Section J Venn diagrams

Transcription

1 Section J Venn diagrams A Venn diagram is a wa of using regions to represent sets. If the overlap it means that there are some items in both sets. Worked eample J. Draw a Venn diagram representing the relationship between isosceles triangles (I), right-angled triangles (R) and equilateral triangles (E). E I R T Notice that the bo around the circles represents everthing else that is relevant; the universal set. In this case it might be all triangles (T). Interpreting the Venn diagram we could sa that some isosceles triangles are right-angled, all equilateral triangles are isosceles, and some triangles are neither isosceles nor right-angled. Eercise J. Draw Venn diagrams to represent the following relationships (a) Squares, rectangles, parallelograms (Universal set: all quadrilaterals). (b) Even numbers, prime numbers, square numbers (Universal set: all positive whole numbers). (c) Quadrilaterals, heagons, rectangles (Universal set: all polgons). (d) Higher level mathematics students, standard level mathematics students, higher level phsics students (Universal set: all IB students). (e) ZQQ, (Universal set: R).. Draw separate Venn diagrams showing two overlapping regions, A and B, and shade the following areas: (a) A B (b) A B (c) A (d) A B (e) A B (f) A B Cambridge Mathematics for the IB Diploma Standard Level Cambridge Universit Press, 0 Prior learning: J Venn diagrams

2 Section K Algebra of epressions Can ou write down the rules of algebra? Do ou have to be able to describe something to do it? The rules of algebra are difficult to describe, so we shall just outline some of the most important things to remember. Rules of arithmetic: All the rules of arithmetic appl equall to algebra. In particular, the same order of operations applies. Combining terms: If we are adding terms together, we can group together terms which are alike. This means that, for eample, and 5 can be collected to form 7, but and cannot be collected together. Dealing with brackets: If we multipl a sum in brackets b we, multipl each term in the sum b. Most other operations, such as square rooting or finding the sine, ou do NOT do separatel to each term. Multipling together two sets of brackets: If we are multipling a sum b another sum, we need to multipl each term in the first sum b each term in the second sum. The difference between epressions and equations: With an equation, we can do the same thing to both sides and maintain the truth of the equation. With an epression we cannot do anthing which will change the epression s value. We can check whether we have succeeded b substituting values for an unknown elements. The epression should have the same value after simplification as it had beforehand. Worked eample K. Epand and simplif ( ( + + ). Multipl everthing in the second bracket b 6 + Multipl everthing in the second bracket b Add the results together, collecting terms which are alike (6 and ) = Eercise K. Write in as few terms as possible: (a) + (b) + a a+ a (c) 5 (d) (z z ( ) + z z (e) ( + ) ( ) (f) + + z) + z ( + ) Cambridge Mathematics for the IB Diploma Standard Level Cambridge Universit Press, 0 Prior learning: K Algebra of epressions

3 . Multipl out and simplif: (a) ( + )( + ) (b) (z + )(zz ) (c) ( a )( a ) (d) ( + )( + ) (e) ( ) (f) ( + ) (g) (a )( a + a + ) (h) ( + )( + )( + ) (i) + ( ). Check the following simplifications b putting = and = into both forms: (a) + (b) ( ) is the same as ( ) is the same as 9 (c) is the same as (d) is the same as (e) (f) (g) is the same as is the same as is the same as + eam hint Just because a simplification works for some values ou tr does not guarantee that it is true. However, if ou can find an value for which it fails to work then ou can be sure the simplification is wrong (h) ( ) is the same as (i) + is the same as (j) (5 i h 5 Cambridge Mathematics for the IB Diploma Standard Level Cambridge Universit Press, 0 Prior learning: K Algebra of epressions

4 Section L Solving equations and inequalities The most important thing to remember when solving equations is that ou have to do the same thing to both sides. A good tactic is to get all the unknowns on one side and everthing else on the other side and then divide. Do not epect all the answers to be whole numbers. Worked eample L. Solve + =. 5 Getting rid of fractions is a good idea. Remember to multipl the whole of each side, not just the terms ou are most interested in! Get all terms containing on one side and everthing else on the other Tr to get a positive coefficient of Multipl b 5: 5 + =0 5 Add 5 : Subtract 5 : 6 5 = 5 6 Linear inequalities can be solved in a similar wa, with one important eception: when multipling or dividing b a negative number, we must reverse the inequalit sign. Worked eample L. Solve the inequalit 4 +. Getting rid of fractions is a good idea Multipl b : 8 + Get all terms containing on one side and everthing else on the other Subtract : 9 Subtract : 9 0 Divide b 9 and remember to change the sign 0 Cambridge Mathematics for the IB Diploma Standard Level Cambridge Universit Press, 0 Prior learning: L Solving equations and inequalities

5 The issue of changing the sign can be avoided if we alwas tr to get a positive coefficient for : Worked eample L. Solve the inequalit 4 +. Get rid of fractions 8 + Get all terms containing on one side and everthing else on the other Tr to get a positive coefficient of Add 8: 9 + Subtract : 0 9 Divide 0 Conventionall is written first, so we need to rewrite this 0 Eercise L. Solve for : (a) (b) 7 5 = 9 (c) (d) = (e) = ( ) + (f) + =. Solve these inequalities: (a) (b) 5 4 (c) < 4 (d) > Cambridge Mathematics for the IB Diploma Standard Level Cambridge Universit Press, 0 Prior learning: L Solving equations and inequalities

6 Section M Working with formulae A formula is a set of instructions for finding one variable if ou know some others. If we treat a formula as an equation we can change the subject of the formula from one variable to another. There are a few useful ideas we need to bear in mind when we are tring to do this. If the new subject occurs in an squares or square roots, isolate these before undoing them. Get all the terms involving the new subject on one side and everthing else on the other side, then factorise. Worked eample M. Make the subject of the formula = + 5. Get rid of fractions and epand brackets ( 5 ) Get everthing involving on one side, everthing else on the other side, then factorise ( + ) 5 + = Another important tool is substituting one formula into another. Worked eample M. If A r + πrm and r m, find A in terms of m. Wherever r appears in the formula for A, replace with (m), then epand all brackets A ( m ) + π( m) m = 9 m + 6πmm = 5πm Cambridge Mathematics for the IB Diploma Standard Level Cambridge Universit Press, 0 Prior learning: M Working with formulae

7 Eercise M. (a) If A +, find A when 6 nd =. (b) If B p), find B when p =. (c) If C ( + ), find C when = 9 and =.. Make the subject of each of these formulae: (a) A + 7 (b) B (c) C = + (d) D (e) E = (f) F = + (g) G = + + (h) H = a+ a b (i) I = 8 + c d. Write z in terms of, giving our answer in a form without brackets: (a) z 5 + 5, (b) z 6, + (c) z, (d) z = = + (e) z = +, (f) z = +, Cambridge Mathematics for the IB Diploma Standard Level Cambridge Universit Press, 0 Prior learning: M Working with formulae

8 Section N Factorisation Factorisation means taking a sum and re-epressing it as a product. This is often done b finding a factor common to each term in the sum. Worked eample N. Factorise ab + 9ab. The largest factor in both terms is ab = ( 4b + a ) We must also be able to factorise quadratic epressions like b + c. If the coefficient of is we look for a factorisation looking like ( p)( q). We tr to find p and q such that their product is c and the add up to b. If there is a common term throughout the whole epression, do not forget to take that out first. Worked eample N. Factorise 7., 4 are two numbers which add to give 7 and multipl to give = ( )() 4) A special case of quadratic factorisation is the difference of two squares: a b = (a b) (a + b). It is ver useful to be able to recognise this. Worked eample N. Factorise 5 9. This is the difference of two squares, a b, with a = 5, b = = (5) () = (5 ) (5 + ) Sometimes it is possible to factorise an epression b first splitting it up into two parts and factorising each one separatel. This sometimes reveals a common factor in both parts. Worked eample N.4 Factorise a + b + a + b. = ( + ) + ( + b) A factor of (a + b) can be taken out = ( + ) ( + ) Cambridge Mathematics for the IB Diploma Standard Level Cambridge Universit Press, 0 Prior learning: N Factorisation

9 Eercise N. Factorise the following: (a) 8 (b) 5 0 (c) 8 4 (d) 4ab 4 7ab (e) + 4 (f) 5 ab + 9 a ab. Factorise the following: (a) + (b) 5 (c) a 8 a 0 (d) b 8 (e) 0 (f) Factorise the following: (a) (b) (c) ab + a b+ 6 (d) 6 b 8a a b 4 (e) pq p q+ (f) 6pq 4pq 5p + 0 Cambridge Mathematics for the IB Diploma Standard Level Cambridge Universit Press, 0 Prior learning: N Factorisation