OCR C1 (Core Mathematics 1) 2011 January

1 The points \(A\) and \(B\) have coordinates \(( 6,1 )\) and \(( - 2,7 )\) respectively.

1. Find the length of \(A B\).
2. Find the gradient of the line \(A B\).
3. Determine whether the line \(4 x - 3 y - 10 = 0\) is perpendicular to \(A B\).

2 Given that $$( x - p ) \left( 2 x ^ { 2 } + 9 x + 10 \right) = \left( x ^ { 2 } - 4 \right) ( 2 x + q )$$ for all values of \(x\), find the constants \(p\) and \(q\).

3 Express each of the following in the form \(8 ^ { p }\) :

1. \(\sqrt { 8 }\),
2. \(\frac { 1 } { 64 }\),
3. \(2 ^ { 6 } \times 2 ^ { 2 }\).

4 By using the substitution \(u = ( 3 x - 2 ) ^ { 2 }\), find the roots of the equation $$( 3 x - 2 ) ^ { 4 } - 5 ( 3 x - 2 ) ^ { 2 } + 4 = 0$$

5

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

6 Given that \(y = \frac { 5 } { x ^ { 2 } } - \frac { 1 } { 4 x } + x\), find

1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).

7

1. Express \(4 x ^ { 2 } + 12 x - 3\) in the form \(p ( x + q ) ^ { 2 } + r\).
2. Solve the equation \(4 x ^ { 2 } + 12 x - 3 = 0\), giving your answers in simplified surd form.
3. The quadratic equation \(4 x ^ { 2 } + 12 x - k = 0\) has equal roots. Find the value of \(k\).

8

1. Find the equation of the tangent to the curve \(y = 7 + 6 x - x ^ { 2 }\) at the point \(P\) where \(x = 5\), giving your answer in the form \(a x + b y + c = 0\).
2. This tangent meets the \(x\)-axis at \(Q\). Find the coordinates of the mid-point of \(P Q\).
3. Find the equation of the line of symmetry of the curve \(y = 7 + 6 x - x ^ { 2 }\).
4. State the set of values of \(x\) for which \(7 + 6 x - x ^ { 2 }\) is an increasing function.

9 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 8 x - 2 y - 3 = 0\).

1. Find the coordinates of \(C\) and the radius of the circle.
2. Find the values of \(k\) for which the line \(y = k\) is a tangent to the circle, giving your answers in simplified surd form.
3. The points \(S\) and \(T\) lie on the circumference of the circle. \(M\) is the mid-point of the chord \(S T\). Given that the length of \(C M\) is 2 , calculate the length of the chord \(S T\).
4. Find the coordinates of the point where the circle meets the line \(x - 2 y - 12 = 0\).