OCR C1 (Core Mathematics 1)

1. Solve the inequality

$$x ( 2 x + 1 ) \leq 6 .$$

1. Differentiate with respect to \(x\)

$$3 x ^ { 2 } - \sqrt { x } + \frac { 1 } { 2 x }$$

1. The straight line \(l\) has the equation \(x - 2 y = 12\) and meets the coordinate axes at the points \(A\) and \(B\).

Find the distance of the mid-point of \(A B\) from the origin, giving your answer in the form \(k \sqrt { 5 }\).

4. (i) By completing the square, find in terms of the constant \(k\) the roots of the equation $$x ^ { 2 } + 2 k x + 4 = 0 .$$ (ii) Hence find the exact roots of the equation $$x ^ { 2 } + 6 x + 4 = 0 .$$

1. The curve with equation \(y = \sqrt { 8 x }\) passes through the point \(A\) with \(x\)-coordinate 2 .

Find an equation for the tangent to the curve at \(A\).

6. $$f ( x ) = x ^ { \frac { 3 } { 2 } } - 8 x ^ { - \frac { 1 } { 2 } }$$

1. Evaluate \(\mathrm { f } ( 3 )\), giving your answer in its simplest form with a rational denominator.
2. Solve the equation \(\mathrm { f } ( x ) = 0\), giving your answers in the form \(k \sqrt { 2 }\).

7. Solve the simultaneous equations $$\begin{aligned} & x - 3 y + 7 = 0
& x ^ { 2 } + 2 x y - y ^ { 2 } = 7 \end{aligned}$$

8. The diagram shows the circle with equation \(x ^ { 2 } + y ^ { 2 } - 2 x - 18 y + 73 = 0\) and the straight line with equation \(y = 2 x - 3\).

1. Find the coordinates of the centre and the radius of the circle.
2. Find the coordinates of the point on the line which is closest to the circle.

9. \(f ( x ) = 2 x ^ { 2 } + 3 x - 2\).

1. Solve the equation \(\mathrm { f } ( x ) = 0\).
2. Sketch the curve with equation \(y = \mathrm { f } ( x )\), showing the coordinates of any points of intersection with the coordinate axes.
3. Find the coordinates of the points where the curve with equation \(y = \mathrm { f } \left( \frac { 1 } { 2 } x \right)\) crosses the coordinate axes. When the graph of \(y = \mathrm { f } ( x )\) is translated by 1 unit in the positive \(x\)-direction it maps onto the graph with equation \(y = a x ^ { 2 } + b x + c\), where \(a , b\) and \(c\) are constants.
4. Find the values of \(a , b\) and \(c\).

10. The curve with equation \(y = ( 2 - x ) ( 3 - x ) ^ { 2 }\) crosses the \(x\)-axis at the point \(A\) and touches the \(x\)-axis at the point \(B\).

1. Sketch the curve, showing the coordinates of \(A\) and \(B\).
2. Show that the tangent to the curve at \(A\) has the equation $$x + y = 2$$ Given that the curve is stationary at the points \(B\) and \(C\),
3. find the exact coordinates of \(C\).