CONTENTS CHECK LIST ACCURACY FRACTIONS INDICES SURDS RATIONALISING THE DENOMINATOR SUBSTITUTION

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CONTENTS CHECK LIST - - ACCURACY - 4 - FRACTIONS - 6 - INDICES - 9 - SURDS - - RATIONALISING THE DENOMINATOR - 4

- FACTORISING QUADRATIC EXPRESSIONS - 0 - COMPLETING THE SQUARE - - SOLVING LINEAR EQUATIONS - 4 - SOLVING

1 CONTENTS CHECK LIST - - ACCURACY FRACTIONS INDICES SURDS - - RATIONALISING THE DENOMINATOR SUBSTITUTION REMOVING BRACKETS FACTORISING COMMON FACTORS DIFFERENCE OF TWO SQUARES FACTORISING QUADRATIC EXPRESSIONS COMPLETING THE SQUARE - - SOLVING LINEAR EQUATIONS SOLVING QUADRATIC EQUATIONS SOLVING SIMULTANEOUS EQUATIONS ANSWERS - - Trinity Academy Department of Mathematics - -

2 You need to complete this booklet before your first Maths lesson. Make sure you fill in the checklist. You might find a GCSE higher tier textbook helpful. If any section causes problems the website is very useful. The booster sections C to B and A to A* are especially good. If you got a B at GCSE some of these topics may be very challenging. Do have a go at the work, You may be surprised at what you can achieve. Trinity Academy Department of Mathematics - -

3 CHECK LIST TOPIC I AM FINE ON THIS TOPIC I NEED TO DO SOME MORE PRACTICE I MUST GET HELP AT THE BEGINNING OF TERM Accuracy Fractions Indices Numerical Algebraic Rules Evaluating Surds Substitution Removing Brackets Common Factors Factorising Difference of Squares x + bx + c ax + bx + c Completing the square Solving Linear Equations Solving Quadratic Equations Factorising Formula Linear Simultaneous Equations Non-Linear Simultaneous Equations Trinity Academy Department of Mathematics - -

4 ACCURACY You may be asked to give answers correct to so many decimal places or so many significant figures. Decimal Places Examples i..746 to decimal places.746 if this number is 5 or over, round up.75 i.e.,.746 is closer to.75 than.74 ii to decimal places iii. To decimal places = = = 6.0 The zeros at the end MUST be included to show you =.00 have corrected to dp Significant Figures Start counting the significant figures from the first non-zero digit. The rule for rounding up or not is the same as for decimal places. Examples i..76 to 4 significant figures ii to significant figures iii. to significant figures: 8.4 = = = Trinity Academy Department of Mathematics - 4 -

5 In Pure Mathematics we always give answers to significant figures unless otherwise stated. Check if this true in other subjects. Obviously, in practical work this may not be the case as the degree of accuracy depends on how accurate your measurements are. IMPORTANT If you are to give an answer to significant figures it is very important that any intermediate answers you obtain are given to at least 4 significant figures. If you use the Memory on your calculator efficiently this should not be a problem, but if you write intermediate answers down, take care. Exercise. Write the following numbers correct to decimal places: i) ii). 6.4 iii) iv) v)..09 vi)..90 vii) Write the following numbers correct to significant figures: i) ii) iii) iv)..999 v) vi)..004 vii) viii) Do you understand the difference between giving an answer as or.00? Trinity Academy Department of Mathematics - 5 -

6 FRACTIONS The following examples should refresh your memory about the rules of working with fractions and remind you of all those phrases and commands we associate with fractions. Example = = find the Lowest Common Denominator 4 8 write as equivalent fractions Example = = = Example = = make into improper fractions, i.e., top heavy cancel when appropriate Example 4 = = = 7 4 turn second fraction upside down and multiply Trinity Academy Department of Mathematics - 6 -

7 Example 5 = = = do the part in brackets first Exercise Carry out the following without using a calculator and then check your answers using the fraction key on your calculator Note: In the C module you are not allowed a calculator. In all other modules you are. So it is important you can do fractions both with and without a calculator. Trinity Academy Department of Mathematics - 7 -

8 Example Express as a single fraction The LCM of and 6 is 6: x x 6 x x 6 x x as equivalent fraction 6 6 x x as a single fraction 6 x x 6 multiplying out brackets x 4 6 collecting like terms Exercise Express each of the following as a single fraction and simplify where possible:. x x 6. x x x x. a a x x x x 4 x 6 x x a a 5a x x 5 4. x x 4 9. y y y y 6y y x x y y a 5a a Trinity Academy Department of Mathematics - 8 -

9 RULES OF INDICES BASE NUMBER INDICES 7 INDEX NUMBER (or POWER) x = ( x ) x ( x x ) = 5 so a m x a n = a m+n so a m a n = a m n a a m n a mn (4 5 ) = 4 5 x 4 5 x 4 5 = = 4 5 so (a m ) n = a mn 8 8 also, 0 0 = so a 0 = Examples. p p = p + = p 4 NB: p = p. (x) (x) = 4x 7x = 08x NB: on C Module, you will NOT have a calculator. Trinity Academy Department of Mathematics - 9 -

10 Exercise 4 Simplify:. x x x x. a a a a a a. 4 a a a 4. 6 a a 5 b b 5. x x 7 6. y y 7 y 7. y 4y 5 8. y y 4y 4 9. a 4b 5b a 0. 6a 4b a b. 5p p q q 4 q. 5p 5 p. 6(p ) 4p 4 4. a 4ab 6bc c 4 5. a 5 (a ) 4 6. a bc 4a bc a b 4 c Trinity Academy Department of Mathematics - 0 -

11 EVALUATING EXPRESSIONS WITH INDICES A fractional power indicates a root e.g a power of means square root so 5 = 5 = 5 e.g a power of means cube root so 7 = 7 = A negative power indicates a reciprocal e.g 6 means = 6 6 e.g 5 means = 5 5 Examples. 00 = 00 = 0. 4 = 4 = = 44 = 4. 4 = 4 = 4 = = 8 NB: on C Module, you will NOT have a calculator. Trinity Academy Department of Mathematics - -

12 Exercise 5 Evaluate: Trinity Academy Department of Mathematics - -

13 SURDS A surd is an IRRATIONAL ROOT e.g.,,, etc., but not 4 because 4 = SIMPLIFY SURDS ab = a b e.g., 80 = (6 5) = 6 5 = 4 5 a b a b e.g., a c + b c = (a + b) c e.g., = (5 ) + (6 ) 5 (4 ) = 5 + ( 6 ) 5( 4 ) = = Exercise 6 Simplify: Trinity Academy Department of Mathematics - -

14 Rationalising the Denominator Examples. i.e., multiply top and bottom by Exercise 7 Rationalise the denominators: Trinity Academy Department of Mathematics - 4 -

15 SUBSTITUTION Substitution into Formulae Don t be tempted to skip this section. Many errors at A level are due to careless substitution or misunderstanding of substitution. Don t be tempted to just write down an answer you need to practise first to gain understanding and confidence. REMEMBER x means lots of x squared so if x = 5, x = lots of 5 = 50 BUT (x) means the term x is squared so if x = 5, (x) = ( 5) all squared = 0 = 00 Example Evaluate the following function: A. f(x) = x 5x + when x = f( ) = ( ) 5x + = = 5 g x x when 4 B. x x g Trinity Academy Department of Mathematics - 5 -

16 Exercise 8. If x = 5, y = and z =, evaluate the following: a) 0 x b) (x) y c) xy d) 7 x e) 4xyz 4y z f) xy 4z 4y z. The volume of a box is given by (x + ) (x ) (x 4). Find the volume if x = 7 cm.. One of the equations of motion when acceleration is constant can be written v = u + as. Find v if u =., a = 0.8 and s = is the formula which changes Fahrenheit into Centigrade, find the 9 Centigrade equivalent of 0 o F to the nearest degree. 4. If C F 5. When x = find the value of the following: a) x + x b) x 4x c) x (x) d) x 6. p = q = r = a) 5 (p + ) = b) p (q + r ) = c) (pq) = r q q r d) p p 7. Work out the value of x + x x 6 when i) x = 0 ii) x = iii) x = iv) x = v) x = 8. What value of x makes the expression x x equal to zero? Trinity Academy Department of Mathematics - 6 -

17 REMOVING BRACKETS Example Remove the brackets and simplify: a) (x y) (x 4y) b) a(a + b) b(a + b) a) (x y) (x 4y) = 6x y x + 8y = 4x + 5y b) a(a + b) b(a + b) = a + ab ba b = a b (since ba = ab) Example Expand (x 4)(x ) (x 4)(x ) = 6x 8 9x + (x 4) (x ) = 6x 7x + Exercise 9 Remove the brackets and simplify: Remove the brackets from the following:. 4(5x + y) 6. x(x y + z). (x + 5)(x 5) 6. (x ) (x ). (a 4b) 7. a(a + b c). (x 4)(x + 4) 7. (x + )(x + 4). (p 6q) 8. x(x + x + ). (x + )(x + ) 8. (x + )(x ) 4. 5(x y + z) 9. 4(x + y) (x y) + (x y) 4. (x )(x ) 9. (x + )(x + ) 5. (a b c) 0. xy(x + y) + x(xy y ) 5. (x + )(x + ) 0. (x )(4x + ) Trinity Academy Department of Mathematics - 7 -

18 FACTORISING Common Factors Example Factorise 6p + q + 9r The factor which common to each term is 6p + q + 9r = (p + q + r) Example Factorise x + xy + 6x The factor which is common to each term is x x + xy + 6x = x(x + y + 6) Example Factorise x 4xy This expression has more than one common factor. Both and x are common factors. x 4xy = x(x y) x 4xy therefore has three factors:, x and (x y) Exercise 0 Factorise the following expressions:. 4x + y 6. s + 0t. 6xy + x. p 6q 7. xy + xz. 5x 0x. 5a xy + x. a + 8b 4c 4. 0b 5 9. y y 4. x y + xyz + xy 5. 4m n 0. y 4y 5. 4pqr p q Trinity Academy Department of Mathematics - 8 -

19 Difference of Two Squares Example Factorise x 9 x 9 = (x )(x + ) Example Factorise 9x 6 9x 6 = (x) (4) = (x 4)(x + 4) Example Factorise 8x 8x and are not perfect squares, but is a common factor of the expression. 8x = (4x ) 4x = (x) which is the difference of two squares 8x = (x )(x + ) (Note: If a quadratic has a common factor, always take out the common factor before factorising the quadratic into two brackets.) Exercise Factorise:. x 4. 4x x 0. x 8. x x x. 9x 6. x x x. x 75 Trinity Academy Department of Mathematics - 9 -

20 Factorising Quadratic Expressions When the coefficient of x is unity Example Factorise x + 5x + 6 Reversing the process of multiplying out brackets, we can see that x + 5x + 6 = (x +?) (x +?) and 6 must factorise as 6 or. The key to factorising the quadratic expression is the x term. 5x is the sum of two x terms and the coefficients of these two terms must be the factors of 6. = 6 and x + x = 5x x + 5x + 6 = (x + ) (x + ) Example Factorise x x 6 The factors of 6, when added together, must equal the coefficient of x, which is. The factors of 6 are 6, 6,,. Only + = x x 6 = (x )(x + ) Example Factorise x + 6x This is a much simpler quadratic expression to factorise than the standard type because it has a common factor which is x. x + 6x = x(x + 6) Exercise Factorise the following:. x + 8x x x 5. 6x x. x + 7x x 6x x 5x 6. x + x 6 9. x + 6x 5. x 6x 6 4. x 5x x 4x 6. x x 5. x 7x. x 7x + 7. x + x x + x 8. x + x 8. x 8x + 6 Trinity Academy Department of Mathematics - 0 -

21 When the coefficient of x is not unity Example Factorise x + x + The method previously used must be adapted to take into account the coefficient. As will only factorise as, it can be see that x + x + = (x + a)(x + b) As before, a b =, but now we require a + b = (not a + b as before) Factors of are a b = and a + b = + 4 = 5 6 = + = 4 4 = + 8 = 4 = = 0 etc. x + x + = (x + )(x + 4) Alternatively list all possible solutions and select the one which works. (x + )(x + ) = x + 5x + (x + )(x + 6) = x + 4x + (x + )(x + 4) = x + x + Exercise Factorise:. x + 5x + 7. x 8x + 4. x + 7x x x 0. x + 9x 5 9. x x + 4. x x 7 0. x + 0x + 5. x + 8x + 5. x x 6 6. x 6x 9. x x 4 Trinity Academy Department of Mathematics - -

22 Some expressions can be factorised as e.g. x 6x 9 = x 0x 5 = ( x 5) COMPLETING THE SQUARE ( x a) or These expressions are called perfect squares. ( x a) ( x ) - check that you agree!! For expressions which are not perfect squares, we complete the square, which means adjusting the constant term: Example x in the completed square form x a b Express 6x. Firstly, halve the coefficient of the x term for inside the bracket: so we have ( x ) If we multiply out, this gives x 6x 9 so we need to add to obtain x 6x. x = x So 6x Example Express 0x x in the completed square form x a b. Again, halve the coefficient of the x term to give ( x 5) If we multiply out, this gives x 0x 5 so we need to subtract to obtain x 0x. So 0x x = x 5 In general terms, the formula for completing the square for x px q is: x p p q Trinity Academy Department of Mathematics - -

23 Exercise 4 Write the following in completed square form:. x + 8x x x 5. x x. x + 6x x 0x + 9. x + x x + x 6 8. x + 6x - 5. x 6x 6 4. x 4x x 4x 4. x + 5x x 8x 0. x 8x + 5. x + x + 0 Trinity Academy Department of Mathematics - -

24 SOLVING LINEAR EQUATIONS Example Solve the equation 5x 4 = x + In this type of equation, the x terms should be collected on one side of the equation and the numerical terms on the other, i.e., x must be eliminated from the RHS and 4 from the LHS. 5x 4 = x + Transposing x and 4 gives: 5x x = + 4 x = 6 x = 8 Answers to algebraic equation should always be checked by substituting back into the LHS and RHS of the original equation, as shown below: LHS = 5x 4 = = 6 RHS = x + = 8 + = 6 LHS = RHS, so the solution is correct. Example Solve 4(x + ) (x 5) = 46 4(x + ) (x 5) = 46 Multiply out the brackets: 4x + x + 0 = 46 Collect terms: x + = 46 Transpose : x = 4 Divide by : x = Exercise 5 Solve the following equations to find the value of x:. 7x + = 5x + 5. (x 5) = 9. (x ) = 7(x ). 5x = x (x + ) = 0. (x + 5) + (x ) = 8. 6x 4 = 0 x 7. (x + 7) + 4 = 8. 4(x + ) + (x + ) = 4. x = 4 5x 8. (x 4) = (x + ). (x + ) (x 4) = 40 Trinity Academy Department of Mathematics - 4 -

25 SOLVING QUADRATIC EQUATIONS. By Factorising Example Solve the equation x 5x 4 = 0 x 5x 4 = 0 Factorising gives: (x 7)(x + ) = 0 either (x 7) = 0 or (x + ) = 0 either x = 7 or x = The solution is x = 7 or This can be shown graphically: Note: A quadratic equation always has two possible solutions. If the quadratic is a perfect square, the two solutions will be the same and they are called repeated roots. Example Solve the equation x 5x + = 0 x 5x + = 0 Factorising gives: (x )(x ) = 0 so either (x ) = 0 or x = 0 x = x = / or x = Trinity Academy Department of Mathematics - 5 -

26 Exercise 6 Solve the following equations:. x 5x + 4 = 0 5. x + 7x + = 0 9. x 0x + = 0. x + x + 8 = 0 6. x + x = 0 0. x + x 4 = 0. x 49 = 0 7. x x 8 = 0. x x 40 = 0 4. x + 6x + 9 = x x = 0. 6x + x = 0 Trinity Academy Department of Mathematics - 6 -

27 . Using the Formula To solve ax + bx + c = 0 b b 4ac x a Example Solve x + x + = 0 Compare x + x + = 0 with ax + bx + c = 0 Then a = +, b = +, c = + and substituting for a, b and c in the formula gives x 94 x 5 AS you can t use a calculator in C, answers must be left in surd form Example Solve x x = 0 In this equation, a =, b =, c = and substituting in the formula gives: x 94 x 7 4 Exercise 7 Solve the following equations using the quadratic formula.. x + x = 0 5. x + 6x 0 = 0 9. x + 5x = 6. x + 4x + = 0 6. x 7x + 9 = 0 0. x 0x = 5. x + x = x 8x 6 = 0. x(x + ) = 5 4. x x = 0 8. x 6x + = 0. x (x ) = Trinity Academy Department of Mathematics - 7 -

28 Solving Simultaneous Equations. Linear Example Solve the equations x + 4y = 8 () 4x y = () Neither x nor y has the same coefficient in each equation, so this needs to be remedied first. Method Working i) Decide which variable is to be eliminated. Eliminate y ii) Multiply one or both equations so that this Multiply equation () by variable has the same coefficient in each and equation () by 4 equation (not counting the signs). 9x y 54 6x y 4 iii) Add or subtract the equations, depending As the coefficients of y have opposite signs, on the signs of the variable to be eliminated. add the equations: 9x y 54 6x y 4 5x 50 iv)solve for the remaining variable. x = v) Substitute into one of the original Substitute x = into () equations to find the eliminated variable and 4y 8 hence the complete solution. 4y y Solution is x =, y = vi) Substitute into the other original equation Substitute into () to check the solution. LHS = 4 = 8 9 = = RHS You can see this on the following graph: Trinity Academy Department of Mathematics - 8 -

29 Exercise 8 Solve the following pairs of simultaneous equations:. x + y = 8 x + y = 6 5. x + y = 6 x + y = 7 9. x y = x + y = 8. x + y = x = y = x + y = 0 x + 5y = 0. 7x + y = 9 x + y =. x + y = x y = 7. x + y = 8 4x y =. x + 5y = 5 / x 4y = 5 / 4. x y = 9 x + y = 5 8. x y = 0 x + 4y = 5. 6x + 5y = 7 x + y = 8 Trinity Academy Department of Mathematics - 9 -

30 . One Linear One Non-Linear Using Substitution method Example y = x + 5x 5x y + 5 = 0 Stage : Make x or y the subject of the linear equation i.e., 5x y + 5 = 0 y = 5x + 5 It s easier to make y the subject here Stage : Substitute this rearranged equation into the other one y = x + 5x 5x + 5 = x + 5x Stage : Rearrange and solve the quadratic 0 = x 8 0 = x 4 0 = (x )(x + ) x = or x = Stage 4: Find the corresponding values of y using the simplest equation x = y = = 5 x = y = = 5 You can see this on the following graph. Trinity Academy Department of Mathematics - 0 -

31 Example Solve simultaneously y x = () x + y = 0 () y x = is the linear equation so re=write as y = x + Now substitute for y in the non-linear equation x + (x + ) = 0 x + x + 4x + 4 = 0 x + 4x 6 = 0 which is a quadratic in x only x + x = 0 (x + )(x ) = 0 so x = or Substitute x = in () y + = therefore y = Substitute x = in () y = therefore y = So the solutions are x =, y = or x =, y = CHECK: in equation (): ( ) + ( ) = 9 + = 0 + = + 9 = 0 Exercise 9 Solve the simultaneous equations:. x + y = 6x + y = 65. y x = x + xy + y = 8 5. x + y = 9 x xy + y = 0 7. y = x + y = 4x 9. x + y = x x + y =. x + y = x + y = 4. x y = 7 x + 4y = 7 6. x = y x + xy = 0 8. u v = u + v = y x = 4 x + xy + y = 8 Trinity Academy Department of Mathematics - -

32 ANSWERS Exercise. (i) (i) 86.4 (ii) 6. (ii) 87 (iii) 0.09 (iii) 0.09 (iv) 0.0 (iv).00 (v).0 (v) 70 (vi).9 (vi).00 (vii).00 (vii) (vii) Exercise Check these yourself. Exercise x a x x 4 7x Exercise x 5 x a 4 y 0. 5 y x x ( x 5) 0 y 0a. x 4. a 6. 4a 4. 0a b 5. x 0 6. y 7. y y a 5 b a 5 b 5. 0p 5 q 9. 5p a 4 b c 7 5. a 7 6a c 6. b Exercise Trinity Academy Department of Mathematics - -

33 Exercise Exercise ½ 6. / Exercise 8 a) 5 b) 88 c) 0 d) e) 9 / 6 f) 5 Exercise 9. 0x + 8y 6. x xy + xz. 4x x x a b 7. a ab + ac. 9x 6 7. x + 9x p + q 8. x + x + x. x + 4x x 4x x 5y + 0z 9. x + 5y 4. x x x + 7x + 5. a + b + c 0. 5x y + xy 5. 9x + 6x + 0. x + x 6 Exercise 0. 4(x + y) 6. 4(s + 5t). x (y + ). (p q) 7. x (y + z). 5x ( x). 5(a + ) 8. x(y + x). (a + 4b c) 4. 5(b ) 9. y(y ) 4. xy(x + z + y) 5. 7(m n) 0. y(y ) 5. 4pq(r p) Exercise. (x )(x + ) 5. (x )(x + ) 9. (5 7x)(5 + 7x). (x 4)(x + 4) 6. (4x 5)(4x + 5) 0. (x )(x + ). (x 5)(x + 5) 7. (7 x)(7 + x). 9(x )( x + ) 4. (x )(x + ) 8. (6 5x)(6 + 5x). (x 5)(x + 5) Trinity Academy Department of Mathematics - -

34 Exercise. (x + )(x + 7) 7. (x + )(x 5). x (x ). (x + )(x + 5) 8. (x )(x ) 4. (x 6)(x + ). (x )(x + ) 9. x (x + ) 5. (x 8)(x + ) 4. (x )(x ) 0. (x 6)(x + ) 6. x ( x) 5. x (x 7). (x )(x 4) 7. (x + 6)(x + 6) 6. (x + 4)(x ). (x + )(x ) 8. (x - 4)(x - 4) Exercise. (x + )(x + ) 5. (x + 5)(x + ) 9. (x )(x 4). (x + )(x + ) 6. (x + )(x ) 0. (x + )(x + 6). (x )(x + 5) 7. (x )(x ). (x + )(x 8) 4. (x + )(x 7) 8. (x + )(x 5). (x 7)(x + ) Exercise 4. (x + 4) (x - ) x 4. (x + ) + 7. (x - 5) 6. (x + 6) (x + ) (x + ) -4. (x - ) (x ) + 9. (x - ) 6 5. (x - 4) (x - 4) - 5 Exercise Exercise 6 / x x. x = 4 or 5. x = or / 9. x = / or 6. x = 7 or 4 6. = 0 or 0. x = 7 / or. x = 7 or 7 7. x = 4 / or. x = 5 or 4 4. x = or 8. x = 0 or / 5. x = / or 4 4 Trinity Academy Department of Mathematics - 4 -

35 Exercise note: can cancel by Exercise 8. x = y = 4 5. x = y =. x = 9 6. x = y = y =. x = 7 7. x = y = 5 y = 6 4. x = 6 8. x = 5 y = 4 / y = 0 9. x = 5 y = 6 0. x = y =. x = / y = /. x = y = Exercise 9. (, ), ( / 7, 49 / 7 ) 5. (6, ), (4 /, 4 / ) 9. (, ), ( / 7, / 7 ). (0, ), ( 4 / 5, / 5 ) 6. (, ), (, ) 0. (, ), (, ). (, 0), ( /, 7 / ) 7. (, ), (, 4) 4. (, ), (6, / ) 8. (8, +5),( 5, 8) Trinity Academy Department of Mathematics - 5 -

Mathematics: Year 12 Transition Work

Mathematics: Year 12 Transition Work There are eight sections for you to study. Each section covers a different skill set. You will work online and on paper. 1. Manipulating directed numbers and substitution