OCR C1 (Core Mathematics 1) Specimen

1 Write down the exact values of

1. \(4 ^ { - 2 }\),
2. \(( 2 \sqrt { } 2 ) ^ { 2 }\),
3. \(\left( 1 ^ { 3 } + 2 ^ { 3 } + 3 ^ { 3 } \right) ^ { \frac { 1 } { 2 } }\).

2

1. Express \(x ^ { 2 } - 8 x + 3\) in the form \(( x + a ) ^ { 2 } + b\).
2. Hence write down the coordinates of the minimum point on the graph of \(y = x ^ { 2 } - 8 x + 3\).

3 The quadratic equation \(x ^ { 2 } + k x + k = 0\) has no real roots for \(x\).

1. Write down the discriminant of \(x ^ { 2 } + k x + k\) in terms of \(k\).
2. Hence find the set of values that \(k\) can take.

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

1. \(y = 4 x ^ { 3 } - 1\),
2. \(y = x ^ { 2 } \left( x ^ { 2 } + 2 \right)\),
3. \(y = \sqrt { } x\)

5

1. Solve the simultaneous equations $$y = x ^ { 2 } - 3 x + 2 , \quad y = 3 x - 7$$
2. What can you deduce from the solution to part (i) about the graphs of \(y = x ^ { 2 } - 3 x + 2\) and \(y = 3 x - 7\) ?
3. Hence, or otherwise, find the equation of the normal to the curve \(y = x ^ { 2 } - 3 x + 2\) at the point ( 3,2 ), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.

6

1. Sketch the graph of \(y = \frac { 1 } { x }\), where \(x \neq 0\), showing the parts of the graph corresponding to both positive and negative values of \(x\).
2. Describe fully the geometrical transformation that transforms the curve \(y = \frac { 1 } { x }\) to the curve \(y = \frac { 1 } { x + 2 }\). Hence sketch the curve \(y = \frac { 1 } { x + 2 }\).
3. Differentiate \(\frac { 1 } { x }\) with respect to \(x\).
4. Use parts (ii) and (iii) to find the gradient of the curve \(y = \frac { 1 } { x + 2 }\) at the point where it crosses the \(y\)-axis.

7
\includegraphics[max width=\textwidth, alt={}, center]{5fa27228-37b2-45d9-a8dc-355b2f7f6fa4-3_757_810_1050_680} The diagram shows a circle which passes through the points \(A ( 2,9 )\) and \(B ( 10,3 ) . A B\) is a diameter of the circle.

1. Calculate the radius of the circle and the coordinates of the centre.
2. Show that the equation of the circle may be written in the form \(x ^ { 2 } + y ^ { 2 } - 12 x - 12 y + 47 = 0\).
3. The tangent to the circle at the point \(B\) cuts the \(x\)-axis at \(C\). Find the coordinates of \(C\).

8

1. Find the coordinates of the stationary points on the curve \(y = 2 x ^ { 3 } - 3 x ^ { 2 } - 12 x - 7\).
2. Determine whether each stationary point is a maximum point or a minimum point.
3. By expanding the right-hand side, show that $$2 x ^ { 3 } - 3 x ^ { 2 } - 12 x - 7 = ( x + 1 ) ^ { 2 } ( 2 x - 7 )$$
4. Sketch the curve \(y = 2 x ^ { 3 } - 3 x ^ { 2 } - 12 x - 7\), marking the coordinates of the stationary points and the points where the curve meets the axes.