Edexcel A Level Maths – Pure A+ Graded 100% Verified

Edexcel A Level Maths – Pure A+ Graded 100% Verified Natural Numbers ANS: The set of numbers 1, 2, 3, 4, ... Also called counting numbers. Integers ANS: The set of whole numbers and their opposites Z⁺ Z⁺₀ Rational Number ANS: set of all numbers that can be written as a fraction of two integers Q irrational numbers ANS: Numbers that can't be written as a fraction... examples √2 and π Universal Set ANS: A set that includes all the objects being discussed. - Ex: U = {5, 6, 7, 8, 9, 10, 11, 12, 13, 14} Perfect Squares ANS: ax²+bx+c a & c must be squares √a x√c =b/2 then perfect square (a+b)²=a²+2ab+b² Real Numbers ANS: are the set of rational and irrational numbers How to find parameters ANS: 1) a(x+p)²+q 2) sub (-p,q) VERTEX 3) when x=0, y=c 4) rearrange to get a 5) Expand 2(x+p)² to equate to b Quadratics in disguise ANS: When one power of x is double the other E.G. x⁶+7x³-8 u=x³ u²+7u-8 solve inflection point ANS: Between TP Minimum number: Turning point - 1 (degree - 2) stationary point ANS: a point on the function where the slope (derivative) is zero degree - 1 (max) turning point ANS: A turning point on a graph is the point on the graph at which the function changes from increasing to decreasing or vice-versa. vertex degree - 1 (max) stationary inflection point ANS: degree-2 / 2 Even and odd degree ANS: even: opens on same side odd: opens on opposite side Hyperbola ANS: Exponential graph ANS: Conjecture ANS: Mathematical statement yet to be proven true Proof by deduction E.G. Prove sum of odd number are even ANS: x=2a+1 y=2b+1 x+y = 2a+2b+2 2(a+b+1) divisible by 2, therefore even proving identities ANS: Make LHS look like RHS Proof by exhaustion E.G. Prove no square number ends in 7 ANS: Look at all possible values within range that conjecture is true for 1) known theorem: all numbers ending in same digits have square numbers that end in same digit: 4²=16, 14²=196 (end in 6) 2) list: 0²=0, 1²=1, 2²=4, ... 9²=81 3) none end in 7 therefore no square number does Proof by counter example ANS: Disprove given statement cannot possibly be correct by showing an instance that contradicts a universal statement Proof by contradiction E.G Prove by contradiction that there is no largest even number ANS: 1) Assume it is not true ∴ 2n=L (largest even number) 2) Add 2 L+2 = 2n+2 =2(n+1) 3) This is even and larger than L This is a contradiction to the original assumption, since there is an even number greater than the "largest even number". Hence, the statement is true. Graphs: Axis of symmetry Vertex ANS: x= -b/2a (-b/2a, y) y=a(x+p)²+q Graph Transformations f(x)±a f(x+a) f(x-a) ANS: Up or down Move left Move right -f(x) E.G. f(x) = 1, -1, 11 ANS: Horizontal reflection E.G -f(x)= -1, 1, -11 f(-x) E.G ANS: Vertical Reflection E.G. x=-3, -x=3 therefore f(-x)= 3 subbed in 1/a f(x) ANS: squashed towards horizontal a f(x) ANS: Stretch away from horizontal f(x/a) ANS: Stretch away from vertical f(ax) ANS: Squash towards vertical 1) Multiply x term first 2) Sub into f(x) Straight lines ax+by+c E.G. y=-5/2x +4/3 y-y₁=m(∞) ANS: 1) Multiply by 2 2y=-5x+8/3 2) Multiply by 3 6y=-15x+8 3) Rearrange 6y+15x-8=0 Straight lines y-y₁/y₂-y₁ = x-x₁/x₂-x₁ E.G. (-3, 1) (3, -2) ANS: 1) Sub in y-1/-2-1 = x+3/3+3 2) Simplify and rearrange Circles Equation Area ANS: (x-a)²+(y-b)²=r² πr² Chord Segment ANS: Intersect at 2 points, do not extend beyond Intersect and extends Tangent ANS: Line from centre of a circle will meet tangent at a right angle Radius line will bisect chord Perpindicular to radius Circumcircle EG. Equation of circle with points A (6, -4) B (1, -5) C (0, 0) ANS: Triangle vertices lie on circle Each side of triangle is also a chord 1) Midpoint of AB (3.5, -4.5) 2) Gradient of perpendicular bisector (radius) y= 5x+13 3) Midpoint of BC (1/2, -5/2) 4) Equation of BC y=1/5x+13/5 5) Solve simultaneously (two lines that pass through centre) giving us centre 6) To find r, sub in any point from triangle Circumcircle AB is the diameter ANS: Right angle triangle, making AB (diameter) the hypotenuse) Therefore, distance between two points is √(x₂-x₁)²+(y₂-y₁)² Intersection points and solving equations ANS: Set equal to each other, solve Intersection points and parametrics ANS: 1) Sub parametrics into cartesian to get value for t 2)sub t value into parametrics for x and y Differentiation ANS: Multiply coefficient by power Subtract 1 from power E.G. Gradient of curve y=4x² with y coordinate 100? ANS: 1) 100=4x² 2) 25=x² 3) ±5=x 4) when x=5, gradient = 8x = 8(-5) = -40 etc. Stationary Point ANS: dy/dx = 0 What does classifying points mean? ANS: Looking at the gradient just before and after a point. Always use the lower value first. e.g. (1,-2) x = 1 Put x = 0.9 into dy/dx then put x=1.1 into dy/dx If -, - then point of inflection if -, + then minimum if +, - then maximum Arc Length ANS: Radius x θ (radians) Radians to degrees ANS: x × 180/π Degrees to Radians ANS: x × π/180 Area of sector ANS: Degrees: πr²(θ/360) Radian: 1/2 × r²θ CAST ANS: Sine differentiated d/dx sin(kx) ANS: kcos(kx) Cos differentiated d/dx cos(kx) ANS: -ksin(kx) First principles E.G. f(x)=x³-6x ANS: E.G. = lim→0 [(x+h)³-6(x+h)] - (x³-6x) = lim→0 x³+3xh²+h³-6x-6h-x³+6x = lim→0 3x²h+3xh²+h²-6h / h = lim→0 3x²+3xh+h²-6 = 3x²-6 d(e^kx)/dx ANS: ke^kx d(a^kx)/dx Proof ANS: a^kx(k ln a) 1) Let y=a^x and knowing e^ln f(x)=f(x) 2) y=e^ln(a^x) = e^x(ln a) 3)dy/dx = ln(a)×e^(ln a)x (this then = ln (a) e^x which is just a^x) 4) Simplifying to ln(a) × a^x AS REQUIRED Chain Rule dy/dx form E.G. e^5x-cos(x) ANS: dy/du × du/dx Composite functions y=e^u u=5x-cos(x) dy/du= e^u du/dx=5+sin(x) 5e^u+e^u sin(x) Chain rule y=[f(x)]^n E.G. (sin x+cos x)^5 ANS: n(f(x))ⁿ-¹ × f'(x) 5(sin x+cos x)^4 × (cos x -sin x Chain rule y=f[g(x)] E.G. e^(x^2-4x+2) ANS: f'[g(x)]×g'(x) f(x)=e^x g(x)=x^2-4x+2 f'(x)=e^x g'(x)=2x-4 e^x[g(x)] ×g'(x) e^(x^2-4x+2)×(2x-4) Product Rule E.G. x²cos(3x) ANS: u=x² v=cos(3x) u'=2x v'=-3sin(3x) -3x²sin(3x)+cos(3x)2x x(-3xsin(3x)+2cos(3x) Quotient Rule (product rule can be used instead if expression rewritten) E.G. x³/x+4 ANS: u=x³ v=x+4 u'=3x² v'=1 (3x²)(x+4)-(x³)(1)/(x+4)² SIMPLIFY Exponential functions ANS: No variable in base Crosses at y=1 (as a⁰=1) -∞<x<∞ Domain f(x)>0 As base increases, graph becomes tighter 0≤a≤1 ANS: a=0 ANS: Horizontal line across x axis a<0 ANS: Undefined Log functions E.G. log₆(x)=4 E.G. ⁴√81 ANS: 6⁴ 3 log graph y= logₙ(x) ANS: domain: x>0 range-∞≤x≤∞ asymptote: y-axis as a increases, becomes steeper 0<a<1 then reflected in x-axis a=1 undefined a≤1 undefined Exponential graph reflected in y=x Log product laws ANS: Log quotient rule ANS: logₙ(n) ANS: 1 klogₙ(x) ANS: logₙ(x^k)) Logs and quadratics E.G. 5(3²ⁿ⁺¹)-25(3ⁿ)-9=0 ANS: 1) 3²ⁿ⁺¹ into 3¹×3²ⁿ (as to add bases, you've originally multiplied) 2) Rewrite 5(3¹×3²ⁿ )-25(3ⁿ)-9=0 15(3²ⁿ)-25(3ⁿ)-9=0 3) let u =3ⁿ 4) 15u²-25u-9=0 Natural log Solving Eq E.G. 5e^(x/2)=19 ANS: 1) Divide by 5 e^(x/2)=19/5 2) take ln ln(e^(x/2))=ln(19/5) x/2=ln(19/5) Log graph E.G. y=5e^x ANS: 1) Take logs log(y)=log(5e^x) 2) Separate and rearrange log(y)=log(5)+log(e^x) log(y)=xlog(e)+log(5) translates to y=mx+c Plotting log(y) E.G. y at (7, 3) ANS: log(y)=3 y=10³ as this is base log to the power of 3 log(y) log(x) graph coordinates (3, 4) ANS: Coordinates are for the log graph x=10³ y=10⁴ log(y)=-3log(x)+log(7) in form ax^b ANS: y=7x^-3 Exponential graph axis ANS: Log(y) plotted against x Algebraic curve axis ANS: log(y) against log9x) Coefficients in exponent graph E.G. y=4e^(-5x) ANS: log(y)=-5log(e)x)+log(4) Inequalities and solution set E.G. 2x-3>20 ANS: {x:x>23/2} Inequalities and sign change E.G. 5<7 ANS: When multiplied by a negative -5>-7 What does the asymptote show ANS: limit of a term (the number it will never reach) Horizontal asymptotes occur when ANS: lim (as x→±∞), fn tends towards some number: the limit Vertical asymptotes occur when ANS: lim (as x→±a), fn tends towards ±∞ w/out limit Factorising cubics ANS: 1) factor theorem 2) polynomial division Cosine Rule ANS: a² = b² + c² - 2bcCosA 3 sides, 1 Angle Sine Rule ANS: (a ÷ SinA) = (b ÷ SinB) = (c ÷ SinC) Trigonometric identities Pythag ANS: sin²x + cos²x = 1 → sinθ=y, cosθ=x (x,y) This shows tan would tell us gradient as sin/cos=y/x E.G. 3sinθ-5cosθ=0 ANS: 1) Divide by cosθ to get tanθ 2) Solve If output is not 0, use... ANS: Pythag identity When input isn't θ E.G. cos(10x)=0.85 ANS: 10x=31.8 x=3.18 Solve sin(4y)=7cos(4y) -90≤y≤270 ANS: 1) Divide by cos(4y) = sin(4y)/cos(4y)=7 2) tan(4y)=7 3) Multiply interval by coefficient of variable -360°≤y≤1080 4) 4y= -278.1, -98.1 etc 5) y= -69.5, -24.5 etc Compound angle identity SINE COS ANS: Sin(A±B)≡sin(A)cos(B)±cos(A)sin(B) Compound Angle identity TAN ANS: Minimum, Maximum, PoI ANS: Key words telling us to differentiate Max: +ve 0 -ve Min: _ve 0 +ve PoI: ±ve 0 ±ve Simultaneous trig equations E.G. ysin(x)=38, ycos(x)=6 ANS: 1) Divide ysin(x)/ycos(x)=38/6 2) Simplify tan(x)=38/6 3) Solve for x = 1.141 rad 4) sub x into one equation to get y Equating trig equations E.G. 9/2 sin(x)-3cos(x)≡(A+4B)sin(x)-(6A+8B)cos(x) ANS: 1) equate 9/2=(A+4B) -3=-6A-8B 2) Solve simultaneously Rsin(x±a) Rcos(x±b) E.G. 6sin(x)+7cos(x)≡Rsin(x+a) ANS: a sin(x)±b cos(x) 1) Rewrite using compound angle Rsin(x)cos(a)+Rcos(x)sin(a) 2) Equate Rsin(a)=7 Rcos(a)=6 3)Find a tan(a)=7/6, a=49.4 4) Find r=√7²+6²=√85 Proof of tan(A+B) ANS: Double Angle Identity Sine Proof ANS: Sin(A+A)=sin(A)cos(A)+cos(A)sin(A) =2sin(A)cos(A) Double angle identity Cosine Proof ANS: cos(A+A) cos(A)cos(A)-sin(A)sin(A) cos²(A)-sin²(A) using pythag identity: sin²(x)=1-cos²(x) cos²(A)-sin²(A) cos²(A)-(1-cos²A) 1-2cos²A SIMILAR FOR SIN Double angle idenity Tan Proof ANS: tan(2A) tan(A+A) tan(A)+tan(A)/1-tan(A)tan(B) Half angle identity Sine ANS: Proof: cos(2x)=1-2sin²(x) cos(x)=1-2sin²(x/2) cos(x)+2sin²(x/2)=1 sin²(x/2)=1-cos(x)/2 sin(x/2)=√1-cos(x)/2 Half angle identity Cosine Proof ANS: cos(2x)=2cos²x-1 cos(x)=2cos²(x/2)-1 cos(x)+1=2cos²(x/2) cos(x)+1 /2=cos²(x/2) square root sec(x) ANS: 1/cos(x) undefined when denominator is 0 Domain:x∈R, not odd multiple of π/2 Range:y≤-1 y≥1 cosec(x) ANS: 1/sin(x) undefined when denominator is 0 Domain:x∈R, not odd multiple of π Range:y≤-1 y≥1 cot(x) ANS: 1/tan(x) undefined when denominator is 0 Domain:x∈R, not odd multiple of π Range:y∈R cosec²(x) proof ANS: 1 + cot²x 1) sin²x+cos²x=1 2) ÷sin²x 1+cot²x=1/sin²x 3) 1+cot²x=cosec²x sec²x ANS: 1+tan²x 1)sin²x+cos²x=1 2)÷cos²x sin²x/cos²x+1=1/cos²x 3)tan²x+1=sec²x Arithmetic nth Term ANS: uₙ = a + (n-1)d du+(a-d) Arithmetic Series ANS: Sₙ = n/2 [2a + (n-1)d] OR n/2(a+L) sum from term 50 to term 72 S₇₂-s₄₉ Prove sum of first n numbers of an arithmetic series is Sₙ = n/2 (2a + (n-1)d) ANS: 1) write sum normal and in reverse a +(a+d)+(a+2d)+(a+3d)+...+[a(n-2)d]+[a(n-1)d [a(n-1)d]+[a(n-2)d]+...+(a+2d)+(a+d)+a 2) Add these together 2Sₙ = n(2a + (n-1)d) 3) Divide by 2 Sₙ = n/2 [2a + (n-1)d] 3D vectors ANS: x, y, z ai, bj, ck How to know if vectors are parallel ANS: multiple of each other Distance between two points ANS: d = √(x₂ - x₁)² + (y₂ - y₁)² Vectors: Distance from origin to (x, y, z) ANS: √(x² + y² + z²) Vectors: Distance between (x₁, y₁, z₁) and (x₂, y₂, z₂) ANS: √(x₁ - x₂)² + (y₁ - y₂)² + (z₁ - z₂)² Angle between vector and axis for a vector a: xi + yj + zk ANS: x axis : cosθ = (x ÷ |a|) y axis : cosθ = (y ÷ |a|) z axis : cosθ = (z ÷ |a|) Vector: AB→ ANS: OB→ - OA→ Perpendicular vectors ANS: Flip vector around, change sign of one of them How to get original function from derivative ANS: Integrate Area of region between curve and x-axis ANS: Area between curve and straight line E.G y=2x y=x(x-5) ANS: 1) Equate to find points of intersection 2x=x²-5x x=0, 7 ∴ y=14 (7,14) 2) Find area to points on intersection (triangle) 7×14/2=49 3) Find area of disregarded piece (we minus this from total area which is 49) When curve intersects x axis, x=5 Therefore limits are 5 and 7 ∫⁷₅ x(x-5) dx = A 4) Subtract A from 49 Area between curves ANS: 1) Area under both curves 2) Substract Trapezium rule ANS: Quadrilateral At least one pair of parallel sides Two right angles Reverse chain rule ∫kf'(x)/f(x) dx