OCR C1 (Core Mathematics 1) 2013 January

1

1. Solve the equation \(x ^ { 2 } - 6 x - 2 = 0\), giving your answers in simplified surd form.
2. Find the gradient of the curve \(y = x ^ { 2 } - 6 x - 2\) at the point where \(x = - 5\).

2 Solve the equations

1. \(3 ^ { n } = 1\),
2. \(t ^ { - 3 } = 64\),
3. \(\left( 8 p ^ { 6 } \right) ^ { \frac { 1 } { 3 } } = 8\).

3

1. Sketch the curve \(y = ( 1 + x ) ( 2 - x ) ( 3 + x )\), giving the coordinates of all points of intersection with the axes.
2. Describe the transformation that transforms the curve \(y = ( 1 + x ) ( 2 - x ) ( 3 + x )\) to the curve \(y = ( 1 - x ) ( 2 + x ) ( 3 - x )\).

4

1. Solve the simultaneous equations $$y = 2 x ^ { 2 } - 3 x - 5 , \quad 10 x + 2 y + 11 = 0$$
2. What can you deduce from the answer to part (i) about the curve \(y = 2 x ^ { 2 } - 3 x - 5\) and the line \(10 x + 2 y + 11 = 0\) ?
3. Simplify \(( x + 4 ) ( 5 x - 3 ) - 3 ( x - 2 ) ^ { 2 }\).
4. The coefficient of \(x ^ { 2 }\) in the expansion of $$( x + 3 ) ( x + k ) ( 2 x - 5 )$$ is - 3 . Find the value of the constant \(k\).

6

1. The line joining the points ( \(- 2,7\) ) and ( \(- 4 , p\) ) has gradient 4 . Find the value of \(p\).
2. The line segment joining the points \(( - 2,7 )\) and \(( 6 , q )\) has mid-point \(( m , 5 )\). Find \(m\) and \(q\).
3. The line segment joining the points \(( - 2,7 )\) and \(( d , 3 )\) has length \(2 \sqrt { 13 }\). Find the two possible values of \(d\).

7 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in each of the following cases:

1. \(y = \frac { ( 3 x ) ^ { 2 } \times x ^ { 4 } } { x }\),
2. \(y = \sqrt [ 3 ] { x }\),
3. \(y = \frac { 1 } { 2 x ^ { 3 } }\).

8 The quadratic equation \(k x ^ { 2 } + ( 3 k - 1 ) x - 4 = 0\) has no real roots. Find the set of possible values of \(k\).

9 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 2 x + 10 y - 19 = 0\).

1. Find the coordinates of \(C\) and the radius of the circle.
2. Verify that the point \(( 7 , - 2 )\) lies on the circumference of the circle.
3. Find the equation of the tangent to the circle at the point \(( 7 , - 2 )\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

10 Find the coordinates of the points on the curve \(y = \frac { 1 } { 3 } x ^ { 3 } + \frac { 9 } { x }\) at which the tangent is parallel to the line \(y = 8 x + 3\).