OCR C1 (Core Mathematics 1) 2009 June

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

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

2 Express \(\frac { 8 + \sqrt { 7 } } { 2 + \sqrt { 7 } }\) in the form \(a + b \sqrt { 7 }\), where \(a\) and \(b\) are integers.

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

1. \(\frac { 1 } { 9 }\),
2. \(\sqrt [ 3 ] { 3 }\),
3. \(3 ^ { 10 } \times 9 ^ { 15 }\).

4 Solve the simultaneous equations $$4 x ^ { 2 } + y ^ { 2 } = 10 , \quad 2 x - y = 4$$

5

1. Expand and simplify \(( 2 x + 1 ) ( x - 3 ) ( x + 4 )\).
2. Find the coefficient of \(x ^ { 4 }\) in the expansion of $$x \left( x ^ { 2 } + 2 x + 3 \right) \left( x ^ { 2 } + 7 x - 2 \right) .$$

6

1. Sketch the curve \(y = - \sqrt { x }\).
2. Describe fully a transformation that transforms the curve \(y = - \sqrt { x }\) to the curve \(y = 5 - \sqrt { x }\).
3. The curve \(y = - \sqrt { x }\) is stretched by a scale factor of 2 parallel to the \(x\)-axis. State the equation of the curve after it has been stretched.

7

1. Express \(x ^ { 2 } - 5 x + \frac { 1 } { 4 }\) in the form \(( x - a ) ^ { 2 } - b\).
2. Find the centre and radius of the circle with equation \(x ^ { 2 } + y ^ { 2 } - 5 x + \frac { 1 } { 4 } = 0\).

8 Solve the inequalities

1. \(- 35 < 6 x + 7 < 1\),
2. \(3 x ^ { 2 } > 48\).
   \(9 \quad A\) is the point \(( 4 , - 3 )\) and \(B\) is the point \(( - 1,9 )\).

1. Calculate the length of \(A B\).
2. Find the coordinates of the mid-point of \(A B\).
3. Find the equation of the line through \(( 1,3 )\) which is parallel to \(A B\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

10

1. Solve the equation \(9 x ^ { 2 } + 18 x - 7 = 0\).
2. Find the coordinates of the stationary point on the curve \(y = 9 x ^ { 2 } + 18 x - 7\).
3. Sketch the curve \(y = 9 x ^ { 2 } + 18 x - 7\), giving the coordinates of all intercepts with the axes.
4. For what values of \(x\) does \(9 x ^ { 2 } + 18 x - 7\) increase as \(x\) increases?

11 The point \(P\) on the curve \(y = k \sqrt { x }\) has \(x\)-coordinate 4 . The normal to the curve at \(P\) is parallel to the line \(2 x + 3 y = 0\).

1. Find the value of \(k\).
2. This normal meets the \(x\)-axis at the point \(Q\). Calculate the area of the triangle \(O P Q\), where \(O\) is the point \(( 0,0 )\). RECOGNISING ACHIEVEMENT