1

How many points do the line $y = -2x + 3$ and the curve $y = x^2 + 3$ have in common? [3]

2

Solve the simultaneous equations $$\begin{aligned} y &= x^2 \\ y &= -7x - 12 \end{aligned}$$ Give the smaller of the two $x$-coordinates. [4]

3

Solve the simultaneous equations $$\begin{aligned} y &= x^2 \\ y &= -5x - 6 \end{aligned}$$ Give the smaller of the two $x$-coordinates. [4]

4

The graph shows the circle $x^2 + y^2 = 25$ (radius about 5) and the line $y = 2x + 5$.The line meets the circle at two points. Use the graph to find the smaller of the two $x$-coordinates of these points. [3]

5

The line $y = -2x + k$ is a tangent to the curve $y = x^2 - 2$. Work out the value of $k$. [4]

6

The line $y = -2x + k$ is a tangent to the curve $y = x^2 + 2$. Work out the value of $k$. [4]