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Year 1/AS Maths

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PURE MATHS Contents

Simultaneous equations/ the discriminant/ quadratic inequalities – SLIDE 3

Modelling with quadratics/ vertex points – SLIDE 5

Equation of a straight line – SLIDE 7

Cubic function – factor theorem/ factorise/ sketch – SLIDE 9

Proof – SLIDE 11

Binomial Expansion – SLIDE 13

Equation of a circle/ tangent to a circle – SLIDE 15

Transformation of a function – SLIDE 17

Trigonometry in triangles – SLIDE 19

Trigonometric identities & exact values – SLIDE 21

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Solving trigonometric equations – SLIDE 23

Vectors – SLIDE 25

Differentiation from first principles – SLIDE 27

Differentiation: Equation of a normal – SLIDE 29

Differentiation: Stationary points – SLIDE 31

Finding equation of curve given gradient function – SLIDE 33

Area between curve and x axis – SLIDE 35

Exponential modelling – SLIDE 37

Solving equations using logarithms – SLIDE 39

Reducing logarithms to a linear form – SLIDE 41

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Solution:

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CORRECT SOLUTION:

When t=0,

Height = 1.2 m

b) Was correct but might be easier/ less chance of making a mistake if the function is rearranged rather than expanding and simplifying.

When h(t) = 0,

1.25(t-2)² = 6.2

(t-2)² = 6.2/1.25 …

t = 4.23 seconds (3.s.f)

c) Maximum height = 6.2 m

Time = 2 seconds

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Using completed square form.

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Solution:

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A straight line passes through the points (-2, 8) and (6, -10).

It crosses the x axis at point A and the y axis at point B.

Find the area of triangle AOB, where O is the origin.

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A straight line passes through the points (-2, 8) and (6, -10).

It crosses the x axis at point A and the y axis at point B.

Find the area of triangle AOB, where O is the origin.

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Prove that n² - 10n + 28 is positive for all values of n.

b) Always, sometimes or never true?

“ If I subtract 4 from a number and square the result, the answer is smaller than the cube of the original number.”

c) Prove that the difference of the squares of two consecutive even numbers is always divisible by 4.

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Prove that n² - 10n + 28 is positive for all values of n.

b) Always, sometimes or never true?

“ If I subtract 4 from a number and square the result, the answer is smaller than the cube of the original number.”

c) Prove that the difference of the squares of two consecutive even numbers is always divisible by 4.

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Solution:

x co-ordinates get multiplied by 3

y co-ordinates get moved up by 1/2

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The function f(x) is shown below.

Sketch the graph of

Label the transformed co-ordinate points of those shown above.

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Question:

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The function f(x) is shown below.

Sketch the graph of

Label the transformed co-ordinate points of those shown above.

CORRECT SOLUTION:

x co-ordinates get multiplied by 1/3

y co-ordinates get multiplied by 1/2

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The perimeter of triangle ABC is 20 cm.

Given that AB = 9 cm and angle BAC is 60°,

Find the exact lengths of AC, BC, and the exact area of the triangle.

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The perimeter of triangle ABC is 20 cm.

Given that AB = 9 cm and angle BAC is 60°,

Find the exact lengths of AC, BC, and the exact area of the triangle.

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Solution:

a)

b)

c)

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The curve C has equation

It crosses the x axis at points L and M.

Find the area of region R.

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The curve C has equation

It crosses the x axis at points L and M.

Find the area of region R.

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STATISTICS Contents

Sampling methods – SLIDE 44

Mean, Standard deviation & Coding – SLIDE 46

Box plots & comparing data – SLIDE 48

Grouped data: Histogram, averages & measures of dispersion – SLIDE 50

Correlation – SLIDE 52

Probability – SLIDE 54

Probability distribution table – SLIDE 56

Binomial probability – SLIDE 58

Hypothesis testing – SLIDE 60

Hypothesis testing: Critical regions – SLIDE 62

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Question:

A lake contains 3 species of fish. There are estimated to be 1400 trout, 600 bass and 450 pike in the lake. A survey of the health of the fish in the lake is carried out and a sample of 30 fish is chosen.

Give a reason why stratified random sampling cannot be used.

State an appropriate sampling method for the survey.

Give one advantage and one disadvantage of this sampling method.

Explain how this sampling method could be used to select the sample of 30 fish. You must show your working.

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Solution:

You can’t randomly choose a fish as you can’t see them all.

Non – random sampling.

Advantage: Easy to do

Disadvantage: Might not be representative of the population.

d) Just survey the first 30 fish that you fish out of the water.

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Question:

A lake contains 3 species of fish. There are estimated to be 1400 trout, 600 bass and 450 pike in the lake. A survey of the health of the fish in the lake is carried out and a sample of 30 fish is chosen.

Give a reason why stratified random sampling cannot be used.

State an appropriate sampling method for the survey.

Give one advantage and one disadvantage of this sampling method.

Explain how this sampling method could be used to select the sample of 30 fish. You must show your working.

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A sampling frame within each species of fish in the lake is impossible to obtain.

Quota sampling.

Advantage: Any from - Sample can be obtained quickly, costs are kept to a minimum, administration of survey is easy, no sampling frame is required, allows a small sample to still be representative of the population, allows for easy comparison between different groups.

Disadvantage: Any from – Can introduce bias (process not random), not possible to estimate sampling errors, surveyor may not be able to identify species of fish easily.

d) Fish are caught from the lake until the quota of 17 trout, 7 bass and 6 pike are reached. If a fish is caught and the species quota is full, then this is ignored.

CORRECT SOLUTION:

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Solution:

It shows data in a picture form.

All students had completed by 75 minutes.

They are outliers.

d) N

e) School B on average score higher suggesting that they are better runners.

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CORRECT SOLUTION:

a)

37 minutes – this is called the upper quartile.

c) These points are outliers – these two children took a lot longer (possibly walked).

d)

e)

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CORRECT SOLUTION:

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Solution:

1.797 is the gradient of the line.

b) w = 1.797 X 60 – 60.445 = 47.375

c) It seems reliable as the weight is around 10-20 less than the length.

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Solution:

148 students

in total so

50/148

b) 80% + 75% + 70% = 225% or 2.25

0.8 X 0.75 X 0.7 = 0.42

In a school there are 148 students in Years 12 and 13 studying Science, Humanities or Arts subjects. Of these students, 89 wear glasses and the others do not. There are 30 Science students of whom 18 wear glasses. The corresponding figures for the Humanities students are 68 and 44 respectively.

A student is chosen at random.

Find the probability that this student

is studying Arts subjects and doesn’t wear glasses.

Amongst the Science students, 80% are right-handed. Corresponding percentages for Humanities and Arts students are 75% and 70% respectively.

A student is again chosen at random.

(b) Find the probability that this student is right-handed.

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Question:

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In a school there are 148 students in Years 12 and 13 studying Science, Humanities or Arts subjects. Of these students, 89 wear glasses and the others do not. There are 30 Science students of whom 18 wear glasses. The corresponding figures for the Humanities students are 68 and 44 respectively.

A student is chosen at random.

Find the probability that this student

is studying Arts subjects and doesn’t wear glasses.

Amongst the Science students, 80% are right-handed. Corresponding percentages for Humanities and Arts students are 75% and 70% respectively.

A student is again chosen at random.

(b) Find the probability that this student is right-handed.

or

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Solution:

a) Follow the pattern, c = 0.4.

b) None of the numbers are bigger or equal to 8 so probability is 0.

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	1	2	3	4
3Y + 2	5	8	11	14
P(Y = y)	0.1	0.4	0.3	0.2

CORRECT SOLUTION:

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35% of the chocolates manufactured by a company are made with white chocolate and the remainder are made with milk chocolate. The company sells boxes of mixed chocolates in batches of 12. A box is selected at random.

Calculate the probability this box contains

(a) exactly 5 white chocolates

(b) more white chocolates than milk ones.

The company takes a random sample of 15 boxes of chocolates.

(c) Find the probability that exactly 5 of these boxes contain more white chocolates than milk ones.

Solution:

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35% of the chocolates manufactured by a company are made with white chocolate and the remainder are made with milk chocolate. The company sells boxes of mixed chocolates in batches of 12. A box is selected at random.

Calculate the probability this box contains

(a) exactly 5 white chocolates

(b) more white chocolates than milk ones.

The company takes a random sample of 15 boxes of chocolates.

(c) Find the probability that exactly 5 of these boxes contain more white chocolates than milk ones.

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Past records suggest that 30% of customers who buy baked beans from a large supermarket buy them in single tins. A new manager questions whether or not there has been a change in the proportion of customers who buy baked beans in single tins. A random sample of 20 customers who had bought baked beans was taken. The manager found that 11 customers from the sample of 20 had bought baked beans in single tins.

Stating your hypotheses clearly, and using a 10% level of significance, test whether or not there has been a change in the proportion of customers buying baked beans in single tins.

Solution:

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Past records suggest that 30% of customers who buy baked beans from a large supermarket buy them in single tins. A new manager questions whether or not there has been a change in the proportion of customers who buy baked beans in single tins. A random sample of 20 customers who had bought baked beans was taken. The manager found that 11 customers from the sample of 20 had bought baked beans in single tins.

Stating your hypotheses clearly, and using a 10% level of significance, test whether or not there has been a change in the proportion of customers buying baked beans in single tins.

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MECHANICS Contents

Constant acceleration – SLIDE 65

Vertical motion – SLIDE 67

Motion –Time graphs – SLIDE 70

Magnitude and direction of a resultant force – SLIDE 72

Connected particles (horizontal plane) – SLIDE 74

Connected particles (pulley) – SLIDE 76

Variable acceleration – SLIDE 78

Variable acceleration – SLIDE 80

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Solution:

From P to Q: From Q to R:

s = 45 s = 120

u = u u = u

v v

a = a a = a

t = 2 t = 4

Using s = ut + ½ at²

45 = 2u + 2a 120 = 4u + 8a

Solve simultaneously: u = 15 and a = 7.5

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CORRECT SOLUTION:

From P to Q: From P to R:

s = 45 s = 165

u = u u = u

v v

a = a a = a

t = 2 t = 6

Using s = ut + ½ at²

45 = 2u + 2a 165 = 6u + 18a

Solve simultaneously: u = 20 and a = 2.5

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Solution:

b) Using v = u + at

T= 3.33 seconds

c) Using F = ma,

F = 0.6 x 9.8

F = 5.88 N

d) The ball might not have been weighed accurately.

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CORRECT SOLUTION:

Solution continued on next slide….

b) From A to ground:

s = -1.5

u = 22.4

v

a = -9.8

t = T

Using s = ut + ½ at²

4.9 T² - 22.4T – 1.5 = 0

T = 4.64 seconds

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CORRECT SOLUTION:

c) continued…

Using F = ma,

mg – F = ma

0.6 x 9.8 – F = 0.6 x -10646.5

F = 6390 N (3.s.f.)

d) Air resistance or variable F.

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Solution:

c)

s u = 25

v = 75

a = a

t = 4

Using

v= u + at,

a = 12.5

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CORRECT SOLUTION:

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Question:

Two forces P and Q act on a particle.

The force P has magnitude 7 N and acts due north.

The resultant of P and Q is a force of magnitude 10 N acting in a direction with bearing 120°.

Find

The magnitude of Q

The direction of Q, giving your answer as a bearing.

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Solution:

P + Q = resultant

So 7 + Q = 10

Q has magnitude of 3 N

tanx = 3/7

x = 23.2°

So bearing is

157°

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Question:

Two forces P and Q act on a particle.

The force P has magnitude 7 N and acts due north.

The resultant of P and Q is a force of magnitude 10 N acting in a direction with bearing 120°.

Find

The magnitude of Q

The direction of Q, giving your answer as a bearing.

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CORRECT SOLUTION:

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Solution:

Using F = ma,

2400 = 1200 x a

a = 2

b) Using the car,

F = 800 x 2

F = 1600 N

s v = u + at

u = 20 t = 4 seconds

v = 28 which is approximately

a = 2 6 seconds.

t = t

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Solution:

d) Integrate v to get s

s = 16t - t²

When t = 10,

s = 60 m

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CORRECT SOLUTION: