This is a page with advice about what it takes to hit the top grades - 6s and 7s in the Maths Applications exams. Everyone has different starting points and different goals, but this page is specifically about the things to watch out for to help you reach these tougher target grades. There is no substitute for lots of hard work and practice and you will, of course, need a good result in your Internal assessment to support this. Its hard to make a comprehensive list, but here are the kinds of things that I think make a difference in the end.
Read the page and then have a go at this quiz to see how you are doing with the tricky questions!
START QUIZ!
The population of town A was 352800 in 2016 and 388962 in 2018, and is said to have been growing according to a geometric sequence since 2014
a) What was the population of the town in 2014
If growth continues in the same way,
b) What will te population of the twon be in 2020 (to the nearest person)
c) In what year will the population of the town first be more than double the 2014 population
a) 2014
b) 2020
c) More than double in
a) \(2016\quad population\quad =\quad 352800\\ 2018\quad population\quad =\quad 388962\\ so,\quad 352800\times { r }^{ 2 }=388962\\ r=\sqrt { \frac { 388962 }{ 352800 } } =1.05\)
b) \(pop\quad in\quad 2014=\quad \frac { 352800 }{ { 1.05 }^{ 2 } } =320000\)
c)\(If\quad 2014\quad is\quad { U }_{ 1 }=320000\\ Then\quad { U }_{ n }={ U }_{ 1 }\times { r }^{ n-1 }=320000\times { 1.05 }^{ n-1 }\\ Enter\quad in\quad to\quad GDC\quad to\quad find\quad first\quad value\quad of\quad n\quad where\quad pop>640000\\ n=16\\ If\quad n=1\quad in\quad 2014\\ Then\quad n=16\quad in\quad 2029\)
A ball is dropped from a height of 1m. On each bounce, it reaches 80% of its previous height. How far has the ball travelled when it hits the ground for the 6th time?
Total distance travelled = m
A diagram like the following is probably essential in trying to answer this kind of question.
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\(Modelling\quad as\quad a\quad geometric\quad sequence\\ { U }_{ 1 }=1,\quad r=0.8,\quad so\\ { U }_{ n }={ U }_{ 1 }\times { 0.8 }^{ n-1 }\\ On\quad the\quad sixth\quad bounce,\quad the\quad first\quad 5\quad terms\\ have\quad been\quad travelled\quad twice\quad (except\quad the\quad first\quad one)\\ so,\quad we\quad need\quad twice\quad the\quad sum\quad of\quad the\quad first\quad 5\quad terms,\quad subtract\quad the\quad first\quad term\\ Total\quad distance\quad =\quad 2\times { S }_{ 5 }-{ U }_{ 1 }\\ { S }_{ 5 }=1\times \left( \frac { { 0.8 }^{ 5 }-1 }{ 0.8-1 } \right) ,\quad { U }_{ 5 }=1\times { 0.8 }^{ 4 }\\ Distance\quad =\quad 2\times \left( \frac { { 0.8 }^{ 5 }-1 }{ 0.8-1 } \right) -1\times { 0.8 }^{ 4 }\\ =6.3136\\ =6.31m\quad (3sf)\)
For another example, see the Geometric Sequences chapter and exam question 3.
Consider the function below and two given points.
\(f(x)=a{ x }^{ 2 }+bx-8,\quad goes\quad through\quad the\quad points\quad (-2,-10)\quad and\quad (4,-4)\)
Work out the values of a and b.
a= , b=
\(From\quad the\quad point\quad (-2-10),\\ f(-2)=-10\quad and\quad substituting\quad x=-2\\ 4a-2b+8=-10\quad so\quad 4a-2b=-18\\ From\quad the\quad point\quad (4,-4)\\ f(4)=-4\quad and\quad substituting\quad x=4\\ 16a+4b+8=-4\quad so\quad 16a+4b=-12\\ Solve\quad these\quad two\quad equations\quad simultaneously\\ 4a-2b=-18\quad and\quad 16a+4b=-12\\ to\quad get\quad a=-2\quad and\quad b=5\)
Consider the right pyramid in the diagram ABCDE
a) What is the length of the edge EC?
b) What is the angle ECD?
c) What is the area of the triangle ECD?
d) What is the perpendicular height of triangle EBC?
e) What is the angle between the line EB and the plane ABCD?
f) What is the surface area of the pyramid?
g) What is the volume of the pyramid?
a) Length EC=
b) Angle ECD=
c) Area of ECD=
d) Perpendicular height of EBC=
e) Angle=
f) Surface area =
g) Volume=
Consider 2 events A and B, \(P(A)=0.2,\quad P(B)=0.5,\quad P(A\cup B)=0.6\)
a) Work out the exact value of \(P(A\cap B)\)
b) Are these events mutually exclusive? (Y or N)
c) Are these events mutually exclusive? (Y or N)
a) \(P(A\cap B)\)=
b) (enter Y or N)
c) (enter Y or N)
a)\(Since\quad P(A\cup B)=P(A)+P(B)-P(A\cap B)\\ 0.6=0.2+0.5-P(A\cap B)\\ so\quad P(A\cap B)=0.1\)
b) NO, because for Mutually exclusive events
\(P(A\cup B)=P(A)+P(B)\)
c) Yes because for independent events,
\(P(A\cap B)=P(A)\times P(B)\)
A sample of people had their heights measured and the results were said to be normally distributed with
\(\mu =175\quad and\quad \sigma =4\)
The probability of a randomly selected person having a height between 'M' and 180cm is 0.3,
a) What is the value of x
a) M= cm (3sf)
a)See diagram below
\(P(x<M)+P(M<x<180)=P(x<180)\\ P(M<x<180)=0.3\\ So,\quad P(x<M)+0.3=P(x<180)\\ Using\quad NCD\quad on\quad GDC\\ \mu =175\quad and\quad \sigma =4,\quad upper\quad limit\quad 180,\quad lower\quad limit\quad 0\\ P(x<180)\quad =\quad 0.89435\\ SO,\quad P(x<M)+0.3=0.89435\\ and\quad then\\ P(x<M)=0.89435-0.3=0.59345\\ Using\quad Inverse\quad Normal\quad on\quad GDC\\ Tail\quad -\quad left,\quad Area=0.59345,\quad \mu =175\quad and\quad \sigma =4\\ Gives,\quad M=175.954997,\quad =\quad 176cm\quad (3sf)\)
Topic Specific advice
Number and Algebra
See all the chapters in the Number and Algebra section
General speaking, the topics like Standard form, accuracy and estimation tend to be quite manageable and it is just a case of paying careful attention.
The trickier questions come up under the sequences and finances section.
- Make sure you are happy with using the compound interest formula with different compounding periods.
- IF you want to use the finance app, then make sure you know exactly how!!! See this page on StudyIB.net
- Know how to use your table function to model different sequences or investments so you can find out questions like 'after which year/term is sequence 1 bigger than sequence 2. Write the nth term of the sequence, then substitute U1 for y and n for x.
- Be careful with time units in finance questions. When you set up your models or use the finance app, then be aware what 'x' st or 'n' stands for (months, years, quarters etc) and remember to pay attention to the time unit requested in the question.
- Understand the key differences between loans, savings accounts and annuities and how this plays out with your finance app!
The following refer to the kinds of things you can find on the Geometric Sequences page
Example 1 - Looking for common ratios in sequences
One of the questions that keeps catching people out goes like this... The population of town A is growing like a geometric sequence. The population in 1990 was..., in 1993 it was.... and then asks you to find the population in 1985.
This can be tricky because you can get focussed on what the first term is. The key is to use the first 2 bits of information to work out the common ratio is. The population has been multiplied by 'r' 3 times between 1990 and 1993. THEN you work backwards to get 1985
Example 2 - Recognising geometric sequences
A question might talk about a regular, recurring percentage increase or decrease. For example, 'a population increases by 7% each year.' or 'The value of a car decreases by 15% each year'
The thing here is to recognise that the percentage increase or decrease gives us the common ratio of a geometric sequence. in the first case, for a 7% increase, you would have r = 1.07. In the second case for a 15% decrease, you would have r = 0.15
Example 3 - Looking at the specific details of the problem
Something like 'a ball is dropped from a height of 1m and reaches 80% of its height on each bounce. How far has it travelled before it hits the ground on the 6th bounce?
This can be tricky, because the respective heights are a geometric sequence with r = 0.8 (like example 2) but every term except the first should count twice since the ball bounces up and then down. In the first case it only travels that distance (1m) once once
Functions
Check out all of the examples on the Functions section for a recap.
- There is very little in the formula booklet, just the formula for the axis of symmetry. You just have to remember to use it! When the question says 'show that the axis of symmetry is x=?' it expects you to substitute the values, not draw an arrow on your graph.
- Know your general expressions for linear, quadratic and exponential models and the role of each part - remember the constant is not the y-intercept for exponentials.
- GDC skills
Example 1 - Details on graph drawing
Graph drawing questions - there is always one and they are worth quite a few marks that you MUST get.
These are easily lost if you are not careful. Pay careful attention to scales on axes, labels for axes, precision with point plotting, given domains for function plotting. Remember to try and draw smooth curves
Example 2 - Working backwards
I am talking about the kind of questions where a quadratic is given as f(x) = x2+ bx + 5, and early in the question you are supposed to work out the value of 'b'. There will be some other piece of information like, the axis of symmetry or a given point that means you can substitute some values and create an equation that can be solved for 'b'. Imagine that you are given the axis of symmetry at x = 5. From your formula booklet, you have the axis as x =-b/2a, in this case you have a=1, so 5 = -b/2 and so b = 10.
In some cases you might even be expected to generate 2 equations. For example, the quadratic ax2 + bx + 5 goes through the points (1,4) and (-2,19), what are the values of a and b?
Substitute x = 1 and y = 4 to get a + b +5 = 4, and so a + b = -1
Substitute x = -2 and y = 19 to get 4a -2b + 5 = 19, and so 4a - 2b = 14
Then solve the two equations simultaneously to get a and b
Example 3 - Simultaneous equations
There are some questions that give you information about 2 different points on a curve and the basic function of the curve. In this case, you are expected to construct a system of simultaneous equations and use your GDC to solve them so that you can deduce the model.
Geometry and Trigonometry
Get everything you need on these ideas on the Geometry and Trigonometry page
Angles of elevation and depression - you need to know what these mean, otherwise these are easily lost marks
Example 1 - Angles and slopes in 3D shapes
This can be hard to visualise and so it can be easy to confuse. For example - Consider a square based pyramid. The four triangular faces are all congruent - It can easy to confuse 'the perpendicular height' of those triangles with the height of the pyramid. Also, when being asked for the angle between and a line and a plane, be careful to pay attention to the diagram carefully.
A good rule to follow is that for every part of a question about 3D geometry, you should always draw a small sketch of the 2D triangle that shows all of the details you are working with.
Example 2 - The Easiest route - which rule?
Sure, its fine, if it works, but at the top level, you need to be efficient with time. So don't waste time using the cosine rule to solve for a length in a right angled triangle and SOHCAHTOA will do. Often students will over look a simple Pythagoras's theorem calculation for something that takes a lot longer.
Statistics and probability
GDC skills - This is a big one. You MUST know how to find things quickly on your GDC. See the Your Graphical Display Calculator page and the Statistics and Probability pages for all the help you need.
- 1 variable data stats
- Using frequency tables - you must be sure how to set your GDC to read list 2 as a frequency and know how to tell it when you want it to and when not!
- 2 variable data stats - when you need summary stats about the 2 variables you might be using for a regression.
- Linear regression
- Entering data in a matrix for a chi squared test
- Knowing how to do Normal CDF and INV normal calculations
- Know what to about choosing left or right tail.
Example 1 - Inverse normal with 2 boundaries
This can be the trickier kind of question. Imagine a question about normally distributed heights. You are told that the probability of a height being between 'x' and 175 is 0.3 and then asked to find the value of 'x'
- First - Find the probability that the result is less than 175, using a normal CDF
- Second - Subtract 0.3 from that value, this will give you the probability that the result is less than x
- Third, then solve for 'x' using an Inverse normal calculation.
This is just the extra thinking that can be used to separate out the top students.
Example 2 - Conditional probabilities from tree diagrams
Check out the Probability page and particularly part 2 on conditional probability
These questions tend to be OK, when working from a table or a sample space, BUT can be a lot trickier when talking about a tree diagram. See the second to last slide on this Focus on Probability page. Essentially, when the word 'Given' has a bearing on the question, then you know you are then excluding everything else.
Example - laws of probability
Check out the last slide on the Probability page about the laws.
These are available in your formula booklet and very occasionally, a question will ask you give you some probabilities and ask you to decide if the events are independent or not. In this case you are expected to substitute the vales you have in to the 'laws' to test.
Calculus
See details on the Calculus pages
- X value, function, derivative - Know that for any given function, we can be talking about each of them. The x value, the value of the function at that x value, the value of the gradient at that 'x' value
- GDC settings - It is very helpful to have the derivative turned 'ON' so that you can always see the three things above in the table. This also helps you to check your answers.
- Differentiating negative indices - You need to be comfortable expressing terms with negative indices so that they can be derived.
Example 1 - Optimisation with dependent models
See Calculus Part 3 for help with this!
Without a doubt, the business end of the exam! A good calculus question based on some 3D shapes (spheres and cones, so expressions involve pi)
The model is usually...
- Here is a picture,
- Show that volume is given by.... (using the given values and variables in the question)
- The surface area is fixed at...
- Hence show that the Volume can be given by..... (usually designed to eliminate variables
- Thus show that the surface area is....
- Find the value of 'x' that gives the max/min
The joy of the show that parts is that they do give you the answer so...
1. You can check how you have done and spot your mistakes
2. You can, if you have to, leave out a part you can't do and still move through the question.
How much of Sixes & sevens have you understood?
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