Parametric equationsAQA A-Level Maths: Topic test

1. 1
   A curve is given by the parametric equations $x=3t-1$, $y=t^2+2t$, where $t$ is a real number.

   (a)

   Which is the Cartesian equation of the curve?

   [1 mark]

   • A$y=\dfrac{x^2+8x+7}{9}$
   • B$y=\dfrac{x^2+2x+1}{9}+2x$
   • C$y=\dfrac{(x-1)^2}{9}+\dfrac{2(x-1)}{3}$
   • D$y=(3x+1)^2+2(3x+1)$

   (b)

   At which point does the curve cross the $y$-axis?

   [1 mark]

   • A$(0,\tfrac13)$
   • B$(0,-\tfrac59)$
   • C$(0,\tfrac79)$
   • D$(0,\tfrac73)$

   (c)

   Find the coordinates of the points where the curve crosses the $x$-axis.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A curve is given by the parametric equations $x=\sin\theta$, $y=\cos2\theta$ for $-\dfrac{\pi}{2}\le\theta\le\dfrac{\pi}{2}$.

   (a)

   Which is the Cartesian equation of the curve?

   [1 mark]

   • A$y=2x^2-1$
   • B$y=1-2x^2$
   • C$y=1-x^2$
   • D$y=2x^2$

   (b)

   What is the range of $y$ for the curve?

   [1 mark]

   • A$0\le y\le1$
   • B$-1\le y\le0$
   • C$-2\le y\le1$
   • D$-1\le y\le1$

   (c)

   Find the coordinates of the points where the curve crosses the $x$-axis.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A curve $C$ has parametric equations $x=\dfrac{t}{t+1}$, $y=\dfrac{1}{t+1}$, where $t\neq-1$.

   (a)

   Show that $C$ is part of the line $x+y=1$ and state which point of that line is not on $C$.

   [3 marks]

   (b)

   The line $y=2x$ meets $C$ at one point. Find the value of $t$ and the coordinates of that point.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A stone is thrown from the top of a cliff $30$ m above the sea. At time $t$ seconds after it is thrown, its horizontal distance from the foot of the cliff is $x$ m and its height above the sea is $y$ m, where $x=12t$ and $y=30+16t-4.9t^2$. The sea is modelled as level ground at $y=0$.

   (a)

   Find the time at which the stone enters the sea and its horizontal distance from the foot of the cliff at that time. Find also the maximum height of the stone above the sea.

   [6 marks]

   (b)

   (i) Show that $y=30+\dfrac{4}{3}x-\dfrac{49}{1440}x^2$. (ii) A tree $32$ m high stands at a horizontal distance of $40$ m from the foot of the cliff, with its base at sea level. Determine whether the stone hits the tree. (iii) State one limitation of the model.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A robot moves on a flat floor. Its position in metres relative to a charging point, $t$ seconds after it starts, is $x=3t$, $y=24-2t^2$ for $0\le t\le3$.

   (a)

   What is the robot's distance from the charging point when $t=2$, to $3$ significant figures?

   [1 mark]

   • A$22.0$ m
   • B$16.0$ m
   • C$14.0$ m
   • D$17.1$ m

   (b)

   At what time is the robot at $y=6$?

   [1 mark]

   • A$1.5$ s
   • B$3$ s
   • C$9$ s
   • D$2.4$ s

   (c)

   Show that the robot's path lies on a curve with Cartesian equation $y=24-\dfrac{2}{9}x^2$ and state the range of $x$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A wheel of radius $0.35$ m rolls along level ground. The valve on the rim has position $x=0.35(\theta-\sin\theta)$, $y=0.35(1-\cos\theta)$ metres, where $\theta$ is the angle in radians through which the wheel has turned.

   (a)

   What is the height of the valve above the ground when $\theta=\pi$?

   [1 mark]

   • A$0$ m
   • B$0.35$ m
   • C$0.70$ m
   • D$1.10$ m

   (b)

   How far has the wheel travelled horizontally when $\theta=2\pi$?

   [1 mark]

   • A$2.20$ m
   • B$0.70$ m
   • C$1.10$ m
   • D$4.40$ m

   (c)

   Find the first positive value of $\theta$ at which the valve is $0.175$ m above the ground, and the horizontal distance travelled at that point.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A swimmer crosses a river $40$ m wide, starting at the origin $O$ on one bank. $t$ seconds after starting, the swimmer is $x$ m across the river and $y$ m downstream of $O$, where $x=0.8t$ and $y=0.3t+0.02t^2$.

   (a)

   Find the time taken to reach the far bank and how far downstream the swimmer is when reaching it.

   [3 marks]

   (b)

   Find the Cartesian equation of the swimmer's path and use it to find the value of $x$ when the swimmer is $20$ m downstream.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The edge of a circular skating rink is modelled by $x=\dfrac{40t}{1+t^2}$, $y=\dfrac{20(1-t^2)}{1+t^2}$ for $t\in\mathbb{R}$, where $x$ and $y$ are in metres relative to the centre of the rink.

   (a)

   (i) Show that $x^2+y^2=400$. (ii) A skater stands on the edge at the point $(16,12)$. Find the value of $t$ at that point.

   [6 marks]

   (b)

   A spotlight stands at $S(30,0)$. Show that for a point $P(x,y)$ on the edge, $SP^2=1300-60x$. Hence find the least and greatest distances from $S$ to the edge, and the value of $t$ at the nearest point.

   [6 marks]

   Total for question 8: 12 marks

End of questions