Mathematics (MEI) Advanced Subsidiary GCE Core 1 (4751) May 2010

Size: px
Start display at page:

Download "Mathematics (MEI) Advanced Subsidiary GCE Core 1 (4751) May 2010"

Transcription

1 Link to past paper on OCR website: These solutions are for your personal use only. DO NOT photocopy or pass on to third parties. If you are a school or an organisation and would like to purchase these solutions please contact Chatterton Tuition for further details. Section A Question 1 The gradient of the line y = 3x + 1 is 3. So the gradient of the new line will also be 3. We now have the gradient and a point (4, 5) that the line goes through so we can work out the equation of the line using the formula y y 1 = m(x x 1 ) y - 5 = 3(x 4) expand the brackets y 5 = 3x 12 add 5 to both sides y = 3x Page 1

2 Question 2 i) 5 3 x a 2x3 x b 3 x 2b 4 = 125 x a 6 x b 3 x 2 x b x 2 x a 6 x b 3+4 = 250a 6 b 7 ii) a negative power has the effect of turning the fraction upside down and changing the negative power to a positive power ( )1 = 16 iii) is the same as x /2 = (16 1/2 ) 3 = ( 16) 3 = 4 3 = 64 Question 3 multiply both sides by c ac = - 5 add 5 to both sides ac + 5 = = ac + 5 square both sides y = (ac + 5) Page 2

3 Question 4 i) first expand the brackets 2 2x 6x + 5 add 2x to both sides 2 8x + 5 subtract 5 from both sides -3 8x divide both sides by 8 x rewrite the other way around x ii) no need to expand the brackets as we want to find when the inequality equals 0 it has already been factorised If the quadratic were equal to 0 then the two x values would be -4 and Now we need to decide if we want x to be between these two values or either side of them Try x = 0 (which is in between), when x = 0: we get -4 which is less than 0 so we do want x to be between the two values Alternatively if we know the way that a quadratic graph lies (like a big ) then we can also see that this curve dips below 0 when x is between the two values -4 x Page 3

4 Question 5 i) 48 = 16 x 3 = 16 x 3 = 4 3 and 27 = 9 x 3 = 9 x 3 = 3 3 so we have = 7 3 ii) we need to rationalise the denominator, so multiply the numerator and the denominator by (3-2) x = when we expand the denominator we get = = 7 expanding the numerator we get 5 2 (3 + 2) = (5 x 3 x 2) + (5 x 2 x 2) = (5 x 2) = so we have Page 4

5 Question 6 (5 + 2x 2 )(x 3 + kx + m) I have shown the two terms that give x 3 with a curved link (shown above) We have 5x 3 + 2kx 3 and we know this should equal 29x k = 29 subtract 5 from both sides 2k = 24 divide both sides by 2 k = 12 As we are dividing by (x 3) we need to input x = 3 into x 3 + kx + m and the remainder should be 59 Using the remainder theorem f(3) = 59 (3 3 ) + 3k + m = k + m = 59 subtract 27 from both sides 3k + m = 32 we already know that k = 12 (3 x 12) + m = m = 32 subtract 36 from both sides m = -4 k = 12 and m = Page 5

6 Question 7 First set out pascal s triangle. The power is 4 so we only need 5 rows (one more than the power) The final row of the triangle corresponds to the coefficients that we need to use in our expansion I find it easier to work down the page. The first term (1) increases in powers from 0 to 4 whilst the second term ( ) decreases in powers from 4 to 0. On any row the sum of the powers should be 4. 1 x (1) 0 x ( )4 + 4 x (1) 1 x ( )3 + 6 x (1) 2 x ( )2 + 4 x (1) 3 x ( )1 + 1 x (1) 4 x ( )0 now we need to tidy up each term (1 x 1 x ( )) + (4 x 1 x ( )) + (6 x 1 x ( )) +(4 x 1 x ( )) + (1 x 1 x 1) x Page 6

7 Question 8 first we need to factorise out the 5 (no need to factorise the whole thing, just the first two terms will do) 5(x 2 + 4x) + 6 now complete the square on the bracket 5((x + 2) 2 4) + 6 now multiply by the 5 again 5(x + 2) group the units 5(x + 2) 2-14 Question 9 means that one side implies the other means the left implies the right x 5 = 0 x = 5 x 2 = 25 but this doesn t work the other way if x 5 = 0, then x must be 5 and so x 2 = 5 2 = 25 x 2 = 25 x = 5 (not just +5) this does not imply that x 5 = Page 7

8 Section B Question 10 i) we need to first multiply 2 by -3 to get -6 then we need to find two numbers that multiply to give -6 but add to give -1 (the coefficient of x) these two numbers are -3 and +2 rewrite the equation splitting the middle x term into -3x and +2x 2x 2 3x + 2x factorise in pairs x(2x 3) + 1(2x 3) = 0 factorise again (2x 3)(x + 1) = 0 So (2x 3) = 0 or (x + 1) = 0 2x = 3 or x = -1 x = 1.5 or x = -1 ii) we know the values of x where the curve meets the x axis from part i) when x = 0, y = 2(0) = -3 we have the three coordinates (-1, 0), (0, -3) and (1.5, 0) the curve is a quadratic (with a leading positive x 2 )so will have the usual shape Page 8

9 iii) the discriminant tells us the number of real roots there are the discriminant is part of the quadratic formula and is equal to b 2 4ac in this case a = 1, b = -5 and c = 10 discriminant = b 2 4ac = (-5) 2 (4 x 1 x 10) = = -15 discriminant 0 so there are no real roots as this is negative there are no real roots (as in the quadratic formula we would need to square root this and we cannot square root a negative) iv) this is simultaneous equations both equations are equal to y, so put them equal to each other 2x 2 x 3 = x 2 5x + 10 subtract x 2 from both sides x 2 x 3 = -5x + 10 add 5x to both sides x 2 + 4x 3 = 10 subtract 10 to both sides x 2 + 4x - 13 = 0 we can t factorise this so we could use the quadratic formula or complete the square I shall complete the square as the numbers are fairly easy for this (x + 2) = 0 (x + 2) 2-17 = 0 add 17 to both sides (x + 2) 2 = 17 square root both sides x + 2 = 17 subtract 2 from both sides x = The question only asked for the x values not the y values so we are done Page 9

10 Question 11 i) first we can find the gradient of the line through A and B gradient = gradient AB = = = - We now have the gradient and a point (A or B) that the line goes through so we can work out the equation of the line AB using the formula y y 1 = m(x x 1 ) (I will use point A) y - 3 = - (x-- 1) multiply both sides by 3 3y 9 = -(x + 1) 3y 9 = -x 1 add x to both sides x + 3y 9 = -1 add 1 to both sides x + 3y - 8 = 0 ii) area of a triangle is ½ base x height we need to find where the line crosses the x and the y axis so we can determine the size of the base and the height of the triangle line AB meets the x axis when y = 0 x + (3 x 0) 8 = 0 x 8 = 0 add 8 to both sides x = 8 base = 8 line AB meets the y axis when x = y 8 = 0 3y 8 = 0 add 8 to both sides 3y = 8 divide both sides by 3 y = height = area of triangle = x 8 x = square units Page 10

11 iii) the gradient of the perpendicular bisector will be the negative reciprocal of the gradient of line AB gradient of bisector will be 3 midpoint of AB is (, ) = (2, 2) We now have the gradient and a point that the line goes through so we can work out the equation of the line using the formula y y 1 = m(x x 1 ) y 2 = 3(x 2) expand the brackets y -2 = 3x 6 add 2 to both sides y = 3x - 4 iv) we need to find where the line x = 3 meets the perpendicular bisector of AB substitute x =3 into y = 3x - 4 y = (3 x 3) 4 = 9 4 = 5 centre is (3, 5) length from centre to A will be the radius of the circle length 2 = (-1 3) 2 + (3 5) 2 = (-4) 2 + (-2) 2 = = 20 r = 20 equation of circle is (x 3) 2 + (y -5) 2 = ( 20) 2 = 20 (x 3) 2 + (y - 5) 2 = Page 11

12 Question 12 i) the factor theorem states that if (x a) is a factor then f(a) = 0 we need to try some simple small numbers (such as 1, 2, 3, -1, -2, -3) to see which gives f(x) = 0 try x = 1 f(1) = (6 x 1 2 ) 1 30 = -24 try x = 2 f(2) = (6 x 2 2 ) 2 30 = = 0 f(2) = 0 so that means that (x 2) is a factor now we know that (x 2) is a factor we can work out how many times this goes into the cubic equation we can do this by inspection, by algebra or by polynomial division algebraically the quadratic equation that we will get will be of the form Ax 2 + Bx + C we can easily see that A must be 1 as the leading term is x 3 (x 2)(x 2 +Bx + C) = x 3 + 6x 2 x - 30 Just looking at the x 2 terms (x 2)(x 2 +Bx + C) = x 3 + 6x 2 x 30 Bx 2 2x 2 = 6x 2 B 2 = 6 add 2 to both sides B = 8 just looking at the x terms (x 2)(x 2 +Bx + C) = x 3 + 6x 2 x Bx +Cx = -x -2B + C = -1 we know that b = C = -1 add 16 to both sides C = Page 12

13 check with the units (x 2)(x 2 +Bx + C) = x 3 + 6x 2 x C = - 30 divide both sides by -2 C = 15 A = 1, B = 8, C = 15 (x 2)(x 2 + 8x + 15) now we can factorise the quadratic Two numbers multiply to give 15 and add to give 8. The two numbers are 3 and 5. (x 2)(x + 3)(x + 5) By Polynomial division x 2 + 8x + 15 x - 2 x 3 + 6x 2 x - 30 x 3 2x 2 8x 2 - x 8x 2 16x 15x x (x 2)(x 2 + 8x + 15) now we can factorise the quadratic Two numbers multiply to give 15 and add to give 8. The two numbers are 3 and 5. (x 2)(x + 3)(x + 5) Page 13

14 ii) now we have factorised we know the three points where the graph crosses the x axis they are -5, -3 and 2 if we put x as 0 then we can also see that the graph crosses the y axis at Page 14

15 iii) this means that the graph is picked up and moved 1 unit in the positive x direction we can replace all the x values with x 1 (when we change the x values we do them in the opposite way to what we would expect hence we subtract 1 rather than adding 1) f(x - 1) = (x - 1) 3 + 6(x - 1) 2 (x - 1) 30 y = (x 1)(x 1)(x 1) + 6(x -1)(x 1) (x 1) 30 expand the brackets to get y = (x 1)(x 2 2x + 1) + 6(x 2 2x + 1) (x 1) 30 expand again y = x 3-3x 2 + 3x x 2-12x + 6 x group terms y = x 3 + 3x 2-10x - 24 If you found these solutions helpful and would like to see some more then visit our website It should be noted that Chatterton Tuition is responsible for these solutions. The solutions have not been produced nor approved by OCR. In addition these solutions may not necessarily constitute the only possible solutions Page 15

Mathematics (MEI) Advanced Subsidiary GCE Core 1 (4751) June 2010

Mathematics (MEI) Advanced Subsidiary GCE Core 1 (4751) June 2010 Link to past paper on OCR website: www.ocr.org.uk The above link takes you to OCR s website. From there you click QUALIFICATIONS, QUALIFICATIONS BY TYPE, AS/A LEVEL GCE, MATHEMATICS (MEI), VIEW ALL DOCUMENTS,

More information

Mathematics AQA Advanced Subsidiary GCE Core 1 (MPC1) January 2010

Mathematics AQA Advanced Subsidiary GCE Core 1 (MPC1) January 2010 Link to past paper on AQA website: http://store.aqa.org.uk/qual/gce/pdf/aqa-mpc1-w-qp-jan10.pdf These solutions are for your personal use only. DO NOT photocopy or pass on to third parties. If you are

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

Mathematics Edexcel Advanced Subsidiary GCE Core 1 (6663) January 2010

Mathematics Edexcel Advanced Subsidiary GCE Core 1 (6663) January 2010 Link to past paper on Edexcel website: http://www.edexcel.com/quals/gce/gce08/maths/pages/default.aspx These solutions are for your personal use only. DO NOT photocopy or pass on to third parties. If you

More information

Mathematics (MEI) Advanced Subsidiary GCE Core 2 (4752) June 2010

Mathematics (MEI) Advanced Subsidiary GCE Core 2 (4752) June 2010 Link to past paper on OCR website: www.ocr.org.uk The above link takes you to OCR s website. From there you click QUALIFICATIONS, QUALIFICATIONS BY TYPE, AS/A LEVEL GCE, MATHEMATICS (MEI), VIEW ALL DOCUMENTS,

More information

OCR Mathematics Advanced Subsidiary GCE Core 4 (4724) June 2010

OCR Mathematics Advanced Subsidiary GCE Core 4 (4724) June 2010 Link to past paper on OCR website: www.ocr.org.uk The above link takes you to OCR s website. From there you click QUALIFICATIONS, QUALIFICATIONS BY TYPE, AS/A LEVEL GCE, MATHEMATICS, VIEW ALL DOCUMENTS,

More information

Edexcel Mathematics Higher Tier, November 2010 (1380/3H) (Paper 3, non-calculator)

Edexcel Mathematics Higher Tier, November 2010 (1380/3H) (Paper 3, non-calculator) Link to examining board: www.edexcel.com This question paper is not currently available to download for free from the Edexcel website. You can purchase your own copy by phoning the Edexcel order line on

More information

OCR Mathematics Advanced Subsidiary GCE Core 4 (4724) January 2010

OCR Mathematics Advanced Subsidiary GCE Core 4 (4724) January 2010 Link to past paper on OCR website: www.ocr.org.uk The above link takes you to OCR s website. From there you click QUALIFICATIONS, QUALIFICATIONS BY TYPE, AS/A LEVEL GCE, MATHEMATICS, VIEW ALL DOCUMENTS,

More information

OCR (MEI) Mathematics Advanced Subsidiary GCE Core 2 (4752) January 2010

OCR (MEI) Mathematics Advanced Subsidiary GCE Core 2 (4752) January 2010 Link to past paper on OCR website: www.ocr.org.uk The above link takes you to OCR s website. From there you click QUALIFICATIONS, QUALIFICATIONS BY TYPE, AS/A LEVEL GCE, MATHEMATICS (MEI), VIEW ALL DOCUMENTS,

More information

Edexcel Mathematics Higher Tier, November 2009 (1380/3H) (Paper 3, non-calculator)

Edexcel Mathematics Higher Tier, November 2009 (1380/3H) (Paper 3, non-calculator) Link to examining board: http://www.edexcel.com/migrationdocuments/qp%20current%20gcse/nov09-qp/1380_3h_que_20091105.pdf As at the time of writing you can download this paper for free from the Edexcel

More information

Mathematics IGCSE Higher Tier, June /4H (Paper 4H)

Mathematics IGCSE Higher Tier, June /4H (Paper 4H) Link to examining board: http://www.edexcel.com The question paper associated with these solutions is available to download for free from the Edexcel website. The navigation around the website sometimes

More information

Mathematics IGCSE Higher Tier, June /3H (Paper 3H)

Mathematics IGCSE Higher Tier, June /3H (Paper 3H) Link to examining board: http://www.edexcel.com The question paper associated with these solutions is available to download for free from the Edexcel website. The navigation around the website sometimes

More information

Mathematics IGCSE Higher Tier, November /3H (Paper 3H)

Mathematics IGCSE Higher Tier, November /3H (Paper 3H) Link to examining board: http://www.edexcel.com The question paper associated with these solutions is available to download for free from the Edexcel website. The navigation around the website sometimes

More information

Mathematics IGCSE Higher Tier, November /4H (Paper 4H)

Mathematics IGCSE Higher Tier, November /4H (Paper 4H) Link to examining board: http://www.edexcel.com The question paper associated with these solutions is available to download for free from the Edexcel website. The navigation around the website sometimes

More information

OCR (MEI) Mathematics Advanced Subsidiary GCE Core 3 (4753) January 2010

OCR (MEI) Mathematics Advanced Subsidiary GCE Core 3 (4753) January 2010 Link to past paper on OCR website: www.ocr.org.uk The above link takes you to OCR s website. From there you click QUALIFICATIONS, QUALIFICATIONS BY TYPE, AS/A LEVEL GCE, MATHEMATICS (MEI), VIEW ALL DOCUMENTS,

More information

Mathematics Edexcel Advanced Subsidiary GCE Core 3 (6665) January 2010

Mathematics Edexcel Advanced Subsidiary GCE Core 3 (6665) January 2010 Link to past paper on Edexcel website: www.edexcel.com The above link takes you to Edexcel s website. From there you click QUALIFICATIONS, GCE from 2008, under the subject list click MATHEMATICS, click

More information

Mathematics Higher Tier, June /1H (Paper 1, non calculator)

Mathematics Higher Tier, June /1H (Paper 1, non calculator) Link to past paper on AQA website: www.aqa.org.uk The question paper associated with these worked answers is available to download for free from the AQA website. You can navigate around the website in

More information

Edexcel Mathematics Higher Tier, November 2011 (1380/3H) (Paper 3, non-calculator)

Edexcel Mathematics Higher Tier, November 2011 (1380/3H) (Paper 3, non-calculator) Link to examining board: www.edexcel.com This question paper is not yet available to download for free from the Edexcel website. You can purchase your own copy by phoning the Edexcel order line on 01623

More information

Edexcel Mathematics Higher Tier, May 2009 (1380/4H) (Paper 4, calculator)

Edexcel Mathematics Higher Tier, May 2009 (1380/4H) (Paper 4, calculator) Link to examining board: http://www.edexcel.com/migrationdocuments/qp%20current%20gcse/june%202009/1380_4h_que_20090601.pdf You will be able to download this paper for free from the website. These solutions

More information

Core 1 Module Revision Sheet J MS. 1. Basic Algebra

Core 1 Module Revision Sheet J MS. 1. Basic Algebra Core 1 Module Revision Sheet The C1 exam is 1 hour 0 minutes long and is in two sections Section A (6 marks) 8 10 short questions worth no more than 5 marks each Section B (6 marks) questions worth 12

More information

OCR (MEI) Mathematics Advanced Subsidiary GCE Core 4 (4754) January 2011

OCR (MEI) Mathematics Advanced Subsidiary GCE Core 4 (4754) January 2011 Link to past paper on OCR website: www.ocr.org.uk The above link takes you to OCR s website. From there you click QUALIFICATIONS, QUALIFICATIONS BY TYPE, AS/A LEVEL GCE, MATHEMATICS (MEI), VIEW ALL DOCUMENTS,

More information

Mathematics Higher Tier, June /2H (Paper 2, calculator)

Mathematics Higher Tier, June /2H (Paper 2, calculator) Link to past paper on AQA website: www.aqa.org.uk The associated question paper is available to download freely from the AQA website. To navigate around the website, choose QUALIFICATIONS, GCSE, MATHS,

More information

OCR Mathematics Advanced Subsidiary GCE Core 3 (4723) June 2012

OCR Mathematics Advanced Subsidiary GCE Core 3 (4723) June 2012 Link to past paper on OCR website: www.ocr.org.uk The above link takes you to OCR s website. From there you click QUALIFICATIONS, QUALIFICATIONS BY TYPE, AS/A LEVEL GCE, MATHEMATICS, VIEW ALL DOCUMENTS,

More information

Edexcel Mathematics Higher Tier, June 2011 (1380/3H) (Paper 3, non-calculator)

Edexcel Mathematics Higher Tier, June 2011 (1380/3H) (Paper 3, non-calculator) Link to examining board: www.edexcel.com This question paper is not currently available to download for free from the Edexcel website. You can purchase your own copy by phoning the Edexcel order line on

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

Mathematics Higher Tier, November /2H (Paper 2, calculator)

Mathematics Higher Tier, November /2H (Paper 2, calculator) Link to past paper on AQA website: www.aqa.org.uk This question paper is available to download freely from the AQA website. To navigate around the website, you want QUALIFICATIONS, GCSE, MATHS, MATHEMATICS,

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

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

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

Mathematics Higher Tier, November /1H (Paper 1, non calculator)

Mathematics Higher Tier, November /1H (Paper 1, non calculator) Link to past paper on AQA website: www.aqa.org.uk The question paper associated with these worked answers is available to download for free from the AQA website. You can navigate around the website in

More information

Mathematics IGCSE Higher Tier, November /3H (Paper 3H)

Mathematics IGCSE Higher Tier, November /3H (Paper 3H) Link to examining board: www.edexcel.com This question paper associated with this paper is not currently available to download for free from the Edexcel website. You can purchase your own copy of the question

More information

Mathematics Higher Tier, June /1H (Paper 1, non-calculator)

Mathematics Higher Tier, June /1H (Paper 1, non-calculator) Link to past paper on AQA website: www.aqa.org.uk The associated question paper is available to download freely from the AQA website. To navigate around the website, choose QUALIFICATIONS, GCSE, MATHS,

More information

Chapter Five Notes N P U2C5

Chapter Five Notes N P U2C5 Chapter Five Notes N P UC5 Name Period Section 5.: Linear and Quadratic Functions with Modeling In every math class you have had since algebra you have worked with equations. Most of those equations have

More information

Π xdx cos 2 x

Π xdx cos 2 x Π 5 3 xdx 5 4 6 3 8 cos x 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

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

The Not-Formula Book for C1

The Not-Formula Book for C1 Not The Not-Formula Book for C1 Everything you need to know for Core 1 that won t be in the formula book Examination Board: AQA Brief This document is intended as an aid for revision. Although it includes

More information

AS PURE MATHS REVISION NOTES

AS PURE MATHS REVISION NOTES AS PURE MATHS REVISION NOTES 1 SURDS A root such as 3 that cannot be written exactly as a fraction is IRRATIONAL An expression that involves irrational roots is in SURD FORM e.g. 2 3 3 + 2 and 3-2 are

More information

Mathematics GCSE Higher Tier Taster Pages

Mathematics GCSE Higher Tier Taster Pages Question 14 (June 2011 4306/1H) a) on the cumulative frequency diagram you can work out the lower quartile, median and upper quartile. These have been got by using the dashed red lines. Draw them across

More information

DEPARTMENT OF MATHEMATICS

DEPARTMENT OF MATHEMATICS DEPARTMENT OF MATHEMATICS AS level Mathematics Core mathematics 1 C1 2015-2016 Name: Page C1 workbook contents Indices and Surds Simultaneous equations Quadratics Inequalities Graphs Arithmetic series

More information

2009 A-level Maths Tutor All Rights Reserved

2009 A-level Maths Tutor All Rights Reserved 2 This book is under copyright to A-level Maths Tutor. However, it may be distributed freely provided it is not sold for profit. Contents indices 3 laws of logarithms 7 surds 12 inequalities 18 quadratic

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

C-1. Snezana Lawrence

C-1. Snezana Lawrence C-1 Snezana Lawrence These materials have been written by Dr. Snezana Lawrence made possible by funding from Gatsby Technical Education projects (GTEP) as part of a Gatsby Teacher Fellowship ad-hoc bursary

More information

Mathematics 1 Lecture Notes Chapter 1 Algebra Review

Mathematics 1 Lecture Notes Chapter 1 Algebra Review Mathematics 1 Lecture Notes Chapter 1 Algebra Review c Trinity College 1 A note to the students from the lecturer: This course will be moving rather quickly, and it will be in your own best interests to

More information

Algebra. Mathematics Help Sheet. The University of Sydney Business School

Algebra. Mathematics Help Sheet. The University of Sydney Business School Algebra Mathematics Help Sheet The University of Sydney Business School Introduction Terminology and Definitions Integer Constant Variable Co-efficient A whole number, as opposed to a fraction or a decimal,

More information

Higher Mathematics Course Notes

Higher Mathematics Course Notes Higher Mathematics Course Notes Equation of a Line (i) Collinearity: (ii) Gradient: If points are collinear then they lie on the same straight line. i.e. to show that A, B and C are collinear, show that

More information

MS 2001: Test 1 B Solutions

MS 2001: Test 1 B Solutions MS 2001: Test 1 B Solutions Name: Student Number: Answer all questions. Marks may be lost if necessary work is not clearly shown. Remarks by me in italics and would not be required in a test - J.P. Question

More information

S4 (4.3) Quadratic Functions.notebook February 06, 2018

S4 (4.3) Quadratic Functions.notebook February 06, 2018 Daily Practice 2.11.2017 Q1. Multiply out and simplify 3g - 5(2g + 4) Q2. Simplify Q3. Write with a rational denominator Today we will be learning about quadratic functions and their graphs. Q4. State

More information

King s Year 12 Medium Term Plan for LC1- A-Level Mathematics

King s Year 12 Medium Term Plan for LC1- A-Level Mathematics King s Year 12 Medium Term Plan for LC1- A-Level Mathematics Modules Algebra, Geometry and Calculus. Materials Text book: Mathematics for A-Level Hodder Education. needed Calculator. Progress objectives

More information

Quadratics. SPTA Mathematics Higher Notes

Quadratics. SPTA Mathematics Higher Notes H Quadratics SPTA Mathematics Higher Notes Quadratics are expressions with degree 2 and are of the form ax 2 + bx + c, where a 0. The Graph of a Quadratic is called a Parabola, and there are 2 types as

More information

Paper 1 (Edexcel Version)

Paper 1 (Edexcel Version) AS Level / Year 1 Paper 1 (Edexcel Version) Set A / Version 1 017 crashmaths Limited 1 y = 3x 4 + x x +1, x > 0 (a) ydx = 3x 3 3 3 + x 3 / x + x {+c} Attempts to integrate, correct unsimplified integration

More information

CfE Higher Mathematics Course Materials Topic 4: Polynomials and quadratics

CfE Higher Mathematics Course Materials Topic 4: Polynomials and quadratics SCHOLAR Study Guide CfE Higher Mathematics Course Materials Topic 4: Polynomials and quadratics Authored by: Margaret Ferguson Reviewed by: Jillian Hornby Previously authored by: Jane S Paterson Dorothy

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

3 Polynomial and Rational Functions

3 Polynomial and Rational Functions 3 Polynomial and Rational Functions 3.1 Polynomial Functions and their Graphs So far, we have learned how to graph polynomials of degree 0, 1, and. Degree 0 polynomial functions are things like f(x) =,

More information

Algebra 2 Segment 1 Lesson Summary Notes

Algebra 2 Segment 1 Lesson Summary Notes Algebra 2 Segment 1 Lesson Summary Notes For each lesson: Read through the LESSON SUMMARY which is located. Read and work through every page in the LESSON. Try each PRACTICE problem and write down the

More information

Math 0031, Final Exam Study Guide December 7, 2015

Math 0031, Final Exam Study Guide December 7, 2015 Math 0031, Final Exam Study Guide December 7, 2015 Chapter 1. Equations of a line: (a) Standard Form: A y + B x = C. (b) Point-slope Form: y y 0 = m (x x 0 ), where m is the slope and (x 0, y 0 ) is a

More information

Partial Fractions. June 27, In this section, we will learn to integrate another class of functions: the rational functions.

Partial Fractions. June 27, In this section, we will learn to integrate another class of functions: the rational functions. Partial Fractions June 7, 04 In this section, we will learn to integrate another class of functions: the rational functions. Definition. A rational function is a fraction of two polynomials. For example,

More information

Algebra 2 Summer Work Packet Review and Study Guide

Algebra 2 Summer Work Packet Review and Study Guide Algebra Summer Work Packet Review and Study Guide This study guide is designed to accompany the Algebra Summer Work Packet. Its purpose is to offer a review of the nine specific concepts covered in the

More information

Algebraic. techniques1

Algebraic. techniques1 techniques Algebraic An electrician, a bank worker, a plumber and so on all have tools of their trade. Without these tools, and a good working knowledge of how to use them, it would be impossible for them

More information

SOLUTIONS FOR PROBLEMS 1-30

SOLUTIONS FOR PROBLEMS 1-30 . Answer: 5 Evaluate x x + 9 for x SOLUTIONS FOR PROBLEMS - 0 When substituting x in x be sure to do the exponent before the multiplication by to get (). + 9 5 + When multiplying ( ) so that ( 7) ( ).

More information

1 Solving equations 1.1 Kick off with CAS 1. Polynomials 1. Trigonometric symmetry properties 1.4 Trigonometric equations and general solutions 1.5 Literal and simultaneous equations 1.6 Review 1.1 Kick

More information

Partial Fraction Decomposition

Partial Fraction Decomposition Partial Fraction Decomposition As algebra students we have learned how to add and subtract fractions such as the one show below, but we probably have not been taught how to break the answer back apart

More information

PARTIAL FRACTIONS AND POLYNOMIAL LONG DIVISION. The basic aim of this note is to describe how to break rational functions into pieces.

PARTIAL FRACTIONS AND POLYNOMIAL LONG DIVISION. The basic aim of this note is to describe how to break rational functions into pieces. PARTIAL FRACTIONS AND POLYNOMIAL LONG DIVISION NOAH WHITE The basic aim of this note is to describe how to break rational functions into pieces. For example 2x + 3 = + x 3 x +. The point is that we don

More information

Chapter 1- Polynomial Functions

Chapter 1- Polynomial Functions Chapter 1- Polynomial Functions Lesson Package MHF4U Chapter 1 Outline Unit Goal: By the end of this unit, you will be able to identify and describe some key features of polynomial functions, and make

More information

Answers to the problems will be posted on the school website, go to Academics tab, then select Mathematics and select Summer Packets.

Answers to the problems will be posted on the school website, go to Academics tab, then select Mathematics and select Summer Packets. Name Geometry SUMMER PACKET This packet contains Algebra I topics that you have learned before and should be familiar with coming into Geometry. We will use these concepts on a regular basis throughout

More information

Algebra of. polynomials2

Algebra of. polynomials2 polynomials2 Algebra of Polynomial functions are used to model many physical situations. One such example, the effect of gravity on falling objects, was investigated experimentally by Galileo Galilei (1564

More information

Polynomial and Synthetic Division

Polynomial and Synthetic Division Polynomial and Synthetic Division Polynomial Division Polynomial Division is very similar to long division. Example: 3x 3 5x 3x 10x 1 3 Polynomial Division 3x 1 x 3x 3 3 x 5x 3x x 6x 4 10x 10x 7 3 x 1

More information

Year 12 Maths C1-C2-S1 2016/2017

Year 12 Maths C1-C2-S1 2016/2017 Half Term 1 5 th September 12 th September 19 th September 26 th September 3 rd October 10 th October 17 th October Basic algebra and Laws of indices Factorising expressions Manipulating surds and rationalising

More information

Pure Mathematics Year 1 (AS) Unit Test 1: Algebra and Functions

Pure Mathematics Year 1 (AS) Unit Test 1: Algebra and Functions Pure Mathematics Year (AS) Unit Test : Algebra and Functions Simplify 6 4, giving your answer in the form p 8 q, where p and q are positive rational numbers. f( x) x ( k 8) x (8k ) a Find the discriminant

More information

Core Mathematics 3 Algebra

Core Mathematics 3 Algebra http://kumarmathsweeblycom/ Core Mathematics 3 Algebra Edited by K V Kumaran Core Maths 3 Algebra Page Algebra fractions C3 The specifications suggest that you should be able to do the following: Simplify

More information

To get horizontal and slant asymptotes algebraically we need to know about end behaviour for rational functions.

To get horizontal and slant asymptotes algebraically we need to know about end behaviour for rational functions. Concepts: Horizontal Asymptotes, Vertical Asymptotes, Slant (Oblique) Asymptotes, Transforming Reciprocal Function, Sketching Rational Functions, Solving Inequalities using Sign Charts. Rational Function

More information

King Fahd University of Petroleum and Minerals Prep-Year Math Program Math Term 161 Recitation (R1, R2)

King Fahd University of Petroleum and Minerals Prep-Year Math Program Math Term 161 Recitation (R1, R2) Math 001 - Term 161 Recitation (R1, R) Question 1: How many rational and irrational numbers are possible between 0 and 1? (a) 1 (b) Finite (c) 0 (d) Infinite (e) Question : A will contain how many elements

More information

Pre-Algebra 2. Unit 9. Polynomials Name Period

Pre-Algebra 2. Unit 9. Polynomials Name Period Pre-Algebra Unit 9 Polynomials Name Period 9.1A Add, Subtract, and Multiplying Polynomials (non-complex) Explain Add the following polynomials: 1) ( ) ( ) ) ( ) ( ) Subtract the following polynomials:

More information

Partial Fractions. Combining fractions over a common denominator is a familiar operation from algebra: 2 x 3 + 3

Partial Fractions. Combining fractions over a common denominator is a familiar operation from algebra: 2 x 3 + 3 Partial Fractions Combining fractions over a common denominator is a familiar operation from algebra: x 3 + 3 x + x + 3x 7 () x 3 3x + x 3 From the standpoint of integration, the left side of Equation

More information

*P43632A0120* Algebra Level 3 Calculator NOT allowed. Pearson Edexcel Award AAL30/01. P43632A 2014 Pearson Education Ltd.

*P43632A0120* Algebra Level 3 Calculator NOT allowed. Pearson Edexcel Award AAL30/01. P43632A 2014 Pearson Education Ltd. Write your name here Surname Other names Pearson Edexcel Award Algebra Level 3 Calculator NOT allowed Centre Number Candidate Number Monday 12 May 2014 Morning Time: 2 hours Paper Reference AAL30/01 You

More information

Math 1 Unit 1 EOC Review

Math 1 Unit 1 EOC Review Math 1 Unit 1 EOC Review Name: Solving Equations (including Literal Equations) - Get the variable to show what it equals to satisfy the equation or inequality - Steps (each step only where necessary):

More information

MEI STRUCTURED MATHEMATICS 4751

MEI STRUCTURED MATHEMATICS 4751 OXFORD CAMBRIDGE AND RSA EXAMINATIONS Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education MEI STRUCTURED MATHEMATICS 75 Introduction to Advanced Mathematics (C)

More information

SCIE 4101 Spring Math Review Packet #2 Notes Algebra I

SCIE 4101 Spring Math Review Packet #2 Notes Algebra I SCIE 4101 Spring 011 Math Review Packet # Notes Algebra I I consider Algebra and algebraic thought to be the heart of mathematics everything else before that is arithmetic. The first characteristic of

More information

1.4 Techniques of Integration

1.4 Techniques of Integration .4 Techniques of Integration Recall the following strategy for evaluating definite integrals, which arose from the Fundamental Theorem of Calculus (see Section.3). To calculate b a f(x) dx. Find a function

More information

Core Mathematics 2 Algebra

Core Mathematics 2 Algebra Core Mathematics 2 Algebra Edited by: K V Kumaran Email: kvkumaran@gmail.com Core Mathematics 2 Algebra 1 Algebra and functions Simple algebraic division; use of the Factor Theorem and the Remainder Theorem.

More information

CHAPTER 2 POLYNOMIALS KEY POINTS

CHAPTER 2 POLYNOMIALS KEY POINTS CHAPTER POLYNOMIALS KEY POINTS 1. Polynomials of degrees 1, and 3 are called linear, quadratic and cubic polynomials respectively.. A quadratic polynomial in x with real coefficient is of the form a x

More information

PARTIAL FRACTIONS AND POLYNOMIAL LONG DIVISION. The basic aim of this note is to describe how to break rational functions into pieces.

PARTIAL FRACTIONS AND POLYNOMIAL LONG DIVISION. The basic aim of this note is to describe how to break rational functions into pieces. PARTIAL FRACTIONS AND POLYNOMIAL LONG DIVISION NOAH WHITE The basic aim of this note is to describe how to break rational functions into pieces. For example 2x + 3 1 = 1 + 1 x 1 3 x + 1. The point is that

More information

Introduction to Advanced Mathematics (C1) WEDNESDAY 9 JANUARY 2008

Introduction to Advanced Mathematics (C1) WEDNESDAY 9 JANUARY 2008 ADVANCED SUBSIDIARY GCE 475/0 MATHEMATICS (MEI) Introduction to Advanced Mathematics (C) WEDNESDAY 9 JANUARY 008 Additional materials: Answer Booklet (8 pages) MEI Examination Formulae and Tables (MF)

More information

Math Precalculus I University of Hawai i at Mānoa Spring

Math Precalculus I University of Hawai i at Mānoa Spring Math 135 - Precalculus I University of Hawai i at Mānoa Spring - 2013 Created for Math 135, Spring 2008 by Lukasz Grabarek and Michael Joyce Send comments and corrections to lukasz@math.hawaii.edu Contents

More information

1 Quadratic Functions

1 Quadratic Functions Unit 1 Quadratic Functions Lecture Notes Introductory Algebra Page 1 of 8 1 Quadratic Functions In this unit we will learn many of the algebraic techniques used to work with the quadratic function fx)

More information

Outline schemes of work A-level Mathematics 6360

Outline schemes of work A-level Mathematics 6360 Outline schemes of work A-level Mathematics 6360 Version.0, Autumn 013 Introduction These outline schemes of work are intended to help teachers plan and implement the teaching of the AQA A-level Mathematics

More information

Review exercise 2. 1 The equation of the line is: = 5 a The gradient of l1 is 3. y y x x. So the gradient of l2 is. The equation of line l2 is: y =

Review exercise 2. 1 The equation of the line is: = 5 a The gradient of l1 is 3. y y x x. So the gradient of l2 is. The equation of line l2 is: y = Review exercise The equation of the line is: y y x x y y x x y 8 x+ 6 8 + y 8 x+ 6 y x x + y 0 y ( ) ( x 9) y+ ( x 9) y+ x 9 x y 0 a, b, c Using points A and B: y y x x y y x x y x 0 k 0 y x k ky k x a

More information

You must have: Ruler graduated in centimetres and millimetres, pair of compasses, pen, HB pencil, eraser.

You must have: Ruler graduated in centimetres and millimetres, pair of compasses, pen, HB pencil, eraser. Write your name here Surname Other names Pearson Edexcel Award Algebra Level 3 Calculator NOT allowed Centre Number Candidate Number Thursday 12 January 2017 Morning Time: 2 hours Paper Reference AAL30/01

More information

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

Maintaining Mathematical Proficiency

Maintaining Mathematical Proficiency Chapter Maintaining Mathematical Proficiency Simplify the expression. 1. 8x 9x 2. 25r 5 7r r + 3. 3 ( 3x 5) + + x. 3y ( 2y 5) + 11 5. 3( h 7) 7( 10 h) 2 2 +. 5 8x + 5x + 8x Find the volume or surface area

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

CONTENTS CHECK LIST ACCURACY FRACTIONS INDICES SURDS RATIONALISING THE DENOMINATOR SUBSTITUTION

CONTENTS CHECK LIST ACCURACY FRACTIONS INDICES SURDS RATIONALISING THE DENOMINATOR SUBSTITUTION CONTENTS CHECK LIST - - ACCURACY - 4 - FRACTIONS - 6 - INDICES - 9 - SURDS - - RATIONALISING THE DENOMINATOR - 4 - SUBSTITUTION - 5 - REMOVING BRACKETS - 7 - FACTORISING - 8 - COMMON FACTORS - 8 - DIFFERENCE

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

Coach Stones Expanded Standard Pre-Calculus Algorithm Packet Page 1 Section: P.1 Algebraic Expressions, Mathematical Models and Real Numbers

Coach Stones Expanded Standard Pre-Calculus Algorithm Packet Page 1 Section: P.1 Algebraic Expressions, Mathematical Models and Real Numbers Coach Stones Expanded Standard Pre-Calculus Algorithm Packet Page 1 Section: P.1 Algebraic Expressions, Mathematical Models and Real Numbers CLASSIFICATIONS OF NUMBERS NATURAL NUMBERS = N = {1,2,3,4,...}

More information

A Level Summer Work. Year 11 Year 12 Transition. Due: First lesson back after summer! Name:

A Level Summer Work. Year 11 Year 12 Transition. Due: First lesson back after summer! Name: A Level Summer Work Year 11 Year 12 Transition Due: First lesson back after summer! Name: This summer work is compulsory. Your maths teacher will ask to see your work (and method) in your first maths lesson,

More information

An equation is a statement that states that two expressions are equal. For example:

An equation is a statement that states that two expressions are equal. For example: Section 0.1: Linear Equations Solving linear equation in one variable: An equation is a statement that states that two expressions are equal. For example: (1) 513 (2) 16 (3) 4252 (4) 64153 To solve the

More information

SCIE 4101 Fall Math Review Packet #2 Notes Patterns and Algebra I Topics

SCIE 4101 Fall Math Review Packet #2 Notes Patterns and Algebra I Topics SCIE 4101 Fall 014 Math Review Packet # Notes Patterns and Algebra I Topics I consider Algebra and algebraic thought to be the heart of mathematics everything else before that is arithmetic. The first

More information

5.3. Polynomials and Polynomial Functions

5.3. Polynomials and Polynomial Functions 5.3 Polynomials and Polynomial Functions Polynomial Vocabulary Term a number or a product of a number and variables raised to powers Coefficient numerical factor of a term Constant term which is only a

More information

Chapter 1- Polynomial Functions

Chapter 1- Polynomial Functions Chapter 1- Polynomial Functions Lesson Package MHF4U Chapter 1 Outline Unit Goal: By the end of this unit, you will be able to identify and describe some key features of polynomial functions, and make

More information

Twitter: @Owen134866 www.mathsfreeresourcelibrary.com Prior Knowledge Check 1) Factorise each polynomial: a) x 2 6x + 5 b) x 2 16 c) 9x 2 25 2) Simplify the following algebraic fractions fully: a) x 2

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