P4: Coordinate geometry in the (x, y) planeEdexcel International A Level Maths: Topic test

1. 1
   A curve $C$ has parametric equations $x=4t$, $y=\frac{2}{t}$, where $t$ is a non-zero real parameter.

   (a)

   Find the coordinates of the point on $C$ where $t=-\frac12$.

   [1 mark]

   • A$(-2,4)$
   • B$(2,-4)$
   • C$(-2,-4)$
   • D$(-8,-1)$

   (b)

   Which of the following is a cartesian equation of $C$?

   [1 mark]

   • A$xy=8$
   • B$y=\frac2x$
   • C$y=\frac x2$
   • D$y=8x$

   (c)

   The line with equation $y=\frac12x$ meets $C$ at two points. Find the coordinates of these points.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A circle $D$ has parametric equations $x=1+2\cos t$, $y=-3+2\sin t$, for $0\leq t<2\pi$.

   (a)

   Which of the following is a cartesian equation of $D$?

   [1 mark]

   • A$(x+1)^2+(y-3)^2=4$
   • B$(x-1)^2+(y+3)^2=2$
   • C$(x-1)^2-(y+3)^2=4$
   • D$(x-1)^2+(y+3)^2=4$

   (b)

   Find the greatest value of $y$ on $D$.

   [1 mark]

   • A$-3$
   • B$-1$
   • C$2$
   • D$-5$

   (c)

   Find the values of $t$ at which $D$ meets the line $x=1+\sqrt3$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A curve $E$ has parametric equations $x=\frac{1}{t-1}$, $y=t+2$, for $t>1$.

   (a)

   Find a cartesian equation of $E$, in the form $y=f(x)$.

   [3 marks]

   (b)

   The line $y=2x+3$ meets $E$ at the point $P$. Find the exact coordinates of $P$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A curve $C$ has parametric equations $x=\cos2t$, $y=\sin t$, for $0\leq t<2\pi$.

   (a)

   (i) Show that a cartesian equation of $C$ is $x=1-2y^2$.
   (ii) State the range of possible values of $x$ and of $y$.
   (iii) Find the exact coordinates of the points where $C$ crosses the $y$-axis.

   [6 marks]

   (b)

   The line $y=x$ meets $C$ at two points. Find the coordinates of these points and the values of $t$ at each of them.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A curve $P$ has parametric equations $x=3t^2$, $y=6t$, where $t$ is a real parameter.

   (a)

   Which of the following is a cartesian equation of $P$?

   [1 mark]

   • A$y^2=36x$
   • B$y^2=12x$
   • C$y^2=\frac{x}{12}$
   • D$y^2=18x$

   (b)

   Find the coordinates of the point on $P$ where $y=-12$.

   [1 mark]

   • A$(-12,-12)$
   • B$(6,-12)$
   • C$(-12,12)$
   • D$(12,-12)$

   (c)

   Show that the point $(20,12)$ does not lie on $P$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A curve $H$ has parametric equations $x=t+\frac1t$, $y=t-\frac1t$, where $t>0$.

   (a)

   Which of the following is a cartesian equation of $H$?

   [1 mark]

   • A$x^2-y^2=4$
   • B$x^2+y^2=4$
   • C$x^2-y^2=2$
   • D$x^2-y^2=0$

   (b)

   Find the value of $y$ at the point of $H$ where $x=\frac52$ and $y>0$.

   [1 mark]

   • A$-\frac32$
   • B$\frac94$
   • C$\frac32$
   • D$\frac12$

   (c)

   Show that $x\geq2$ for every point on $H$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A Ferris wheel has radius $12$ m and its lowest point is $2$ m above the ground. A passenger's position $t$ seconds after the wheel starts turning is $x=12\sin\left(\frac{\pi t}{30}\right)$, $y=14-12\cos\left(\frac{\pi t}{30}\right)$, where $x$ m is the horizontal distance from the centre of the wheel and $y$ m is the height above the ground, for $0\leq t<60$.

   (a)

   Find a cartesian equation of the path of the passenger.

   [3 marks]

   (b)

   Find the times at which the passenger is $20$ m above the ground.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A curve $C$ has parametric equations $x=\frac{t}{1+t}$, $y=\frac{t^2}{1+t}$, for $t>-1$.

   (a)

   (i) Show that $t=\frac{x}{1-x}$.
   (ii) Hence show that a cartesian equation of $C$ is $y=\frac{x^2}{1-x}$.
   (iii) State the range of possible values of $x$.

   [6 marks]

   (b)

   The line $y=4x$ meets $C$ at two points. Find the values of $t$ at these points and the coordinates of the points.

   [6 marks]

   Total for question 8: 12 marks

End of questions