2.1 Equation of a straight lineIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   A line $L$ passes through the points $A(1, 4)$ and $B(5, 12)$.

   (a)

   Find the gradient of $L$.

   [1 mark]

   • A$\frac12$
   • B$2$
   • C$\frac83$
   • D$-2$

   (b)

   Which of the following is the equation of $L$?

   [1 mark]

   • A$y=2x+2$
   • B$y=2x+4$
   • C$y=2x+6$
   • D$y=\frac12x+\frac72$

   (c)

   Find the equation of the line parallel to $L$ that passes through the point $(3, -1)$. Give your answer in the form $ax+by+d=0$, where $a$, $b$ and $d$ are integers.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The line $l_1$ has equation $3x+4y-24=0$.

   (a)

   Find the gradient of $l_1$.

   [1 mark]

   • A$\frac34$
   • B$-\frac43$
   • C$-\frac34$
   • D$6$

   (b)

   Find the gradient of a line perpendicular to $l_1$.

   [1 mark]

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

   (c)

   Find the coordinates of the points where $l_1$ crosses the $x$-axis and the $y$-axis.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A mountain road climbs from a village at a height of 420 m above sea level to a viewpoint at a height of 570 m. The horizontal distance between the village and the viewpoint is 2.5 km. The road may be modelled as a straight line.

   (a)

   (i) Find the gradient of the road.
   (ii) Write your answer to (i) as a percentage and explain what it means for a driver.

   [3 marks]

   (b)

   Let $h$ be the height of the road, in metres, at a horizontal distance of $d$ kilometres from the village. Find an equation for $h$ in terms of $d$, and use it to find the height of the road when $d=1.8$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A garden is a triangle with corners $P(1, 2)$, $Q(7, 6)$ and $R(5, 9)$, where one unit on each axis represents 1 metre.

   (a)

   (i) Find the gradient of $PQ$ and the gradient of $QR$.
   (ii) Hence show that $PQ$ is perpendicular to $QR$.
   (iii) Find the equation of the line $QR$ in the form $ax+by+d=0$, where $a$, $b$ and $d$ are integers.

   [6 marks]

   (b)

   A straight path starts at $P$ and is parallel to $QR$. Another straight path starts at $R$ and is parallel to $PQ$. The two paths meet at the point $S$.
   (i) Find the equation of each path.
   (ii) Use your GDC or algebra to find the coordinates of $S$.

   [6 marks]

   Total for question 4: 12 marks

End of questions