Quadratic equations

The point A lies on the curve with equation

Given that A has coordinates (– 5, 37), show that

5 p – q = 30

Given that A is a stationary point, show that

10 p – q = 75

Hence find the value of p and the value of q

The curve with equation

has a second stationary point B

Find the coordinates of B

Fully justify your answer.

Show that the equation

2 log₁₀ x log₁₀ 4 log₁₀ (x 8)

has exactly one solution.

Fully justify your answer.

Sketch the curve with equation

The polynomial p (x) is given by

It is given that x 3 is a factor of p(x)

Use the factor theorem to show a 2

State the transformation which maps the curve with equation

onto the curve with equation

The polynomial x²(2x 2) 36 can be written as (x 3)(2x² bx c)

Without finding the values of b and c, use your answers to parts (a) and (b)(ii) to explain why

The polynomial p (x) is given by

where b is a constant.

Use the factor theorem to prove that (x 2) is a factor of p(x) for all values of b.

The graph of y p(x) meets the x-axis at exactly two points.

Sketch a possible graph of y p(x)

Given p (x) can be written as

find the value of b.

Fully justify your answer.

Figure 2 shows the approximate shape of the vertical cross section of the entrance to a cave. The cave has a horizontal floor.

The entrance to the cave joins the floor at the points O and P.

Garry models the shape of the cross section of the entrance to the cave using the

where a is a constant, and x and y are the horizontal and vertical distances respectively, in metres, measured from O.

The distance OP is 16 metres.

Find the value of a that Garry should use in the model.

Show that the maximum height of the cave above OP is approximately 10.5 metres.

Suggest one limitation of the model Garry has used.

The function f is defined by

where f has its maximum possible domain.

The curve y f (x) intersects the line y x at the points P and Q as shown below.

State the value of x which is not in the domain of f.

Explain how you know that the function f is many-to-one.

Show that the x-coordinates of P and Q satisfy the equation

Hence, find the exact x-coordinate of P and the exact x-coordinate of Q.

Show that P and Q are stationary points of the curve.

Fully justify your answer.

Using set notation, state the range of f.

Section B

all questions in the spaces provided.

Four possible sketches of y ax² bx c are shown below.

Given b² 4ac 0 and a, b and c are non-zero constants, which sketch is the only one that could possibly be correct?

Tick () one box.

A robotic arm which is attached to a flat surface at the origin O, is used to draw a graphic design.

The arm is made from two rods OP and PQ, each of length d, which are joined at P.

A pen is attached to the arm at Q.

The coordinates of the pen are controlled by adjusting the angle OPQ and the angle between OP and the x-axis.

For this particular design the pen is made to move so that the two angles are always equal to each other with 0 as shown in Figure 2.

Show that the x-coordinate of the pen can be modelled by the equation

Hence, show that

It can be shown that

State the greatest possible value of x and the corresponding value of cos

Figure 3 below shows the arm when the x-coordinate is at its greatest possible value.

Find, in terms of d, the exact distance OQ.

p(x) 4x³ 15x² 48x 36

Use the factor theorem to prove that x 6 is a factor of p(x).

Prove that the graph of y p(x) intersects the x-axis at exactly one point.

State the coordinates of this point of intersection.

The function f is defined by

Find the range of f.

f ¹ x 6 (b) (i) Find ()

State the range of f ¹(x)

State the transformation which maps the graph of y f (x) onto the graph of

Find the coordinates of the point of intersection of the graphs of y f (x) and

A curve, C, has equation y x² 6x k, where k is a constant.

The equation x² 6x k 0 has two distinct positive roots.

Sketch C on the axes below.

Find the range of possible values for k.

Fully justify your answer.