Parametric equations

The curve C has parametric equations

The point P with x coordinate –3 lies on C.

Find the y coordinate of P.

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

Hence, or otherwise, find the gradient of C at the point P.

Figure 4 shows a sketch of the curve C with parametric equations

The point P with coordinates (4, 2) lies on C.

Using parametric differentiation, show that the tangent to C at P has equation

The curve C is used to model the profile of a slide at a water park.

Units are in metres, with y being the height of the slide above water level.

Find, according to the model, the greatest height of the slide above water level.

The curve C has parametric equations

Show that a Cartesian equation for C is

where A and B are integers to be found. The curve C cuts the y-axis at the point P

Show that the equation of the tangent to C at P can be written in the form ax + by = c ln 5 where a, b and c are integers to be found.

Figure 6 shows a sketch of the curve C with parametric equations

The line l is the normal to C at the point P where t =

Using parametric differentiation, show that an equation for l is

Show that all points on C satisfy the equation

The straight line with equation

intersects C at two distinct points.

Find the range of possible values for k.

A curve C has parametric equations

Show that all points on C satisfy

A curve C has parametric equations

Show that all points on C satisfy

The curve shown in Figure 3 has parametric equations

The region R, shown shaded in Figure 3, is bounded by the curve and the x-axis.

Hence show, by algebraic integration, that the area of R is exactly 20

Part of the curve is used to model the profile of a small dam, shown shaded in Figure 4. Using the model and given that ● x and y are in metres ● the vertical wall of the dam is 4.2 metres high ● there is a horizontal walkway of width MN along the top of the dam

calculate the width of the walkway.

The curve C₁ with parametric equations

meets the circle C₂ with equation

at four distinct points as shown in Figure 2. Given that one of these points, S, lies in the 4th quadrant, find the Cartesian coordinates of S.

A curve C has parametric equations

Show that all points on C satisfy y 6(x 3)²

Sketch the curve C.

Explain briefly why C does not include all points of y 6(x 3)², x

The line with equation x y k, where k is a constant, intersects C at two distinct points.

State the range of values of k, writing your answer in set notation.

The curve C has parametric equations

The point P lies on C where t =

Find the equation of the tangent to C at the point P. Give your answer in the form y = px + q, where p and q are integers to be determined.

Show that the cartesian equation for C can be written in the form

where a and b are integers to be determined.

Figure 4 shows a sketch of part of the curve C with parametric equations

The point P (k, 8) lies on C, where k is a constant.

Find the exact value of k.

The finite region R, shown shaded in Figure 4, is bounded by the curve C, the y-axis, the x-axis and the line with equation x = k.

Show that the area of R can be expressed in the form

where , and are constants to be determined.

Hence use integration to find the exact value of the area of R.

A curve C has parametric equations

Find the value of at the point on C where t = 2, giving your answer as a fraction in its simplest form.

Show that the cartesian equation of the curve C can be written in the form

where a and b are integers to be determined.

Figure 3 shows a sketch of the curve C with parametric equations

Show that

Show that a cartesian equation of C is

where a and b are integers to be determined.

Figure 4 shows a sketch of part of the curve C with parametric equations

The point P lies on C and has coordinates (3, 2). The line l is the normal to C at P. The normal cuts the x-axis at the point Q.

Find the x coordinate of the point Q.

The finite region S, shown shaded in Figure 4, is bounded by the curve C, the x-axis, the y-axis and the line l. This shaded region is rotated 2 radians about the x-axis to form a solid of revolution.

Find the exact value of the volume of the solid of revolution, giving your answer in the form + q², where p and q are rational numbers to be determined. [You may use the formula V πr²h for the volume of a cone.]

The curve shown in Figure 3 has parametric equations

The point A, with coordinates (k, 1), lies on the curve. Given that k > 0

find the exact value of k,

find the gradient of the curve at the point A.

There is one point on the curve where the gradient is equal to −

Find the value of t at this point, showing each step in your working and giving your answer to 4 decimal places. [ Solutions based entirely on graphical or numerical methods are not acceptable. ]

A curve C has parametric equations

Find at the point where t =

Find a cartesian equation for C in the form

stating the value of the constant k.

Write down the range of f(x).

Figure 2 showsa sketch of the curve C with parametric equations

Show that the cartesian equation of C may be written in the form

stating the values of a and b. (3)

shown shaded in Figure 3. This region is rotated through 27 radians about the x-axis to form a solid of revolution.

Use calculus to find the exact value of the volume of the solid of revolution. (5)

Figure 2 shows a sketch of the curve C with parametric equations

Show that = kt, where k is a constant to be determined.

Find an equation of the tangent to C at the point where t = Give your answer in the form y = ax + b, where a and b are constants.

Find a cartesian equation of C.

Figure 3 shows part of the curve C with parametric equations

The point P lies on C and has coordinates ˲ 3, 3.

Find the value of at the point P.

The line l is a normal to C at P. The normal cuts the x-axis at the point Q.

Show that Q has coordinates k ˲ 3, 0, giving the value of the constant k.

The finite shaded region S shown in Figure 3 is bounded by the curve C, the line x ˲3 and the x-axis. This shaded region is rotated through 2 radians about the x-axis to form a solid of revolution.

Find the volume of the solid of revolution, giving your answer in the form p ¥ 3 + q ², where p and q are constants.

The curve C has parametric equations

Find

a cartesian equation of C.

The finite area R, shown in Figure 1, is bounded by C, the x-axis, the line x ln 2 and the line x ln 4. The area R is rotated through360 about the x-axis.

Use calculus to find the exact volume of the solid generated.

Figure 2 shows a sketch of the curve C with parametric equations

The curve C cuts the x-axis at the points A and B.

Find the x-coordinate at the point A and the x- coordinate at the point B.

The region R, as shown shaded in Figure 2, is enclosed by the loop of the curve.

Use integration to find the area of R.

The curve C shown in Figure 3 has parametric equations

where t is a parameter. Given that the point A has parameter t = –1,

find the coordinates of A.

The line l is the tangent to C at A.

Show that an equation for l is 2 x – 5 y – 9 = 0.

The line l also intersects the curve at the point B.

Find the coordinates of B.

Figure 3 shows the curve C with parametric equations

The point P lies on C and has coordinates (4, 2√3).

Find the value of t at the point P.

The line l is a normal to C at P.

Show that an equation for l is y = –x√3 + 6√3.

The finite region R is enclosed by the curve C, the x-axis and the line x = 4, as shown shaded in Figure 3.

Show that the area of R is given by the integral.²64 sin² t cost dt

Use this integral to find the area of R, giving your answer in the form a + b√3, where a and b are constants to be determined.

The curve C has parametric equations

The finite region R between the curve C and the x-axis, bounded by the lines with equations x = ln 2 and x = ln 4, is shown shaded in Figure 3.

Show that the area of R is given by the integral

Hence find an exact value for this area.

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

State the domain of values for x for this curve.