Quadratic equations

The diagram shows the curve y = ax (x - b), where a and b are constants and b > 0.

Given that the curve has a stationary point at x = 3, state the value of b.

Given also that the stationary point at x = 3 is a maximum, state what can be deduced about

the value of a.

Find the y-coordinate of the stationary point, giving your answer in terms of a.

State the range of values of x for which the curve is increasing.

In this question you must show detailed reasoning.

Solve the following equations.

e^3y = 2

x⁴ - 3x² - 4 = 0

Express 3x² - 12x + 17 in the form a(x - b)² + c where a, b and c are constants.

State the coordinates of the minimum point of the curve y = 3x² - 12x + 17.

State the equation of the normal to the curve y = 3x² - 12x + 17 at its minimum point.

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.

The diagram shows the curve 5x - 2xy + 2y² - k = 0, where k is a positive integer.

At the points P and Q on the curve, the tangents to the curve are parallel to the y-axis.

Given that the difference in the y-coordinates of P and Q is 3, determine the x-coordinates of

P and Q.

Show that + can be written in the form , where a, b and c are

Hence solve the equation + = 2.

In this question you must show detailed reasoning.

Solve the equation 2^2y - 7 × 2^y - 8 = 0.

It is given that ABCD is a quadrilateral. The position vector of A is i + j, and the position vector of B is 3 i + 5 j.

Find the length AB.

The position vector of C is p i + p j where p is a constant greater than 1.

Given that the length AB is equal to the length BC, determine the position vector of C.

The point M is the midpoint of AC.

Given that = , determine the position vector of D.

State the name of the quadrilateral ABCD, giving a reason for your answer.

Find the first three terms in the expansion of (4 + 3x in ascending powers of x.

State the range of values of x for which the expansion in part (a) is valid.

Determine the possible values of the constant a.

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.

Write 2x² + 6x + 7 in the form p(x + q)² + r, where p, q and r are constants.

State the coordinates of the minimum point on the graph of y = 2x² + 6x + 7.

Hence deduce

• the minimum value of 2 tan²θ + 6 tan θ + 7,

• the smallest positive value of θ, in degrees, for which the minimum value occurs.

The graph of y = 2^x can be transformed to the graph of y = 2^x+4 either by a translation or by a stretch.

Give full details of the translation.

Give full details of the stretch.

In this question you must show detailed reasoning.

Solve the equation log₂(8x) = 1 - log₂(1 - x).

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.

Solve the following equations.

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.

The positive integers x, y and z are the first, second and third terms, respectively, of an arithmetic progression with common difference - 4.

Show that y satisfies the equation y⁴ - 16y² - 225 = 0.

Hence determine the sum to infinity of the geometric progression.

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 particle moves in the x - y plane so that at time t seconds, where t ⩾ 0, its coordinates are given by

x = e^2t - 4e^t + 3, y = 2e^-3t.

Explain why the path of the particle never crosses the x-axis.

Determine the exact values of t when the path of the particle intersects the y-axis.

Hence find the coordinates of the particle when its path is parallel to the y-axis.

In this question you must show detailed reasoning.

Solve the equation 3sin⁴φ + sin²φ = 4, for 0 ⩽ φ < 2π, where φ is measured in radians.

The function f is defined by f(x) = (x - 3)² - 17 for x ⩾ k, where k is a constant.

Given that f^-1(x) exists, state the least possible value of k.

Evaluate ff(5).

Solve the equation f (x) = x.

Explain why your solution to part (c) is also the solution to the equation f(x) = f^-1(x).

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.

In this question you must show detailed reasoning.

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

Write down expressions for

fg (x),

gf (x).

Hence find the values of x for which fg (x) - gf (x) = 24.