OCR C1 (Core Mathematics 1) 2009 January

1 Express \(\sqrt { 45 } + \frac { 20 } { \sqrt { 5 } }\) in the form \(k \sqrt { 5 }\), where \(k\) is an integer.

2 Simplify

1. \(( \sqrt [ 3 ] { x } ) ^ { 6 }\),
2. \(\frac { 3 y ^ { 4 } \times ( 10 y ) ^ { 3 } } { 2 y ^ { 5 } }\).

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

4

1. Sketch the curve \(y = \frac { 1 } { x ^ { 2 } }\).
2. The curve \(y = \frac { 1 } { x ^ { 2 } }\) is translated by 3 units in the negative \(x\)-direction. State the equation of the curve after it has been translated.
3. The curve \(y = \frac { 1 } { x ^ { 2 } }\) is stretched parallel to the \(y\)-axis with scale factor 4 and, as a result, the point \(P ( 1,1 )\) is transformed to the point \(Q\). State the coordinates of \(Q\).

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

1. \(y = 10 x ^ { - 5 }\),
2. \(y = \sqrt [ 4 ] { x }\),
3. \(y = x ( x + 3 ) ( 1 - 5 x )\).

6

1. Express \(5 x ^ { 2 } + 20 x - 8\) in the form \(p ( x + q ) ^ { 2 } + r\).
2. State the equation of the line of symmetry of the curve \(y = 5 x ^ { 2 } + 20 x - 8\).
3. Calculate the discriminant of \(5 x ^ { 2 } + 20 x - 8\).
4. State the number of real roots of the equation \(5 x ^ { 2 } + 20 x - 8 = 0\).

7 The line with equation \(3 x + 4 y - 10 = 0\) passes through point \(A ( 2,1 )\) and point \(B ( 10 , k )\).

1. Find the value of \(k\).
2. Calculate the length of \(A B\). A circle has equation \(( x - 6 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 25\).
3. Write down the coordinates of the centre and the radius of the circle.
4. Verify that \(A B\) is a diameter of the circle.

8

1. Solve the equation \(5 - 8 x - x ^ { 2 } = 0\), giving your answers in simplified surd form.
2. Solve the inequality \(5 - 8 x - x ^ { 2 } \leqslant 0\).
3. Sketch the curve \(y = \left( 5 - 8 x - x ^ { 2 } \right) ( x + 4 )\), giving the coordinates of the points where the curve crosses the coordinate axes.

9 The curve \(y = x ^ { 3 } + p x ^ { 2 } + 2\) has a stationary point when \(x = 4\). Find the value of the constant \(p\) and determine whether the stationary point is a maximum or minimum point.

10 A curve has equation \(y = x ^ { 2 } + x\).

1. Find the gradient of the curve at the point for which \(x = 2\).
2. Find the equation of the normal to the curve at the point for which \(x = 2\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
3. Find the values of \(k\) for which the line \(y = k x - 4\) is a tangent to the curve.