Paper 2 Question 11 - Linear Programming

Here are the notes on Linear Programming:

Introduction - here and here
Looking at part a) half-planes
Worked example of a part b).

NB: when answering part b) (i) which is usually "write down two inequalities in x and y and illustrate these on graph paper" you must
a) use graph paper and
b) shade in the region which represents all the inequalities (including x≥0 and y≥0) or at least indicate the half planes with arrows. If you just draw the lines you could lose 3 marks per inequality ... or 1%!

Paper 2 Question 7 - Statistics

Here are the links to the statistics notes on this blog.

Introduction - here
A follow up with notes on an exam question - here

Paper 2 Question 6 - Probability

Here are the links to notes on Probability:

A Tricky probability question done using rules of probability and sample space technique.

Paper 2 Question 5 - Trigonometry

Here are the links to Trigonometry
The main points you need to know to cover 90% or questions asked in exams are listed here.
A one-page overview and a subtlety of finding the inverse Sine of an angle > 90° is here.

Paper 2 Question 2 and 3 - Co-ordinate Geometry of the Line and Circle

Here are the links related to Co-ordinate Geometry of the Line and Circle.

A one-page overview of the topic here.
The key points you need to know for coordinate geometry of the circle here.

Paper 2 Question 1 - Perimeter Area and Volume

These links are related to Perimeter, Area and Volume.
An overview here
Lots of examples of how to tackle different types of Simpson's rule questions here.

Paper 1 Question 6, 7 and 8 - Functions and Differentiation

Here are the links relating to Functions and Differentiation:

An overview of functions and graphs here.
Some notes on differentiating from first principles here.
Some notes on how you lose marks when differentiating from first principles here.
One page overview of differentiation here.

Paper 1 Question 4 - Complex Numbers

This is the only blog entry on complex numbers. It gives the 1-page overview.

Paper 1 Question 2 and 3 - Algebra

Here are all the links to algebra entries on this blog.

There is an overview of algebra here.
An example of simultaneous equations one linear one quadratic and awkward algebraic fractions can be found here. (sorry about the formatting in the second part, if anyone can tell me why " " doesn't work, I'm all ears).
There are some notes on numberlines and manipulating indices here.

Paper 1 Question 1 Arithmetic

Here are the links to arithmetic to help you revise for this question:

There is an overview of the arithmetic question and in particular compound interest here.

Linear Programming Class - Thursday morning

There will be a special Linear Programming class on Thursday (22nd May) morning for anyone who is interested. The class will be in Room 1 at 8:40 (normal maths class time).
If you would like a class focussing on another topic, or just want to have a meeting with me, contact me at the school asap.

Checklist for exam

Make sure you have your exam strategy ready long before the exam. Be familiar with which questions you are planning to answer.

Have your pens, pencil, calculator (check battery), geometry set ready well before the exam time.

2½ hours for exam = 25 mins per question. Exam starts at 9:30, you should be well into your 2nd question by 10am. If you have choices, do not spend too long deciding which questions you are going to do. Circle the questions you are going to do and draw a pencil line through the questions you are definitely not going to attempt.

Write clearly. Number all questions and parts of questions. Don't waste time rewriting the question. Highlight your answer clearly.

Keep all parts of each question together. It is a good idea to keep questions in order. If you have to come back to re-attempt a question later in the exam, it will be easier to find it.

Read questions carefully, do your work and then read question again to make sure you have done what was asked and to make sure you format your answer properly (correct to 2 decimal places, as a fraction, in surd form, in km/h etc.).

Attempt all questions. Where there is a part (i), (ii), (iii) etc., don't assume that if you cannot finish part (i) that you cannot attempt part (ii), (iii). If you are running out of time remember how to get attempt marks - “any correct substitution”.

Only attempt a 7th question after you have checked the first 6.

Some students find it reassuring to write all the formulae you have memorised down as soon as you are told that you may start.

Remember, for every blunder (-3 marks) you lose ½%, so be careful!

How NOT to lose marks in your LC exam

Question		How I lose marks ...		How I get them back.
Arithmetic	▼	Not giving answer in correct format.	▲	Recheck what was asked.
	▼	Not handling percentages, percentage error and taxes properly.	▲	Practice all type of percentage questions
Algebra	▼	Failing to square an expression properly (x+y)² = x² + y² + 2xy	▲	Practice
	▼	Getting sloppy towards the end of a simplification, mixing up signs, adding instead of multiplying....	▲	Practice and be careful. Then, substitute test values to verify results.
	▼	Failing to find corresponding y-values after solving for x, esp when one simultaneous equation is quadratic.	▲	Pay attention to steps of solution – always include the checking answers step.
	▼	Not handling laws of indices properly	▲	Learn these – check them on your calculator! Make sure you can enter complicated indices on your calculator correctly.
Complex numbers	▼	Failing to set i² = -1	▲	Practice.
	▼	Not recognising symbols and their meanings: complex conjugate z-bar, modulus |2+3i|	▲	Go over chapter – it is quite self-contained.
	▼	Not handling substitution properly w=2+3i, what is w²	▲	Look at past exam questions.
	▼	Not knowing quadratic formula properly	▲	Write it down and check it every day.
Functions	▼	Not showing marks on graph or understanding how to use graph,	▲	Use a pencil and ruler to show how you interpret graph.
	▼	Uneven scale on x- or y-axis	▲	Use graph paper, watch that all intervals are even, including 0 to 1, 1 to 2 etc.
Differentiation	▼	Not taking care applying rules for differentiation, or not knowing chain rule (it is not in tables)	▲	Learn it and practice.
	▼	Leaving out LHS or Limit idea in first principles.	▲	Practice.
Perimeter, area & volume	▼	Not knowing simpson's rule inside out.	▲	Practice, especially with unknown heights etc.
	▼	Not handling Π as requested.	▲	Re-check what you are asked.
	▼	General errors applying formulae.	▲	Practice using tables for formulae, and sanity check your answers.
Co-ordinate geometry of line	▼	Not knowing formulae and how to apply them.	▲	As for quadratic formula, write them out each day – make sure you understand logic of slope, midpoint and distance formulae.
	▼	Uneven scale on x- and y-axis (must be both to same scale)	▲	Use graph paper and check this.
Co-ordinate geometry of circle	▼	Not taking minus signs into account in formula for equation of circle (e.g. (x-2)² + (y+3)² = 8 means centre of circle is (2,-3) and radius = √8	▲	Practice.
	▼	Failing to understand geometry of circle (tangent at a point perpendicular to radius, tangents at opp end of diameter are parallel)	▲	Go over JC geometry if necessary.
Trigonometry	▼	Not knowing or correctly using Sin= 0pp/Hyp etc	▲	Learn mnemonic.
	▼	Failing to “fill out” the sine rule/ cosine rule correctly.	▲	Practice.
	▼	Being caught out by inverse sine of an angle that is >90º	▲	Be alert.
	▼	Having calculator set to radians or not knowing how to use degrees/minutes function.	▲	Check it and practice.
Probability	▼	Not showing your work	▲	Can be difficult, but write something in case your answer is wrong.
	▼	Failing to see that there are two ways a combined event can happen and adding probabilities of each.	▲	Use a sample space to check answers
Statistics	▼	Forgetting to divide by sum of frequencies	▲	Sanity check answers
	▼	Not showing marks on graph (eg median from ogive)	▲	Indicate how you use graph clearly
Linear programming	▼	Not indicating half-plane when graphing inequalities	▲	Include it in your graph.
	▼	Leaving out the “obvious” inequalities, x≥0	▲	Make sure you give the number of inequalities you were asked for.
	▼	Not using simultaneous equations to calculate point of intersection.	▲	You must not use your graph for this part.
	▼	Not writing english sentence at the end	▲	Explain what x=10, y=7 means, 10 what, 7 what and why (“to maximise profit” or whatever).

Probability Question

While the probability question can be very straightforward, you need to pay attention to the detail of what you are being asked. I gave this question (1999) in the mini-mock exam and nobody got it right.

In your exam, you can answer questions like this using the sum or product of probabilities as appropriate or using a sample space. While the sample space might take some time to set up, it is a foolproof method. To illustrate how the probabilities work, I have created the matching sample space for each part of the question. (you will need to click on the image to be able to read the text within the sample space diagram)

Note that there are 66 possible combinations, as you cant pick the same exact sample twice and selecting e.g. A1 and B2 is the same as selecting B2 and A1.


Looking at part (i):From the sample space you can count the 10 cases where this can happen. So the probability is 10/66 = 5/33


Using probabilities only, you work it out like this:

P(1st is type A) = 5/12

P(2^nd is type A) = 4/11 [If the first one was an A, then there will only be 4 As left. Also, there will only be 11 samples left, as the first sample isn't replaced.]

P(1st is type A AND 2^nd is type A) = 5/12 x 4/11 = 20/132 = 5/33

Looking at part (ii):

Counting from the sample space you get 12/66 = 2/11.
Working this out using probabilities is trickier.
Here you have to remember that there are two ways this could happen. The first way is that you get a B first, then an O, OR the second is the other way around. In both cases you have to remember that you are choosing from 12 the first time you pick and from 11 the second time.
The probabilities work out like this:
P(B then O OR O then B) :
Work out P(B then O) first:
P(1st is type B) = 4/12
P(2nd is type O) = 3/11
So P(B AND O) = 4/12 x 3/11 = 12/132
Then work out P(O then B)
P(1st is type O) = 3/12
P(2nd is type B) = 4/11
So P(O AND B) = 3/12 x 4/11 = 12/132
(Note you could have assumed that you "B followed by O" is equally likely as "O followed by B" so that probabilities would be the same).
Now work out P(B then O OR O then B)
= P(B then O) + P(O then B)
= 12/132 + 12/132
= 24/132
= 12/66 (same as sample space above)
Simplify to 2/11.

Looking at part (iii):From the sample space you can see that this can occur 19 ways out of 66 so P(both the same type) = 19/66.
Using probabilities, you need to consider what is a successful outcome. If you write out what "of the same blood type" means, you will see that a successful outcome is "both A OR both B OR both O", so you need to add P(both A) + P(both B) + P(both O).
You have already found the probabilty of both samples being type A in part (i).
The answer was 10/66 or 5/33.
Now you need to find the probability of both being B and the probability of both being O.
P(Both B) = 4/12 x 3/11 = 12/132 = 6/66
P(Both O) = 3/12 x 2/11 = 6/132 = 3/66
(I am keeping all fractions with 66 as common denominator on purpose).
Now add
P(Both A OR both B OR both O) = 10/66 + 6/66 + 3/33 = 19/66

Mini-mock Paper 2 Feedback

For the Trigonometry question, you just had to know your Trig ratios sin=opp/hyp etc and apply them to the triangle. It wasn't necessary to work out the exact answer. You lost marks if you didn't give the answer as a fraction.

The question on Co-ordinate Geometry of the circle was well answered. You had to use the midpoint formula to find the centre of the circle.

Centre of S is midpoint of diameter. (1,-2)

Radius = distance fom either endpoint of diameter (e.g. (-1,-1)) to the centre (1,-2). (or half the diameter)

= √(1+1)² + (-2+1)²

= √5

Now use radius √5 and centre (1,-2) to form equation of circle:

(x – 1)² + (y + 2)² = 5

The simpson's rule was straightforward and was well answered in general.

Nobody got the probability question right - I am going to create a full post about that.

The Linear programming question was tricky, because it was easy to overlook the "number of drivers available" as a limitation. Most people who got this second inequality right got the correct solution.
NB: Don't forget that you must use simultaneous equations to find the vertex at the intersection of the two lines.

By the way, it looks like we will have only one class left before Friday, so have your questions ready.

Paper 1 feedback

Following the mini-mock exam yesterday, there were a few noteworthy common errors.

Firstly, lots of students lost marks by not following instructions exactly.

• The solution set of an inequality isn't n<4. You have to list the elements as {0,1,2,3} or graph this on a number line.
• The cooking instructions question asked for the time in hours and minutes - if you gave it in minutes only, you lost marks.
• On the functions question, you were asked for the co-ordinates of the local max and min - if you only gave the values of x for which the max and min occur, you lose marks.

Every time you get a blunder (-3) you are effectively losing ½%

In the differentiation question there are still issues.
Right throughout the first principles question there are opportunities to lose marks. The main one that we've covered before is not having a left-hand-side, and not mentioning the concept of a limit.
Also lots of students cancelled
3h² + 6xh + h
--------------------
h
to get
3h + 6x
forgetting that h/h = 1

The complex numbers question was poorly answered. This needs to be revised. The previous entry on complex numbers is here.

Also, the performance on the laws of indices question was weak.


Almost everyone got caught on the second question where each part of the equation was already in the form 2 to the power of something. This was a part (c) question so it is unlikely to be so simple. You must work out the right-hand-side
(64 - 32 = 32 = 2^5) and then equate the powers.
This is also a good place to demonstrate the value of checking your answer.
If you do the question wrong:
3p-7 = 6-5
3p = 8
p = 8/3
Try substituting this back in to the original and you will see that it doesn't balance.
The correct way is to recognise that the RHS ends up as 2^5 and then equating the powers correctly, you get:
3p -7 = 5
3p = 12
p = 4

Also, don't forget ... always make an attempt. Many attempts are work 3 marks - which is ½%

In tomorrow's double, we will take a general look at Paper 2 and then we'll do a Paper 2 mini-mock in our second period.

Last bits of revision

In tomorrow's class we will have a Paper 1 mini-mock exam.
On Thursday we will do a proper revision of co-ordinate geometry of the circle as this topic seems to be still causing problems for many.
On Friday we will have a Paper 2 mini-mock and then resume revision, including probability.

We'll also cover general exam approach and prep over the next few days.

Perimeter Area and Volume

For my one-page overview you can look at the previous entry.

In class on Thursday, we looked at the different types of questions you can get asked for perimeter, area and volume.

Finding area and perimeter of rectangular shapes and triangles
Irregular rectangular shapes: break them into smaller rectangles.
Triangles: half the base by the perpendicular height.
[Area of a triangle of where base = 8cm and perpendicular height = 3cm multiply as follows
1/2 x 8 x 3 = 12cm²]

Volume of a prism (any rectangular solid with a uniform cross-section)
Find the area of the uniform cross-section and multiply this by the length of the prism.

Circular shapes
The first question is how to handle π, pi.
The most important thing is to follow instructions, using 22/7, 3.14 or giving your answer in terms of π.
Know where to find the formulae you need.

Trickier questions
Changing shapes
If a solid in one shape is being melted down and recast into another shape you need to set up an equation. If both shapes are circular (eg a sphere being melted down into a cylinder, or liquid in a cylindrical jug being poured into smaller cylindrical tumblers) then the πs on both sides of the equation will cancel. In questions like this you will not be told how to handle π.
Liquid in a pipe
Water flows in a pipe at the rate if 50cm/sec. The pipe has a radius of 2cm. How long will it take to fill a tank measuring 30cm x 30cm x 100cm.
To answer a question like this, consider a single molecule of H2O in the pipe. It takes 1 second to travel 50cm - that means that in the space of 1 second all the molecules are replaced. If you find the volume of 50cm of pipe you will have 1 second's worth of water.
[This is π x 4² x 50]
Then find the volume of the tank
[This is 30 x 30 x 50]
Finally divide the first answer into the second answer to find the time in seconds.
Then convert to minutes or hours as required.