1. 1
   The line $L_1$ has equation $3x - 4y + 12 = 0$.

   (a)

   Find the gradient of $L_1$.

   [1 mark]

   • A$-\frac34$
   • B$\frac34$
   • C$\frac43$
   • D$3$

   (b)

   Find the $y$-intercept of $L_1$.

   [1 mark]

   • A$3$
   • B$-3$
   • C$12$
   • D$-4$

   (c)

   The line $L_2$ is perpendicular to $L_1$ and passes through the point $(6, -1)$. Find the equation of $L_2$, giving your answer in the form $ax + by + d = 0$, where $a, b, d\in\mathbb{Z}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The points $P(-2, 5)$ and $Q(4, -7)$ lie on the straight line $L$.

   (a)

   Find the gradient of $L$.

   [1 mark]

   • A$2$
   • B$-\frac12$
   • C$\frac12$
   • D$-2$

   (b)

   Find the coordinates of the point where $L$ crosses the $x$-axis.

   [1 mark]

   • A$\left(-\frac12, 0\right)$
   • B$(0, 1)$
   • C$\left(\frac12, 0\right)$
   • D$(2, 0)$

   (c)

   Determine whether the point $T(10, -19)$ lies on $L$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A straight section of mountain road rises 84 m vertically over a horizontal distance of 1.2 km. The section starts at an altitude of 350 m. Let $x$ be the horizontal distance in metres from the start of the section and $y$ the altitude in metres.

   (a)

   Find the gradient of the road, and express it as a percentage.

   [3 marks]

   (b)

   (i) Write down the equation of the road in the form $y = mx + c$.
   (ii) Find the horizontal distance from the start at which the road reaches an altitude of 500 m. Give your answer to three significant figures.
   (iii) Lorries are banned from roads with a gradient steeper than 8%. State, with a reason, whether lorries may use this road.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The line $L_1$ passes through the points $P(1, 2)$ and $Q(7, 5)$. The line $L_2$ has equation $2x + y - 19 = 0$.

   (a)

   (i) Find the equation of $L_1$, giving your answer in the form $ax + by + d = 0$, where $a, b, d\in\mathbb{Z}$.
   (ii) Show that $L_1$ and $L_2$ are perpendicular.

   [6 marks]

   (b)

   (i) Find the coordinates of the point of intersection of $L_1$ and $L_2$.
   (ii) Hence find the shortest distance from $P$ to $L_2$, justifying your method.
   (iii) The line $L_3$ is parallel to $L_2$ and passes through $P$. Find the coordinates of the point where $L_3$ meets the $x$-axis.

   [6 marks]

   Total for question 4: 12 marks

End of questions