OCR C1 (Core Mathematics 1) 2011 June

1 Express \(3 x ^ { 2 } - 18 x + 4\) in the form \(p ( x + q ) ^ { 2 } + r\).

2

1. Sketch the curve \(y = \frac { 1 } { x }\).
2. Describe fully the single transformation that transforms the curve \(y = \frac { 1 } { x }\) to the curve \(y = \frac { 1 } { x } + 4\).

3 Simplify

1. \(\frac { ( 4 x ) ^ { 2 } \times 2 x ^ { 3 } } { x }\),
2. \(\left( 36 x ^ { - 2 } \right) ^ { - \frac { 1 } { 2 } }\).

4 Solve the simultaneous equations $$y = 2 ( x - 2 ) ^ { 2 } , \quad 3 x + y = 26$$

5

1. Express \(\sqrt { 300 } - \sqrt { 48 }\) in the form \(k \sqrt { 3 }\), where \(k\) is an integer.
2. Express \(\frac { 15 + \sqrt { 40 } } { \sqrt { 5 } }\) in the form \(a \sqrt { 5 } + b \sqrt { 2 }\), where \(a\) and \(b\) are integers.

6 Solve the equation \(3 x ^ { \frac { 1 } { 2 } } - 8 x ^ { \frac { 1 } { 4 } } + 4 = 0\).

7 Solve the inequalities

1. \(- 9 \leqslant 6 x + 5 \leqslant 0\),
2. \(6 x + 5 < x ^ { 2 } + 2 x - 7\).

8

1. Find the coordinates of the stationary point on the curve \(y = 3 x ^ { 2 } - \frac { 6 } { x } - 2\).
2. Determine whether the stationary point is a maximum point or a minimum point.

9 The points \(A ( 1,3 ) , B ( 7,1 )\) and \(C ( - 3 , - 9 )\) are joined to form a triangle.

1. Show that this triangle is right-angled and state whether the right angle is at \(A , B\) or \(C\).
2. The points \(A , B\) and \(C\) lie on the circumference of a circle. Find the equation of the circle in the form \(x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0\).

10 A curve has equation \(y = ( 2 x - 1 ) ( x + 3 ) ( x - 1 )\).

1. Sketch the curve, indicating the coordinates of all points of intersection with the axes.
2. Show that the gradient of the curve at the point \(P ( 1,0 )\) is 4 .
3. The line \(l\) is parallel to the tangent to the curve at the point \(P\). The curve meets \(l\) at the point where \(x = - 2\). Find the equation of \(l\), giving your answer in the form \(y = m x + c\).
4. Determine whether \(l\) is a tangent to the curve at the point where \(x = - 2\).