Parametric equations – Year 13 A level maths worksheet with answers to download as PDF

A curve is defined by \(x=2t+1\), \(y=t^2-3\).

1. Complete a table of values for \(t=-2,-1,0,1,2\) and plot the points. [3 marks]
2. Find the value of \(t\) for which the point is \((7,6)\). [2 marks]
3. Explain why the point \((0,-3)\) is not on the curve. [2 marks]

Find the Cartesian equation of each curve.

1. \(x=3t-2\), \(y=t^2+1\) [3 marks]
2. \(x=2t\), \(y=\dfrac{8}{t}\) [2 marks]

Find the Cartesian equation of each curve and describe its shape.

1. \(x=4\cos t\), \(y=3\sin t\) [3 marks]
2. \(x=2+3\cos t\), \(y=-1+3\sin t\) [3 marks]

A curve has parametric equations \(x=\sin t\), \(y=\cos2t\) for \(0\le t\le2\pi\).

1. Show that the Cartesian equation is \(y=1-2x^2\). [3 marks]
2. State the range of values of \(x\) and of \(y\) on the curve. [2 marks]

The curve \(C\) has parametric equations \(x=t^2+1\), \(y=2t\). It meets the line \(y=x-4\) at two points.

1. Show that the values of \(t\) at the intersections satisfy \(t^2-2t-3=0\). [2 marks]
2. Solve this equation and find the coordinates of the two points. [3 marks]

A curve has parametric equations \(x=t^2-4\), \(y=t^3-4t\).

1. Find the coordinates of the points where the curve meets the \(x\)-axis. [3 marks]
2. Find the coordinates of the points where the curve meets the \(y\)-axis. [2 marks]
3. What does your answer tell you about the shape of the curve? [1 mark]

A footballer takes a free kick. The path of the ball is modelled by \(x=15t\), \(y=1+12t-4.9t^2\), where \(x\) is the horizontal distance in metres, \(y\) the height in metres and \(t\) the time in seconds.

1. Find the greatest height reached by the ball. [3 marks]
2. Find the horizontal distance from the kick when the ball lands. [3 marks]
3. A wall of defenders, 2.2 m tall, stands 9 m from the kick. Does the ball clear the wall? [3 marks]
4. State two assumptions of the model. [2 marks]

A Ferris wheel at a fair turns once every 40 seconds. A passenger car has position \(x=10\sin\left(\dfrac{\pi t}{20}\right)\), \(y=12-10\cos\left(\dfrac{\pi t}{20}\right)\) where \(x\) and \(y\) are in metres relative to the ground level foot of the wheel and \(t\) is in seconds from the lowest point.

1. Show that the car moves on a circle and state its centre and radius. [3 marks]
2. Find the lowest and highest heights of the car. [2 marks]
3. Find the height of the car when \(t=5\). [2 marks]
4. For how many seconds in each turn is the car more than 17 m above the ground? [4 marks]

A drone flies round an elliptical circuit given by \(x=40\cos t\), \(y=25\sin t\) (metres from the control tower at the origin), where \(t\) is a parameter.

1. Find the Cartesian equation of the circuit. [2 marks]
2. Show that the squared distance from the tower is \(625+975\cos^2t\). [3 marks]
3. Hence find the greatest and least distance from the tower. [3 marks]
4. Find the position when \(t=\dfrac\pi3\). [2 marks]

A firework rocket is launched from the ground and follows the path \(x=5t\), \(y=60t-4.9t^2\) (metres, seconds) until it explodes at the top.

1. Find the Cartesian equation of the path. [2 marks]
2. Find the maximum height and the time at which it occurs. [3 marks]
3. Find how far from the launch point the rocket lands if it does not explode. [3 marks]
4. The rocket explodes at the highest point. How far is it then, horizontally, from the launch point? [1 mark]

A reflector is attached to the rim of a bicycle wheel of radius 0.35 m, which rolls along a flat road. The position of the reflector is \(x=0.35(t-\sin t)\), \(y=0.35(1-\cos t)\) in metres, where \(t\) is the angle turned through by the wheel in radians.

1. Complete the table to 2 decimal places. [4 marks]
2. State the greatest height of the reflector and the distance travelled in one complete turn of the wheel. [2 marks]
3. Explain why the reflector is momentarily at rest relative to the road when \(t=0\) and \(t=2\pi\). [1 mark]

A skier leaves the end of a horizontal ramp 30 m above a flat landing area. The path is \(x=20t\), \(y=30-5t^2\).

1. Show that the Cartesian equation is \(y=30-\dfrac{x^2}{80}\). [2 marks]
2. Find the horizontal distance of the landing point from the ramp. [3 marks]
3. Find the height of the skier when 30 m has been covered horizontally. [2 marks]
4. The landing area is a 25 degree slope falling away at the end of the ramp instead. Explain whether the skier would land earlier or later. [1 mark]

A small satellite follows the elliptical path \(x=4\cos t\), \(y=3\sin t\) (units of 1000 km). A tracking beam is the line \(y=x\).

1. Show that the beam meets the path where \(\tan t=\dfrac43\). [3 marks]
2. Find the coordinates of both points of intersection. [4 marks]
3. Calculate the distance between the two points. [2 marks]

A boat sails so that its position relative to a lighthouse at the origin is \(x=3t-6\), \(y=4t-5\) (km, with \(t\) in hours).

1. Show that the square of the distance from the lighthouse is \(25t^2-76t+61\). [3 marks]
2. Find the time at which the boat is closest to the lighthouse and the minimum distance. [4 marks]
3. The lighthouse light can be seen up to 2 km away. Can the sailors see it at their closest approach? [1 mark]

Find the Cartesian equation of each curve and state the part of that curve which is traced out.

1. \(x=e^t\), \(y=e^{2t}+1\) for all real \(t\). [4 marks]
2. \(x=\dfrac{1}{1+t}\), \(y=\dfrac{t}{1+t}\) for \(t\ne-1\). [4 marks]
3. \(x=t^2\), \(y=t^4-2\) for \(-1\le t\le2\). [4 marks]

A curve is given by \(x=t^2-1\), \(y=t^3-t\).

1. Show that the curve passes through the origin for two different values of \(t\). [2 marks]
2. Find the coordinates where the curve crosses the \(x\)-axis. [2 marks]
3. The curve forms a loop for \(-1\le t\le1\). Show that the area enclosed is \(\dfrac8{15}\) square units. [6 marks]
4. Describe the symmetry of the curve. [1 mark]

The curve \(C\) has parametric equations \(x=t^2\), \(y=t^3\).

1. Show that the Cartesian equation is \(y^2=x^3\). [2 marks]
2. Find the gradient of the tangent at the point where \(t=2\). [3 marks]
3. Find the equation of this tangent. [2 marks]
4. Show that the tangent meets the curve again at the point \((1,-1)\). [4 marks]

A drone is tracked and its position (in metres) recorded every second.

1. Show that the data fit the model \(x=2+3t\), \(y=1+4t-0.5t^2\). [3 marks]
2. Find the Cartesian equation of the path. [4 marks]
3. Use the model to predict the position at \(t=8\) and comment on whether the model is realistic. [3 marks]

A player shoots at a hoop 3.05 m above the ground and 4.5 m horizontally from the player. The ball leaves the hand at a height of 2 m. Its path is modelled by \(x=5.625t\), \(y=2+5.25t-4.9t^2\) (metres and seconds).

1. Find the time at which the ball is 4.5 m from the player horizontally. [2 marks]
2. Find the height of the ball at this time. [2 marks]
3. The hoop has a diameter of 0.45 m (take its centre at height 3.05 m). Decide whether the shot goes in. [2 marks]
4. Find the greatest height of the ball. [3 marks]
5. State two limitations of the model. [2 marks]

A curve has parametric equations \(x=\dfrac{1-t^2}{1+t^2}\), \(y=\dfrac{2t}{1+t^2}\) for all real \(t\).

1. Show that \(x^2+y^2=1\). [3 marks]
2. Find the points of the circle corresponding to \(t=0,1,-1\). [2 marks]
3. Explain why the point \((-1,0)\) is never reached, although it lies on the circle. [3 marks]

A curve has parametric equations \(x=\sin t\), \(y=\sin2t\) for \(0\le t\le2\pi\).

1. Show that \(y^2=4x^2(1-x^2)\). [3 marks]
2. Find the values of \(t\) for which the curve passes through the origin and explain what this means about its shape. [3 marks]
3. Find the greatest value of \(y\) and the corresponding \(x\) for \(0\le t\le\dfrac\pi2\). [3 marks]