Higher Tier - Algebra revision

Size: px
Start display at page:

Download "Higher Tier - Algebra revision"

Transcription

1

2 Higher Tier - Algebra revision Contents: Indices Epanding single brackets Epanding double brackets Substitution Solving equations Solving equations from angle probs Finding nth term of a sequence Simultaneous equations 2 linear Simultaneous equations 1 of each Inequalities Factorising common factors Factorising quadratics Factorising grouping & DOTS Solving quadratic equations Using the formula Completing the square Rearranging formulae Algebraic fractions Curved graphs Graphs of y = m + c Graphing inequalities Graphing simultaneous equations Graphical solutions to equations Epressing laws in symbolic form Graphs of related functions Kinematics

3 Indices (F 2 ) 4 a 2 a 3 c 0 a 4 2e 7 3ef 2 t 2 t y 3 2y b 1 5p 5 qr 6p 2 q 6 r

4 Epanding single brackets Remember to multiply all the terms inside the bracket by the term immediately in front of the bracket e.g. 4(2a + 3) = 8a + 12 If there is no term in front of the bracket, multiply by 1 or -1 Epand these brackets and simplify wherever possible: 1. 3(a - 4) = 2. 6(2c + 5) = 3. -2(d + g) = 4. c(d + 4) = 5. -5(2a - 3) = 6. a(a - 6) = 3a c d - 2g cd + 4c -10a + 15 a 2-6a 7. 4r(2r + 3) = 8. -(4a + 2) = (t + 5) = 8r r -4a - 2-2t (2a + 4) + 4(3a + 6) = 16a p(3p + 2) - 5(2p - 1) = 6p 2-6p + 5

5 Epanding double brackets Split the double brackets into 2 single brackets and then epand each bracket and simplify (3a + 4)(2a 5) = 3a(2a 5) + 4(2a 5) = 6a 2 15a + 8a 20 = 6a 2 7a 20 3a lots of 2a 5 and 4 lots of 2a 5 If a single bracket is squared (a + 5) 2 change it into double brackets (a + 5)(a + 5) Epand these brackets and simplify : 1. (c + 2)(c + 6) = c 2 + 8c (c + 7) 2 = c c (2a + 1)(3a 4) = 6a 2 5a 4 6. (4g 1) 2 = 16g 2 8g (3a 4)(5a + 7) = 15a 2 + a (p + 2)(7p 3) = 7p p 6

6 Substitution If a = 5, b = 6 and c = 2 find the value of : 3a c 2 4b 2 ac ab 2c c(b a) a 2 3b 4bc a (3a) (5b 3 ac) Now find the value of each of these epressions if a = - 8, b = 3.7 and c = 2 / 3

7 Solving equations Solve the following equation to find the value of : = = = = 3 18 = 3 18 = 3 6 = = 6 Take 4 from both sides Add 1 to both sides Divide both sides by 3 Now solve these: = = = ( + 3) = 20 5 Some equations cannot be solved in this way and Trial and Improvement methods are required Find to 1 d.p. if: = 200 Try Calculation Comment = 10 (10 10)+(3 10) = 130 Too low = 13 (13 13)+(3 13) = 208 Too high etc.

8 Solving equations from angle problems Find the size of each angle 2y 4y y Rule involved: Angles in a quad = y + 2y + y = 360 7y = 360 7y = y = 210 y = 210/7 y = 30 0 Angles are: 30 0,60 0,120 0, v + 5 Find the value of v 4v + 5 = 2v v - 2v + 5 = 39 2v + 39 Rule involved: Z angles are equal 2v + 5 = 39 2v = v = 34 v = 34/2 v = 17 0 Check: (4 17) + 5 = 73, (2 17) + 39 = 73

9 Finding nth term of a simple sequence Position number (n) This sequence is the 2 times table 5, 7, 9, 11, 13, 15,. shifted a little Each term is found by the position number times 2 then add another 3. So the rule for the sequence is n th term = 2n th term = = 203 Find the rules of these sequences 1, 3, 5, 7, 9, 6, 8, 10, 12,. 3, 8, 13, 18, 20,26,32,38, 7, 14, 21,28, 2n 1 2n + 4 5n 2 6n n And these sequences 1, 4, 9, 16, 25, 3, 6,11,18,27. 20, 18, 16, 14, 40,37,34,31, 6, 26,46,66, n 2 n n n n - 14

10 Finding nth term of a more comple sequence n = , 13, 26, 43, 64, nd difference is 4 means that the first term is 2n 2 2n = 2, 8, 18, 32, 50,. This sequence has a rule = 3n - 1 What s left = 2, 5, 8, 11, 14,. So the nth term = 2n 2 + 3n - 1 Find the rule for these sequences (a) 10, 23, 44, 73, 110, (b) 0, 17, 44, 81, 128, (c) 3, 7, 17, 33, 55, (a) nth term = 4n 2 + n + 5 (b) nth term = 5n 2 + 2n 7 (c) nth term = 3n 2 5n + 5

11 Simultaneous equations 2 linear equations 1 Multiply the equations up until the second unknowns have the same sized number in front of them 2 4a + 3b = 17 6a 2b = Eliminate the second unknown by combining the 2 equations using either SSS or SDA 8a + 6b = 34 18a 6b = 18 26a = 52 a = a = 2 3 Find the second unknown by substituting back into one of the equations Put a = 2 into: 4a + 3b = 17 Now solve: 5p + 4q = 24 2p + 5q = b = 17 3b = b = 9 b = 3 + So the solutions are: a = 2 and b = 3

12 Simultaneous equations 1 linear and 1 quadratic Sometimes it is better to use a substitution method rather than the elimination method described on the previous slide. Follow this method closely to solve this pair of simultaneous equations: 2 + y 2 = 25 and + y = 7 Step 1 Rearrange the linear equation: = 7 - y Step 2 Substitute this into the quadratic: (7 - y) 2 + y 2 = 25 Step 3 Epand brackets, rearrange, (7 - y)(7 - y) + y 2 = 25 factorise and solve: 49-14y + y 2 + y 2 = 25 Step 4 Substitute back in to find other unknown: y = 3 in + y = 7 = 4 y = 4 in + y = 7 = 3 2y 2-14y + 49 = 25 2y 2-14y + 24 = 0 (2y - 6)(y - 4) = 0 y = 3 or y = 4

13 Inequalities Inequalities can be solved in eactly the same way as equations Add 8 to both sides Divide both sides by 2 Remember to turn the sign round as well The difference is that inequalities can be given as a range of results Here can be equal to : 11, 12, 13, 14, 15, Or on a scale: Find the range of solutions for these inequalities : > < < 10 X > 1 X 3 X < 2 or or or X = 2, 3, 4, 5, 6 X = 3, 2, 1, 0, -1 X = 1, 0, -1, -2, -4 X < 4 or X = -4, -3, -2, -1, 0, 1, 2, 3

14 Factorising common factors Factorising y = 5( + 2y) Epanding Factorising is basically the reverse of epanding brackets. Instead of removing brackets you are putting them in and placing all the common factors in front. Factorise the following (and check by epanding): 15 3 = 2a + 10 = ab 5a = a 2 + 6a = = 3(5 ) 2(a + 5) a(b 5) a(a + 6) 4(2 1) 10pq + 2p = 20y 16 = 24ab + 16a 2 = r r = 3a 2 9a 3 = 2p(5q + 1) 4(5y - 4) 8a(3b + 2a) r(r + 2) 3a 2 (1 3a)

15 Factorising quadratics Factorise To help use a 2 2 bo 2-22 Find the pair which add to give Factor pairs of - 22: -1, 22-22, 1-2, 11-11, Answer = ( + 2)( 11) Here the factorising is the reverse of epanding double brackets Factorising = ( + 7)( 3) Epanding Factorise the following: = = = = = ( + 3)( + 1) ( 2)( 1) ( + 10)( 3) ( + 2)( 6) ( + 2)( + 5)

16 Factorising - quadratics Factorise Find the pair which add to give + 5 (-1, 6) When quadratics are more difficult to factorise use this method Write out the factor pairs of 6 (from 2 multiplied 3) -1, 6-6, 1-2, 3-3, 2 Rewrite as Factorise (2 1) + 3(2 1) in 2 parts Rewrite as ( + 3)(2 1) double brackets Now factorise these: (a) 25t 2 20t + 4 (b) 4y y + 5 (c) g 2 g 20 (d) (e) 8t 4 2t 2 1 Answers: (a) (5t 2)(5t 2) (b) (2y + 1)(2y + 5) (c) (g 5)(g + 4) (d) (3 2)(2 + 5) (e) (4t 2 + 1)(2t 2 1)

17 Factorising grouping and difference of two squares Grouping into pairs Fully factorise this epression: 6ab + 9ad 2bc 3cd Factorise in 2 parts 3a(2b + 3d) c(2b + 3d) Rewrite as double brackets (3a c)(2b + 3d) Fully factorise these: (a) w + z + wy + yz (b) 2w 2z wy + yz (c) 8fh 20fi + 6gh 15gi Answers: (a) ( + y)(w + z) (b) (2 y)(w z) (c) (4f + 3g)(2h 5i) Difference of two squares Fully factorise this epression: Look for 2 square numbers separated by a minus. Simply Use the square root of each and a + and a to get: (2 + 5)(2 5) Fully factorise these: (a) (b) ¼ t 2 (c) 16y Answers: (a) (9 + 1)(9 1) (b) (½ + t)(½ t) (c) 16(y 2 + 4)

18 Solving quadratic equations (using factorisation) Solve this equation: = 0 ( + 7)( 2) = = 0 or 2 = 0 Factorise first Now make each bracket equal to zero separately = - 7 or = 2 2 solutions Solve these: = = =0 25t 2 20t + 4 = = =0 ( + 3)(2 1)=0 = -3 or =1/2 ( 5)( 2)=0 = 5 or = 2 ( + 7)( + 5)=0 = -7 or = -5 (5t 2)(5t 2)=0 t = 2/5 ( + 3)( 2)=0 = -3 or = 2 (2 8)(2 + 8)=0 = 4 or = -4

19 Solving quadratic equations (using the formula) The generalization of a quadratic equations is: a 2 + b + c = 0 The following formula works out both solutions to any quadratic equation: Solve = 0 using the quadratic formula a = 6, b = 17, c = 12 = -b ± b 2 4ac 2a = -17 ± = -17 ± = or = = -b ± b 2 4ac 2a Now solve these: = = = = = =0 Answers: (1) -0.23, (2) 3.7, -2.7 (3) 1.77, (4) 0.29, (5) 0.19, (6) 1.89, 0.1 = or = -1.5

20 Solving quadratic equations (by completing the square) Another method for solving quadratics relies on the fact that: ( + a) 2 = 2 + 2a + a 2 (e.g. ( + 7) 2 = ) Rearranging : 2 + 2a = ( + a) 2 a 2 (e.g = ( + 7) 2 49) Eample Rewrite in the form ( + a) 2 b. Hence solve the equation = 0 (1 d.p.) Step 1 Write the first two terms as a completed square = ( + 2) 2 4 Step 2 Now incorporate the third term 7to both sides = ( + 2) = ( + 2) 2 11 (1 st part answered) Step 3 When = 0 then ( + 2) 2 11 = 0 ( + 2) 2 = = 11 = 11 2 = 1.3 or = - 5.3

21 Rearranging formulae Now rearrange these P = 4a + 5 A = be r Rearrange the following formula so that a is the subject a a V = u + at t +u t -u V V D = g 2 + c B = e + h a = V - u t 5. E = u - 4v d Answers: 1. a = P e = Ar b 4. h = (B e) 2 5. u = d(e + 4v) 6. Q = 4cp - st 3. g = D c 6. p = Q + st 4c

22 3. Rearranging formulae Rearrange to make g the subject: (r t) = 6 2s g g(r t) = 6 2gs gr gt = 6 2gs gr gt + 2gs = 6 g(r t + 2s) = 6 g = 6 r t + 2s s(t r) = 2(r 3) Multiply all by g When the formula has the new subject in two places (or it appears in two places during manipulation) you will need to factorise at some point Multiply out bracket Collect all g terms on one side of the equation and factorise r = st s Now rearrange these: ab = 3a + 7 a = e h e + 5 e = u 1 d a = 7 b 3 e = h 5a a 1 d = u e + 1

23 Algebraic fractions Addition and subtraction Like ordinary fractions you can only add or subtract algebraic fractions if their denominators are the same Show that can be written as ( + 1) 3 + 4( + 1) ( + 1) ( + 1) ( + 1) ( + 1) ( + 1) ( + 1) Simplify Multiply the top and bottom of each fraction by the same amount ( 4) 6( 1). ( 1)( 4) ( 1)( 4) ( 1)( 4) ( 1)( 4)

24 Algebraic fractions Multiplication and division Simplify: ( + 1) ( + 4) ( + 1) 2 4( + 4) 3( + 1) 2( + 4) Again just use normal fractions principles Algebraic fractions solving equations Solve: = Multiply all by ( 2)( + 1) 4( + 1) + 7( 2) = 2( 2)( + 1) = 2( ) = = = 0 (2 1) 6(2 1) = 0 (2 1)( 6) = = 0 or 6 = 0 = ½ or = 6 Factorise

25 Curved graphs y There are four specific types of curved graphs that you may be asked to recognise and draw. y = 2 y = 2 3 y y = y = 3 Any curve starting with 2 is U shaped y y = 5/ y = 1/ Any curve starting with 3 is this shape 2 + y 2 = 16 Any curve with a number / If you are asked to draw is this an shape accurate curved All graph circles (eg have y = an 2 + equation 3-1) simply substitute values to find y values and like the this co-ordinates 16 = radius 2 y

26 Graphs of y = m + c In the equation: c y Y = y = m + c m = the gradient (how far up for every one along) m c = the intercept (where the line crosses the y ais)

27 Graphs of y = m + c y Write down the equations of these lines: Answers: y = y = + 2 y = y = y = = 4 y = - 3

28 Graphing inequalities = -2 y y = y > 3 y = 3 Find the region that is not covered by these 3 regions -2 y y > 3-2 y <

29 Graphing simultaneous equations Finding co-ordinates for 2y + 6 = 12 using the cover up method: y = 0 2y + 6 = 12 = 2 (2, 0) = 0 2y + 6 = 12 y = 6 (0, 6) Finding co-ordinates for y = = 0 y = (20) + 1 y = 1 (0, 1) = 1 y = (21) + 1 y = 3 (1, 3) = 2 y = (22) + 1 y = 5 (2, 5) The co-ordinate of the point where the two graphs cross is (1, 3). Therefore, the solutions to the simultaneous equations are: = 1 and y = 3 Solve these simultaneous equations using a graphical method : 2y + 6 = 12 y = y 2y + 6 = 12 y =

30 Graphical solutions to equations If an equation equals 0 then its solutions lie at the points where the graph of the equation crosses the -ais. e.g. Solve the following equation graphically: = 0 y y = All you do is plot the equation y = and find where it crosses the -ais (the line y=0) There are two solutions to = 0 = - 3 and =2

31 Graphical solutions to equations If the equation does not equal zero : Draw the graphs for both sides of the equation and where they cross is where the solutions lie e.g. Solve the following equation graphically: 2 2 Be prepared 11 = 9 to solve 2 simultaneous equations y graphically where one is linear (e.g. + Plot the following y = y = 2 7) 2 and 11the other is equations and find a circle (e.g. 2 + y 2 = 25) where they cross: y = 9 y = y = There are 2 solutions to = 9 = - 4 and = 5

32 Epressing laws in symbolic form In the equation y = m + c, if y is plotted against the gradient of the line is m and the intercept on the y-ais is c. Similarly in the equation y = m 2 + c, if y is plotted against 2 the gradient of the line is m and the intercept on the y-ais is c. And in the equation y = m + c, if y is plotted against 1 the gradient of the line is m and the intercept on the y-ais is c. y c m y = m + 2 c+ c 1 2

33 Epressing laws in symbolic form e.g. y and are known to be connected by the equation y = a + b. Find a and b if: y Find the 1/ values Plot y against 1/ y / b = a = = 12 So the equation is: y = /

34 Transformation of graphs Rule 1 The graph of y = - f() is the reflection of the graph y = f() in the - ais y = f() y y = - f()

35 Transformation of graphs Rule 2 The graph of y = f(-) is the reflection of the graph y = f() in the y - ais y = f() y y = f(-)

36 Transformation of graphs Rule 3 The graph of y = f() + a is the translation of the graph y = f() vertically by vector y [ o ] a y = f() y = f() + a

37 Transformation of graphs Rule 4 The graph of y = f( + a) is the translation of the graph y = f() horizontally by vector -a o [ ] y y = f() y = f( + a) -a

38 Transformation of graphs Rule 5 y y = kf() y = f() The graph of y = kf() is the stretching of the graph y = f() vertically by a factor of k

39 Transformation of graphs Rule 6 y y = f() y = f(k) The graph of y = f(k) is the stretching of the graph y = f() horizontally by a factor of 1/k

40 Straight Distance/Time graphs D Straight Velocity/Time graphs V V = 0 Remember the rule S = D/T The gradient of each section is the average speed for that part of the journey The horizontal section means the vehicle has stopped The section with the negative gradient shows the return journey and it will have a positive speed but a negative velocity T Remember the rule A = V/T The gradient of each section is the acceleration for that part of the journey The horizontal section means the vehicle is travelling at a constant velocity The sections with a negative gradient show a deceleration The area under the graph is the distance travelled T

Intermediate Tier - Algebra revision

Intermediate Tier - Algebra revision Intermediate Tier - Algebra revision Contents : Collecting like terms Multiplying terms together Indices Expanding single brackets Expanding double brackets Substitution Solving equations Finding nth term

More information

Algebra Revision Guide

Algebra Revision Guide Algebra Revision Guide Stage 4 S J Cooper 1st Edition Collection of like terms... Solving simple equations... Factorisation... 6 Inequalities... 7 Graphs... 9 1. The straight line... 9. The quadratic curve...

More information

Quadratics NOTES.notebook November 02, 2017

Quadratics NOTES.notebook November 02, 2017 1) Find y where y = 2-1 and a) = 2 b) = -1 c) = 0 2) Epand the brackets and simplify: (m + 4)(2m - 3) To find the equation of quadratic graphs using substitution of a point. 3) Fully factorise 4y 2-5y

More information

12. Quadratics NOTES.notebook September 21, 2017

12. Quadratics NOTES.notebook September 21, 2017 1) Fully factorise 4y 2-5y - 6 Today's Learning: To find the equation of quadratic graphs using substitution of a point. 2) Epand the brackets and simplify: (m + 4)(2m - 3) 3) Calculate 20% of 340 without

More information

Rearrange m ore complicated formulae where the subject may appear twice or as a power (A*) Rearrange a formula where the subject appears twice (A)

Rearrange m ore complicated formulae where the subject may appear twice or as a power (A*) Rearrange a formula where the subject appears twice (A) Moving from A to A* A* Solve a pair of simultaneous equations where one is linear and the other is non-linear (A*) Rearrange m ore complicated formulae may appear twice or as a power (A*) Simplify fractions

More information

Maths A Level Summer Assignment & Transition Work

Maths A Level Summer Assignment & Transition Work Maths A Level Summer Assignment & Transition Work The summer assignment element should take no longer than hours to complete. Your summer assignment for each course must be submitted in the relevant first

More information

QUADRATIC GRAPHS ALGEBRA 2. Dr Adrian Jannetta MIMA CMath FRAS INU0114/514 (MATHS 1) Quadratic Graphs 1/ 16 Adrian Jannetta

QUADRATIC GRAPHS ALGEBRA 2. Dr Adrian Jannetta MIMA CMath FRAS INU0114/514 (MATHS 1) Quadratic Graphs 1/ 16 Adrian Jannetta QUADRATIC GRAPHS ALGEBRA 2 INU0114/514 (MATHS 1) Dr Adrian Jannetta MIMA CMath FRAS Quadratic Graphs 1/ 16 Adrian Jannetta Objectives Be able to sketch the graph of a quadratic function Recognise the shape

More information

ILLUSTRATIVE EXAMPLES

ILLUSTRATIVE EXAMPLES CHAPTER Points to Remember : POLYNOMIALS 7. A symbol having a fied numerical value is called a constant. For e.g. 9,,, etc.. A symbol which may take different numerical values is known as a variable. We

More information

Mathematics Revision Guide. Algebra. Grade C B

Mathematics Revision Guide. Algebra. Grade C B Mathematics Revision Guide Algebra Grade C B 1 y 5 x y 4 = y 9 Add powers a 3 a 4.. (1) y 10 y 7 = y 3 (y 5 ) 3 = y 15 Subtract powers Multiply powers x 4 x 9...(1) (q 3 ) 4...(1) Keep numbers without

More information

SOLVING QUADRATICS. Copyright - Kramzil Pty Ltd trading as Academic Teacher Resources

SOLVING QUADRATICS. Copyright - Kramzil Pty Ltd trading as Academic Teacher Resources SOLVING QUADRATICS Copyright - Kramzil Pty Ltd trading as Academic Teacher Resources SOLVING QUADRATICS General Form: y a b c Where a, b and c are constants To solve a quadratic equation, the equation

More information

Linear And Exponential Algebra Lesson #1

Linear And Exponential Algebra Lesson #1 Introduction Linear And Eponential Algebra Lesson # Algebra is a very powerful tool which is used to make problem solving easier. Algebra involves using pronumerals (letters) to represent unknown values

More information

6.6 General Form of the Equation for a Linear Relation

6.6 General Form of the Equation for a Linear Relation 6.6 General Form of the Equation for a Linear Relation FOCUS Relate the graph of a line to its equation in general form. We can write an equation in different forms. y 0 6 5 y 10 = 0 An equation for this

More information

Algebra Skills Required for Entry to a Level Two Course in Mathematics

Algebra Skills Required for Entry to a Level Two Course in Mathematics Algebra Skills Required for Entr to a Level Two Course in Mathematics This is a list of Level One skills ou will be required to demonstrate if ou are to gain entr to the Level Two Achievement Standard

More information

3 6 x a. 12 b. 63 c. 27 d. 0. 6, find

3 6 x a. 12 b. 63 c. 27 d. 0. 6, find Advanced Algebra Topics COMPASS Review revised Summer 0 You will be allowed to use a calculator on the COMPASS test Acceptable calculators are basic calculators, scientific calculators, and approved models

More information

Chapter 2 Analysis of Graphs of Functions

Chapter 2 Analysis of Graphs of Functions Chapter Analysis of Graphs of Functions Chapter Analysis of Graphs of Functions Covered in this Chapter:.1 Graphs of Basic Functions and their Domain and Range. Odd, Even Functions, and their Symmetry..

More information

THOMAS WHITHAM SIXTH FORM

THOMAS WHITHAM SIXTH FORM THOMAS WHITHAM SIXTH FORM Algebra Foundation & Higher Tier Units & thomaswhitham.pbworks.com Algebra () Collection of like terms. Simplif each of the following epressions a) a a a b) m m m c) d) d d 6d

More information

SCHOOL OF MATHEMATICS MATHEMATICS FOR PART I ENGINEERING. Self-paced Course

SCHOOL OF MATHEMATICS MATHEMATICS FOR PART I ENGINEERING. Self-paced Course SCHOOL OF MATHEMATICS MATHEMATICS FOR PART I ENGINEERING Self-paced Course MODULE ALGEBRA Module Topics Simplifying expressions and algebraic functions Rearranging formulae Indices 4 Rationalising a denominator

More information

ACCUPLACER MATH 0310

ACCUPLACER MATH 0310 The University of Teas at El Paso Tutoring and Learning Center ACCUPLACER MATH 00 http://www.academics.utep.edu/tlc MATH 00 Page Linear Equations Linear Equations Eercises 5 Linear Equations Answer to

More information

PLC Papers. Created For:

PLC Papers. Created For: PLC Papers Created For: Algebra and proof 2 Grade 8 Objective: Use algebra to construct proofs Question 1 a) If n is a positive integer explain why the expression 2n + 1 is always an odd number. b) Use

More information

THE DISTRIBUTIVE LAW. Note: To avoid mistakes, include arrows above or below the terms that are being multiplied.

THE DISTRIBUTIVE LAW. Note: To avoid mistakes, include arrows above or below the terms that are being multiplied. THE DISTRIBUTIVE LAW ( ) When an equation of the form a b c is epanded, every term inside the bracket is multiplied by the number or pronumeral (letter), and the sign that is located outside the brackets.

More information

Pure Core 2. Revision Notes

Pure Core 2. Revision Notes Pure Core Revision Notes June 06 Pure Core Algebra... Polynomials... Factorising... Standard results... Long division... Remainder theorem... 4 Factor theorem... 5 Choosing a suitable factor... 6 Cubic

More information

ACCUPLACER MATH 0311 OR MATH 0120

ACCUPLACER MATH 0311 OR MATH 0120 The University of Teas at El Paso Tutoring and Learning Center ACCUPLACER MATH 0 OR MATH 00 http://www.academics.utep.edu/tlc MATH 0 OR MATH 00 Page Factoring Factoring Eercises 8 Factoring Answer to Eercises

More information

Key Facts and Methods

Key Facts and Methods Intermediate Maths Key Facts and Methods Use this (as well as trying questions) to revise by: 1. Testing yourself. Asking a friend or family member to test you by reading the questions (on the lefthand

More information

Polynomials. Exponents. End Behavior. Writing. Solving Factoring. Graphing. End Behavior. Polynomial Notes. Synthetic Division.

Polynomials. Exponents. End Behavior. Writing. Solving Factoring. Graphing. End Behavior. Polynomial Notes. Synthetic Division. Polynomials Polynomials 1. P 1: Exponents 2. P 2: Factoring Polynomials 3. P 3: End Behavior 4. P 4: Fundamental Theorem of Algebra Writing real root x= 10 or (x+10) local maximum Exponents real root x=10

More information

AQA Level 2 Further mathematics Number & algebra. Section 3: Functions and their graphs

AQA Level 2 Further mathematics Number & algebra. Section 3: Functions and their graphs AQA Level Further mathematics Number & algebra Section : Functions and their graphs Notes and Eamples These notes contain subsections on: The language of functions Gradients The equation of a straight

More information

Answers for NSSH exam paper 2 type of questions, based on the syllabus part 2 (includes 16)

Answers for NSSH exam paper 2 type of questions, based on the syllabus part 2 (includes 16) Answers for NSSH eam paper type of questions, based on the syllabus part (includes 6) Section Integration dy 6 6. (a) Integrate with respect to : d y c ( )d or d The curve passes through P(,) so = 6/ +

More information

GCSE Mathematics Non-Calculator Higher Tier Free Practice Set 1 1 hour 45 minutes ANSWERS. Grade Boundaries A* A B C D E.

GCSE Mathematics Non-Calculator Higher Tier Free Practice Set 1 1 hour 45 minutes ANSWERS. Grade Boundaries A* A B C D E. MathsMadeEasy GCSE Mathematics Non-Calculator Higher Tier Free Practice Set 1 1 hour 45 minutes ANSWERS Grade Boundaries A* A B C D E 88 71 57 43 22 13 3 Authors Note Every possible effort has been made

More information

January Core Mathematics C1 Mark Scheme

January Core Mathematics C1 Mark Scheme January 007 666 Core Mathematics C Mark Scheme Question Scheme Mark. 4 k or k (k a non-zero constant) M, +..., ( 0) A, A, B (4) 4 Accept equivalent alternatives to, e.g. 0.5,,. M: 4 differentiated to give

More information

Brief Revision Notes and Strategies

Brief Revision Notes and Strategies Brief Revision Notes and Strategies Straight Line Distance Formula d = ( ) + ( y y ) d is distance between A(, y ) and B(, y ) Mid-point formula +, y + M y M is midpoint of A(, y ) and B(, y ) y y Equation

More information

Quadratic Inequalities in One Variable

Quadratic Inequalities in One Variable Quadratic Inequalities in One Variable Quadratic inequalities in one variable can be written in one of the following forms: a b c + + 0 a b c + + 0 a b c + + 0 a b c + + 0 Where a, b, and c are real and

More information

Mesaieed International School

Mesaieed International School Mesaieed International School SUBJECT: Mathematics Year: 10H Overview of the year: The contents below reflect the first half of the two-year IGCSE Higher course which provides students with the opportunity

More information

Algebra. Topic: Manipulate simple algebraic expressions.

Algebra. Topic: Manipulate simple algebraic expressions. 30-4-10 Algebra Days: 1 and 2 Topic: Manipulate simple algebraic expressions. You need to be able to: Use index notation and simple instances of index laws. Collect like terms Multiply a single term over

More information

Fundamentals of Algebra, Geometry, and Trigonometry. (Self-Study Course)

Fundamentals of Algebra, Geometry, and Trigonometry. (Self-Study Course) Fundamentals of Algebra, Geometry, and Trigonometry (Self-Study Course) This training is offered eclusively through the Pennsylvania Department of Transportation, Business Leadership Office, Technical

More information

Coordinate geometry. + bx + c. Vertical asymptote. Sketch graphs of hyperbolas (including asymptotic behaviour) from the general

Coordinate geometry. + bx + c. Vertical asymptote. Sketch graphs of hyperbolas (including asymptotic behaviour) from the general A Sketch graphs of = a m b n c where m = or and n = or B Reciprocal graphs C Graphs of circles and ellipses D Graphs of hperbolas E Partial fractions F Sketch graphs using partial fractions Coordinate

More information

Pure Core 1. Revision Notes

Pure Core 1. Revision Notes Pure Core Revision Notes Ma 06 Pure Core Algebra... Indices... Rules of indices... Surds... 4 Simplifing surds... 4 Rationalising the denominator... 4 Quadratic functions... 5 Completing the square....

More information

ZETA MATHS. Higher Mathematics Revision Checklist

ZETA MATHS. Higher Mathematics Revision Checklist ZETA MATHS Higher Mathematics Revision Checklist Contents: Epressions & Functions Page Logarithmic & Eponential Functions Addition Formulae. 3 Wave Function.... 4 Graphs of Functions. 5 Sets of Functions

More information

Algebra Review C H A P T E R. To solve an algebraic equation with one variable, find the value of the unknown variable.

Algebra Review C H A P T E R. To solve an algebraic equation with one variable, find the value of the unknown variable. C H A P T E R 6 Algebra Review This chapter reviews key skills and concepts of algebra that you need to know for the SAT. Throughout the chapter are sample questions in the style of SAT questions. Each

More information

AMB111F Notes 3 Quadratic Equations, Inequalities and their Graphs

AMB111F Notes 3 Quadratic Equations, Inequalities and their Graphs AMB111F Notes 3 Quadratic Equations, Inequalities and their Graphs The eqn y = a +b+c is a quadratic eqn and its graph is called a parabola. If a > 0, the parabola is concave up, while if a < 0, the parabola

More information

Day 3: Section P-6 Rational Expressions; Section P-7 Equations. Rational Expressions

Day 3: Section P-6 Rational Expressions; Section P-7 Equations. Rational Expressions 1 Day : Section P-6 Rational Epressions; Section P-7 Equations Rational Epressions A rational epression (Fractions) is the quotient of two polynomials. The set of real numbers for which an algebraic epression

More information

Further factorising, simplifying, completing the square and algebraic proof

Further factorising, simplifying, completing the square and algebraic proof Further factorising, simplifying, completing the square and algebraic proof 8 CHAPTER 8. Further factorising Quadratic epressions of the form b c were factorised in Section 8. by finding two numbers whose

More information

Module 2, Section 2 Solving Equations

Module 2, Section 2 Solving Equations Principles of Mathematics Section, Introduction 03 Introduction Module, Section Solving Equations In this section, you will learn to solve quadratic equations graphically, by factoring, and by applying

More information

Pure Mathematics P1

Pure Mathematics P1 1 Pure Mathematics P1 Rules of Indices x m * x n = x m+n eg. 2 3 * 2 2 = 2*2*2*2*2 = 2 5 x m / x n = x m-n eg. 2 3 / 2 2 = 2*2*2 = 2 1 = 2 2*2 (x m ) n =x mn eg. (2 3 ) 2 = (2*2*2)*(2*2*2) = 2 6 x 0 =

More information

4.5 Rational functions.

4.5 Rational functions. 4.5 Rational functions. We have studied graphs of polynomials and we understand the graphical significance of the zeros of the polynomial and their multiplicities. Now we are ready to etend these eplorations

More information

A-Level Notes CORE 1

A-Level Notes CORE 1 A-Level Notes CORE 1 Basic algebra Glossary Coefficient For example, in the expression x³ 3x² x + 4, the coefficient of x³ is, the coefficient of x² is 3, and the coefficient of x is 1. (The final 4 is

More information

Newbattle Community High School Higher Mathematics. Key Facts Q&A

Newbattle Community High School Higher Mathematics. Key Facts Q&A Key Facts Q&A Ways of using this booklet: 1) Write the questions on cards with the answers on the back and test yourself. ) Work with a friend who is also doing to take turns reading a random question

More information

(1) Assignment # 1 Absolute Value. (2) Assignment # 2 Compound Absolute Values. (3) Assignment # 3 Exponents. (4) Assignment # 4 Simplifying Radicals

(1) Assignment # 1 Absolute Value. (2) Assignment # 2 Compound Absolute Values. (3) Assignment # 3 Exponents. (4) Assignment # 4 Simplifying Radicals Alg_0 Packet # The beginning of our Quest () Assignment # Absolute Value () Assignment # Compound Absolute Values () Assignment # Eponents () Assignment # Simplifying Radicals (5) Assignment # 5 Fractional

More information

2. Which of the following expressions represents the product of four less than three times x and two more than x?

2. Which of the following expressions represents the product of four less than three times x and two more than x? Algebra Topics COMPASS Review You will be allowed to use a calculator on the COMPASS test. Acceptable calculators are: basic calculators, scientific calculators, and graphing calculators up through the

More information

Algebra. CLCnet. Page Topic Title. Revision Websites. GCSE Revision 2006/7 - Mathematics. Add your favourite websites and school software here.

Algebra. CLCnet. Page Topic Title. Revision Websites. GCSE Revision 2006/7 - Mathematics. Add your favourite websites and school software here. Section 2 Page Topic Title 54-57 12. Basic algebra 58-61 13. Solving equations 62-64 14. Forming and solving equations from written information 65-67 15. Trial and improvement 68-72 16. Formulae 73-76

More information

Review: Properties of Exponents (Allow students to come up with these on their own.) m n m n. a a a. n n n m. a a a. a b a

Review: Properties of Exponents (Allow students to come up with these on their own.) m n m n. a a a. n n n m. a a a. a b a Algebra II Notes Unit Si: Polynomials Syllabus Objectives: 6. The student will simplify polynomial epressions. Review: Properties of Eponents (Allow students to come up with these on their own.) Let a

More information

Section 5.1 Model Inverse and Joint Variation

Section 5.1 Model Inverse and Joint Variation 108 Section 5.1 Model Inverse and Joint Variation Remember a Direct Variation Equation y k has a y-intercept of (0, 0). Different Types of Variation Relationship Equation a) y varies directly with. y k

More information

Algebra Concepts Equation Solving Flow Chart Page 1 of 6. How Do I Solve This Equation?

Algebra Concepts Equation Solving Flow Chart Page 1 of 6. How Do I Solve This Equation? Algebra Concepts Equation Solving Flow Chart Page of 6 How Do I Solve This Equation? First, simplify both sides of the equation as much as possible by: combining like terms, removing parentheses using

More information

Goal: To graph points in the Cartesian plane, identify functions by graphs and equations, use function notation

Goal: To graph points in the Cartesian plane, identify functions by graphs and equations, use function notation Section -1 Functions Goal: To graph points in the Cartesian plane, identify functions by graphs and equations, use function notation Definition: A rule that produces eactly one output for one input is

More information

Section 3.3 Graphs of Polynomial Functions

Section 3.3 Graphs of Polynomial Functions 3.3 Graphs of Polynomial Functions 179 Section 3.3 Graphs of Polynomial Functions In the previous section we eplored the short run behavior of quadratics, a special case of polynomials. In this section

More information

Chapter XX: 1: Functions. XXXXXXXXXXXXXXX <CT>Chapter 1: Data representation</ct> 1.1 Mappings

Chapter XX: 1: Functions. XXXXXXXXXXXXXXX <CT>Chapter 1: Data representation</ct> 1.1 Mappings 978--08-8-8 Cambridge IGCSE and O Level Additional Mathematics Practice Book Ecerpt Chapter XX: : Functions XXXXXXXXXXXXXXX Chapter : Data representation This section will show you how to: understand

More information

CP Algebra 2. Summer Packet. Name:

CP Algebra 2. Summer Packet. Name: CP Algebra Summer Packet 018 Name: Objectives for CP Algebra Summer Packet 018 I. Number Sense and Numerical Operations (Problems: 1 to 4) Use the Order of Operations to evaluate expressions. (p. 6) Evaluate

More information

Algebraic Expressions and Identities

Algebraic Expressions and Identities 9 Algebraic Epressions and Identities introduction In previous classes, you have studied the fundamental concepts of algebra, algebraic epressions and their addition and subtraction. In this chapter, we

More information

3.1 Functions. We will deal with functions for which both domain and the range are the set (or subset) of real numbers

3.1 Functions. We will deal with functions for which both domain and the range are the set (or subset) of real numbers 3.1 Functions A relation is a set of ordered pairs (, y). Eample: The set {(1,a), (1, b), (,b), (3,c), (3, a), (4,a)} is a relation A function is a relation (so, it is the set of ordered pairs) that does

More information

The region enclosed by the curve of f and the x-axis is rotated 360 about the x-axis. Find the volume of the solid formed.

The region enclosed by the curve of f and the x-axis is rotated 360 about the x-axis. Find the volume of the solid formed. Section A ln. Let g() =, for > 0. ln Use the quotient rule to show that g ( ). 3 (b) The graph of g has a maimum point at A. Find the -coordinate of A. (Total 7 marks) 6. Let h() =. Find h (0). cos 3.

More information

OCR Mathematics Advanced Subsidiary GCE Core 1 (4721) January 2012

OCR Mathematics Advanced Subsidiary GCE Core 1 (4721) January 2012 Link to past paper on OCR website: www.ocr.org.uk The above link takes you to the OCR website. From there you click QUALIFICATIONS, QUALIFICATIONS BY TYPE, AS/A LEVEL GCE, MATHEMATICS, VIEW ALL DOCUMENTS,

More information

2 2xdx. Craigmount High School Mathematics Department

2 2xdx. Craigmount High School Mathematics Department Π 5 3 xdx 5 cosx 4 6 3 8 Help Your Child With Higher Maths Introduction We ve designed this booklet so that you can use it with your child throughout the session, as he/she moves through the Higher course,

More information

Unit 3: Number, Algebra, Geometry 2

Unit 3: Number, Algebra, Geometry 2 Unit 3: Number, Algebra, Geometry 2 Number Use standard form, expressed in standard notation and on a calculator display Calculate with standard form Convert between ordinary and standard form representations

More information

St Peter the Apostle High. Mathematics Dept.

St Peter the Apostle High. Mathematics Dept. St Peter the postle High Mathematics Dept. Higher Prelim Revision 6 Paper I - Non~calculator Time allowed - hour 0 minutes Section - Questions - 0 (40 marks) Instructions for the completion of Section

More information

Basic ALGEBRA 2 SUMMER PACKET

Basic ALGEBRA 2 SUMMER PACKET Name Basic ALGEBRA SUMMER PACKET This packet contains Algebra I topics that you have learned before and should be familiar with coming into Algebra II. We will use these concepts on a regular basis throughout

More information

Twitter: @Owen134866 www.mathsfreeresourcelibrary.com Prior Knowledge Check 1) Simplify: a) 3x 2 5x 5 b) 5x3 y 2 15x 7 2) Factorise: a) x 2 2x 24 b) 3x 2 17x + 20 15x 2 y 3 3) Use long division to calculate:

More information

Multi-Step Equations and Inequalities

Multi-Step Equations and Inequalities Multi-Step Equations and Inequalities Syllabus Objective (1.13): The student will combine like terms in an epression when simplifying variable epressions. Term: the parts of an epression that are either

More information

A summary of factoring methods

A summary of factoring methods Roberto s Notes on Prerequisites for Calculus Chapter 1: Algebra Section 1 A summary of factoring methods What you need to know already: Basic algebra notation and facts. What you can learn here: What

More information

The Grade Descriptors below are used to assess work and student progress in Mathematics from Year 7 to

The Grade Descriptors below are used to assess work and student progress in Mathematics from Year 7 to Jersey College for Girls Assessment criteria for KS3 and KS4 Mathematics In Mathematics, students are required to become familiar, confident and competent with a range of content and procedures described

More information

1. (a) 10, 5, 1, 0, 2, 3 (b) 12, 5, 3, 0, 3, 10 (c) 7, 3, 1, 2, 4, 10 (d) 31, 21, , 21, 41

1. (a) 10, 5, 1, 0, 2, 3 (b) 12, 5, 3, 0, 3, 10 (c) 7, 3, 1, 2, 4, 10 (d) 31, 21, , 21, 41 Practice Book UNIT 0 Equations Unit 0: Equations 0 Negative Numbers (a) 0 0 (b) 0 0 (c) 7 4 0 (d) 4 (a) 4 (b) 0 (c) 6 4 (d) 4 (a) < (b) < 4 (c) > 8 (d) 0 > 0 (a) 0 (b) 0 (c) July (d) 0 Arithmetic with

More information

Core Mathematics 2 Trigonometry

Core Mathematics 2 Trigonometry Core Mathematics 2 Trigonometry Edited by: K V Kumaran Email: kvkumaran@gmail.com Core Mathematics 2 Trigonometry 2 1 Trigonometry Sine, cosine and tangent functions. Their graphs, symmetries and periodicity.

More information

MEI Core 1. Basic Algebra. Section 1: Basic algebraic manipulation and solving simple equations. Manipulating algebraic expressions

MEI Core 1. Basic Algebra. Section 1: Basic algebraic manipulation and solving simple equations. Manipulating algebraic expressions MEI Core Basic Algebra Section : Basic algebraic manipulation and solving simple equations Notes and Examples These notes contain subsections on Manipulating algebraic expressions Collecting like terms

More information

PLC Papers. Created For:

PLC Papers. Created For: PLC Papers Created For: Algebra and proof 2 Grade 8 Objective: Use algebra to construct proofs Question 1 a) If n is a positive integer explain why the expression 2n + 1 is always an odd number. b) Use

More information

Algebra y funciones [219 marks]

Algebra y funciones [219 marks] Algebra y funciones [9 marks] Let f() = 3 ln and g() = ln5 3. a. Epress g() in the form f() + lna, where a Z +. attempt to apply rules of logarithms e.g. ln a b = b lna, lnab = lna + lnb correct application

More information

Quadratic Functions Objective: To be able to graph a quadratic function and identify the vertex and the roots.

Quadratic Functions Objective: To be able to graph a quadratic function and identify the vertex and the roots. Name: Quadratic Functions Objective: To be able to graph a quadratic function and identif the verte and the roots. Period: Quadratic Function Function of degree. Usuall in the form: We are now going to

More information

Maths Higher Prelim Content

Maths Higher Prelim Content Maths Higher Prelim Content Straight Line Gradient of a line A(x 1, y 1 ), B(x 2, y 2 ), Gradient of AB m AB = y 2 y1 x 2 x 1 m = tanθ where θ is the angle the line makes with the positive direction of

More information

National 5 Course Notes. Scientific Notation (or Standard Form) This is a different way of writing very large and very small numbers in the form:-

National 5 Course Notes. Scientific Notation (or Standard Form) This is a different way of writing very large and very small numbers in the form:- National 5 Course Notes Scientific Notation (or Standard Form) This is a different way of writing very large and very small numbers in the form:- a x 10 n where a is between 1 and 10 and n is an integer

More information

Mathematics. Polynomials and Quadratics. hsn.uk.net. Higher. Contents. Polynomials and Quadratics 1. CfE Edition

Mathematics. Polynomials and Quadratics. hsn.uk.net. Higher. Contents. Polynomials and Quadratics 1. CfE Edition Higher Mathematics Contents 1 1 Quadratics EF 1 The Discriminant EF 3 3 Completing the Square EF 4 4 Sketching Parabolas EF 7 5 Determining the Equation of a Parabola RC 9 6 Solving Quadratic Inequalities

More information

MEI Core 2. Sequences and series. Section 1: Definitions and Notation

MEI Core 2. Sequences and series. Section 1: Definitions and Notation Notes and Eamples MEI Core Sequences and series Section : Definitions and Notation In this section you will learn definitions and notation involving sequences and series, and some different ways in which

More information

Algebra Final Exam Review Packet

Algebra Final Exam Review Packet Algebra 1 00 Final Eam Review Packet UNIT 1 EXPONENTS / RADICALS Eponents Degree of a monomial: Add the degrees of all the in the monomial together. o Eample - Find the degree of 5 7 yz Degree of a polynomial:

More information

Maths Revision. Book 2. Name:.

Maths Revision. Book 2. Name:. Maths Revision Book 2 Name:. Number Fractions Calculating with Fractions Addition 6 + 4 2 2 Subtraction 2 Subtract wholes first 0 2 + 9 2 2 2 Change one whole into thirds 9 2 + + 2 7 2 2 Multiplication

More information

Algebra Year 10. Language

Algebra Year 10. Language Algebra Year 10 Introduction In Algebra we do Maths with numbers, but some of those numbers are not known. They are represented with letters, and called unknowns, variables or, most formally, literals.

More information

Question Scheme Marks AOs. 1 Area( R ) ( ) 2 = 6(2) + ("1.635") = B1ft 2.2a 1

Question Scheme Marks AOs. 1 Area( R ) ( ) 2 = 6(2) + (1.635) = B1ft 2.2a 1 9MA0/01: Pure Mathematics Paper 1 Mark scheme 1 (a) 1 Area( R ) 0.5 0.5 ( 0.674 0.884 0.9686) 1.0981 + + + + B1 1.1b (c)(i) (c)(ii) 1 = 6.5405 = 1.63515 = 1.635 (3 dp) 4 Any valid reason, for example Increase

More information

A polynomial expression is the addition or subtraction of many algebraic terms with positive integer powers.

A polynomial expression is the addition or subtraction of many algebraic terms with positive integer powers. LEAVING CERT Honours Maths notes on Algebra. A polynomial expression is the addition or subtraction of many algebraic terms with positive integer powers. The degree is the highest power of x. 3x 2 + 2x

More information

Math 154 :: Elementary Algebra

Math 154 :: Elementary Algebra Math 4 :: Elementary Algebra Section. Additive Property of Equality Section. Multiplicative Property of Equality Section.3 Linear Equations in One-Variable Section.4 Linear Equations in One-Variable with

More information

Higher. Differentiation 28

Higher. Differentiation 28 Higher Mathematics UNIT OUTCOME Differentiation Contents Differentiation 8 Introduction to Differentiation 8 Finding the Derivative 9 Differentiating with Respect to Other Variables 4 Rates of Change 4

More information

What you may need to do: 1. Formulate a quadratic expression or equation. Generate a quadratic expression from a description or diagram.

What you may need to do: 1. Formulate a quadratic expression or equation. Generate a quadratic expression from a description or diagram. Dealing with a quadratic What it is: A quadratic expression is an algebraic expression containing an x 2 term, as well as possibly an x term and/or a number, but nothing else - eg, no x 3 term. The general

More information

Twitter: @Owen134866 www.mathsfreeresourcelibrary.com Prior Knowledge Check 1) Find the point of intersection for each pair of lines: a) y = 4x + 7 and 5y = 2x 1 b) y = 5x 1 and 3x + 7y = 11 c) 2x 5y =

More information

MATHS PAMPHLET Year 7

MATHS PAMPHLET Year 7 MATHS PAMPHLET Year 7 TABLE OF CONTENTS A- Course Synopsis B- Practice Questions C- Answer sheets A- Course Synopsis Revision Guide for year 7 Set 3 set will be attempting a calculator and non calculator

More information

Bishop Kelley High School Summer Math Program Course: Algebra 2 A

Bishop Kelley High School Summer Math Program Course: Algebra 2 A 06 07 Bishop Kelley High School Summer Math Program Course: Algebra A NAME: DIRECTIONS: Show all work in packet!!! The first 6 pages of this packet provide eamples as to how to work some of the problems

More information

STUDY KNOWHOW PROGRAM STUDY AND LEARNING CENTRE. Functions & Graphs

STUDY KNOWHOW PROGRAM STUDY AND LEARNING CENTRE. Functions & Graphs STUDY KNOWHOW PROGRAM STUDY AND LEARNING CENTRE Functions & Graphs Contents Functions and Relations... 1 Interval Notation... 3 Graphs: Linear Functions... 5 Lines and Gradients... 7 Graphs: Quadratic

More information

COUNCIL ROCK HIGH SCHOOL MATHEMATICS. A Note Guideline of Algebraic Concepts. Designed to assist students in A Summer Review of Algebra

COUNCIL ROCK HIGH SCHOOL MATHEMATICS. A Note Guideline of Algebraic Concepts. Designed to assist students in A Summer Review of Algebra COUNCIL ROCK HIGH SCHOOL MATHEMATICS A Note Guideline of Algebraic Concepts Designed to assist students in A Summer Review of Algebra [A teacher prepared compilation of the 7 Algebraic concepts deemed

More information

NATIONAL QUALIFICATIONS

NATIONAL QUALIFICATIONS Mathematics Higher Prelim Eamination 04/05 Paper Assessing Units & + Vectors NATIONAL QUALIFICATIONS Time allowed - hour 0 minutes Read carefully Calculators may NOT be used in this paper. Section A -

More information

Math 030 Review for Final Exam Revised Fall 2010 RH/ DM 1

Math 030 Review for Final Exam Revised Fall 2010 RH/ DM 1 Math 00 Review for Final Eam Revised Fall 010 RH/ DM 1 1. Solve the equations: (-1) (7) (-) (-1) () 1 1 1 1 f. 1 g. h. 1 11 i. 9. Solve the following equations for the given variable: 1 Solve for. D ab

More information

Higher. Polynomials and Quadratics. Polynomials and Quadratics 1

Higher. Polynomials and Quadratics. Polynomials and Quadratics 1 Higher Mathematics Contents 1 1 Quadratics EF 1 The Discriminant EF 3 3 Completing the Square EF 4 4 Sketching Parabolas EF 7 5 Determining the Equation of a Parabola RC 9 6 Solving Quadratic Inequalities

More information

AP Calculus AB Summer Assignment

AP Calculus AB Summer Assignment AP Calculus AB Summer Assignment Name: When you come back to school, it is my epectation that you will have this packet completed. You will be way behind at the beginning of the year if you haven t attempted

More information

Summer Review Packet for Students Entering AP Calculus BC. Complex Fractions

Summer Review Packet for Students Entering AP Calculus BC. Complex Fractions Summer Review Packet for Students Entering AP Calculus BC Comple Fractions When simplifying comple fractions, multiply by a fraction equal to 1 which has a numerator and denominator composed of the common

More information

A-LEVEL MATHS Bridging Work 2017

A-LEVEL MATHS Bridging Work 2017 A-LEVEL MATHS Bridging Work 017 Name: Firstly, CONGRATULATIONS for choosing the best A-Level subject there is. A-Level Maths at Wales is not only interesting and enjoyable but is highly regarded by colleges,

More information

January Further Pure Mathematics FP1 Mark Scheme

January Further Pure Mathematics FP1 Mark Scheme January 6667 Further Pure Mathematics FP Mar Q z + 8i + i (a) z i + i + i + 8i 8 + 5i z (b) ( ) + 5 z (or awrt 5.8) () ft () (c) 5 5 tan α or z arg π.... z + i (a) and attempt to multiply out for + i -

More information

Algebra Review. 1. Evaluate the expression when a = -3 and b = A) 17 B) 1 C) Simplify: A) 17 B) 29 C) 16 D)

Algebra Review. 1. Evaluate the expression when a = -3 and b = A) 17 B) 1 C) Simplify: A) 17 B) 29 C) 16 D) Algebra Review a b. Evaluate the epression when a = - and b = -. A) B) C). Simplify: 6 A) B) 9 C) 6 0. Simplify: A) 0 B) 8 C) 6. Evaluate: 6z y if =, y = 8, and z =. A) B) C) CPT Review //0 . Simplify:

More information

Mark Scheme (Results) January 2007

Mark Scheme (Results) January 2007 Mark Scheme (Results) January 007 GCE GCE Mathematics Core Mathematics C (666) Edecel Limited. Registered in England and Wales No. 4496750 Registered Office: One90 High Holborn, London WCV 7BH January

More information

MATHEMATICS Higher Grade - Paper I (Non~calculator)

MATHEMATICS Higher Grade - Paper I (Non~calculator) Prelim Eamination 006 / 007 (Assessing Units & ) MATHEMATICS Higher Grade - Paper I (Non~calculator) Time allowed - hour 0 minutes Read Carefully. Calculators may not be used in this paper.. Full credit

More information