OCR C1 (Core Mathematics 1) 2015 June

1 Express \(\frac { 8 } { \sqrt { 3 } - 1 }\) in the form \(a \sqrt { 3 } + b\), where \(a\) and \(b\) are integers.

2

1. Sketch the curve \(y = - \frac { 1 } { x }\).
2. The curve \(y = - \frac { 1 } { x }\) is translated by 2 units parallel to the \(x\)-axis in the positive direction. State the equation of the transformed curve.
3. Describe a transformation that transforms the curve \(y = - \frac { 1 } { x }\) to the curve \(y = - \frac { 1 } { 3 x }\).

3 Express each of the following in the form \(5 ^ { k }\).

1. \(25 ^ { 4 }\)
2. \(\frac { 1 } { \sqrt [ 4 ] { 5 } }\)
3. \(( 5 \sqrt { 5 } ) ^ { 3 }\)

4 Solve the equation \(x ^ { \frac { 2 } { 3 } } - x ^ { \frac { 1 } { 3 } } - 6 = 0\).

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

1. Find the length of \(A B\).
2. Find an equation of the line through the mid-point of \(A B\) which is perpendicular to \(A B\), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.

6 Solve the simultaneous equations $$2 x + y - 5 = 0 , \quad x ^ { 2 } - y ^ { 2 } = 3$$

7

1. Given that \(\mathrm { f } ( x ) = \left( x ^ { 2 } + 3 \right) ( 5 - x )\), find \(\mathrm { f } ^ { \prime } ( x )\).
2. Find the gradient of the curve \(y = x ^ { - \frac { 1 } { 3 } }\) at the point where \(x = - 8\).

8

1. Sketch the curve \(y = 2 x ^ { 2 } - x - 3\), giving the coordinates of all points of intersection with the axes.
2. Hence, or otherwise, solve the inequality \(2 x ^ { 2 } - x - 3 > 0\).
3. Given that the equation \(2 x ^ { 2 } - x - 3 = k\) has no real roots, find the set of possible values of the constant \(k\).

9 The curve \(y = 2 x ^ { 3 } - a x ^ { 2 } + 8 x + 2\) passes through the point \(B\) where \(x = 4\).

1. Given that \(B\) is a stationary point of the curve, find the value of the constant \(a\).
2. Determine whether the stationary point \(B\) is a maximum point or a minimum point.
3. Find the \(x\)-coordinate of the other stationary point of the curve.

10 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 10 x + 4 y + 4 = 0\).

1. Find the coordinates of \(C\) and the radius of the circle.
2. Show that the tangent to the circle at the point \(P ( 8,2 )\) has equation \(3 x + 4 y = 32\).
3. The circle meets the \(y\)-axis at \(Q\) and the tangent meets the \(y\)-axis at \(R\). Find the area of triangle \(P Q R\).