A Level Mathematics for AQA Student Book 1 (AS/Year 1) with Cambridge Elevate Edition (2 Years) (AS/A Level Mathematics for AQA) [Student ed.] 1316644685, 9781316644683, 9781316644225, 9781316644591, 9781316644607

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In this chapter you will learn how to: use appropriate terms to describe mathematical objects, such as identity and equation use a counter example to disprove a mathematical idea apply some techniques for proving a mathematical idea – deduction and exhaustion.

a

b

a

b

The missing symbol is a So is only one solution – there is also the possibility that .

a

b

Ø

Let the even number be , for some integer . Let the odd number be some integer .

, for

the product of an even and an odd number is even.

Let the smaller odd number be Let the larger odd number be

. .

the difference between the squares of consecutive odd numbers is always a multiple of .

is not divisible by , , , or . Therefore it must be a prime number.

Let be a whole number. Then must be: a multiple of , or one more than a multiple of , or two more than a multiple of . If

then:

which is a multiple of . If

then:

which is one more than a multiple of . If

then:

which is one more than a multiple of . So either there is no remainder or the remainder is .

In this chapter you will learn how to: use laws of indices work with expressions involving square roots (called surds).

In this chapter you will learn how to: apply your knowledge of factorisation and the quadratic formula for solving quadratic equations recognise the shape and main features of graphs of quadratic functions complete the square solve quadratic inequalities identify the number of solutions of a quadratic equation solve disguised quadratic equations.

Graph B shows a positive quadratic, so graph B corresponds to equation a. Graph A has a positive -intercept, so graph A corresponds to equation c. Graph C corresponds to equation b.

Sketch the graph of When

,

When

:

Repeated root at is a repeated factor.

So the equation is:

a From Worked example

:

∴ the coordinates of the turning point are

b From Worked example

:

the coordinates of the turning point are

The turning point is at so the function must be of the form When

,

:

So the equation is



So:

£

£

If

for all there are no real roots

In this chapter you will learn how to: define a polynomial find the product of two polynomials find the quotient of two polynomials quickly find factors of a polynomial sketch polynomials.



Sketch the graph of

.

There are single roots at and are factors. There is a double root at factor.

and is a

So the equation is

.

In this chapter you will learn how to: link solving simultaneous equations and the intersection of graphs determine the number of intersections between a line and a curve use transformations of graphs use direct and inverse proportion illustrate two-variable inequalities on a graph.

The coordinates of the points of intersection are and .

Equation of the line is:

Substitute into the equation of the curve:

Two solutions

:

Let:

So when

the graph of has the same height as the graph of . This occurs when is units to the left of the equivalent point on .

Then the new graph is:

. Then:

It is a horizontal stretch with scale factor .

The transformation taking to is the reflection in the -axis. The maximum point is

.

∝ ∝

a

Calculate the time, in hours:

(about hours minutes) b The speed on the motorway is not constant. It doesn’t take into account the time

from getting off the motorway in York.

The miles distance is probably not exact; it doesn’t specify where in York Ben is going.

£

£

£

a

b

The largest -value corresponds to the point labelled .

occurs where:

So the largest value of is

.

£ £

£

In this chapter you will learn how to: find the distance between two points and the midpoint of two points find the equation of a straight line using determine whether two straight lines are parallel or perpendicular find the equation of a circle with a given centre and radius solve problems involving intersections of lines and circles.

a The distance is

b The midpoint is

The distance between the points is given by

So

So the points are

or

Gradient:

Equation of the line:

When

:

The -intercept is

a

So Equation of :

b The -intercept is the point on the line where

The -intercept is the point on the line where

For the distance between and

:

For the line through

:

Gradient of the line segment:

Perpendicular gradient:

Midpoint of the segment joining to :

Equation of the line:

a

b i For point :

Point lies on the circle. ii For point :

Point lies outside the circle.

The centre is

and the radius

is

a

so is perpendicular to BC. Since , is the diameter of the circle. b Find the centre.

Find the radius.

Equation of the circle: Equation of the line: Intersection:

Discriminant must be zero:

Coordinates of the centre:

Gradient of the radius:

Gradient of the tangent:

Tangent passes through

Normal:

:

a

The first circle has centre and radius The second circle has centre and radius The distance between the centres is:

So the two circles intersect. b

From the first equation: Substitute into the second equation:

The coordinates of the points of intersection are and

In this chapter you will learn how to: use an operation called a logarithm to undo exponential functions use the laws of logarithms use logarithms to find exact solutions of some exponential equations use the number e.

a b c

a b

a b

When

:

LHS = is not real so this solution does not work. When

:

Let

.

When solutions. When

there are no

,

In this chapter you will learn how to: recognise and use graphs of exponential functions use exponential functions in modelling use logarithms to transform curved graphs into straight lines.

Consider the graphs of and with tangents drawn at and respectively.

The gradient of

Since of

at the point

is

is a horizontal stretch by factor ,

So, the gradient of

at

is

But also

So the gradient of the red line when So, in general the gradient of

a ,

b

is

is

c

a

b

a b

The population will first reach million at

.

c The model predicts that the population will grow indefinitely. This is not realistic, as the growth will eventually be limited by lack of food or space.

a b

The rate of change is:

The mass is decreasing at the rate of grams per second.

a

b The model predicts that there will be bacteria.

c The model predicts that the bacteria population will continue growing indefinitely, but it will eventually slow down as food and space become limiting factors. The information given in the model is only approximate so in hours errors in this information may cause the prediction to be very different from the correct value.

When

,

, so

The rate of increase is

£

, so:

£

£

£

If

then:

Comparing this to

:

The gradient is The -intercept is

So

£

£

In this chapter you will learn how to: expand an expression of the form for any positive integer find individual terms in the expansion of for any positive integer use partial expansions of to find an approximate value for a number raised to a positive integer power understand and use the notations and

a

b

The required term is

The term is the coefficient is

Comparing coefficients of :

as must be positive. Comparing coefficients of :

So:

a

b

The term is: So the coefficient of is

a

b

are:

In this chapter you will learn how to: use the definitions of the sine, cosine and tangent functions, their basic properties and their graphs solve equations with trigonometric functions use the relationships (called identities) between different trigonometric functions use identities to solve more complicated equations.

a

b

a

b

Also,

a

b





There are two solutions.

There are three solutions.

There are four solutions.

Let If

then

There are four solutions.

‚

Let If So 

then

There are two solutions.



There are two solutions in each case.





There are two solutions. is impossible.





When

:

When



:







is impossible.



There are two solutions.







When

When

:

:



In this chapter you will learn how to: use the sine rule to find sides and angles of any triangle use the cosine rule to find sides and angles of any triangle use a formula for the area of a triangle when you don’t know the perpendicular height.



or So the length is

or

Using the sine rule in triangle , let angle :

there is only one solution:

Using to sine rule in triangle :



a

b

In this chapter you will learn how to: sketch the gradient function for a given curve find the gradients of curves from first principles differentiate use differentiation to decide whether a function is increasing or decreasing.

So the gradient is

Since Hence

,

for all

is always increasing.

When

a

b c So the function is decreasing.

So the gradient is increasing.

In this chapter you will learn how to: find the equations of tangents and normals to curves at given points find maximum and minimum points on curves solve problems which involve maximising or minimising quantities.

a At

:

Equation of tangent:

b Gradient of normal:

Equation of normal:

Let be the point with coordinates Then

When

:

Equation of tangent:

Since the line passes through :

When

,

When

,

So the coordinates of are or

Equation of normal at :

So, gradient of curve at is:

When

:

For stationary points

When

:

When

:

:

Therefore, the stationary points are:

The stationary points are

is a maximum

is a minimum.

Stationary points:

End points:

and

So, in the specified region, the largest value of is and the smallest value of is or

a

Stationary points:

Nature of stationary point:

When

:

it is a maximum.

b Maximum value of :

a Let the length of the rectangle be . Then:

Since the perimeter

:

So:

b Stationary points:

Nature of stationary point: the point is a maximum. The maximum value of is:

a



So

b Stationary points:

Nature of stationary point:

it is a maximum. Maximum value of is:

a The coordinates of are The square of the distance between and is:

b

Stationary points:

Nature of stationary point:

the point is a minumum. The minimum value of is:

c From part b, the minimum value of occurs when , so the point is

In this chapter you will learn how to: reverse the process of differentiation (this process is called integration) find the equation of a curve, given its derivative and a point on the curve find the area between a curve and the -axis find the area between a curve and a straight line.

When

,

:

When

,

When

,

:

:

For intersection points:

Area of When When

Let be the radius and the height of the cylinder. Surface area:

Volume:

The graph shows that the minimum value of the surface area is

In this chapter you will learn how to: represent two-dimensional vectors using the base vectors and find the magnitude and direction of a vector add and subtract vectors, and multiply vectors by a scalar recognise when two vectors are parallel find unit vectors work with positions and displacement of points in the plane use vectors to solve problems about geometrical figures.

a

b

 

  

a

b The direction is the direction of

above



The required angle is

a

is a unit vector. b



a

for some scalar

b

is parallel to



Let

Then



:

a i

ii b

a

b

c

a i

ii

b

The coordinates of are

a

b

a

b



a

If

Since and are parallel and contain a common point , then , and lie on a straight line. b

a

b

c

a

b

So is parallel to its length.

and half

In this chapter you will learn how to: use mathematical models to simplify mechanical situations use the basic concepts in kinematics – displacement, distance, velocity, speed and acceleration use differentiation and integration to relate displacement, velocity and acceleration represent motion on a travel graph solve more complicated problems in kinematics, for example, involving two objects or several stages of motion.

GCSE

You should be able to find the gradient of a straight line connecting two points.

1 Consider the points

GCSE

You should be able to find areas of triangles and trapeziums.

2 Find the areas of the shaded regions marked and

GCSE

You should be able to interpret displacement–time and velocity–time graphs.

3 Use this velocity–time graph to find: a the acceleration of the object during the first seconds b the distance travelled during the whole seconds.

,

and

Find the gradient of the straight line connecting: a and b and

Chapter 12

You should be able to differentiate polynomials.

4 Given that

, find:

a b the gradient of the curve when Chapter 13

You should be able to find stationary points.

Chapter 14

You should be able to use integration to find an area under a graph.

6 Find the area enclosed by the graph of and the -axis.

Chapter 14

You should be able to find the constant of integration.

7 A curve has gradient

5 Find the coordinates of the maximum point on the graph of

passes through the point equation of the curve.

and Find the

a

b This is a small difference , so the straight-line model is appropriate.

a b

c

a

b

a

b

a

b

c

a

b

c

a

ii

c

d

a The boat starts the lighthouse.

from

b For the first seconds, the boat moves away from the lighthouse, slowing down. For the next seconds, the boat is stationary. Its velocity is zero. From seconds the boat moves back towards the lighthouse with constant velocity. It passes the lighthouse at seconds and continues to move away from it. c

d

a The velocities are, in order:

So the maximum velocity is

b The maximum speed is



a

b

So the stationary point is a maximum. The maximum velocity is:

c

When

:

The boat passes the rock again after seconds.

b

In this chapter you will learn how to: derive equations for motion with constant acceleration use constant acceleration equations for horizontal motion apply constant acceleration equations to vertical motion under gravity solve multi-stage problems.

a

b

So,

a

b It would take longer as air resistance would decrease the acceleration.

The velocity of the stone is

The maximum height of the stone is

.

the speed is

From the first to the second camera:

From the first to the third camera:

Solving simultaneously:

First stage:

Second stage:

a

 First stage:

 Second stage:

  

b Second stage:



  The total time is



In this chapter you will learn how to: understand what causes motion and the concept of a force (Newton’s first law) relate force to acceleration (Newton’s second law) work with situations where several forces act on an object work with different types of forces, including gravity determine whether a particle is in equilibrium.

a

b

a

b

The ball travels to rest.

before coming

The direction is to the left.

The magnitude is:

So, the resultant force acts to the left at an angle above the horizontal.

a

b

The magnitude of is and its direction is to the left.

a No force, so constant speed.

b

a

b

c

The force in the stick is a thrust

of magnitude

a

b

c



Vertically:

Horizontally:

In this chapter you will learn how to: use Newton’s third law: that two objects always exert equal and opposite forces on each other calculate the contact force between two objects find the tension in a string or rod connecting two objects analyse the motion of particles connected by a string passing over a pulley.

Skater A:

Skater B:

a 

The forces on the table are the table’s weight, the normal reaction from the book and normal reactions from the ground (the thrust in the legs).

b Forces on the book:

Forces on the table:

c The force on the ground from the table is , directed downwards.

Forces on the book:

Forces on the table:

a

b Newton’s second law for the two objects together:

c For the person:

a

b

a  i

 

b 



a  For the car and trailer together:

b  For the trailer:

The mass of the cable wasn't included in the total mass of the two objects. Both the car and the trailer have the same acceleration

For the car:

For the trailer:

For the crate:

a 

For particle :

For particle :

b 

reaches the pulley hits the ground.

seconds after

Time taken for to reach the ground:

In this chapter you will learn how to: interpret statistical diagrams including histograms, scatter diagrams, cumulative frequency curves and box-and whisker plots calculate mean, median and mode and standard deviation for data understand correlation and use a regression line clean data to remove outliers.

⩽

Group Width Frequency density Frequency So the total frequency is The area from to is probability is

Weight, , of egg

Frequency

people, so the

Median



Upper quartile



Lower quartile



a

From the diagram the approximately

b

Eggs with weights in the range are classified as extra large.

percentile is

to

The median income in the UK is slightly higher than in the USA, so people get paid a little more, on average. Both the range and the interquartile range for the USA incomes is larger, so they have a larger spread of incomes. The highest incomes in the USA are higher than the highest incomes in the UK.

£

£

The range is Data in order:

Therefore

These answers are only estimates because you have assumed that all the data in each group is at the centre, rather than using the actual data values.

Lower quartile: Upper quartile: Interquartile range: Upper quartile plus

:

Lower quartile minus

:

The smallest value is which is not an outlier. The largest value is which is an outlier.

£

£

Two standard deviations from the mean is

so £

is an outlier.

In this chapter you will learn how to: work out combined probabilities when you are interested in more than one outcome work out the probability of a sequence of events occurring construct and use a table showing probabilities of all possible outcomes in a given situation (probability distribution) calculate probabilities in a situation when an experiment is repeated several times (binomial distribution).

a

and are mutually exclusive.

b

and are not mutually exclusive.

Let the two events be:

Then: and are independent, so:

Possible outcomes for the total:

The probability distribution of the total: Total

Probability

Let

So the distribution is:

a

Not binomial; the number of trials is not constant.

b

Binomial,

c

Not binomial; the probability of



success is not constant.

d

Not binomial; the trials are not independent.

e

Binomial,

Let be the number of times Anna hits the target. Then:

So:

⩽

In this chapter you will learn how to: understand the difference between a sample and a population understand different types of sampling methods understand and use the vocabulary associated with hypothesis tests conduct a hypothesis test using the binomial distribution to test if a proportion has changed.

a

The sample mean is equally likely to be larger or smaller than the true population mean.

b

For example, the college might be located in an area populated by an ethnic group that is, on average, taller or shorter than the whole population.

c

It is unlikely that both the shortest and the tallest -year-old girls in the country go to this particular college. So Priya is right, the sample range will be smaller than the population range.

a b

Opportunity sample i

The sample may not be representative because people who use public transport are

more likely to have ‘green’ attitudes. ii

The sample could be representative, as there is no obvious link between use of public transport and football.

a

Systematic sample

b

Not all samples are equally likely, for example, the sample with all the people at the bottom of each page has zero probability of being selected.

c

Many people may not be home at this time of the day, particularly perhaps people without children who may be working, so they will not be able to answer. This would mean the calculated mean is higher than it should be.

The proportion of the school which is girls is

of is

a

Quota sampling

b

The researcher would have to know in advance who was going to be shopping on that day to create a random sample, and this is not feasible.

c

The people who do stop to talk to the researcher might not be representative.

a

Cluster sampling

b

Only some countries are chosen in this sample. For a stratified sample values would be chosen from all countries and combined in proportion to the size of the country.

Let be the proportion of boys in the population.

Let be the number of boys in a sample of babies. If

is correct then

Test statistic:

So there is sufficient evidence to reject

At the significance level, there is evidence to support the scientist’s theory.

Let be the proportion of voters supporting Party Z.

Let be the number of supporters of party Z in sample of people. If

is true,

Test statistic:

so there is sufficient evidence to reject

There is evidence, at the significance level, that the proportion of voters who support party Z has changed.

Let be the number of heads out of coin tosses. If

is true,

The significance level is number such that

, so look for a

In order to have sufficient evidence against at the significance level, Robert would need to observe or fewer heads out of coin tosses.

Let be the proportion of A grades.

Let be the proportion of A grades. If is true then

The critical region is

a (where is the proportion of faulty parts) b

The test statistic is not in the critical region, so there is not sufficient evidence to reject There is not sufficient evidence that the proportion of faulty parts has increased.

c

If then the probability of rejecting is:

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Message from AQA This textbook has been approved by AQA for use with our qualification. This means that we have checked that it broadly covers the specification and we are satisfied with the overall quality. Full details of our approval process can be found on our website. We approve textbooks because we know how important it is for teachers and students to have the right resources to support their teaching and learning. However, the publisher is ultimately responsible for the editorial control and quality of this book. Please note that when teaching the A/AS Level Mathematics (7356, 7357) course, you must refer to AQA’s specification as your definitive source of information. While this book has been written to match the specification, it cannot provide complete coverage of every aspect of the course. A wide range of other useful resources can be found on the relevant subject pages of our website: www.aqa.org.uk IMPORTANT NOTE AQA has not approved any Cambridge Elevate content