/ f is the sum of the frequency of the data values its the same as saying the total number of data values. ii) f(x 2), labelling the coordinates of the minimum and where the curve meets the x-axis. || | | | | | || | | | | | | | | | | | | | | | | | | | | | | || | | | | | | | | | | | | | | | | | | | | | | | 13) Solve the following by factorising: a) x2 3x + 2 = 0 b) x2 + 6x + 5 = 0 c) 2x2 3x 5 = 0 || | | | || || | | | | | | | | | | | | | | | | | | | | | | | | | | | | The formula is: - b ! This is outside the interval so add 360 to get the first solution in the interval: 44.42 + 360 = 315.57 Using the sketch above, you can see that the other solution is the same distance to the right of 180 as the first solution is to the left of 360. b) cos1(0.1) = 84.26, which gives one solution. Including a quick test to check how much you remember Head Start to A-Level Maths Tantalising teasers of new A-Level concepts Its true get a first look at how to use them Straightforward notes and practice questions Yep, well give those brain cells a real workout This isnt just any old ALevel Maths book, this is a CGP ALevel Maths book! EXAMPLE A group of students were asked which of biology (B), chemistry (C) and physics (P) they like. 1 | || | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | If the first solution you find is outside the interval youre interested in, add or subtract multiples of 36 0 to find a solution inside the interva l. | || | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | Sketch the graph of y = sin x to find the other solution: NEW CONTEN T y 0.5 0 x 30 90 180 270 360 0.5 To find the other one you use the 1 symmetry of the graph. 2 people bought both types, so 2 goes in the overlap. domains and ranges in the NEW CONTEN The set of values output from a function is called the range. || | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | 5 3 Find 1 6 - 4 . E.g. revision tips gcse; Engineering; Head Start on A levels; BSc - Natural Science (Q64) - Biology; Hello, my name is ccasper! Quadratic graphs are symmetrical, so the x-coordinate of the turning point of the graph is halfway between 2 and 3. er by Check your answ brackets out multiplying the ht , youll end again if its rig started u up back where yo | | | | | | | | | | | | | | | | | | | | | | | | | | | | EXAMPLES | | | | | | | | | | || | | | | | || | | | | | | | | | | | | | | | | | | | | || NEW CONTEN T EXAMPLE Factorise 3x(x + 2) 4(x + 2). (The plural of index is indices.) But when it comes to exam revision, CGP are the undisputed champions. This is because you can rearrange the formula for frequency density to give: Frequency = Frequency Density Class Width. NEW CONTEN T The unit circle is a circle with radius 1 centred on the origin. | || | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | 2) The composite function of f(x) and g(x) is written Composite functions were covered at GCSE, but theres fg(x) which means do g first, then apply f some trickier stuff to learn about them in Year2 of 2 to your answer. 5+6 Page 55 Averages 1 a) The total of the frequencies is 30, so the median is halfway between the 15th and 16th values. Amazing course!! b) Rearrange the cosine rule into the correct form b2 + c2 - a2 to find an angle: cos A = 2bc 72 + 32 - 52 This gives cos x = 2 7 3 = 0.7857 x = cos1 (0.7857) = 38.21 = 38.2 (1 d.p.) Find the probability that I roll: a) 1 or 2 b) higher than 2. You should factorise to solve the quadratic at this step if possible you can use the quadratic formula if not. There are three trials and you can define two results for each trial even number (E), 1 and odd number (O), where each result has a probability of 2 . Youll need to be really confident with quadratics for A-level they crop up everywhere, including in later work on polynomials and graphs of functions. 17 Factorising Quadratics Factorising a Quadratic is Trickier when a 1 When a 1, the basic method is still the same as before, but its just a bit more fiddly. b) Find fg(x). | | | | | | | || | | | | | | || || | || | | | | | | | | | | | | | | | | | | | | | | | 53 Histograms Histograms Show Frequency Density | | | | | | | | | | | | || Youll have seen histograms at GCSE but they rear their ugly heads again at A-Level. The roots of f(x) are x = 2 and x = 4. The set of real numbers is represented by the symbol R. There are also numbers called imaginary numbers. So the line crosses the x-axis at the point (1.5, 0). I~lg :)\Db6{c&8JZ]5fiIz[~[2L At the end of each branch, draw a set for the second Friday. || | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | EXAMPLES This table shows information about the widths of cardboard boxes. a) 5x + y > 17 b) 3 y 4x + 2 c) x 8 < 2y 2) Shade the region that satisfies the following inequalities: x < 8 x + y > 2 x4 x2 4 and 3x + 1 y. 2 | | | | | | | || y > x 4 0 > 4 which is true. If the signs are opposites, then add. from us, online or by phone: Unbeatable books and practice papers for A-Level Maths and Further Maths revision essentials for AQA, Edexcel, OCR and OCR MEI www.cgpbooks.co.uk 0800 1712 712 cgpbooks.co.uk Or youll find our range in any good bookshop, including: 0521 - 22597 Head Start to A-Level Maths Looking to start your A-Level Maths studies early? 2 Using the sketch, you can see that there is another solution at: 360 45 = 315 There are also solutions at 45 and 315 because the graph is symmetrical about the y-axis. Youll need to try each number in 2 positions because the brackets are different. 3+2 5 Although this looks harder, just do the same thing. Vectors are written as a lowercase bold letter (a) or a lower case underlined letter (a). Turning off the personalised advertising setting wont stop you from seeing Etsy ads or impact Etsy's own personalisation technologies, but it may make the ads you see less relevant or more repetitive. TICE PRAC ONS TI QUES 1 1 4 4 b 1 l = 1 1 = 256 256 4 As an editor I know just how bad rewriting an index is 1) Simplify the following expressions: 1 2 b) (4x)x c) 3m 92 d) b a l a -3 a) a0 b 2 2) Evaluate the following powers without using a calculator: 2 1 a) 4 2 b) 27 3 1 2 3 3) Express (5 4 ) 2 (5 3 ) - 4 (5 -1) -2 1 125 3 d) b 8 l as 5k, where k is an integer. The y-intercept of f(x) is when x = 0. f(0) = (0)2 + 2(0) 8 = 8, so is at (0, 8). Rational Numbers can be Written as Fractions | | || || | | | | | || | | 1) A rational number is any number which can be written as a fraction with integers on the top and bottom. rzGy)YDE"9NzQNW}vLKqv$nfyAX_f!V"'Wr9A&P>t2;fy;DA4QBV The first thing you need to do is add a third column to the table for cumulative frequency. are x = 84.3, x = 84.3, x = 275.7 and x = 275.7. 2 The centre of this circle is at (1, 2). Alternatively, unlimited access to our complete catalogue of Maths resources is included with a Premium Membership subscription. | | | | | | | || 3x + 1 y 1 0 which is true. The upper class boundaries and lower class boundaries can be defined with inequalities to cover all possible values. a) x2 + 2x 10 = 0 b) 2x2 5x 1 = 0 d) 3x2 13x = 12 15) Solve the following by completing the square. <>/Metadata 285 0 R/ViewerPreferences 286 0 R>> | | | | | | | | || 1) Its really important that a sample is similar to the population. EXAMPLE Factorise x2 3y2. ax2 + bx + c < d), its a bit trickier to solve. /GSa 3 0 R M, N, P and Q are the midpoints of the sides. EXAMPLE Find angle x to 3 s.f. y Solutions: x = 4, y = 20 and x = 3, y = 8 Answers 70 Answers Page 33 Functions 1 g(2) = (11 2) + 13 = 35 = 5 7 21 - (7 2) 2 f(3) = (5 32) + (12 3) 7 = 45 + 36 7 = 74 3 a) f(2) = (6 2) + 4 = 12 + 4 = 16 28 b) g(4) = 4 = 7 168 + 28 l l b c) fg(x) = f b 28 4 x = 6 x + 4 = x 28 = 14 d) gf(x) = g(6x + 4) = 6x + 4 3x + 2 4 a) g(0) = (6 0) 6 = 6 fg(0) = f(6) = (4 (6)2) + 3 = 147 b) f(0) = (4 02) + 3 = 3 gf(0) = g(3) = (6 3) 6 = 12 c) f(0) = 3 f2(0) = ff(0) = f(3) = (4 32) + 3 = 39 d) g(0) = 6 g2(0) = gg(0) = g(6) = (6 6) 6 = 42 9-x 9-x 5 a) x = 9 11y y = 11 so f1(x) = 11 y-4 b) x = 2 y = 2x + 4 so g1(x) = 2x + 4 Section 5 Graphs Page 34 Straight Lines 1 Rearrange the equation of the line into y = mx + c form to find the gradient: 2x + y = 2 y = 2x 2, so the gradient is 2. You can use the origin for 3x < 6 + y and 2 + y x , but the origin is on the line y = x so use (1, 2) for y x: 3x < 6 + y 0 < 6 which is true. 3) Then replace y with f1(x). Ill be back soon! So the solution is 5 < x < 1. The opposite is the side opposite the angle youre working with (x). || | | | || | | | | | | | | | | | | | | | | | | | | | | | | Solving Equations and Rearranging Formulas x + 2 + 2x = + x 2 3 5 d) 2x(x + 1) = 2x + 18 12) Make x the subject of the following formulas: 3x + 2 a) y = mx + c b) y = 5 c) y = 2x2z + 1 3x + 1 x-2 d) y = | | | | | || Quadratic Equations Quadratics are covered in Section 3 p.16-21. Try these number pairs out in the brackets until you find a pair that gives 7x. a) (x + 4)(x 6) b) (x + 5)2 c) (2x 1)(x + 3) 8) Factorise the following: a) 5x + 20 b) 3a + 12ab c) x2 4 d) 9x2 36 e) x2 5 | | | | || | | | | | | | | | | | | | | | | | | | | | | | | | | | | This topic is covered on p.12-13. Find the values of m and n. 31) F ind the equation of the tangent to x2 + y2 = 25 at the point (3, 4). Find both times at which the temperature is 0 C. You need to Know about Two Types of Frequency Table 1) Frequency tables show how many times a data value occurs. This seller usually responds within a few hours. You can order any of our books (with next-day delivery!) t || | | | | | | AD = s + t + (u) = t s u || | | | | | | | | | | | | | | | | | | | | | | | | t The first thing to do is relabel the vectors so theyre nose to tail from A to D like this. So the origin is on the correct side of the line. factorises to the form (x + a)2), this root is the vertex. 3) Cancel the fractions down before you start multiplying if you can itll make things easier. Multiply by 3 and then 4 to cancel the denominators: 3 4 2x + 3 4 ( 4 - x ) = 3 4 7x , so 8x + 3(4 x) = 84x 4 3 2) Next, multiply out the set of brackets: 8x + 12 3x = 84x 3) Now collect the x-terms on one side of the equation and the number terms on the other: 12 = 84x 8x + 3x | || || | | | You dont need ste2p 6 because theres no x term | | | | | | | | | | | | | | | | | | | | | | | | | | 12 5) And divide by 79 to get x = 79 | | | | || | | | | | | | | | | | | || | | | | | | | | | | | | || 4) Combine like terms: 79x = 12 Here are a couple more examples both have a squared term to deal with, so youll need to use step 6 of the method above. The probabilities of all the sequences added together should always be 1. Everyday low prices and free delivery on eligible orders. If theyre all the same, they have a common denominator, so you can just add up the numerators. It's suitable for every exam board, and covers key GCSE topics you'll need with brilliant study notes, examples, practice questions and a handy 'Diagnostic Test' to check how much you remember. Categories: Mathematics - Mathematical Foundations Year: 2021 Language: english Pages: 84 File: PDF, 8.59 MB* For more school books * Visit my shop: (Click on the link)https://www.etsy.com/uk/shop/SchoolBooksCoWe aim to the 100% satisfaction of all our customers.Contact us and well be happy to help. Customers Reviews The go-to shop for any textbooks online! Bridge the gap between GCSE and A-Level CAUTION: Book exterior may be slippery when wet www.cgpbooks.co.uk. | | | | | | | | | | | Double Brackets and Squared Brackets Can Be Expanded | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | || Expand and simplify (2x 4)2. 2) If P(A) = 0.6, P(B) = 0.5 and P(A and B) = 0.3, show that events A and B are independent. Cannot recommend enough! Etsy uses cookies and similar technologies to give you a better experience, enabling things like: Detailed information can be found in Etsys Cookies & Similar Technologies Policy and our Privacy Policy. | | | | | | | | || Find a counter example to disprove the following statement: When x is prime, 2x + 1 is prime. By finding a counter example, disprove the following statement: Given that a and b are integers, if a > b, then a2 + 1 > b2 + 1. At first glance, this might look correct but take a closer look at the statement. So it can tell you where the graph crosses the x-axis (if it crosses the x-axis at all). 2 = 3 + c c = 2 + 3 = 5 So the equation of the perpendicular line is y = x + 5. And there are 15 + 10 + 20 + 15 = 60 children in total. Use features like bookmarks, note taking and highlighting while reading Head Start to A-Level Maths: bridging the gap between GCSE and A-Level (CGP A-Level Maths). To solve an equation, follow the steps below you can ignore any steps you dont need. Step 1: Recap key GCSE concepts (in case youve already forgotten!). Thank you! Theres lots more about the equations of circles on page 39. The known angle is not enclosed by the two known sides, so use the sine rule. So line Qs equation is 4 1 y = - 4 x 9 . The radius of the circle goes from (1, 2) to (3, 4), 4-2 2 so it has gradient - = 2 = 1. 788.750000 0] The first solution is 360 315.57 = 44.42 to the left of 360, so the other solution is 44.42 to the right of 180: 180 + 44.42 = 224.42 So solutions to 1 d.p. 7) If youve got x2 = , square root both sides to get x = . X Draw a diagram and fill in the known values. nno y s of an The square root l. na io at irr e 4) A number which cannot be written exactly as a fraction is irrational. 2 2 3 2 0 1 x 1 y = 1 4x 1 c) Write the inequality as an equation in the form y = : 1 y = 2 x 4. The inverse of a function f(x) is written with the notation f 1(x). Cubic graphs have a characteristic wiggly shape. Then we work our socks off to get them out to you at the cheapest possible prices. Can't complain - free kindle version of this "head start for A'level Maths" book with solutions. 1) To find x, you just need to put the values of a, b and c into the formula. 2) Rewrite each fraction with the common denominator by multiplying the top and bottom by the same thing. Substitute these values into the equation to find k: P = kd 625 = k4 k = TICE PRAC ONS TI QUES 4 625 = 5 0 d What goes up must come down unless its an exponential graph 1) Sketch the graphs of the following cubics: a) y = (x + 1)(x 1)(x + 2) b) y = (x 1)(x 2)(x 3) Section 5 Graphs c) y = x3 5x2 + 6x 39 Harder Graphs Reciprocal Graphs have Equation y = k /x or xy = k 1) Reciprocal graphs have two curves in diagonally opposite quadrants. Ships from United States. The diagram shows the graph of y = g(x). 2 0 obj a) mm 15 19 16 mm x b) Section 6 Trigonometry and Vectors cm 3 x 5c m 7 cm c) 13 cm x 12 cm m 11 c | | | || | | | | 35 | | | | | | | || 2 Y 42 miles y | | | | | | | | | | | | | | | | | | | | | || | Label the sides and angles z is the side youre trying to find. The sections begin on the following pages: 1. Some of the technologies we use are necessary for critical functions like security and site integrity, account authentication, security and privacy preferences, internal site usage and maintenance data, and to make the site work correctly for browsing and transactions. So the origin is on the wrong side of the line. Diagnostic Test Section 1 Number 6 Types of Number First things first, here are some important terms that will crop up throughout A-Level Maths. | || | | | | | | | | | | | | | | | | | | | | | | | | | | | Inequalities in the form ax + b > cx + d are known as linear inequalities. | | | | | || || | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | 13x = 26 x = 2 x = 2 in 2 gives 2x + 3y = 11.5 4 + 3y = 11.5 3y = 7.5 y = 2.5 Now check your answer by putting both these values into equation 1 . You might also see the groups called sets and the objects called elements. Seller is great with communication too. A Q D M B N 2(a b) C M 3a b | | | | Youll find these topics on p.51-53. A-Level Maths New Head Start to A-Level Maths (CGP A-Level Maths) - Link Here Bridging GCSE and A-Level Maths Student Book - Link Here Key Topics to work through before starting college here>> Find the probability that, on a given day, Martha or Sahil plays tennis. Multiply out any brackets (see page 10). EXAMPLE Make a the subject of the formula a2 + b2= (3 + a)(a b). Give your answers to 1 d.p. But there are some differences between the two: 1) cos x is symmetrical about the y-axis you can reflect it in the y-axis and itll look the same. /Title ( { D O W N L O A D } H e a d S t a r t T o A - L e v e l M a t h s e b o o k f r e e d o w n l o a d) Use the method from part a) to find and shade the region: 2 2x 4 < 3x x > 4 5 7x 2 7x 7 x 1 5 y 4 3 2 5 4 3 2 1 0 1 2 1 The set of values where the lines overlap satisfy both inequalities, so 4 < x 1. Its a handy skill thats used all the time in A-Level Maths, e.g. stream >> y + 10 < 14 x 10 < 14 which is true. 1) Scalars are quantities without a direction e.g. Find the combined mean of both sets of numbers. So y = f(x) intersects the y-axis at (0, 8). b) Use the histogram to complete the table. 2 | | | | || | | | | | | | | | | | | | y=x 4 8 Draw and shade the region satisfying the following inequalities: x2 + 3x 2 > y x2 < y + 9 3 y y = x2 9 2 Write each inequality as an equation in the form y = : y = x2 + 3x 2 y = x2 9 Then draw the graphs. Section 3 Quadratic Equations | | | | | | | | Double check that this works by multiplying out the brackets (x + 3)(x + 4) = x2 + 4x + 3x + 12 = x2 + 7x + 12 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | || Write these in the brackets, and then fill in the +/ to make 3 and 4 add to give 7: (x + 3)(x + 4) = 0. Now you just need to find where the graph crosses the axes. b) There are 4 outcomes matching roll higher than 2, and 6 outcomes in total. Super smooth process to download the pdf of the textbook which allows you to copy and paste, and Norman was really helpful in giving me an exchange when I accidently bought the wrong book. So solutions to 1 d.p. When it is negative, the graph moves to the right. Section 7 Statistics and Probability Length (l, mm) Frequency 300 l < 350 6 350 l < 410 10 410 l < 470 24 470 l < 520 14 520 l < 600 6 57 Probability Remember these Important Probability Facts Youll meet trickier probability in A-Level Maths, and build on your knowledge in new topics like probability distributions. The equation of the graph is F = mnt where t is the number of years and m and n are positive constants. EXAMPLE a) The table below shows information about the masses of 100 puppies. | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | || | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | || | | | | | | | | | | | | | | | | | 3 (1 - 2 ) 3-3 2 3 = = + + 1 2 (1 2 ) (1 2 ) 1 2 - ( 2 ) 2 3-3 2 = = 3 2-3 -1 This trick also works when theres a number in front of the root in the denominator. Atomic structure and bonding pg. Give your answer to 2 d.p. x= - b ! At A-Level, youll use sample data to carry out hypothesis tests for populations. So if a = 1 and b = 2, then 1 > 2, so a > b. 1) First, get rid of the fractions. This Guides & How Tos item by SchoolBooksCo has 2 favorites from Etsy shoppers. There are 8 data values and 8 2 = 4. Set where you live, what language you speak and the currency you use. a) Draw a histogram to represent the information. Heres the method for solving simultaneous equations if both equations are linear: || | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | || 1) Rearrange the equations into the form ax + by = c. | In step 2, you can ignore the 2) Make the coefficients of one of the variables the same by signs of the coefficients. At A-Level youll use some new notation for it. % For each of the following, state whether it could be g(x). Etsy is powered by 100% renewable electricity. 526 263 5.26 is rational because it is a terminating decimal b5.26 = 100 = 50 l. All Rational and Irrational Numbers are Real Numbers 1) 2) 3) 4) NEW CONTEN T Any rational or irrational number is a real number. l This type of translation can be described by the vector b 0 a . | | | | | | | | | | | | | | | | | | | || || | | | | | | | | Now, move the adjusting number to the RHS: (x 7) = 13 2 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | || | | | | | | | | | Set the completed square equal to zero to solve (x 7)2 13 = 0 Finally, add seven to each side to get x on its own: x = 7 + 13 and x = 7 13 TICE PRAC ONS TI QUES Complete Trafalgar Square 1 Column + 4 plinths + 510 pigeons 1) Write each of the following in the form (x + m)2 + n: a) x2 + 4x + 1 b) x2 12x + 5 c) x2 20x 10 2) Write x2 + 6x 8 in the form (x + m)2 + n. Hence solve x2 + 6x 8 = 0. d) x2 + 7x 3 Section 3 Quadratic Equations 20 Completing the Square You can Sketch the Graph Using the Completed Square If the adjusting number is 0, the equation has one root . 3) So you need to show that the vectors are scalar multiples of one another. E.g. | || | | | | | | | | | | | | | | | | | | | | | | | || | | | | | | | | | | | | | | | | | | | | | | | | || || x = 3 and y = 1 in 2 gives 32 + 12 = 10 Why did the chickens simultaneously cross the road* 1) Solve the following simultaneous equations: y2 x2 = 0 and 3x y = 3 a) y = 4x + 4 and y = x2 + 3x 8 b) 2 c) x + xy 10 = 0 and y + 2x = 7 * [EDIT dont worry, weve since had words with the Editor responsible for that bad joke] Section 4 More Algebra 30 Proof Remember These Important Facts to Help You Prove Things You came across proof at GCSE but at A-Level youll have to prove some trickier things. Complete the diagnostic test on page 1. . So you have to find the lowest common multiple of all denominators (the lowest common denominator). The turning point is at (1.5, 0.25). The only possible number pairing is 1 and 7, so try each number in both positions: (3x 1)(x 7) multiplies to give 21x and x which add/subtract to give 22x or 20x. The probabilities for the second trial change because the first muffin isnt replaced. class with the highest freq uency. Head Start to A-Level Maths (with Online Edition) Product code: MBR72 ISBN: 9781782947929 (26) Write a review In stock 5.95 Add to Basket Add to Book List Looking to hit the ground running for A-Level Maths? b) Which group contains the median? 3 a) This is a reflection in the x-axis, so the x-coordinate stays the same, but the y-coordinate is multiplied by 1. From the example on the previous page, you know these solutions are x = 7 + 13 and x = 7 13 . shoe size. c) 2x2 = x + 10 Section 3 Quadratic Equations 18 The Quadratic Formula The Quadratic Formula Gives Solutions to Quadratic Equations The solutions to any quadratic equation in the form ax2 + bx + c = 0 are given by this formula: It means you can solve quadratics that are too hard to factorise. So the interquartile range is approximately 482 405 = 77 mm. | | | | | | | || | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | || 3 2 4+3 6+2 7 8 Write the mixed numbers as improper fractions: 1 4 2 3 = 4 3 = 4 3 Now cancel down across your fractions and multiply. Draw the graph of y = 17 5x. 4) The angle a needs to be measured anticlockwise from the positive x-axis. x2 + y2 = 10 (5 2y)2 + y2= 10 3 Then rearrange 3 to make a quadratic equation in the form ax2 + bx + c = 0 and solve it. If youre clear on what they mean now, itll make A-Level life a lot easier. 2a V Section 6 Trigonometry and Vectors 50 Vectors Multiplying a Vector by a Scalar Changes its Size 1) Multiplying a vector by a positive scalar changes the vectors size, but not its direction. If x3 has a negative coefficient, then the graph goes from the top left to the bottom right. A 6 objects are not in group A or group B. EXAMPLE Find the discriminant of 4x2 + 3x + 1. 2) To show three points A, B and C are collinear, you need to show that AB and BC are parallel. There arent any square roots or fractions, so go straight to multiplying out the brackets: a2 + b2 = 3a 3b + a2 ab. You can take common factors outside a set of brackets to rewrite an expression. %PDF-1.7 2 Line P is given by y = 4x + c. Plug in point (1, 2): 2 = 4 + c c = 2, so line Ps equation is y = 4x + 2. || | | | | | | || | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | 1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | This type of translation can be described by the column vector c 0a m. TICE PRAC ONS TI QUES 6 5 4 3 2 1 x y = f(x) NEW CONTEN T I cant find that graph it mustve been lost in translation 1) The diagram on the right shows the graph of y = f(x): a) Sketch the following translations: i) f(x) + 3, labelling the coordinates of the minimum and where the curve meets the y-axis. Length (mm) Frequency ( f ) x fx 0-50 5 27.5 137.5 60-110 7 85 595 110-160 22 135 2970 160-210 16 185 2960 10 235 2350 60 9012.5 Each classs mid-point is halfway between 210-260 the upper and lower class boundaries. Number of outcomes where event happens Total number of possible outcomes 5) Another useful fact is that probability always adds up to 1. 3.0 out of 5 stars Good book but not focussed. Take full advantage of our site features by enabling JavaScript. | | | | | | | | | | | | || | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | At GCSE, you might have rearranged 1) Rearrange the linear equation to get a variable on its own. The National Assessment of Educational Progress found that children who are eligible for school lunch programs in 4th grade score on average 13 points lower on the math portion of the assessment . << 3 13 3 14 Dividing by a Fraction Flip It and Multiply 1) To divide by a fraction, just turn the fraction youre dividing by upside down, and change the divide sign to a multiply. | | | | | || 26) Point A has coordinates (5, 2) and point B has coordinates (2, 4). 3) Put those two numbers in the brackets, and then choose their signs to make it work. 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