The discriminant

Given that

• f(x) = 2x³ + 3x² – 16x + 16

• f(–4) = 0

write f(x) in the form (x + a) Q (x) where a is a constant and Q(x) is a quadratic expression.

Hence prove that –4 is the only real root of the equation f(x) = 0 (Solutions relying on calculator technology are not acceptable.)

The functions f and g are defined by

Find fg(2)

Find g^–1

Find gf(x), giving your answer as a simplified fraction.

Deduce the range of gf(x).

The function h is defined by

where k is a constant.

Find the range of values of k for which the equation

f(x) = h(x)

has no real solutions.

The curve C has equation

Find an expression for in terms of x and y.

The point P(1, 0) lies on C.

Show that the normal to C at P has equation

Prove that the normal to C at P does not meet C again.

You should use algebra for your proof and make your reasoning clear.

A circle C has equation

where k is a constant.

Find in terms of k,

the coordinates of the centre of C

the radius of C

The line with equation y = 2x – 1 intersects C at 2 distinct points.

Find the range of possible values of k.

The function f is defined by

where k is a positive constant.

Show that

where g(x) is a function to be found. Given that the curve with equation y = f(x) has at least one stationary point,

find the range of possible values of k.

The circle C has equation

Find

the coordinates of the centre of C,

the exact radius of C, giving your answer as a simplified surd.

The line l has equation y = 3x + k where k is a constant.

Given that l is a tangent to C,

find the possible values of k, giving your answers as simplified surds.

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.

Calculate f (2)

Write f (x) as a product of two algebraic factors.

Using the answer to (a)(ii),

prove that there are exactly two real solutions to the equation

deduce the number of real solutions, for 7π θ <10π, to the equation

The equation (p – 1)x² + 4x + (p – 5) = 0, where p is a constant has no real roots.

Show that p satisfies p² – 6p + 1 0

Hence find the set of possible values of p.

The curve C has equation y = x² + 8 The line L has equation y = 3 x + k, where k is a positive constant.

Sketch C and L on separate diagrams, showing the coordinates of the points at which C and L cut the axes.

Given that line L is a tangent to C,

find the value of k.

Given the simultaneous equations 2 x + y = 1

where k is a non zero constant,

show that

Given that x² + 8kx + k = 0 has equal roots,

find the value of k.

For this value of k, find the solution of the simultaneous equations.

The equation

has two distinct real solutions for x.

Show that k satisfies

Hence find the set of possible values of k.

4x – 5 – x² = q – (x + p)² where p and q are integers.

Find the value of p and the value of q.

Calculate the discriminant of 4x – 5 – x²

On the axes on page 17, sketch the curve with equation y = 4x – 5 – x² showing clearly the coordinates of any points where the curve crosses the coordinate axes.

Show that x² +6x + 11 can be written as

where p and q are integers to be found.

In the space at the top of page 7, sketch the curve with equation y x² 6x 11, showing clearly any intersections with the coordinate axes.

Find the value of the discriminant of x² +6x + 11

f(x) x² 4kx (311k), where k is a constant.

Express f(x) in the form (x + p)² + q, where p and q are constants to be found in terms of k.

Given that the equation f(x) = 0 has no real roots,

find the set of possible values of k.

Given that k = 1,

sketch the graph of y = f(x), showing the coordinates of any point at which the graph crosses a coordinate axis.

The equation x² + 3px + p = 0, where p is a non-zero constant, has equal roots. Find the value of p.

The equation kx² + 4x + (5 – k) = 0, where k is a constant, has 2 different real solutions for x.

Show that k satisfies

Hence find the set of possible values of k.

Given that the equation 2qx² + qx – 1 = 0, where q is a constant, has no real roots,

Hence find the set of possible values of q.

The equation

has no real solutions for x.

Hence find the set of possible values of k.