P1: Coordinate geometry in the (x, y) planeEdexcel International A Level Maths: Topic test

1. 1
   The points $P(-2,7)$ and $Q(4,-5)$ lie on the line $l$.

   (a)

   Find the gradient of $l$.

   [1 mark]

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

   (b)

   Which of the following is an equation of $l$?

   [1 mark]

   • A$y=-2x+11$
   • B$y=2x+11$
   • C$y=-\frac12x+6$
   • D$2x+y-3=0$

   (c)

   Find the coordinates of the point where $l$ meets the $x$-axis.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The line $m$ has equation $5x-2y+4=0$.

   (a)

   Find the gradient of $m$.

   [1 mark]

   • A$-\frac52$
   • B$\frac25$
   • C$\frac52$
   • D$-\frac25$

   (b)

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

   [1 mark]

   • A$-\frac25$
   • B$\frac25$
   • C$-\frac52$
   • D$\frac52$

   (c)

   Find an equation of the line parallel to $m$ that passes through the point $(2,9)$, giving your answer in the form $ax+by+c=0$, where $a$, $b$ and $c$ are integers.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The points $R(1,-3)$, $S(7,1)$ and $T(3,7)$ form a triangle.

   (a)

   Show that angle $RST$ is a right angle.

   [3 marks]

   (b)

   The point $U$ is such that $RSTU$ is a rectangle. Find the equations of the lines $TU$ and $RU$, and hence find the coordinates of $U$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The line $l$ has equation $5x-3y=30$. It meets the $x$-axis at the point $P$ and the $y$-axis at the point $Q$.

   (a)

   (i) Find the gradient of $l$ and the coordinates of $P$ and $Q$.
   (ii) Find an equation of the line through $P$ that is perpendicular to $l$, giving your answer in the form $ax+by+c=0$, where $a$, $b$ and $c$ are integers.

   [6 marks]

   (b)

   The line $n$ has equation $3x+5y-18=0$. The line $p$ passes through the point $(-4,2)$ and is perpendicular to $l$.
   (i) Find an equation of $p$, giving your answer in the form $ax+by+c=0$, where $a$, $b$ and $c$ are integers.
   (ii) Find the coordinates of the point where $p$ meets the $x$-axis.
   (iii) State the relationship between $p$ and $n$, giving a reason.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A line passes through the points $U(5,-1)$ and $V(-3,3)$.

   (a)

   Which of the following is an equation of the line?

   [1 mark]

   • A$2x+y-9=0$
   • B$x+2y+3=0$
   • C$x-2y-7=0$
   • D$x+2y-3=0$

   (b)

   Find the $y$-coordinate of the point where the line crosses the $y$-axis.

   [1 mark]

   • A$3$
   • B$\frac32$
   • C$-\frac32$
   • D$-3$

   (c)

   The point $(k,-4)$ lies on the line. Find the value of $k$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The line $l_1$ has equation $y=(k+1)x+3$ and the line $l_2$ has equation $2x+3y=6$, where $k$ is a constant.

   (a)

   Find the value of $k$ for which $l_1$ is parallel to $l_2$.

   [1 mark]

   • A$-\frac53$
   • B$\frac13$
   • C$\frac12$
   • D$-\frac23$

   (b)

   Find the value of $k$ for which $l_1$ is perpendicular to $l_2$.

   [1 mark]

   • A$-\frac53$
   • B$-\frac52$
   • C$\frac12$
   • D$\frac32$

   (c)

   Given that $k=\frac12$, find the coordinates of the point where $l_1$ meets the $x$-axis.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The line $l$ has equation $2x-3y+6=0$ and the point $T$ has coordinates $(5,1)$.

   (a)

   Find an equation of the line through $T$ that is parallel to $l$, giving your answer in the form $ax+by+c=0$, where $a$, $b$ and $c$ are integers.

   [3 marks]

   (b)

   Find an equation of the line through $T$ that is perpendicular to $l$, and hence find the coordinates of the point where this line meets $l$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The line $p$ has equation $y=\frac23x-4$. The line $q$ passes through the point $W(2,6)$ and is perpendicular to $p$.

   (a)

   (i) Find an equation of $q$, giving your answer in the form $ax+by+c=0$, where $a$, $b$ and $c$ are integers.
   (ii) The lines $p$ and $q$ meet at the point $X$. Find the coordinates of $X$.

   [6 marks]

   (b)

   The line $p$ meets the $y$-axis at the point $Y$. The line $r$ passes through $Y$ and is parallel to $q$.
   (i) Find an equation of $r$, giving your answer in the form $ax+by+c=0$, where $a$, $b$ and $c$ are integers.
   (ii) Find the coordinates of the point $Z$ where $r$ meets the $x$-axis.
   (iii) Determine whether $p$ and $r$ are perpendicular, showing your working.

   [6 marks]

   Total for question 8: 12 marks

End of questions