The discriminant

Functions f and g are defined for all real values of x by

You are given that the equation f^-1g(x) = 4 - 2kx has real distinct roots.

Show that k satisfies the inequality 2k² - k - 6 > 0.

In this question you must show detailed reasoning.

Hence find the set of values of k. Give your answer in set notation.

Express x² - 8x + 11 in the form (x - a)² + b where a and b are constants.

Hence write down the minimum value of x² - 8x + 11.

A circle has centre C which lies on the x-axis, as shown in the diagram. The line y = x meets the circle at A and B. The midpoint of AB is M.

The equation of the circle is x² - 6x + y² + a = 0, where a is a constant.

In this question you must show detailed reasoning.

Show that the area of triangle ABC is

Find the value of a when the area of triangle ABC is zero.

Give a geometrical interpretation of the case in part (b)(i).

Give a geometrical interpretation of the case where a = 5.

A curve has equation 2x³ + 6xy - 3y² = 2.

Show that there are no points on this curve where the tangent is parallel to y = x.

In this question you must show detailed reasoning.

A curve has equation y = x³ - 3x² + 4x.

Show that the curve has no stationary points.

Show that the curve has exactly one point of inflection.

In this question you must show detailed reasoning.

Show that the equation m sec θ + 3 cos θ = 4 sin θ can be expressed in the form

m tan²θ - 4 tan θ + (m + 3) = 0.

It is given that there is only one value of θ, for 0 < θ < π, satisfying the equation m sec θ + 3 cos θ = 4 sin θ.

Given also that m is a negative integer, find this value of θ, correct to 3 significant figures.

Determine the set of values of k such that the equation x² + 4x + (k + 3) = 0 has two distinct real

roots.

A circle with centre C has equation x² + y² - 6x + 4y + 4 = 0.

Find

the coordinates of C,

the radius of the circle.

Determine the set of values of k for which the line y = kx - 3 does not intersect or touch the

circle.

Express 2x² - 12x + 23 in the form a(x + b)² + c.

Use your result to show that the equation 2x² - 12x + 23 = 0 has no real roots.