MATHEMATICS NATIONAL 5 NOTES

Transcription

1 MATHEMATICS NATIONAL 5 NOTES A = πr 2 ± V = lbh π + -

2 INDEX: page 1 Chapter 1: Fractions page 3 Chapter 2: Percentages page 5 Chapter 3: Surds page 7 Chapter 4: Indices, including Scientific Notation and Significant Figures page 9 Chapter 5: Volumes of Solids page 11 Chapter 6: Pythagoras Theorem page 13 Chapter 7: Similar Shapes page 15 Chapter 8: Angles and Shape page 17 Chapter 9: Properties of the Circle page 19 Chapter 10: Arcs and Sectors page 20 Chapter 11: Algebraic Epressions page 24 Chapter 12: Equations and Inequalities, including Transposing Formulae page 27 Chapter 13: Straight Line, including scatter diagram page 30 Chapter 14: Simultaneous Equations page 33 Chapter 15: Quadratic Equations page 40 Chapter 16: Trigonometry: Right-Angled Triangles page 42 Chapter 17: Trigonometry: Graphs and Equations page 48 Chapter 18: Trigonometry: Triangle Formulae page 51 Chapter 19: Vectors page 53 Chapter 20: Statistics 2013 D R Turnbull This material may be used for non-profit educational purposes only and must include this copyright notice.

3 FORMULAE LIST The roots of a 2 + b + c = 0 are = b ± b2 4ac 2a, a 0 Sine rule: a sina = b sinb = c sinc Cosine rule: a 2 = b 2 + c 2 2bccos A or cosa = b2 + c 2 a 2 2bc Area of a triangle: Area = 1 2 absinc Volume of a sphere: Volume = 4 3 π r 3 Volume of a cone: Volume = 1 3 π r 2 h Volume of a pyramid: Volume = 1 3 Ah Volume of a cylinder: Volume = π r 2 h Standard deviation: s = ( )2 n 1 = 2 ( ) 2 n n 1, where n is the sample size.

4 CHAPTER 1: FRACTIONS Simplify (cancel down): = = 7 9 divide by highest common factor, HCF(28,36) = 4 Mied numbers and improper fractions (top heavy): = = = = 11 4 bottom stays as = 2 R3 Addition and Subtraction requires a common denominator: = = 7 18 least common multiple LCM(6,9)= = = subtract top numbers, still 18ths both fractions now 18ths Multiplication and Division requires no mied numbers: a b c d = a c b d a b c d = a b d c = go ' top heavy' = multiply by reciprocal = = page 1

5 ORDER OF CALCULATION: to change the order use brackets and before + or BRACKETS FIRST! Carry out separate calculations: (1) of multiply first subtraction last = = 22 9 of means multiply fully simplify = = common denominator subtract whole numbers and fractions separately = (2) brackets first = = top-heavy first = 2 6 fully simplify = multiply by reciprocal = 1 3 = 4 21 page 2

6 CHAPTER 2: PERCENTAGES PERCENTAGE CHANGE original changed value value INCREASE: growth, appreciation, compound interest 100% +a% ( a)% DECREASE: decay, depreciation 100% a% ( 100 a)% For eample, 8% increase: 100% +8% 108% = % decrease: 100% 8% multiply quantity by 1 08 for 8% increase 92% = 0 92 multiply quantity by 0 92 for 8% decrease REVERSING PERCENTAGE CHANGE Divide by the factor which produced the increase. (1) Including VAT of 20%, a radio costs 96. Find the original cost eclusive of VAT. 20% VAT added 100% +20% 120% = = 96 = = 80 non-calculator: 120% 12 10% % = = 80 (2) A camera costs 120 after a discount of 25% is applied. Find the original cost. 25% discount subtracted 100% 25% 075% = = 120 = = 160 non-calculator: 75% = = = 160 page 3

7 COMPOUND PERCENTAGE CHANGE appreciation and depreciation (1) A house appreciates in value by 5% in 2007, appreciates 10% in 2008 and depreciates by 15% in Calculate the value of the house at the end of % increase: 100% + 5% = 105% = % increase: 100% +10% = 110% = % decrease: 100% 15% = 085% = = or evaluate year by year year 1 5% of = = year 2 10% of = = year 3 15% of = = compound interest (2) Calculate the compound interest on invested at 5% pa for 3 years ( 1 05) 3 ie or evaluate year by year = compound interest = = EXPRESSING CHANGE AS A PERCENTAGE % change = change start 100% A car is resold for loss = = 3000 Find the percentage loss. % loss = % = 20% page 4

8 CHAPTER 3: SURDS NUMBER SETS: Natural numbers N = {1, 2, 3...} Whole numbers W = {0, 1, 2, 3...} Integers Z = {... -3, -2, -1, 0, 1, 2, 3...} N W Z Q R Rational numbers, Q, can be written as a division of two integers. Irrational numbers cannot be written as a division of two integers. Real numbers, R, are all rational and irrational numbers. SURDS ARE IRRATIONAL ROOTS. For eample, 2, 5 9, 3 16 are surds. whereas 25, 4 9, 3 8 are not surds as they are 5, 2 3 and -2 respectively. SIMPLIFYING ROOTS: RULES: mn = m n m n = m n (1) Simplify 24 3 (2) Simplify = is the largest = 36 2 square number which is a factor of 72 = 6 2 = = = = = page 5

9 (3) Remove the brackets and fully simplify: ( ) 2 (b) ( ) ( 3 2 2) (a) 3 2 = ( 3 2) ( 3 2) = 3 ( 3 2) 2 ( 3 2) = = = = ( ) ( 3 2 2) = 3 2 ( 3 2 2) + 2 ( 3 2 2) = = = 14 RATIONALISING DENOMINATORS: Removing surds from the denominator. Epress with a rational denominator: (1) 4 6 (2) = = = multiply the ' top' and ' bottom' by the surd on the denominator = = = 6 6 page 6

10 CHAPTER 4: INDICES base a n INDICES RULES: require the same base. inde or eponent Eamples: a m a n = a m + n w 2 w 5 a m a n = a m n w 3 = w7 w 3 = w4 a m n = a mn = 3 10 ( ab) n = a n b n 2a 3 b 2 = 2 2 a 6 b 2 = 4a 6 b 2 a 0 =1 5 0 =1 a 1 = a 5 1 = 5 a m n n = a m = n a m a p = a p = 8 =2 4 =16 3 = = 1 16 page 7

11 SCIENTIFIC NOTATION (STANDARD FORM) Writing numbers in the form a 10 n where 1 a <10 place the decimal point after the first non-zero digit and n is an integer numbers...-3, -2, -1, 0, 1, 2, = = = = = Notice for numbers starting 0 the power of 10 is negative (same as 10). SIGNIFICANT FIGURES indicate the accuracy of a measurement. For eample, 3400 cm = 34 m = km same measurement, same accuracy, each 2 significant figures. Count the number of figures used, but not zeros at the end of a whole number or zeros at the start of a decimal. rounding: to 2 significant figures to 2 significant figures (note: wrong) (1) One milligram of hydrogen gas contans molecules. Calculate, to 3 significant figures, the number of molecules in 5 grams of hydrogen = molecules learn to enter standard form in the calculator using the appropriate button EE or EXP or 10 n eg EXP 20 (2) The total mass of argon in a flask is grams. The mass of a single atom of argon is grams. Find, correct to 3 significant figures, the number of argon atoms in the flask = atoms use the (-) button for a minus eg EXP (-) 2 divide top by bottom and write the unrounded answer write the rounded answer page 8

12 CHAPTER 5: VOLUMES OF SOLIDS PRISMS a solid with the same cross-section throughout its length. length l is at right-angles to the area A. V = Al cylinder V = πr 2 h A l A l A l PYRAMIDS h A V = 1 3 A h h r cone V = 1 3 π r2 h SPHERE r V = 4 3 π r 3 EFFECT OF CHANGE Describe the effect on the volume of a cylinder of: (i) trebling the radius (ii) doubling the radius and halving the height V = π( 3r) 2 h = π 9r 2 h = 9πr 2 h 9 times bigger page 9 V = π 2r ( ) 2 1 ( ) 2 h = π 4r h = 2πr 2 h 2 times bigger

13 USING THE FORMULAE (1) Calculate the volume. A = 1 2 bh 8 cm = = 40 cm 2 8 cm 10 cm 20 cm 10 cm V = Al = = 800 cm 3 (2) Calculate the volume correct to 3 significant figures. radius = 20cm 2 = 10cm 25 cm 30 cm V = 1 3 πr 2 h = 1 3 π = cm 3 V = πr 2 h = π = cm 3 20 cm V = 4 3 πr 3 2 = 4 3 π = cm 3 total volume = = cm 3 page 10

14 CHAPTER 6: PYTHAGORAS THEOREM a b c Side length a units is opposite the right angle. It is the longest side, the hypotenuse. For right-angled triangles: a 2 = b 2 + c 2 3D Identify right-angled triangles in 3D diagrams. D V 4 40 A C 4 C 2 M 2 W 24 B 18 face diagonal AC 2 = = = 900 AC = 900 = 30 slant height VM 2 = = 16 4 = 12 VM = 12 = 2 3 space diagonal AD 2 = = = 2500 AD = 2500 = 50 height VC 2 = ( 12) = 12 4 = 8 VC = 8 = 2 2 page 11

15 CONVERSE OF PYTHAGORAS THEOREM a b c Longest side, length a units, is opposite the biggest angle. If a 2 = b 2 + c 2 then the triangle is right-angled Show that ABC is right angled. C 10 m 6 m A 8 m B AB 2 + BC 2 = =100 AC 2 = 10 2 = 100 since AB 2 + BC 2 = AC 2 by the Converse of Pyth. Thm. ΔABC is right - angled at B (ie. ABC = 90 o ) page 12

16 CHAPTER 7: SIMILAR SHAPES Shapes are similar if they are enlargement or reductions of each other. (1) the angles remain unchanged - the shapes are equiangular. and (2) the sides are enlarged or reduced by some scale factor (SF). 6 cm 8 cm 16 cm cm 9 cm 12 cm equiangular and SF = cm 15 cm Triangles are special: Enlarge or reduce sides by some scale factor and the two triangles will be equiangular. If triangles are equiangular then they are similar. 37 SCALING LENGTH length scale factor, SF = image side original side enlargement if SF > 1 reduction if 0 < SF < 1 o Original 16 cm cm 16 cm 9 cm 6 cm 15 cm s equiangular and so similar Find the value of. cm o 6 cm Image shape with the unknown side SF = image original = 6 15 = < SF < 1 as epected for a reduction = = 6 4 smaller than 16 as epected for a reduction page 13

17 SCALING AREA length SF = n area SF = n 2 for a 2D shape both length and breadth must be scaled. 48 cm 2 12 cm A cm 2 15 cm Given that the two shapes shown are similar, find the area of the larger shape. length SF = image original = = 5 4 SF > 1 as epected for an enlargement area SF = A = = = bigger than 48 as epected for an enlargement SCALING VOLUME length SF = n volume SF = n 3 for a 3D shape length, breadth and height must be scaled. 15 cm 12 cm 250 ml V ml Given that the two solids shown are similar find the volume of the smaller solid. length SF = image original = = < SF < 1 as epected for a reduction volume SF = = V = = smaller than 250 as epected for a reduction page 14

18 CHAPTER 8: ANGLES and SHAPE LINES SUPPLEMENTARY angles a + b = 180 b a b VERTICALLY OPPOSITE angles are equal COMPLEMENTARY angles c + d = 90 d c for parallel lines: a b a + b = 180 ALTERNATE angles are equal CORRESPONDING angles are equal ALLIED angles are supplementary TRIANGLES angle sum a + b + c = 180 b 60 a + b c a EXTERIOR ANGLE is the sum of the opposite INTERIOR ANGLES ISOSCELES EQUILATERAL POLYGONS INTERIOR ANGLES sum to (n-2) 180 where n sides EXTERIOR ANGLES sum to 360 all sides and angles equal pentagon interior sum = 540 eterior sum 360 REGULAR pentagon interior angle = 108 page 15

19 QUADRILATERALS KITE * * * * TRAPEZIUM b d * a c a + b = 180, c + d = 180 PARALLELOGRAM a trapezium * * * RHOMBUS a kite and parallelogram * * * * RECTANGLE a parallelogram * * SQUARE a rectangle and rhombus. 45 * * page 16

20 CHAPTER 9: PROPERTIES OF THE CIRCLE angle in a semicircle is a right-angle. the perpendicular bisector of a chord is a diameter. a tangent and the radius drawn to the point of contact form a right-angle. ANGLES (1) A 120 O radius OA = OB so ΔAOB is isosceles and Δ angle sum 180 : OBA = ( ) 2 = 30 C B D tangent CD and radius OB : OBC = 90 Calculate the size of angle ABC. ABC= = 60 (2) A diameter AB bisects chord CD : AMD = C M D ΔAMD angle sum 180 : ADM = = 60 B Calculate the size of angle BDC. angle in a semicircle : ADB = 90 BDC = = 30 page 17

21 PYTHAGORAS THEOREM 16 m 10 m h m A circular road tunnel, radius 10 metres, is cut through a hill. The road has a width 16 metres. Find the height of the tunnel. 10 h 10 8 the diameter drawn is the perpendicular bisector of the chord: is right-angled so can apply Pyth. Thm. 2 = = = 36 = 36 = 6 h = + 10 = h =16 height 16 metres page 18

22 CHAPTER 10: ARCS and SECTORS AOB 360 = arc AB πd = area of sector AOB πr 2 O A B Choose the appropriate pair of ratios based on: (i) the ratio which includes the quantity to be found (ii) the ratio for which both quantities are known (or can be found). (1) Find the eact length of major arc AB. A 6 cm 120 O? cm AOB 360 = arc AB π d 240 arc AB = 360 π 12 B diameter d = 2 6 cm =12 cm refle AOB = = 240 arc AB = π 12 = 8π cm ( ) (2) Find the eact area of sector AOB. A O 6 cm? cm 2 arc AB πd area of sector AOB = πr 2 24 area of sector AOB = π 12 π cm B area of sector AOB = 24 π 6 6 π 12 = 72 cm 2 page 19

23 CHAPTER 11: ALGEBRAIC EXPRESSIONS REMOVING BRACKETS SINGLE BRACKETS (1) 3 ( 2 y + 7) (2) 2 ( 3t + 5) (3) 3w ( w 2 4) = 6 2 3y + 21 = 6t 10 = 3w w 3 2 = y = 3y = +21 Fully simplify: 2 3t = 6t = 10 (4) 2t ( 3 t) + 5t 2 (5) 5 3 ( n 2) = 6t 2t 2 + 5t 2 = 6t + 3t 2 = 5 3n + 6 = n = 11 3n 3w w 2 = 3w 3 3w 4 = +12w DOUBLE BRACKETS (1) ( 3 + 2) ( 2 5) = 3 ( 2 5) + 2 ( 2 5) = = or FOIL ( 3 + 2) ( 2 5) (2) ( 2t 3) 2 (3) ( w + 2) ( w 2 3w + 5) ( )( 2t 3) ( ) 3 ( 2t 3) = 2t 3 = 2t 2t 3 = 4t 2 6t 6t + 9 = 4t 2 12t + 9 = w ( w 2 3w + 5) + 2 ( w 2 3w + 5) = w 3 3w 2 + 5w + 2w 2 6w +10 = w 3 3w 2 + 2w 2 + 5w 6w + 10 = w 3 w 2 w +10 page 20

24 FACTORSATION ab+ac = a( b+c) COMMON FACTORS Highest Common Factors are used to write epressions in fully factorised form. Factorise fully: 4a 2a 2 = 2a( 2 a) 2a 2 2a a using HCF( 4a,2a 2 ) = 2a NOTE: the following answers are factorised but not fully factorised: ( ) 2 2a a 2 ( ) a 4 2a DIFFERENCE OF TWO SQUARES a 2 b 2 = ( a +b) ( a b) Factorise fully: (1) (2) = ( 2) = ( 2 + 3) ( 2 3) = 4( 2 9) common factor first = 4( + 3) ( 3) NOTE: ( 2 + 6) ( 2 6) is factorised but not fully factorised. page 21

25 TRINOMIALS a 2 +b+c, a =1 ie b+c = ( +? )( +? ) The missing numbers are: a pair of factors of c that sum to b Factorise fully: (1) (2) (3) = 2 3 = = 5 use + 2 and + 3 1, 6 or 2, ( ) = 5 use 2 and 3 1,6 or 1, 6 or 2,3 or 2, ( ) = 5 use + 1 and 6 = ( + 2) ( + 3) = ( 2) ( 3) = ( + 1) ( 6) a 2 +b+c, a 1 ie. not 1 2 Try out the possible combinations of the factors which could be in the brackets. Factorise 3t 2 10t ( ) = 24 pairs of factors 1,24 or 2,12 or 3,8 or 4,6 3t 2 10 t 8 factors sum to = 10 12t + 2t = 10t one factor is negative try combinations so that 12t and 2t are obtained factor pairs: 3t t for 3t or 2 4 for 8, one factor negative 3t +1 3t -8 3t +2 3t +4 t -8 1t -24t t +1 t -4-8t 3t 2t -12t ( 3t + 2) ( t 4) t -2 4t -6t page 22

26 ALGEBRAIC FRACTIONS SIMPLIFYING: fully factorise top and bottom and cancel common factors MULTIPLY/DIVIDE: a b c d = a c b d a b c d = a b d c ADD/SUBTRACT: a common denominator is required. (1) (2) 4 ab 8a b 2 ( )( + 3) ( )( + 3) = 3 1 = 3 1 fully factorise cancel common factors = 4 ab b 2 = 4 b2 ab 8a 8a = 4 b b 8 a a b = 1 b 2 a a = b 2a 2 (3) (4) ( ) = 3 6 = 3 2( 3) 6 = = ( 3) 6 notice sign change ( ) ( ) = = ( ) = 3 ( ) = ( ) can simplify page 23

27 CHAPTER 12: EQUATIONS and INEQUALITIES Simplify by following the rules: addition and subtraction + a = b a = b = b a = b + a (1) Solve: 5 4 = 2 19 multiplication and division a = b a = b = b a 3 4 = 19 subtracted 2 from each side 3 = 15 added 4 to each side = 5 divided each side by 3 = ab WITH BRACKETS remove first and simplify (2) Solve: ( 4 + 3) ( 2) = ( 2 3) = removed brackets, fully simplifying 5 6 = subtracted 4 2 from each side 7 6 = + 9 added 12 to each side 7 = 15 added 6 to each side = 15 7 divided each side by 7 WITH FRACTIONS remove first, multiplying by the LCM of the denominators (3) Solve ( ) = ) 6 = 6 6 write with common denominators 3( + 3) + 2 = 6 both sides 6 to remove fractions = 6 5 = 3 = 3 5 page 24

28 INEQUALITIES Follow the rules for equations, ecept: multiply or divide by a negative number, reverse the direction of the inequality sign a > b a > b < b a < ab (1) > 2 (2) 8 3 > 2 +3 > 6 3 > 6 subtracted 8 from each side > 6 +3 divided each side by + 3 notice sign unchanged < 6 3 divided each side by 3 notice sign reversed > 2 < 2 simplified RESTRICTIONS ON SOLUTIONS (1) 5 2 where is a whole number (2) 2 < 2 where is an integer = 0,1,2 = 2, 1,0,1 page 25

29 TRANSPOSING FORMULAE (CHANGE OF SUBJECT) Follow the rules for equations to isolate the target letter. addition and subtraction + a = b a = b = b a = b + a multiplication and division a = b a = b = b a = ab powers and roots 2 = a = a = a = a 2 F = 3r 2 + p Change the subject of the formula to r. r square r 2 3 3r 2 +p F F p 3 F p 3 F p 3 inverse operations in reverse order p F F = 3r 2 + p F p = 3r 2 subtract p from each side F p 3 F p 3 = r 2 divide each side by 3 = r square root both sides r = F p 3 subject of formula now r page 26

30 CHAPTER 13: STRAIGHT LINE GRADIENT The slope of a line is given by the ratio: m = vertical change horizontal change Using coordinates, the gradient formula is m AB = y B y A B A Y R R ( 0,8), S ( 4,2) m y RS = S y R = S 2 8 R 4 0 = 6 4 = P S Q X P ( 3, 1), Q ( 6,1) m PQ = y Q y P Q P = 1 ( 1) 6 3 = 2 3 note: same result for = 2 3 = 2 3 EQUATION OF A STRAIGHT LINE gradient m, y-intercept C units ie. meets the y-ais at (0,C) y=m +C gradient m, through the point (a,b) y b=m ( a ) equation of line RS: m RS = 3 2 R( 0,8) C = 8 y = m + C y = page 27 equation of line PQ: m PQ = 2 3 a b Q ( 6, 1 ) y 1 = 2 3 or use point P y b = m( a) ( 6) 3y 3 = 2( 6) 3y 3 = y = 2 9

31 Rearrange the equation to y = m + C for the gradient and y-intercept y 16 = 0 2y = isolate y - term y = obtain 1y = y = m + C compare to the general equation m = 3 2, C = 8 meets the Y - ais at (0,8) RATE OF CHANGE The gradient is the rate of change. distance/time graphs: speed/time graphs: the gradient is the speed. the gradient is the acceleration. The steeper the graph, the greater the rate of change. A positive gradient is an increase, a negative gradient is a decrease. Gradient zero (horizontal line), no change. RENTAL TAXIS requires a 4 payment plus a charge of 50p per mile. CITY TAXIS charges 1.50 per mile. Advise on the better buy. cost ( ) CITY rate of change 1.50 per mile, m = 1.5 no initial charge, C = 0 equation y = RENTAL rate of change 0.50 per mile, m = 0.5 initial charge 4, C = 4 equation y = same cost for 4 miles, but under 4 miles CITY cheaper over 4 miles RENTAL cheaper miles page 28

32 SCATTER DIAGRAM: LINE OF BEST FIT W grams A salt is dissolved in a litre of solvent. The amount of salt that dissolves at different temperatures is recorded and a graph plotted. The best-fitting straight line through the points is drawn. The equation of the graph is of the form W = mt + C. Find the equation of the line and use the equation to calculate the mass of salt that will dissolve at 30 C T C using two well-separated points on the line m = = = 0 15 substituting for one point on the line a b 16, 6 0 ( ) ( 16, 6 0) ( 12, 5 4) y b = m( a) y 6 0 = 0 15 ( 16) y 6 0 = y = W = 0 15T T = 30 W = = = grams page 29

33 CHAPTER 14: SIMULTANEOUS EQUATIONS The equation of a line is a rule connecting the and y coordinates of any point on the line. y + 2 = 8 For eample, ( 2) = 8 y (-2,12) To sketch a line, find the points where the line meets the aes. SOLVE SIMULTANEOUS EQUATIONS: GRAPHICAL METHOD Sketch the two lines and the point of intersection is the solution. Solve graphically the system of equations: y + 2 = 8 y = 2 y + 2 = 8 (1) y = 8 substituted for = 0 ( ) y = 8 plot 0,8 y = 2 (2) y 0 = 2 substituted for = 0 ( ) y = 2 plot 0,2 y + 2 = 8 (1) = 8 substituted for y = 0 ( ) = 4 plot 4,0 y = 2 (2) 0 = 2 substituted for y = 0 ( ) = 2 plot 2,0 Y y + 2 = 8 8 y = 2 point of intersection ( 2,4) SOLUTION: = 2 and y = 4 2 CHECK: = 2 and y = 4 substituted in both equations X y + 2 = 8 (1) y = 2 (2) = = 2 8 = 8 2 = 2 page 30

34 SOLVE SIMULTANEOUS EQUATIONS: SUBSTITUTION METHOD Rearrange one equation to y = and substitute for y in the other equations ( or = ) Solve algebraically the system of equations: y + 2 = 8 3y =10 y + 2 = 8 (1) can choose to rearrange to y = or = y = 8 2 choosing y = avoids fractions as = y y = 10 (2) ( ) = 10 replace y by = 10 solve 7 = 14 = 2 y = 8 2 (1) can choose either equation (1) or (2) = substituted for = 2 y = 4 SOLUTION: = 2 and y = 4 CHECK: 3y = 10 (2) using the other equation = 10 substituted for = 2 and y = 4 10 = 10 page 31

35 SOLVE SIMULTANEOUS EQUATIONS: ELIMINATION METHOD Can add or subtract multiples of the equations to eliminate either the or y term. Solve algebraically the system of equations: 4 + 3y = 5 5 2y = y = 5 (1) 2 can choose to eliminate or y term 5 2y =12 (2) 3 choosing y term, LCM (3y,2y) = 6y 8 + 6y =10 (3) multiplied each term of (1) by 2 for + 6y 15 6y = 36 ( 4) multiplied each term of (2) by 3 for 6y 23 = 46 ( 3) + (4) added equations, adding "like" terms = 2 + 6y and 6y added to 0 (ie eliminate) 4 + 3y = 5 (1) can choose either equation (1) or (2) y = 5 substituted for = y = 5 3y = 3 y = 1 SOLUTION: = 2 and y = 1 CHECK: 5 2y =12 (2) using the other equation ( 1) =12 substituted for = 2 and y = 1 10 ( 2) =12 12 =12 page 32

36 CHAPTER 15: QUADRATIC EQUATIONS A function pairs one number with another, its IMAGE. It can be defined by a formula. QUADRATIC FUNCTIONS f ( ) = a 2 + b + c, a 0, where a, b and c are constants. f ( ) = ( ) = ( 2) 2 2 ( 2) 3 = = 5, the image of -2 is 5. f f () points (-2, 5) (-1, 0) (0,-3) (1,- 4) (2,-3) (3, 0) (4, 5) If all possible values of are considered, a graph will show the images (the RANGE). The graph is a symmetrical curve called a PARABOLA. Y X The graph meets the -ais where = 0. The zeros of the graph are -1 and 3 which are the roots of the equation. page 33

37 QUADRATIC EQUATIONS An equation of the form a 2 +b+c = 0, a 0, where a, b and c are constants. The values of that satisfy the equation are the roots of the equation. The quadratic formula can be used to solve the equation. QUADRATIC FORMULA = b± b2 4ac 2a, a 0 Find the roots of the equation = 0, correct to two decimal places = 0 a 2 + b + c = 0 a = 3, b = 5, c = 1 b 2 4ac = ( 5) ( 1) = 37 = b ± b2 4ac 2a = 5 ± 37 6 b = ( 5) = +5 2a = 2 3 = 6 = or = or = or roots are 0 18 and 1 85 DISCRIMINANT b 2 4ac gives the nature of the roots (i) b 2 4ac < 0 NO REAL ROOTS ie. negative (i) (ii) (iii) (ii) b 2 4ac = 0 ONE REAL ROOT (repeated) (two EQUAL roots) (iii) b 2 4ac > 0 TWO REAL ROOTS ie. positive (and DISTINCT ) X page 34

38 Find k so that k = 0 has real roots k = 0 a 2 + b + c = 0 a = 2, b = 4, c = k b 2 4ac = 4 ( ) ( k) =16 + 8k for real roots, b 2 4ac k 0 8k 16 k 2 FACTORISATION RATIONAL ROOTS b 2 4ac = a square number ie. 0, 1, 4, 9, (non-surds) the quadratic can be factorised to solve the equation. IRRATIONAL ROOTS (surds) b 2 4ac a square number solve the equation by formula. Solve: (1) 4n 2n 2 = 0 (2) 2t 2 + t 6 = 0 2n( 2 n) = 0 2n = 0 or 2 n = 0 n = 0 or n = 2 ( 2t 3) ( t + 2) = 0 2t 3 = 0 or t + 2 = 0 2t = 3 t = 3 2 or t = 2 The equation may need to be rearranged: (3) ( w + 1) 2 = 2( w + 5) (4) + 2 = 15, 0 w 2 + 2w + 1 = 2w +10 w 2 9 = 0 ( w + 3) ( w 3) = 0 w + 3 = 0 or w 3 = 0 w = 3 or w = 3 ( + 2) = = = 0 ( + 5) ( 3) = = 0 or 3 = 0 = 5 or = 3 page 35

39 GRAPHS A sketch of the graph of a quadratic function should show where the parabola meets the aes and the maimum or minimum turning point. Sketch the graph y = (i) meets the Y-ais where = 0 y = = 3 point (0, -3) Y = 1 y = (ii) meets the X-ais where y = = 0 ( + 1) ( 3) = 0 = 1 or = 3 points (-1, 0) and (3, 0) X (iii) ais of symmetry vertical line half-way between the zeros = = 2 = 1, equation = (1, - 4) (iv) turning point lies on the ais of symmetry =1 y = = 4 point (1, - 4) Note: (1) from the graph, (1,- 4) is a minimum turning point. (2) the minimum value of the function is - 4. page 36

40 APPLICATIONS Problems involving maima or minima which can be modelled by a quadratic equation. A sheet of metal 40 cm. wide is folded cm from each end to form a gutter. To maimise water flow the rectangular cross-section should be as large as possible. 40 Find the maimum cross-sectional area possible A = lb = ( 40 2) sketch the graph A = = Zeros: = 0 2( 20 ) = 0 2 = 0 or 20 = 0 = 0 or = 20 A = 10 (10, 200) A = Turning Point: ( ) 2 = 10 ais of symmetry = 10 y = y = = 200 maimum turning point (10, 200) X Maimum area 200 square centimetres. page 37

41 EQUATION OF A GRAPH y = k( a) ( b) a and b are the zeros of the graph k is a constant. a b X Write the equation of the graph. Y zeros -1 and 3 y = k( a) ( b) y = k( + 1) ( 3) for (0 y, 6), 6 = k 0 +1 ( )( 0 3) ( ) 6 = k = 3k k = 2 y = 2( + 1) ( 3) X Write the equation of the graph. Y y = k 2 (-3,18) for ( 3 y,18), 18 = k 3 18 = 9k k = 2 ( ) 2 y = X page 38

42 COMPLETED SQUARE: Quadratic functions written in the form y = ±1( a ) 2 +b, a and b are constants. ais of symmetry = a turning point a, b ( ), minimum for +1, maimum for 1 y = can be written as y = ( 4) Y = 4 meets the y - ais where = 0 ( ) = = 21 ( ) y = 0 4 point 0, turning point: y= +1( 4) 2 +5 minimum turning point ( 4,5) ais of symmetry = 4 (4, 5) 0 X COMPLETING THE SQUARE: coefficient of, 8, is halved and squared, ( 4) 2 to 16 = add and subtract 16 = ( 4) complete square around trinomial = ( 4) page 39

43 CHAPTER 16: TRIGONOMETRY: RIGHT-ANGLED TRIANGLES SOH-CAH-TOA H a A O The sides of a right-angled triangle are labelled: Opposite: opposite the angle a. Adjacent: net to the angle a. Hypotenuse: opposite the right angle. The ratios of sides O H, A H and O A have values which depend on the size of angle a. These are called the sine, cosine and tangents of a, written sin a, cos a and tan a. For eample, 5 a 4 3 S = O H sin a o = 3 5 C = A H cos a o = 4 5 T = O A tan a o = 3 4 The trig. function acts on an angle to produce the value of the ratio. The inverse trig. function acts on the value of a ratio to produce the angle. For eample, sin 30 = 0.5 sin = 30 ACCURACY Rounding the angle or the value in a calculation can result in significant errors. For eample, 100 tan 69.5 o = tan = tan 70 o = tan = page 40

44 FINDING AN UNKNOWN SIDE (1) Find. 10 m m 40 rearrange for sin sin 40 o = 10 = 10 sin 40 o = ensure calculator set to DEGREES H 40 A O know H, find O SOH-CAH-TOA = 6 4 (2) Find y. rearrange for y cos55 o = 4 y y m 55 4 m cos55 4 y y = 4 cos55 o = cos 55, calculator set to DEGREES H know A, find H O 55 SOH-CAH-TOA A y = 7 0 FINDING AN UNKNOWN ANGLE (3) Find. 7 cm tan o = cm use brackets = tan 1 9 for (9 7), 7 calculator set to DEGREES = H A O know O, know A SOH-CAH-TOA = 52 1 page 41

45 CHAPTER 17: TRIGONOMETRY: GRAPHS & EQUATIONS The cosine graph is the sine graph shifted 90 to the left. Y 1 y = cos X -1 y = sin The graphs have a PERIOD of 360 (repeat every 360 ). Y 1 (90,1) y = sin Y 1 (0,1) y = cos (360,1) X X -1 (270,-1) -1 (180,-1) Turning points: maimum (90,1), minimum (270,-1) maimum (0,1), minimum (180,-1) Y y = tan X The tangent graph has a PERIOD of 180. page 42

46 TRANSFORMATIONS Same rules for y = sin and y = cos. Y-STRETCH y = n sin y-coordinates multiplied by n. amplitude n units maimum value + n, minimum value - n (1) Y 2 1 y = 2 sin y = sin X -1-2 X-STRETCH y = sin n -coordinates divided by n. period 360 n. There are n cycles in 360. (2) Y 1 y = sin 2 y = sin X -1 period 180, 2 cycles in 360 page 43

47 Y-SHIFT y = sin + n add n units to y-coordinates graph shifted n units vertically. (3) Y 3 y = sin y = sin X -1 X-SHIFT y = sin ( + n ) subtract n units from the -coordinates graph shifted - n horizontally. (4) Y 1 y = sin y = sin ( - 20) X -1 graph shifted 20 to the right NOTE: for y = sin ( + 20) the graph y = sin would be shifted 20 to the left. page 44

48 COMBINING TRANSFORMATIONS (1) Y 15 y = 7 sin amplitude 7 oscillates around the line y = period X by Turning points: maimum (30,15), minimum (90,1) y = sin y = 7 sin (90, 1) (30,15) (270, -1) (90, 1) (2) Y 5 y = 5 cos ( - 10) X period Turning points: maimum (10,5), minimum (190,-5) y = cos y = 5 cos (-10) (0, 1) (10, 5) (180, -1) (190,-5) page 45

49 EQUATIONS The graphs with equations y = cos and y = 4 are shown. Find the coordinates of the points of intersection A and B. Y 8 y = cos A B y = X 5 + 3cos = 4 3cos = 1 cos = 1 3 * A, S, T, C is where functions are positive: cosine negative S A cosine positive =109 5 or a = a = cos -1 1/ 3 = a = a = * A all functions are positive S sine function only is positive T tangent function only is positive C cosine function only is positive cosine negative T C cosine positive page 46

50 IDENTITIES sin 2 + cos 2 =1 tan = sin cos Simplify 1 cos 2 sin cos. = = = sin 2 sin cos sin sin sin cos sin cos = tan since sin 2 + cos 2 = 1 sin 2 = 1 cos 2 "cancel" sin EXACT VALUES For eample, sin60 = 3 2, tan 30 = 1 3 cos 45 = 1 2, tan 45 = 1 1 = 1 ANGLES > 90 sin240 = sin( ) a 240 S A a = 60 = sin60 = a = 240 sine negative T C a page 47

51 CHAPTER 18: TRIGONOMETRY: TRIANGLE FORMULAE SINE RULE C b a a sina = b sinb = c sinc A c NOTE: requires at least one side and its opposite angle to be known. B FINDING AN UNKNOWN SIDE C b 82 6 m a A B Find the length of side BC? a sin A = b sin B a sin 43 = a = 6 sin55 6 sin55 sin43 = BC 5 0 m FINDING AN UNKNOWN ANGLE A b 6 m C? 55 5 m a Find the size of angle BAC. B sin A = sin B a b sin A sin 55 = 5 6 sin 55 sin A = 5 6 = A = sin = BAC 43.0 page 48

52 COSINE RULE A b c C a B a 2 = b 2 + c 2 2bc cosa cosa = b2 + c 2 a 2 2bc FINDING AN UNKNOWN SIDE NOTE: requires knowing 2 sides and the angle between them. C 6 m 32 A 9 m Find the length of side BC. a B a 2 = b 2 + c 2 2bc cos A = cos32 a 2 = a = = BC 5 0 m FINDING AN UNKNOWN ANGLE NOTE: requires knowing all 3 sides. A? 6 m C 5 m 9 m Find the size of angle BAC. a B cos A = b2 + c 2 a 2 2bc = cos A = A = cos -1 ( ) = BAC 31 6 page 49

53 AREA FORMULA C b a Area Δ ABC = 1 ab sinc 2 A c NOTE: requires knowing 2 sides and the angle between them. B (1) A C 53 6 m 5 m B Area Δ ABC = 1 ab sinc 2 = sin53 2 = Area 12 0 m 2 Find the area of the triangle. (2) C? 6 m 12 m 2 A Find angle ACB. 5 m B Area Δ ABC = 1 ab sinc 2 12 = sinc double both sides 2 24 = 30 sinc sinc = = 0 8 C = sin -1 ( 0.8) = or ( from as angle could be obtuse) ACB 53 1 from diagram, angle acute page 50

54 CHAPTER 19: VECTORS SCALAR quantities have size(magnitude). VECTOR quantities have size and direction. eg. time, speed, volume eg. force, velocity A directed line segment represents a vector. Vectors can be written in component form as column vectors A H 6 C B 5 I D E AB = 4 0 CD = 5 3 EF = 3 6 GH = 4 IJ = negative vector, opposite direction - 4 G J F HG = 4 6 SIZE follows from Pyth. Thm u = a u = a 2 + b 2 GH = 4 b ( ) = 52 = 2 13 units parallelogram D v C (1, 5) u u A v B (-2,1) AD = BC = 3 4 same size and direction same vector u same component form ADD/SUBTRACT by column vectors add or subtract components = 4 6 by diagram head-to-tail addition PQ + QR = PR Q a b P R resultant vector a + b page 51

55 MULTIPLY by a number u = a ku = ka b kb AB = 2 1 CD = 8 4 D if v = ku then u and v are parallel CD = 4 AB CD is parallel to AB C A B 3D Vectors in 3D operate in the same way as vectors in 2D. Points are plotted on 3 mutually perpendicular aes. P(,y,z) has position vector p = y z Z 0 5 a Y A(5, 2, 7) 5 7 a = X 1 1 If u = 2 and v = find the value of v 2u. v 2u = = = v 2u = ( 3) ( 5) 2 = 50 = 5 2 page 52

56 CHAPTER 20: STATISTICS Studying statistical information, it is useful to consider: (1) typical result: average (2) distribution of results: spread AVERAGES: mean = total of all results number of results median = middle result of the ordered results mode = most frequent result SPREAD: Ordered results are split into 4 equal groups so each contains 25% of the results. The 5 figure summary identifies: L, Q 1, Q 2, Q 3, H (lowest result, 1st, 2nd and 3rd quartiles, highest result) A Bo Plot is a statistical diagram that displays the 5 figure summary: L Q 1 Q 2 Q 3 H range, R = H L interquartile range, IQR = Q 3 Q 1 semi interquartile range, SIQR = Q 3 Q 1 2 NOTE: If Q 1, Q 2 or Q 3 fall between two results, the mean of the two results is taken. For eample, 12 ordered results: split into 4 equal groups of 3 results Q 1 Q 2 Q Q 1 = = 15, Q 2 = = 20, Q 3 = = 25 5 page 53

57 The pulse rates of school students were recorded in Biology class. Pulse rates: 66, 64, 71, 56, 60, 79, 77, 75, 69, 73, 75, 62, 66, 71, 66 beats per minute. 15 ordered results: Q 1 Q 2 Q Figure Summary: L = 56, Q 1 = 64, Q 2 = 69, Q 3 = 75, H = 79 Bo Plot: pulse rate (beats per minute) Spread: R = H L = = 23 IQR = Q 3 Q 1 = = 11 SIQR = Q 3 Q 1 2 = = 11 2 = 5 5 Averages: (total = =1030) MEAN = (Q 2 )MEDIAN = 69 = = 68 7 MODE = 66 page 54

58 STANDARD DEVIATION A measure of the spread of a set of data, giving a numerical value to how the data deviates from the mean. It therefore gives an indication of how good the mean is as a representitive of the data set. Formulae: mean = n standard deviation s = ( ) 2 n 1 or s = 2 n 1 ( ) 2 n Eamples, (1) High Standard Deviation: results spread out Speed (mph) mean = 38, standard deviation = 7.5 (2) Low Standard Deviation: results clustered around the mean the results are more consistent Speed (mph) mean = 38, standard deviation = 3.8 page 55

59 The pulse rates of 8 army recruits: 61, 64, 65, 67, 70, 72, 75, 78 beats per minute. = n = = 69 TOTALS ( ) s = ( ) 2 = n 1 = or = n = = TOTALS s = = 2 = n 1 ( ) 2 n = page 56

60 PROBABILITY The probability of an event A occuring is P( A) = number of outcomes involving A total number of outcomes possible 0 P 1 P = 0 impossible, P = 1 certain P( not A) = 1 P( A) number of epected outcomes involving event A= number of trials P( A) The eperimental results will differ from the theoretical probability. (1) In an eperiment a letter is chosen at random from the word ARITHMETIC and the results recorded. letter vowel consonant frequency total = 300 relative frequency = = 0.63 total = 1 Estimate of probability, P( vowel) = 0 37 (2) A letter is chosen at random from the word ARITHMETIC. 4 vowels out of 10 letters, P( vowel) = 4 10 = 2 5 ie. 0 4 for 300 trials, number of vowels epected = number of trials P( vowel) = = consonants out of 10 letters, P( consonant) = 6 P( not a vowel) = = = 3 5 ie. 0 6 vowel or consonant, vowel and consonant, P( either) = 10 = 1 certain P both 10 ( ) = 0 10 = 0 impossible page 57