Parametric equations

Find an equation of the tangent to C at the point where t = 0

Find a Cartesian equation for C in the form y = f (x)

Fully justify your answer.

A water slide is the shape of a curve PQ as shown in Figure 1 below.

The curve can be modelled by the parametric equations

The horizontal distance from O is x metres.

The vertical distance above the point O at ground level is y metres.

To make sure speeds are safe at Q, the difference in height between P and Q must be less than 7 metres.

Show that the slide meets this safety requirement.

Find an expression for in terms of t

A vertical support, RS, is to be added between the ground and the lowest point on the slide as shown in Figure 2 below.

Find the length of RS

Find the acute angle the slide makes with the horizontal at Q

Give your answer to the nearest degree.

TURN OVER FOR SECTION B

Section B

all questions in the spaces provided.

A curve is defined by the parametric equations

Which of the options shown below is a Cartesian equation for this curve?

Circle your answer.

A design for a surfboard is shown in Figure 1.

The curve of the top half of the surfboard can be modelled by the parametric equations

for 0 t 95 as shown in Figure 2, where x and y are measured in centimetres.

Find the length of the surfboard.

Find an expression for in terms of t.

Hence, show that the width of the surfboard is approximately one third of its length.

The curve C is defined for t 0 by the parametric equations

2 2 3

C is shown in the diagram below.

Find the gradient of C at the point where it intersects the positive x-axis.

The area A enclosed between C and the x-axis is given by

Find the value of b.

Use the substitution y 4t² t³ to show that

2 3 4

Find the value of A.

The curve defined by the parametric equations

is shown in Figure 1 below.

Find a Cartesian equation of the curve in the form y² f (x)

The point A lies on the curve where t a

The tangent to the curve at A is at an angle to a line through A parallel to the x-axis.

The point B has coordinates (1, 0)

The line AB is at an angle to the x-axis.

By considering the gradient of the curve, show that

Find tan in terms of a.

Show that tan 2 tan

At time t hours after a high tide, the height, h metres, of the tide and the velocity, v knots, of the tidal flow can be modelled using the parametric equations

High tides and low tides occur alternately when the velocity of the tidal flow is zero.

A high tide occurs at 2 am.

Use the model to find the height of this high tide.

Find the time of the first low tide after 2 am.

Find the height of this low tide.

Use the model to find the height of the tide when it is flowing with maximum velocity.

Comment on the validity of the model.

A curve is defined by the parametric equations

Find the Cartesian equation of the curve in the form xy ax by c, where a, b and c are integers.