OCR C1 (Core Mathematics 1) 2007 June

1 Simplify \(( 2 x + 5 ) ^ { 2 } - ( x - 3 ) ^ { 2 }\), giving your answer in the form \(a x ^ { 2 } + b x + c\).

2

1. On separate diagrams, sketch the graphs of
   1. \(\mathrm { y } = \frac { 1 } { \mathrm { x } }\),
   2. \(y = x ^ { 4 }\).
2. Describe a transformation that transforms the curve \(y = x ^ { 3 }\) to the curve \(y = 8 x ^ { 3 }\).

3 Simplify the following, expressing each answer in the form \(a \sqrt { 5 }\).

1. \(3 \sqrt { 10 } \times \sqrt { 2 }\)
2. \(\sqrt { 500 } + \sqrt { 125 }\)

4

1. Find the discriminant of \(k x ^ { 2 } - 4 x + k\) in terms of \(k\).
2. The quadratic equation \(k x ^ { 2 } - 4 x + k = 0\) has equal roots. Find the possible values of \(k\)

5 \includegraphics[max width=\textwidth, alt={}, center]{581ef815-59f0-434e-a7ec-9128e74c0323-2_256_1113_1366_516} The diagram shows a rectangular enclosure, with a wall forming one side. A rope, of length 20 metres, is used to form the remaining three sides. The width of the enclosure is x metres.

1. Show that the enclosed area, \(\mathrm { Am } ^ { 2 }\), is given by $$A = 20 x - 2 x ^ { 2 } .$$
2. Use differentiation to find the maximum value of A .

6 By using the substitution \(y = ( x + 2 ) ^ { 2 }\), find the real roots of the equation $$( x + 2 ) ^ { 4 } + 5 ( x + 2 ) ^ { 2 } - 6 = 0$$

7

1. Given that \(f ( x ) = x + \frac { 3 } { x }\), find \(f ^ { \prime } ( x )\).
2. Find the gradient of the curve \(\mathrm { y } = \mathrm { x } ^ { \frac { 5 } { 2 } }\) at the point where \(\mathrm { x } = 4\).

8

1. Express \(x ^ { 2 } + 8 x + 15\) in the form \(( x + a ) ^ { 2 } - b\).
2. Hence state the coordinates of the vertex of the curve \(y = x ^ { 2 } + 8 x + 15\).
3. Solve the inequality \(x ^ { 2 } + 8 x + 15 > 0\).

9 The circle with equation \(x ^ { 2 } + y ^ { 2 } - 6 x - k = 0\) has radius 4 .

1. Find the centre of the circle and the value of k . The points \(\mathrm { A } ( 3 , \mathrm { a } )\) and \(\mathrm { B } ( - 1,0 )\) lie on the circumference of the circle, with \(\mathrm { a } > 0\).
2. Calculate the length of \(A B\), giving your answer in simplified surd form.
3. Find an equation for the line \(A B\).

10

1. Solve the equation \(3 x ^ { 2 } - 14 x - 5 = 0\). A curve has equation \(\mathrm { y } = 3 \mathrm { x } ^ { 2 } - 14 \mathrm { x } - 5\).
2. Sketch the curve, indicating the coordinates of all intercepts with the axes.
3. Find the value of C for which the line \(\mathrm { y } = 4 \mathrm { x } + \mathrm { C }\) is a tangent to the curve.