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Maths Y12 Pure
Maths Autumn Y12
| Question | Answer |
|---|---|
| Probably just skip to the last 100 flashcards of this set. | Do |
| When you factorise something | Check to see if it can be factorised further (same thing for rationalising) |
| For multiple steps with an equation equalling 0 | DO keep the =0 between steps (could do 0 = then repeat the =). For almost everything, it doesn't benefit you to pollute the zero - less simple to have a two-sided equation. |
| When a quadratic (or anything with multiple branches for solutions) has a branch that doesn't give a result | Specifically say NO SOLUTIONS for the branch or lose marks |
| When there are lots of weird indices for a quadratic | Try to make neat quadratics with indices rules |
| When the subject of a quadratic is weird (e.g. x + 3) | Can sub in y |
| When under time pressure | Don't skip steps - write faster instead |
| FOR ABSOLUTELY ALL QUESTIONS' WORKING | YOU NEED TO SHOW ALL OF YOUR WORKING NEATLY AND COHERENTLY |
| What to remember with indices rules going both ways | a^6 = a^3 x a^3 |
| Silly mistake with splitting up 6^(5/2) | 6^(5/2) does not equal 6(1/2) x 6^5 |
| How to split up 6^(5/2) | 2^(5/2) x 3^(5/2) |
| When you see terms like 6, 12, and 18 in a question | Probably a hint towards simplifying by prime factors |
| When cancelling out indices | Still write the cancelled term (e.g. 2^0), then sub in 1. |
| When dealing with multiplying indices with different roots | Split everything into its prime factors so you can have interactions between similar factors |
| When expanding brackets 3(n +1) | Check you've multiplied every term |
| (5 + 3x)/15x^2 | Split the fraction |
| What should all parabola sketches have? | Y intercept (when x = 0) X intercept (when y = 0) Shape (positive/negative a) Check to see if other lines on the graph actually intersect and positioning of turning point |
| How to factorise higher degrees: e.g. x^4 + 8(x^2) - 9 | Also works for e.g. 6 and 3. Sub in x^2 (or 3, etc) as y. Leaves you with a nice quadratic. Remember when subbing in y to keep it when first factorising - don't autopilot and write x in the brackets |
| How to factorise (x^3 + 8) | 'Sum of cubes' technique: a^3 + b^3 = (a + b)(a^2 - ab + b^2) |
| How to factorise (x^4 - 4) | Remember it can be diff of 2 squared |
| What is the discriminant of a quadratic function? | A part of the quadratic formula showing how many roots a function has: b^2 - 4ac |
| What do the different values of the discriminant of a quadratic function mean? | 0 = 1 root (i.e. Equal roots) Negative = No real roots Positive = 2 roots (i.e. Real roots) |
| e.g. Find values of k where x2 + kx + 9 = 0 has equal roots | b2 - 4ac = 0 (remember to put the =0 for equal roots) Then just do algebra to get proper working marks and for harder questions k2 - 36 = 0 k2 = 36 k = +-6 |
| e.g. 9 - 8(-k-1) | BIDMAS lol 9 + (-8(-k-1) 9 + (8k + 8) 17 + 8k |
| What are rational numbers? | Numbers which can be written as a fraction of integers (pretty much means no weird surds and i,e,pi, etc) |
| X > 3 2 > X Find the set of values of x for which both are true | No values of x |
| When multiplying both sides of an inequality by a negative number | Flip the sign |
| Find the set of values for which: 6/x > 2 x =/= 0 | Wrong: Multiplying both sides by x, because x might be negative, so multiplying by x might also flip the sign Instead multiply by x^2 to make sure it is positive Might give same answer but still lose tons of marks |
| When finding solutions to quadratic without factorising | Write quadratic formula - useful for when no nice roots |
| How to show there are no real roots | Discriminant. Should always do to show no real roots |
| If you're drawing a graph for a quadratic inequality of a different letter than x | Use the different letter, not x. The 'x' axis on the graph isn't actually an x |
| e.g. 3^(4x) * 2^(2x) = 2^1 * 3^2 | Not valid to say 4x + 2x = 2 + 1. Just not a real indices rule Instead say 3^4x = 3^2. 2^2x = 2^1. 4x = 2. 2x = 1. Both sides take the form 2^n x 3^2n, so the indices can be equated |
| e.g. Rationalise 1/(1 + ∛2) | Sum of 2 cubes. a3 + b3 = (a + b)(a2 - ab + b2). Notice how a + b is in the form of (1 + ∛2). Use that to find the other bracket to rationalise with: similar in style to diff of 2 squares - the cube part is the rational solution, not the starting point |
| How to find gradient based on angle of line from x axis | y = tanθ x + c tanθ = gradient. Gradient = dy/dx. tan = O/A. |
| How to cancel out a 2^(1/3) on the bottom of a fraction (if it'll rationalise) | x both sides by 2^(2/3) |
| Line passes through (k, k + 1) and (4, 7) and is parallel to x axis | Either normal method (rise over run = 0) or y stays the same always so k + 1 = 7 |
| When doing a show question's final working | Make sure to repeat Q at end of show. e.g. 'therefore the line is colinear'. |
| 'Points can be joined by a straight line' means what | Colinear |
| Be super careful when using rise/run | Not to do run/rise |
| When line parallel to y axis and gradient = (k - 6)/(k - 4) | Remember this is rise/run. If you know x never changes, then k 4 = 0 |
| When finding midpoints | Add the two co-ordinates before dividing by 2. Don't subtract |
| When talking about perpendicular and using variables to stand in for gradients (e.g. m1 m2) | When introducing variables, state what it stands for (gradient for line 1) State the rule (m1 x m2 = -1 if they are perpendicular) Show (2 x -1/3) = -2/3 which does not equal -1 |
| Equation of straight line | y - y1 = m(x - x1) |
| Using equation of straight line to find gradient | Plug in 2 different values of y and corresponding values of x to get gradient Can then use this for finding equation with other method |
| Using equation of straight line to find equation of specific line | Plug in a value of y1 and corresponding x1. Plug in gradient. Expand out and simplify |
| When using equation of straight line to find equation of specific line, how to expand out when gradient is a fraction | Can multiply both sides by denominator so you can easily expand numerator |
| For book working of line equation | Write coords and line equation. |
| For working with formulas | ALWAYS WRITE THE FORMULA/EQUATION OUT |
| Dotted line on inequality graph | > |
| Full line on inequality graph | >= |
| When doing discriminant working. | Write a = b = c = |
| kx^2 + 4kx + 3 = 0 When does this have no real roots? 0 < k < 3/4 is wrong Why? | Should be 0 =< k < 3/4 Using the discriminant relies on it actually being a quadratic (quad formula doesn't work with a = 0). So when k = 0, you find: when kx^2 + 4kx + 3 = y and k = 0, y = 3, which doesn't touch the x axis, so no roots |
| 3x - y - 11 = 0. Does point (2, -5) lie on the line? | Best: Can isolate y, plug in one value of x, add conclusion Can't: Plug in both with the assumption the equal sign is correct. You have to separate the =0 (and write LHS: 3x - y - 11 and RHS: 0), then calculate its value |
| For all graphs, what do you need to add? | Neatness is vital. Add y and x axis labels (because we won't be using them all the time) |
| Show line l (equation given) passes through midpoint of line m (where equation and end points are known) | Can do simultaneous equation to show intersect = midpoint: Maam approves. Quicker is to find m midpoint easily, then plug in values to equation of l (in the specific way in previous flashcard) |
| With a question asking which points lie on a line | Conclude after each point proven and about all points at the end |
| For coordinate questions (especially about finding points in relation to other points), what can you consider | Vectors |
| When writing vectors to find coordinates | Convert back to coords at the end |
| How to write 'implies' symbol | ⇒ |
| For pythagoras with integers, what are the numbers called? | Pythagorean triple |
| x - 6x(0.5) + 2 = 0 x^(2/3) - x^(1/2) - 6 = 0 | Sub in y as x^(1/2) or x^(1/3). You can't just square/cube both sides - nasty answer |
| Show how _____ is calculated | Working out and stating a rule is good, but also actually work it out with the example as simply as possible |
| How to go from an expanded form into normal circle equation | Complete square for each term separately (remember the centre is the opposite of what's with the term in brackets) |
| General advice when working out multiple parts of a question | Use info from the question rather than your own if given a choice in order to avoid ECF |
| Important way of phrasing chord circle theorem | The perpendicular bisector of a chord must pass through the centre of the circle |
| How to find equation of circle with 3 given points | Find 2 chords (connect 2 points twice). Find perpendicular bisector of both chords. Find point of intersection between them = center (because circle theorem). But check gradients betwen points first to check if right angle: 2 points make diameter - mid |
| Does point lie in/out/on circle? | Plug in point to equation. Say value of radius and center. Say that the distance from the point to the center is greater than the radius, so out of circle |
| Find values of k for which line y = 7x + k never insersects equation x2 + y2 = 2 | Plug in in terms of x. Discriminant less than 0. Draw graph if quadratic and find inequalities |
| Checking for other bounds of variable examples for if it was part of a radius | Radius must be greater than (not equal to) zero |
| How to find midrange | 0.5(Max + min) (because it's 0.5(max-min) + min) |
| What does uppercase sigma mean and how to draw it? | Sum. Draw the bottom line flat. Then draw the 2 diagonals. Then draw the top line. |
| For 'show that' questions, how should you use the answer you were given | Don't use it as working |
| When x is 2 times as large as y (this is true for huge equations too) | x = 2y |
| When proving things are kites | Mention symmetry, mentioning specific points |
| When comparing data | Give evidence data |
| Circle with radius 2 and point A and B on lines going up-right. x coord of A = 1. y coord of B = 1. Calculate angle AOB. (call point 2, 0 X) | Find AOX - AOB |
| What does it mean if a line 'touches' | Doesn't cross - tangent to |
| OAB is a 60 degree sector of 12cm radius circle. Find radius of circle center Q, which touches OA, OB and arc AB | Express in terms of r. Answer = 4 |
| A ⇔ B | Imply each other (both sides necessary and sufficient) |
| Sufficient condition | The one that the arrow is pointing from |
| Necessary condition | The one the arrow is pointing to. The one that has to be true for the sufficient to be true |
| Converse | Not necessarily true |
| e.g. x = 4 ⇒ x^2 = 16 | 16 can be 4 or -4 so can't imply |
| e.g. x = 1 or -1 (x + 1)(x -1) = 0 | Both ways arrow (if it was AND rather than or, then arrow would only face right) |
| Are 'lines' infinite | Yes. They would be line segments otherwise |
| At the end of every proof question | Just say the question again |
| Proof by contradiction | Trying (and failing) to prove the opposite. Allowed. e.g. proving root 2 is irrational. say: 'ASSUME it is rational' then prove that it falls apart |
| What to say when introducing variables | LET _____ e.g. Let 3 consecutive integers be: |
| What does rational imply? | Can be expressed as a/b, with a and b being integers in SIMPLEST FORM - *no factors in common* |
| Place following numbers in ascending order (mix of rational and irrational). Fully justify your answer | Should isolate to only one of rational/irrational. X is proportional to x^2 (when x > 0). So order a, b, c, d is the same as the order of their squares. Or could just convert into irrational |
| 4 = p + p^-1 + 3/2 | Multiply all by p. Multiplying both sides by variable can mess with things (adding a solution sometimes) so check your answers fit the info given. |
| Methods of proof | Deduction - normal Exhaustion - for some proofs/steps it's possible to test all possible cases (or at least reduce significantly) Disproving by finding counter example - only need one |
| Main variables used for algebraic proofs | Integers n and m Even number = 2n (n ∈ Z) |
| Prove a number is divisible by 9 if and only if the sum of its digits is divisible by 9 | Let a 4 digit number be abcd Number abcd = 1000a + 100b + 10c + d =999a + 99b + 9c + a + b + c + d Factorise. M of 9, so whole number divisible by 9 if and only if sum of digits multiple of 9 |
| What happens when y = x^2 becomes y = (x^2) x 3 | Stretched scale factor 3 parallel to y axis. Because it's a quadratic, it looks like the x has shrunk |
| what does a cubic y = (x-4)x^2 | Touches at the x^2 |
| x^2 and x^3 on same graph | Should cross twice. Remember x^3 less steep before 1, more steep after one (where they meet) |
| x^2 and x^4 on graph | x^4 starts slower, then gets faster after 1, where they cross |
| 1/x and 1/x^2 on same graph | The x^2 increases far quicker nearer to the y axis, but remains slightly under as x increases to infinity, so crosses at 1. The x^2 doesn't go negative y |
| f(x) = x^2 what happens from f(x-3) | Translation (3/0). Only the x is moving (because changes are inside the brackets), so must equal the same y, so does the opposite of the change. [graph is showing x for given y. Changes to x, but still same given y] |
| sin(x) = 1 sin90 = 1 When sin2x = 1, x must be | 45. Has been stretched by scale factor 0.5 parallel to x axis |
| y = fx + 4 | Works as it should: translation (0/4). If rearranged to y - 4 = fx, it's also doing the opposite. |
| When does order of translations matter | When they affect the same variable. |
| Proofs about proving something is divisible | Think of prime factorisation - divisible by 6 means divisible by both 2 and 3 e.g. Of 3 adjacent integers, one must be a multiple of 3, and similar |
| Show 3^n never has 5 as final digit | Unique prime factorisation - cannot be multiple of 5. |
| Is n(n+1)(n+2)(n+3) divisible by 4? | n(n+1) divis by 2 (n+2)(n+3) divis by 2 No overlapping brackets chosen |
| To show that different versions of equations are equal | Use double sided arrow |
| y = (x+1)(x-2) Sketch both y = fx and y = f(x/3) | Enlargement is from y axis - any given x value is tripled - not distance from line of symmetry. Turning point must also be tripled. |
| y = (2/x) + 1 | Asymptotes. Will be approaching certain values - find them |
| For an asymptote on an axis | Still draw the dotted line |
| 'Comment on reliability of these 2 estimates' | Less and more |
| Causal relationships | If a change in one variable causes a change in the other. Correlation does not imply causation - think about context |
| Very elaborate working for showing possible values of x in 0 > 2x^2 - 5x + 2 | Sketch graph, labelling the line 'y = 2x^2 - 5x + 2' Then say ' y negative when' then say the answer Feels very extra |
| Alan suspects that the variables are linked by an equation of the form . By plotting values of against values of , comment fully on Alan’s suspicions and, if appropriate, suggest values for the constants c and k. | Sub in the values of c and k and reference it in the first part (i.e. strong positive correlation when ______ is plotted) |
| Prove that an integer is divisible by 5 if and only if it is the sum of five consecutive integers. | 'If and only if' means prove both ways. If int is sum of 5 consecutive ints, multiple of 5 AND If int divisible by 5, can be written as sum of 5 consecutive ints Hence, (both statements with double arrow) |
| When in complex question about finding diameter but all the formulae use radius | Consider using r instead of d/2 - simpler Not really a new variable because its value is just a version of d |
| What is the period of a sin/cos/tan wave | How long before repeats self. Sin and cos are 360, tan is 180 |
| On the ACTS circle, explain finding secondary solutions | Theta is from positive x up. Secondary solutions certain amount of theta from x axis of corresponding ACTS sector. By weird method, make whatever is in the sin^-1 brackets +ve, then if the result is +ve vs -ve read the solutions from fitting sectors |
| Tan graphs | Asymptotes |
| Main trig identities | tan = sin/cos (can work out with unit circle) sin2 + cos2 = 1 |
| When asked to state the equation of a line | Include the 'y =' |
| When giving coordinates of points on sin/cos/tan graphs | Say the y value too (e.g. 30, 1) |
| When multiple steps of transforming sin/cos/tan graphs | Do step by step small graphs working for ease |
| When doing proof by exhaustion | Specifically expand on discounting every number that isn't valid. Mention exhaustion |
| Finding remainder and quotient | If the matrix method gives -15 when it should be -3, then the remainder is +12 (what you need to make it normal). The quotient is the other factor you're finding. |
| Factor theorem | If (x-a) is a factor of f(x), then f(a) = 0 Say f(-3) = btw Show all steps or bad - sub then simple State factor theorem for all Qs that have it |
| What does it mean if x+1 and x+3 are factors? | f(-1) = 0 and f(-3) = 0. Useful for simul equations |
| When asked to factorise cubic | Use table function to find some values, might then have to work out the rest. Say you found them 'by inspection' then show they are actually each factors with factor theorem |
| Other way of stating remainders after factorising e.g. x^2 + 7x + 12 divided by x - 1 | Remainder 20, so the original equation equals (the true sum of the factors + 20)/x-1 So that equals x+8 + 20/x-1 |
| Remainder theorem (not in spec but useful) | Any value subbed into the equation gives the remainder of f(x)/x-a |
| Write cos(-120) as cos of an acute angle | Draw sketch. Symmetrical, so cos(-120) is equal to -cos(-60) around 90 degrees, which is equal to cos(60) around 0 degrees |
| Cos(theta) = -3/5 and theta is reflex find value of sin(theta) | Ignore the minus sign and do as normal |
| What is the constant term for a polynomial (what was called c) | q |
| Find y^3 coefficient in expansion (y^3 + 2y^2 - 3y - 4)(y^2 - 2y + 4) | Finding all pairs that give y^3. For the pure y^3, it must be paired with the constant of the other equation, not just left alone |
| Quartics | Many turning points (looks like a crooked smile when positive) or just beefy quadratic |
| For making sketches how to work out what it's doing | Don't stop sketches at axis. Check what values of x and y are valid for: e.g. fractions not divided by zero so asymptote. As x/y tends to infinity/-infinity/very small what happens |
| Differentiate y = 3x^2 + 5 from first principles | For a tiny section, straight line almost. Starting point is (x, 3x^2 + 5) and change in x is h. Sub in x+h for x for both coordinates to find end point dy/dx = lim h-> 0 (new y - old y)/(new x - old x) Simplify. Since h -> 0, (then sub in 0 for h) |
| Remainder theorem better explained and usage | f(x) = q(x)(x-a) + r This means that a function is the quotient times the divisor plus any messy remainders P(a) therefore = r. Usage: if remainders of dividing given f(x) by (x-2) and (x+3) the same, f(2) = f(-3) |
| Differentiating fractions (and anything else with calculations required) | You can't simplify stuff after differentiating - have to remove fraction before differentiating |
| Circle's gradient at x is 3. What is the gradient of the tangent? | 3 The normal is -1/3 |
| Increasing function | Gradient must be greater than or equal to zero for all values of x. A 'strictly increasing' one can't be zero either. Differentiate, then using discriminant and graph shape show always positive. |
| When proving something is an increasing/decreasing function | Must always have the line f'(x) < 0 at some point |
| Point of inflection | Where rate of change of gradient is zero (gradient doesn't have to be zero, just the second derivative). Switches from concave to convex |
| f(x), f'(x), and f''(x) on same graph | f'(x) will cross z axis directly under turning points of f(x) At max point, f'(x) always decreasing, so gradient negative, represented by negative f''(x) Max gradient HAS to be at point of inflection |
| Line to always add when doing second derivative to identify type of turning point | Say 48 > 0, therefore minimum point |
| One angle and one side given of right angle triangle | Use sohcahtoa |
| When drawing graphs in exam | You still need some working for turning points etc - you can't just cheat |
| Let any 3 consecutive integers be | Must be in order |
| When n is odd, let it be represented by (2m + 1) | WHERE M IS ANY INTEGER |
| Discriminant | -11 < 0, so the equation has no ***REAL*** roots |
| For questions where second derivative = zero | MUST show the tiny amounts left and right (preferably in table) |
| x^2 = 1 | Turning points at -1 and +1 |
| Any time max or min mentioned in question | Will need to differentiate |
| One you find max side length for max volume | Then convert back into volume |
| For maximising/min questions if your area is expressed in terms of 2 variables | You HAVE to eliminate one to get area against side length (or equivalent) |
| Even if there is only one solution for a max/min question | Still MUST show it's a maximum. |
| Indefinite integrals | No numbers on the integral sign Add one to the power first, then divide the whole thing by that NEW n THEN ADD C FOR THE LOVE OF GOD DANIEL Correct notation is ∫x^3 dx = x^4/4 + c (the dx means with respects to x) |
| Integrating fractions | Same as with differentiation, namely have to take it out of the fraction (if x is the denominator can make it a negative indices) |
| 1/rootx | x^-0.5 |
| Definite integrals | Shows area between curve and the x axis between the two x values on the integral sign. Proof of result not needed for A level. |
| e.g. working for ∫x^2 + 3 dx with a 3 on top and a 2 on bottom | = [x^3/3 + 3x] then write the 3 on the top right and 2 on the bottom left. MUST have the square brackets. Don't need the c because it cancels out Then sub in the top number and subtract subbing in the bottom number |
| If definite integral gives negative | The area under the graph is the magnitude |
| How to state tending to zero effect | As k -> 0, -1/k tends to zero. Therefore... |
| Finding area between points under curve when it dips below x axis | Write out the definite integrals before and after and absolute both of them (do Mixed on graph calc) |
| Finding area under graph when it's 90 degrees turned | Works the same. Just ignore that it's a swapped y and x to do on calc |
| One way of doing an obtuse angle style trig Q | Could work out primary, secondary, etc angles. Work out which is obtuse, then cos^-1 and tan^-1 |
| For all integration | You HAVE to show the [dqwqdqda] thing, then show subbing in each |
| Vectors | Remember to underline Can be written in component form 2i + 5j or vector form Write magnitude as decimal apparently? |
| Unit vector | Parallel but with magnitude one. Symbol is the letter underlined with a hat Don't need to rationalise - needs to be a precise number so not decimal |
| Coplanar | Lie on same plane |
| Magnitude regardless of dimension | Works the exact same way |
| Find angle a = i + 3j + 2k makes with x axis | Don't 3D it. 2D it - view from z direction |
| Magnitude direction form for vectors | Always from positive x axis anticlockwise. (magnitude, angle in degrees) |
| Given that r = Ai - 2Bj + Ck and r is parallel to (2i - 4j - 3k), show that 3A + 2C = 0 | A/2 = C/-3 A/2 - C/-3 = 0 x6 |
| Decuctive series | ak = k^2 |
| Inductive series | a(k+1) = a(k) + 2 |
| Sequence meaning | 1,2,3,4 |
| Series meaning | 1 + 2 + 3 + 4 |
| What letters to describe the arithmetic series 2, 5, 8, 11, 14 | a = 2 d = 3 n = 5 |
| Length of arc easiest using | Radians |
| Remember to say 'minor sector' etc | |
| sin(x) = | sin(180 - x) and equivalent for radians true too |
| sin(theta) approximately equals what when theta is small and measured in radians | theta |
| cos(theta) approximately equals what when theta is small and measured in radians | 1 - ((theta^2)/2) MAKE SURE TO KEEP IT SQUARED |
| tan(theta) approximately equals what when theta is small and measured in radians | theta |
| How to check an approximation question when theta is small | Sub in a VERY small value (0.005 e.g.) for theta |
| When taking logs of both sides | Can do for whole sides, but make sure that you use proper rules |
| When doing those range change inequalities for angles | Remember to write in the range change Make sure to use 2 theta and then adjust |
| sin(-A) = | -sin(A) graph it |
| cos(-A) = | cosA graph it |
| Sum of sequence 2^r - 12r from 1 to 15 | Mix of geometric and arithmetic Can be split into 2^r sum minus 12r sum |
| For the tan(A+B) formula proof how to remember | Has to have 1 - etc on the bottom, so cosAcosB has to equal 1 |
| If it's sin(A-B) | Sub in B not -B |
| sin2A = | 2sinAcosA |
| cos2A = | (2cos^2A) - 1 1 - 2sin^2A cos^2A - sin^2A |
| Show 3sinx + 4cosx can be expressed in form R(sin(x+a) | Start with the final step for once and rewrite it so you can get coefficients of sinx and cosx in terms of R. Then equate coefficients. Then square both sides and add them together. R = 5 (the R value is essentially seen as a length so always +ve) |
| Minimum value of cos | -1 |
| sin2x = cos2x | tan2x = 1 not 0 |
| tan2A = | 2tanA/(1-Tan^2A) |
| How do the double angle formula proofs work | Just use the other y2 trig formulas for (A+B) but say B = A |
| Given that cosx = 0.75 180 < x < 360 Find exact value of sin2x | sinx = (-sqrt7)/4 maybe triangle it |
| Other way of writing 225 degrees in magnitude-direction form | -135 |
| Position vector | Vector between origin and a point |
| Finding vector between two position vectors | Go back to origin then to other point |
| For weird questions with many vectors affecting an object | Don't do a vector polygon - just add the components |
| Remember when finding components of forces | That the components must be negative if they go against the chosen positive direction. Check specifically |
| How to know when to take logs | Unknowns in powers |
| When you have different coefficients like (x - 4) to different logs on both sides of equation | Can expand brackets to get x coefficients alone, then factorise out x |
| e^(lnx) = | x |
| y^(logy(x)) = | x y^logy cancel out |
| Given that a and b are positive and 4(loga)^2 + (logb)^2 = 1, what is the greatest possible value of a (the logs are base 10) | Looking for max of loga, so (logb)^2 must be as small as possible. Drawing graph, can be zero with b still positive, so (logb)^2 = 0 |
| Log graph | 10^x is exponential passing through 1. logx is reflected in y = x |
| With logs | Watch out for hidden quadratics. Make sure to ln both sides. Don't do the log rules on e unless it's a log. Watch for removable solutions with negative logs. |
| Differentiating 'with respect to x' | Means differentiating the xs (like normal) not the ys |
| P = 100e^3t What is the initial rate of growth? | Differentiate before subbing in |
| Go through formula booklet and make sure you know stuff | e.g. double angle rule |
| Integrating | Remember the dx and to keep everything in brackets |
| Set notation | The stuff like ∪, ∩, A = {1,2,3} For example, the inequalities x < -5 and x > 0.5 (for quadratic), in set notation is {x: x < -5} ∪ {x: x > 0.5}. X: means in respect to x for an expression ∈ = belongs to ⊂ = subset Ø = null set etc |
| Weird N | Natural numbers |
| Weird Z | Integers |
| Weird Q | Rational number |
| Weird R | Real numbers |
| IQR | Needs a unit |
| What radians actually are | Sector with an arc of one radius length |
| How to solve tan^2 = 1/3 | Move the 1/3 to the other side then difference of 2 squares it and solve each = 0 |
| Small angle approximate solutions given how? | In decimals not surds |
| Finding arc length | L = theta/2pi radians x (2 x pi x radius) = theta x radius |
| e.g. 3cos^2 small angle approx in form a + b(theta) + c(theta^2) | MAKE SURE TO USE THE ~ EQUALS Watch out for the squared Can just turn to sin with sin2 + cos2 = 1, or can expand as cos and remove the theta^4 because its so small |
| What is a sector? | Region bound by 2 radii |
| What is a segment? | Line anywhere in circle - can be seen as the region bound between a chord and the circumference. So sector area - triangle area |
| Periodic sequence | Repeats itself at regular intervals. The number of terms in each cycle is called a period |
| Increasing sequence | n + 1 > n |
| Convergent sequence | nth term gets closer and closer to a limit. If geometric, Difference between each term decreases. Magnitude of r is less than 1 |
| Divergent sequence | Does not tend towards a limit. If geometric, Difference between each term increases. Magnitude of r is more than 1 |
| Iterative formula (aka recurrence relation) | Subscript n+1 |
| Sequence definition | Set of numbers in defined order with rule for obtaining the terms |
| Series definition | When terms of sequence added |
| Writing out terms of series | Apparently have to for a working mark Q - should use table function |
| Sequence of 8, 4, 2, 1, 0.5 | 2^3, 2^2, 2^1 nth term of 3, 2, 1, 0 is -n + 4 so answer is 2^(-n+4) |
| General nth term of an arithmetic sequence (not given) | a + (n-1)d |
| What is d? | Common difference |
| How to find S100 (sum of first 100 integers) | Add two of them together, showing them in opposite order lining up the numbers, giving 2S100 = 101 x 100, then divide |
| How to prove general sum of arithmetic sequence formula | Sn = a + (a+d) + (a+2d) +.... + (a+ (n-1)d) Do the S100 thing of flipping another one and simplifying (there are n terms). |
| Linear sequence | Forms an arithmetic series/progression when added |
| When finding first term | Check if the sequence starts at zero |
| If it says r = n+1 under the sigma | Plug in n+1 for a in the formula |
| What is a geometric sequence | One where each term is obtained by multiplying the previous term by a constant number (the common ratio) |
| General nth term of a geometric sequence (not given) | U subscript n = a(r^n-1) |
| Sum to infinity of r^n as r varies | If -1 < r < 1 then r^n -> 0 as n -> infinity, so sum to infinity of the series is S(subscript infinity symbol) = a/(1-r) If the magnitude is greater than 1, then sum to infinity cant be found |
| Common difference | Make sure to make it negative if needed |
| Study loan repayment series - can use the dog-eared Dr Williams book page to help | |
| Interest calculated monthly is charged at 11% p.a. | 11/12% each month |
Created by:
Pyrogearos2