## Algebra & Numeracy 2

ALGEBRA AND NUMERACY 20040307

An answer to one question may well help in another. You are presumed able to think. And to work without a calculator. Working and answers in your exercise book for this one, please.

1  Find two answers to       (x + 3)²  - 9  =  16

2  If \$1.60 = £1 and €1 = 71p what is \$100 in €uros?

3  If x² – 5x + 6  is odd, is x odd or even? Any idea why?

4  If 3 < 2x - 5 ≤ 17  then what values might x have?.

5  Sketch y = 2x – 5 with –5 ≤ y ≤ 20 and on the same pair of axes, draw y = 3, y = 17 and y + 2x = 17 Mark all six intercepts and the intersection of the two sloping lines.

6a) How do you tell if a number is divisible by three?  Is 2047?

b) The four digit number 20x8 is divisible by 3 exactly. Find values for x.

c) Can you find a x so that 20x8 is divisible by nine?

d) Is 2e + 3e + e divisible by 3, even if e = √5 ??

e) If the three digit number pqr is divisible by 9, is √pqr divisible by 3? Evidence?

f) A 3 digit number is divisible by 3 5 and 7. Find several answers, and

g) find the one closest to being a perfect square (an integer squared)

Homework should be 30 minutes of effective work, which is not the same as sitting with this page for half an hour. There are two consequences to this: if you have spent the time struggling (probably not so with this sheet), then write me a note so to tell me you have struggled (but that should be visible on the page already). The other consequence is that I make a judgement about how much you can do; if this is particularly small (or particularly large), you affect my opinion of your abilities – which is what I write about in reports and is the basis for judgements on your ability and effort. Difficult, isn’t it?

I want you to do 30 minutes effective work. However, if you pass 30 minutes significantly, it would help to know where the 30 minutes was up (draw a line, write a note in the work). Working much further than the 30 minutes is your choice, not my demand; it would be sensible, anyway, to stop working on maths at the half-hour and do other homework. By all means come back to it because you’re interested, but don’t put us both in a position where other work isn’t being done because you’re interested, obligated or just plain obstinate. Don’t be frightened to ask, or to stop, or even to stick with it.

1.  x+3 = √25=±5 so x= 2 or -8.  Check answers differ by 10=2x5.

2.  \$100 = £ 100/1.60 = € (100/1.600/0.71 = €88.03

3. On previous page, even

4. You could sketch y=2x-5 and see where y>3 and y≤17. By calculation, this is (4,3) and (11,17), so 4<x≤11. Q5 does all this and more. The two straight lines cross at (5.5, 6); The new line has intercepts at (0,17) and ( 8.5, 0).

6.  add up the digits and see if that has a factor of 3.  2+0+4+7 = 13, so no it isn’t.
20x8  has 3 as a factor in these cases: 2028, 2058, 2088.  2088 has 9 as a factor too.
adds to 6e, so yes and no; 6e/3 = 2e but e=√5 makes the answer irrational (not an integer)
pqr=9xyz  but if any of {p,q,r} is prime or has a prime factor, such as 7 then √pqr is irrational, so no, √pqr is not guaranteed to be an integer.
N has factors 3,5,7. Possibilities are the nine multiples of 105 below 1000. Two of these are 4 different from the perfect squares 841 and 529. Bonus mark for finding both!