Inequalities

In this question you must show detailed reasoning.

Solve the inequality x² + 3x ⩽ 10.

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.

In this question you must show detailed reasoning.

The diagram shows a sector AOB of a circle with centre O and radius (3 x + 1)cm. The angle AOB is 2 radians. The area of sector AOB is less than (44x - 7)cm².

Find the set of possible values of x. Give your answer in set notation.

In this question you must show detailed reasoning.

Solve the following equations.

f ′ (x) = 0

f ′′ (x) = 0

The diagram shows the graph of the gradient function y = f ′ (x) for the domain 0 ⩽ x ⩽ 2 π.

Use your answers to parts (a)(i) and (a)(ii) to find the coordinates of points A, B, C and D.

Explain how to use the graph of the gradient function to find the values of x for which

f(x) is increasing.

Using set notation, write down the set of values of x for which f(x) is increasing in the

domain 0 ⩽ x ⩽ 2 π.

A school contains 500 students in years 7 to 11 and 250 students in years 12 and 13. A random sample of 20 students is selected to represent the school at a parents’ evening. The number of students in the sample who are from years 12 and 13 is denoted by X.

State a suitable binomial model for X.

Use your model to answer the following.

Write down an expression for P(X = x).

State, in set notation, the values of x for which your expression is valid.

Find P(5 ⩽ X ⩽ 9).

State one disadvantage of using a random sample in this context.

The diagram shows the curves y = x² - 2x + 1 and y =-x² + 6x - 5. This diagram is repeated in the Printed Answer Booklet.

On the diagram in the Printed Answer Booklet, draw the line y = 2 x - 2.

Show in a sketch the region of the x - y plane within which all three of the following inequalities hold.

y ⩾ x², x + y ⩽ 2, x ⩾ 0.

You should indicate the region for which the inequalities hold by labelling the region R.

Determine the set of values of n for which and are positive integers.

A ‘Pythagorean triple’ is a set of three positive integers a, b and c such that a² + b² = c².

Prove that, for the set of values of n found in part (a), the numbers n and form

a Pythagorean triple.

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

f(x) = 2x² + 6x and g(x) = 3x + 2.

Find the range of f.

Give a reason why f has no inverse.

Given that fg(-2) = g^-1(a), where a is a constant, determine the value of a.

Determine the set of values of x for which f (x) > g (x). Give your answer in set notation.

In this question you must show detailed reasoning.

Solve the inequality 10x² + x - 2 > 0.

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.

In this question you must show detailed reasoning.

The probability that Paul’s train to work is late on any day is 0.15, independently of other days.

The number of days on which Paul’s train to work is late during a 450-day period is denoted by the random variable Y. Find a value of a such that P(Y > a) ≈

The number of days on which Paul’s train to work is late during a 50-day period is modelled by the random variable X.

Find the values of r for which P (X = r) ⩽ P (X = r + 1).

Hence find the most likely number of days on which the train will be late during a 50-day period.