Revision questions for the Mid-term break

These are the questions you should try to work through over the mid-term break. Attempt as much of it as you can after revising the related chapters.
If you get stuck leave the work and try a different question before you come back to it. Check here over the course of next week, as I will post some tips for each section.
In the two classes after the mid-term break, we will cover as much ground as possible and hopefully resolve any issues that you got stuck on.

I will add some tips for some of the questions over the next few days, in blue text like this.
I have now finished adding my notes - you can add a comment here if you need any more help.


Preparation for paper 1:
Question 1 and 2 - arithmetic, percentages, tax, number theory (primes etc), sets, indices, surds and that type of thing.
(Chapters 11, 13, 16, 17, 18)
Page 150 q5 (note part d on next page!) Remember to multiply across to get rid of the fractions in the ratio part a. In part b, change all the money amounts to either cent or euro and remember to sanity check your answer. For the last part you are trying to find what percentage one number is of another, and the rate of VAT should be reasonable. For part d, remember the layout of these questions that we used in class, and how you can navigate backwards from the net tax, to gross tax, to the 2 tax amounts (you can calculate the amount of tax paid at the lower rate).
Page 192 q2 (You'll need to revise all the symbols in set notation - book 1 ch 15).
Page 238 q4 (in ex 16.5) and q 12 (next page) (Look for factors that are square numbers. In q12, the denominator is the clue to the rest of the question, try expressing the other numbers as something × √2)
Page 247 q1, 2, 3, 7, 8, 9 and 18 (Refer to the rules on page 240/1 and practice using your calculator to check answers).
Page 253 q 17, 18 (Make sure you know how to do these on your calculator!)

Question 3 and 4 and 5- algebra
(Chapters 1 to 7 and 27)
Page 13 q12,15,18,21,24 (Last one has 3 factors. If you have trouble with these, practice a few more from the page.)
Page 19 q25 (Be careful with the "-" sign. The Lowest Common Denominator is the LCM of all 3 fractions. Note this isn't an equation, you are just rewriting a fraction, so you can multiply above and below by the same thing as in e.g. 1/3 = 2/6)
Page 22 q17 (I only set one Long Division question, but if you have trouble, practice some more. Remember that there is never a remainder.)
Page 26 q7, 13 (Another important skill to practice as it comes up on paper 1 and 2. In q13, you need to tidy it up by getting rid of the fractions. As it is an equation, you must multiply the left hand side and right hand side by the same thing.)
Page 38 q4
Page 40 q15, 17 ( Note that q15 doesn't look like a quadratic until you multiply across by everything under the line to get rid of the fractions (***). In q17 "correct to 2 decimal places" means you have to use the quadratic formula.)
Page 46 q12, 14 (First get rid of brackets, then treat it like an equation (adding to from both sides, dividing across by same thing etc) to arrive at e.g. x ≤ a number. Revise the rules for plotting x < or x ≤ and for plotting x ∈ N, Z or R).
Page 50 q9,10
Page 397 q2, 3 and next page q8, 10. (We got quite far in q10 in class.
i) number of rows needed first night = 600/x
ii) number of rows needed second night = 630/(x-2)
iii) the number of rows on first night + 5 = number of rows on second night.
or
the number of rows on first night = number of rows on second night - 5
Giving us
600/x + 5 = 630/(x-2) (this will be a quadratic when you tidy it up and get rid of fractions - see comment above (***))
These questions are tricky, but get easier with practice.

Question 5 and 6 - functions
(Chapters 14 and 15)
Page 226 q15 (If you rearrange f(x) to have the x² term first, then make sure you keep the minus signs in the right place. Your g(x) should be a straight line. For part (vii) you are looking for output values of f(x) that cannot occur. For example, if you look at the graph on page 223, f(x) = 6 or f(x) = 7 has no solution. You can never get those output values from the function f(x). In that case f(x) > 5.1 has no solution.)
Page 230 q 5 (The wording is a bit confusing here - they talk about the width of the rectangle being 10-x where they mean the height according to the diagram. If the perimeter is fixed at 20, then x(10-x) = 10x - x² is the area, regardless of what value x has. So if x = 5, you'd have a square with area = 25. If x = 2 you'd have a 2 × 8 rectangle with area 16. Follow the normal steps you go through for graphing functions. The output values of f(x) will be the various possible areas of the shape.
This is a simpler version of the "A4 page malteaser box" problem we looked at briefly in class.)

Preparation for Paper 2
Question 1 - Perimeter area and volume
(Chapter 12)
Page 178 q3 and page 180 q6 (Q3: Careful with diameter v radius in part a and b. In part b and c you need to set up an equation and solve it. E.g. Vol of cylinder × 6 = Vol of block. For Q6: you need to note what fraction of the pizza 135º represents i.e. simplify 135/360. Note also you are interested in how much pizza is left. For part c remember that the radius, height and slant-height of a cone form a right-angle triangle.)

Question 2 - Coordinate Geometry
(Chapter 10)
Page 123 q12, 13, 14 and page 125 q27 (Revise transformations especially translations. Remember that the corners of a parallelogram are always given in order either clockwise or anti-clockwise. Also remember how to change the slope of a line to find the slope of the perpendicular line - invert and change sign.)

Question 3 and 4 - Geometry
(Chapters 20 to 26 and 28) (NB We will cover ch 26 after the mocks!)
Page 311 q11, 16 (For q11: use the angles you are given and properties of opposite angles and quadrilaterals. For q16, look for which angles are equal and isosceles triangles).
Page 318 q1 (For part v, construct the missing lines and try to prove that the newly formed triangles are congruent and that the sum of alternate angles make the opposite angles equal.)
Page 332 q6, 9 (Note that ∠acb is not shown. Q9 i: ∠bap = 45º then find two angles that measure 90º For part ii, one way to solve it is to construct a line completing the triangle abp. Then you can use the fact that triangle abp is isosceles. Find another isosceles triangle ... hint: radius of a circle is a constant.)
Page 359 to 363 practice all constructions.
Theorems: pages 401 and 408. (Don't forget that the latter theorem boils down to proving that the lines are parallel - then you use the (not examinable) earlier theorem which I gave you a handout for).

Question 5 - Trigonometry
(Chapter 19)
Page 294 q13, 14 (Q13: For part a, try drawing a rough sketch, marking the sides you know, being careful to place them correctly with respect the the angle A which you need to mark. You then have to work out the missing side of a right angle (using pythagoras). sin²A is just the (sinA)². For part b i, note that you need two sides and the angle between them to use ½abSinC. Hint: think radius!!! For part b ii, you need to work out 80/360 times the area of the circle. For part c, draw a rough sketch and label both sides you know. You will need a compass. For part d, no right angles here, so you need to use the Sine rule twice. Remember that the angles in a triangle add to 180°
Q14: Part a, similar to part a in q13. Note that you can find the area of a triangle more easily if you know the base and the perpendicular height. For part c, use pythagoras, then the Sine rule. Don't assume any angles are 90° just because they look like it! Part d, first draw the CAST diagram to work out which quadrants have negative Cos.)

Question 6 - Statistics
(Chapter 9)
Page 92 q6 and page 95 q10 (don't forget part c on next page) (In part a of the first question remember that 45 seconds is ¾ or 0.75 of a minute. These questions will help you practice your statistics, especially histograms and ogives. In each type of graph, make sure you always put the frequency (no. of students or no. of people) on the vertical axis. If you have 100 people then the median person is the 50th, the lower quartile person is the 25th and the upper quartile person is the 75th. Find each of these on the vertical axis, draw a horizontal line across and drop a perpendicular from where it hits the curve.Remember that histograms are a little unintuitive, the area of each block reflects the frequency and there are no gaps between blocks.)

Using Equations to solve problems

Finally, we get to the whole point of algebra.

You are given some text and you have to create an equation and then solve it. Sometimes you can work the easy answers out in your head, but you must write out the equation. That way you practice the technique and develop the skills to work out tougher problems also.
The work for the long weekend included these 2 questions:

p386 - q4 (make sure you only use a single unknown)
p389 - q1 (simultaneous equations)

and I also want you to try to solve the third type of problem you can enounter

p394 - q3 (this is solved using a quadratic equation)

We will look at each of these as well as a much trickier problem of the 3rd type (quadratics) tomorrow.

Coordinate Geometry

P124 q 23

For parts (i) through (iv) first write down the relevant formula, then plug in your values.
Watch out for the negative coordinates - say for example, when you are finding slope x2-x1 is going to be 6 - (-2) = 6+2 = 8

For part (v) remember what "equation of a line" means. It is a rule for being on a line.
For example - is the point (2,1) on the line 2x + 3y = 7?
Sub in for x and y: 2(2) + 3(1) = 7 which is true, so the point (2,1) is on the line.
Is the point (5,-1) on the line. Sub in and make sure you can figure out that yes, it is.

For part (vi) remember the rule for perpendicular slopes. If two lines are perpendicular, their slopes multiply to give -1 (or put another way, the product of their slopes is -1).
So if you have the slope of one line, you find the slope of the perpendicular line by inverting and changing the sign. If the slope of the first line is -2, then the slope of the perpendicular line is ½.

Part (viii) can be done using simultaneous equations.

For part (ix) draw a rough diagram if you haven't already done so. Axial symmetry in the line ab means that the line ab acts as a mirror. Where will the image of d be? A clue is to use translations.

For part (x) you are finding the points where the line crosses the X and Y axes. Remember that every point on the X axis has the coordinate (something,0).
You can work out the "something" by subbing y=0 in the equation of the line ab (see ans to part iv).

Finally, make sure that you use this question as an opportunity to revise all the formulae you need to know.

Really finally, don't forget your geometry sets tomorrow!

Theorems

Don't forget to use the pro-forma (template) layout when proving theorems).

A diagram, labelled followed by
Given:
To prove:
Construction:
Proof:
... and the last line of your Proof: should be the same as your To prove: line.

Look over the first 2 theorems on page 401 and practice writing these out.
Look over the theorem we did today - I will give you a simplified version of it which is from the marking scheme (2004) in class tomorrow.

Then practice your constructions from the Ordinary level course (you'll find more detail on them in book 1).
You should be able to draw parallel lines, bisect an angle, construct the perpendicular bisector of a line and divide a line into 3 or more equal parts.
These constructions take some practice, so it is worthwhile spending time.
Remember that the construction lines aren't supposed to be hidden - so make sure that any arcs or other marks you make on the page are visible.

Don't forget your geometry sets for Tuesday!!
You need at least a compass and ruler, preferably a set square also.

Indices Problems

Taking a closer look at page 248 question 22:

(i) You are asked to express √2 as 2 to the power of something.
Remember that 9 to the power of ½ is the same as √9 which is = 3.
Make sure you know how to check this on your calculator.

(ii) Know your square and cube numbers.
Square numbers are 1,4,9,16,25,36,49,64,81,100,121,144,169,196 (up as far as 14²)
Cube numbers are 1,8,27,64,125,216 (up as far as 6³)
It is also useful to know your powers of 2.
2²=4, 2³=8 2^4 = 16 etc.

(iii)Looking at 8√2 you need to use your answers for the previous two parts to work this out. 8 = 2³ and √2 = 2 to the power of ½
So you need to apply the first rule of indices on the list on page 240, adding the two powers (even if one is a fraction).

Finally solving the equation.
Leave the LHS as it is.
Write down the RHS in the form that you arrived at in part (iii).
Now you have:
2 to the power of something (3x-1) = 2 to the power of something else (ans from part iii)
So you ignore the 2s and equate the powers to solve for x.

Note that you can check your answer by putting
2 ^ (3×x +1) into your calculator (subbing for x) and then check if you get the same result for 8√2.

Work for the weekend

As promised ...!

First, do the correct questions from the surds chapter.
P238 Q 1,2,3,4,5 of the chapter test
In question 1 and 2 you get a clue to the answer in the question - you know that you can rewrite √108 as √something × √3
In question 3 rewrite "Three and one sixteenth" as a top-heavy fraction, then the answer will become clearer.
Remember in all these questions that you need to look out for square numbers: 1,4,9,16 etc.

Next, the new stuff:
Page 247 q 13,14,17,22
In each of these, refer back to the rules and examples from pag 240 and remember how to solve equations with exponents.
In particular q22 is one that you should try hard to work through. Parts (i) (ii) and (iii) are needed to solve the main part of the question.

We will spend most of next week revising geometry.
As preparation for this, try the following questions. Remember how to write out a formal proof, in particular naming angles properly.
Page 310 q 9
Page 319 q7
For coordinate geometry revision, read over chapter 10 and revise all formulae.
Then answer
Page 124 q 23

The last topic to look at is the final chapter in the book - using equations to solve problems.
Have a go at these 2 types of questions:
p386 - q4 (make sure you only use a single unknown)
p389 - q1 (simultaneous equations)


That is the lot!
I will post some help for some of these questions over the weekend.

Revised Plan Leading to Mock Exams

If you have completed the work you were set last week (see the list) that is great. If you haven't, then try to complete it over the next few days, using the tips on here or asking me in class if you are really stuck. We will look at the solutions later in the week.
This week we will wrap up the last two topics in statistics (Histograms and Ogives) and deal with two small topics which look a lot more difficult than they really are.
Next week will be devoted to Geometry and co-ordinate Geometry.
We will use the 2 days after the mid-term break for general revision.
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Mon 2  Tue 3  Wed 4 Thu 5 Fri 6
       Stats Stats Surds Indices
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Mon 9  Tue 10  Wed 11 Thu 12 Fri 13
Theorems Geometry Coordinate Geom
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Mon 16  Tue 17  Wed 18 Thu 19 Fri 20
    M I D T E R M B R E A K
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Mon 23  Tue 24  Wed 25  Thu 26 Fri 27
General revision
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