rr + oT, = 8 term | + Two = 100% term; etc. «Ta = the nth term (the general term) TF you know what Tn is, you can Find any term in a pattern (a sequence). Add (or subtract) the same number each time. Eg. -5; 0; 5; 10;.. or: 5; 2 -k -h =—""” er we’ +5 +5 45 -3 -3 -3 Multiply by the same number each time. Eg. 2, 4; & 16 vi #8 |e; or: 2% 9, 3 de ol X =P Xie X wi ol Powered by (3 CamScannerener eral term (Tn) ji a a stant dif ference ] 10, IS; Constant difference = +3 | 5 4 16; | a Think of squares, cubes or other exponents. How can you use these to write | the original sequence? E.g.: (i Te Ts Te Tn lates 9) 16 You can write each term in the r Bg Be ye pattern (I; 45 9% .) aS @ square, Now find the connection between the number he term and the sequence (V';2° 3°; 4: Tigi 123 Powered by CamScannerAlgebraic expressions and Substitution NN 5 2 2x° 5x? + SEY + ~2Y 12+ yay | Variable Constant: Coefficient: Exponent: aletter of the alphabet, & xory (x or y can be anything, Its value can change.) a number that stands by itself, eg. 5 or ~12 (5 will always be 5. Its value cannot change.) the number (and its sign) which you find in Front of a variable (eg, -x?= (x? »> the coefficient of x? in -x? is -1) The exponent of - x? is 2. Powered by CamScannerYr poor + SKY? +2 og. a+ yrs Like terms: Terms with the same type of variables, eg. Sex? and 2 ( (there is an x? in both) Remember: xy? = yr ary? and yx are therefore like terms too. The order of the letters does not matter. Unlike terms: Terms with different types of variables, eg, Say? and = 2y Remember: 2%07y, 3xy’, - 5% and 2x? are all unlike terms +00. Remember: + You can add or subtract like terms only. ” * Terms are separated by a + oF =. ~ Powered by CamScannerFactorisation « Factorise » write an expression as something x some thing (eg. bracket x bracket) + You can check your answer: Multiply the bracket(s) > you should get the Original tern once again, Type Example /Common factor 2xy + 4x0? | = 2xc(y + 2x0) Difference of xc? - 25y" | | squares (2 terms) = (x - sy)(x + 5y) Trinomial (4 or 5 terms) x? +5x-6 (3 terms) = (x +6) -1) | a —¢ Grouping 2x + 2y-x2-xy ' =an+y)-xlx+y) | j = (x + y(2-x) Z a Powered by CamScanner(a) Gommon factor Always look for a common Factor first, 4axs +2xty = 2x%(2x0 + y) 1, Look for the greatest number that can divide into all terms: 2 2. Look for the letter(s) that appear in all Lerms: x Choose the smallest exponent of this letter (because this power can divide into all terms): x2 = common factor: 2x? 3. What, should you write in the bracket? By what should you multiply 2x? to get the original terms once again? > 20%X2x + y) or go? 2ety) (s+ Be 2a aE st ial Powered by CamScanner4 pif ference of squares (2 terms) 7 4x? -| / \ [| yx? Vl [| = (2x - 1)(ax +1) | Is there a common factor? No, 2, Number of terms: 2 | Ts this a difference (-) of squares? Check: a) Are both terms perfect squares? Yes. b) Is there a minus (a difference)? Yes. : Difference (-) of squares » This becomes!2 brackets: one with a - and one with a + Write - and + in the brackets straight away. Then fill in the rest. i Powered by CamScannerPr +6 Standard form: ay 5 exponent, x2 rm (nt able +3) y Ne constant , mitSE'+6 term Js there @ common Factor? No. t "Number of terms: 3 . yas this a trinomial in standard Form (2+! + number)? Yes. 5) Find the Factors: » Write x's in left column (factors of x). . Write factors of +6 in right column (+6 = factors: same signs > both + or both -). « Cross multiply and write answers below. + Add these janswers. Which combination gives middle term (+52)? = -2 y x -3 fae -3x Powered by CamScanner| nN & oa ping (q or 5 terms) : ax - ay +2n-2y \ = aay) + 2224) =(x-y)la+2) _ Is there a common factor? No, not in all terms. , Number of terms: 4 Is there a common factor in 2 of the terms? Group. Form 2 groups so that you can take out a common Factor from each group. a) ax -ay = common factor: a b) jae 2y = common factor: 2 - Take out common factor from each group: = a(x -y) + lx -y) . Now take out the new common Factor (x y) = alse = y) + lary) —— Everything that is left = (a= ya 2) goes into this bracket. Powered by (3 CamScannerFactorisation ee 320y - lay . = Syl? 4) = 3y(x - 2)(x +2) | Is there a common factor? Yes, = common factor: 34 1, Look at the bracket (x*-4). Can you factorise this further? 3, Number of terms in bracket: 2 « both terms in bracket ore perfect, squares « minus (- ) «. This is a difference of squares. + syle? 4) = syle 2)ee +2) Powered by (9 CamScannerAlgebraic fractions 2-4 KP +2e-3 Ww —_—a —_—_ Tr jerg ax-ma 3 « first step: factorise (common factor, difference of Sum trinomial, grouping) + see what can cancel out then simplify X-1. 6-2K [ez th 3 + make denominators the same (LCM) . then simplify 178 Powered by (9 CamScannert | | Equations with x: 5X-6+X=10-2x0 sn-6 +x =10-20 srtxt2e =l0+6 gx =16 x =2 Equations with x2: |. x's to one side 2. Numbers to other side x?-3x=4 x?-3x =4 I. x2-3x-4 =° . (x-4)la+i) =° x-4=0 or xt+l=0 x=4 or xe -l Everything to one side (get 0 on the other side) Factorise Remember: o x anything = 0 «either (x - 4) must be 0 = (o)(x+)=0 or (xc +1) must be 0 > (x-4)lo)=0 Powered by (9 CamScanneraphs (straight lines) Gr Gon of a straight line y =mx+c Standard form How steep the line is (the inclination) - gradient ‘ey-inbercept Where the line cuts (intersects) the y-axis vt J, NB! yis always on top This is the same as: Ax change in x You need 2 points on a graph to work out the gradient (m). vot Powered by CamScannerie Description ‘Gradient, (m) | Sketch mis 8 + Line lopes: Up (deg) ' : From left {0 right “m>o * « If L walk on the (positive) ‘ graph from left to . d eg yea nig, wo Ae nctencest . Graph is iscend y increases too. (increasing). mis — « Line slopes down (Fall) from right, emo + If Twolk on the | (negative) graph From et to | eg. y= - x4! right, I wolk downhill |” « Graph is descending If x increases, | y decreases. (decreasing). “ m=0 +—+— + Line runs horizontally, / mis | | a 24 Powered by CamScannerdycintercept _ To find the x-intercept: > let y=0 and find x | To find the y- intercept: > let x=0 and find y Horizontal lines Equation: eg. oie -3 « Cuts y-axis + Parallel to x - axis + Gradient = 0 Tip: Lf a line intersects (cuts) the y-axis only > then the equation contains a y only. 203 Powered by CamScannerEquation: og be? + Parallel to y-axis ; . Gradient is undefined Tip: If a line intersects (cuts) the x - axis op| = then the equation contains an x only Parallel (| Fis parallel to g: . » £4 “Fil ke i ° m, =m, (=2) 204 Powered by [@ CamScanner, pendicular (1) lines y cis perpendicular to h: tah m, X m,= -I | | a2x-9 £ . gradient, of F x gradient of h = -1 Collinear points . A,B and C are collinear, + This means you can draw a straight line through all 3 points. A * Mag = Mac = Mac + The gradients of AB, BC and AC are all the same, A, Band C all lie on the same straight line. (‘cot = together; "linear" = on a line) Powered by CamScannergeomet oO gles and lines f straight lines | Tria co Statement Redan A=B=C=6o" | (AARC is equilateral) Cc B an A B=C (Zs Opposite equal sides) OR: - (equal sides; equal zs) 8 A AB=AC {sides opposite equal zs OR: , c (equal 2s, equal sides) J~\ A+B+C=1g0° | (interior Zs of A) 8 c A C,=A+B (exterior £ of A)

you also have +0 say which lines are |I, + Alt. Zs and corresp. 2s are only = when the lines are |I. + Co-int. Zs are only supplementary (= 180°) when the lines are ||. IMPORTANT: |. First state what = what, Write down an equation. 2. Then write down your reason, How do you know these are =? = There is often more than | method, Your reason just needs to match nag Whatever you are stating, Powered by CamScannerf 2D objects Gomer) ol spties oF quadrilaterals | gE re slog” - « both pairs oF opp, sides = | — © both pairs of Opp. sides || ¢ both pairs of Opp. Ls = ° diagonals bisect each other both pairs of Opp. sides 2 « both pairs of opp. sides || + both pairs of Opp. Zs = ° diagonals bisect each other “AND: \s diagonals = |e each interior 2 = 90° Rhombus e both pairs of opp. sides = e both pairs of opp. sides ll ¢ both pairs of opp. 2s = ° diagonals bisect each other AND: | e all 4 sides = . diagonals intersect L | . diagonals bisect ds | Powered by (9 CamScanner ! |Oquare Trapezium + both pairs of “OPP. sid | « both pairs of opp + both pairs of opp. 1, . diagonals bisect each AND: . diagonals = + each interior £ = oe AND: e all 4 sides = . diagonals intersect, | diagonals bisec bisect zs si te, ; | pair of opp. sides les || . diagonals intersect L * both pairs of adjacent sides = * | pair of opp. ds = | + longer diagonal bisects | shorter diag onal + longer dogo bisects 48 Powered by CamScannerem OT ye 4cm % is the hypotenuse: A Re Vy y? (Pyth) is0m x is not the hypobenuse 17? = x? 415? (Pyth) 47cm x? =IF°-15? (Pyth) Ina right - angled A: (gest side)? = (short side)? + (other short side)! In an obtuse - angled A: longest side)? > (short side)? + (other short side)? | In an acute - angled A: Inst ide)? < (short side)? + (other short, side) Remember: — of ten more than | me netho J. Your reasons just need +o match your s statements _—_—— Powered by CamScannerCongruence and similarity « side, side, side (SSS) side, angle, side (SAS or SZS) + angle, angle, side (AAS or ZZS) « 90°, hypotenuse, side (ao"HS or RHS) « angle, angle, angle (AAA or zzz) e sides in proportion ce Powered by CononE (g55)— «iB c | D el The order of the letters is important. A and F both lie opposite the iside with jhash mark (BC and DE, which are equal). |, . Aand F are corresponding Ls ("buddies") as they lie opposite corresponding sides. » Therefore you write A and F in the same position in AABC= AFDE. In the same way: | 8 and B both lie opp. side with 2 hash marks. (ond E both lie opp. side with 3 hash marks, You could also write: ABAC=ADFE, for example. Order of + A's letters doesn't matter, as long as Corresponding letters are in corresponding positions Powered by CamScannerene” 4 x side a es we (= side x side) E- Area = 5 x x diagonal x diagonal 2 Rage “perimeter = 2 + 2b =2(£+b) Ld Areas & x b Perimeter = side + side + side Area = sbase x Lh — 7S Powered by (3 CamScannerCircumference = 27 Area = tr? Tip: Both formulae have a 2 somewhere, Deena eee Circumference: 211r Area: tr? SU EUUDGET AEDST UST SEPSUNE IV, SSS Ipemree ee Measure perimeter in cm, Measure area in cm? j m, etc. (not squared) m?, ete. Just rin formula (not |r? in Formula squared) © Pavallelogram Perimeter = 24 + 2b Area=base x Lh Perimeter =all the sides Area = tb) x ‘Lh 296 Powered by 9 CamScannerperimeter =4 x side A / —h jyea= BASE X ih _#——. | fs yy Area = 5 x diagonal x diagonal 2 Perimeter = (2 x short side) + (2 x long side) fre = , x diagonal x diagonal 2 Kite with 4 equal sides . rhombus Square, rhombus and kite: Area = 5 x diagonal | x diagonal 2 297 Powered by 9 CamScannerace area and volume su Rectangular prism Surface areq Surface area =2 rectangles + 2 rectongles + 2 rectangles = 2(€ x h) + alb x h) + (£ xb) = 2(th) + albh) + 2(£b) Volume | Volume = area of base x h =t{xbxh = tbh 321 Powered by CamScannerCube (a special rectangular prism) Surface area =6 squares 2 =6 x side side Volume =area of base x h 4) = side x side x side = side" side 322 4 i , — Powered by 9 CamScannerTriangular prism pds + rectangle + rectangle + rec tangle aeave A;bxh) + + (€xb) + (xb) + (£xb) 'ealsbh) - th + th +. b Volume = area of base x H =area A xH H = tbxhxH (hed height of A; H = height of prism) 323 Powered by 9 CamScannerCylinder Surface area Think of the label around a tin of sweetcorn (tin = cylinder). If you remove the label and lay it down flat, it looks like this: Label is a rectangle when you remove it, - TSA = circle on top + circle below + rectangle (total surface area) Area of rectangle = length x breadth + Length of rectangle = circumf, (2mn) of circle (as the label Folds around the tin) « Breadth of rectangle = height (h) of tin a8 (cylinder) — Powered by CamScannerr tA-O+O+[ ] eur’ + tr? + “xb! eur? + Tr? + 2tpxh | = 2? +2Trh Volume Jolume = area of base x h =r xh h =arh NB! When you calculate volume: . Triangular prism > base is always a A * Cylinder = base is always circle Note: TSA = total surface area os Powered by CamScanner« Shift up or down > y + Or - eg, (2, - 3) shifts 2 up 2 y- coordinate + 2 = (2, -3+2) = (2-0) Rule: (x; y) > (x; y+2) Tf a point shifts 2 units down = Rule: x. y)> © ShiFt right or left > x: + on - eg. (2; -3) shifts 2 leFt = x-coordinate - 2 > (2-2, -3) = (0; -3) | Rule: (x; y) + (2-2, y) | TF a point Shifts 2 units £0 the night | = Rule: (x, y)>(ee2, y) | Powered by (3 CamScannerProbability Definitions: PIA) probability / = the chance that A will happen => P(A)= a | |= 0«< PIA) < nA) n(S) Outcome “something newer happen ‘the mie of times something \(event A) can happen (the paper of outcomes) sample space > all possible outcomes (everything that could possibly _| happen) the total umber ee ee in the sample space | Randomly bina (without | looking) Powered by CamScanner