\).

5

1. Sketch the curve \(y = \frac { 2 } { x ^ { 2 } }\).
2. The curve \(y = \frac { 2 } { x ^ { 2 } }\) is translated by 5 units in the negative \(x\)-direction. Find the equation of the curve after it has been translated.
3. Describe a transformation that transforms the curve \(y = \frac { 2 } { x ^ { 2 } }\) to the curve \(y = \frac { 1 } { x ^ { 2 } }\).

6 A circle \(C\) has equation \(x ^ { 2 } + y ^ { 2 } + 8 y - 24 = 0\).

1. Find the centre and radius of the circle.
2. The point \(A ( 2,2 )\) lies on the circumference of \(C\). Given that \(A B\) is a diameter of the circle, find the coordinates of \(B\).

7 Solve the inequalities

1. \(3 - 8 x > 4\),
2. \(( 2 x - 4 ) ( x - 3 ) \leqslant 12\).
   \(8 \quad A\) is the point \(( - 2,6 )\) and \(B\) is the point \(( 3 , - 8 )\). The line \(l\) is perpendicular to the line \(x - 3 y + 15 = 0\) and passes through the mid-point of \(A B\). Find the equation of \(l\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

9

1. Sketch the curve \(y = 2 x ^ { 2 } - x - 6\), giving the coordinates of all points of intersection with the axes.
2. Find the set of values of \(x\) for which \(2 x ^ { 2 } - x - 6\) is a decreasing function.
3. The line \(y = 4\) meets the curve \(y = 2 x ^ { 2 } - x - 6\) at the points \(P\) and \(Q\). Calculate the distance \(P Q\).

10 The curve \(y = ( 1 - x ) \left( x ^ { 2 } + 4 x + k \right)\) has a stationary point when \(x = - 3\).

1. Find the value of the constant \(k\).
2. Determine whether the stationary point is a maximum or minimum point.
3. Given that \(y = 9 x - 9\) is the equation of the tangent to the curve at the point \(A\), find the coordinates of \(A\).