Edexcel A Level Maths (A+ Graded 100% Verified)

Edexcel A Level Maths (A+ Graded 100% Verified) Cosine Rule - Know 2 sides, and the angle in between. You want the missing side. ANS: a² = b² + c² - 2bcCosA Cosine Rule - Know 3 sides. You want an angle. ANS: Cos A = (b² + c² - a²) ÷ 2bc Sine Rule - Know 2 angles and a side. You want the missing side. ANS: (a ÷ SinA) = (b ÷ SinB) = (c ÷ SinC) Sine Rule - Know 2 sides and an angle. You want a missing angle. ANS: (Sin A ÷ a) = (Sin B ÷ b) = (Sin C ÷ c) Midpoint of a line segment on a circle. ANS: ( (x₁ + x₂) ÷ 2 ) , ( (y₁ + y₂) ÷ 2 ) Equation of a circle with centre (0,0) ANS: x² + y² = r² Equation of a circle with centre (a,b) ANS: (x - a)² + (y - b)² = r² Arithmetic nth Term ANS: uₙ = a + (n-1)d Arithmetic Series ANS: Sₙ = n/2 (2a + (n-1)d) Geometric nth Term ANS: uₙ = ar^(n-1) Geometric Series ANS: Sₙ = a(1-rⁿ) ÷ (1-r) Equation of a straight line ANS: y - y₁ = m(x - x₁) 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→ Integration by substitution ANS: ∫f(x) = ∫f(x) × (dx÷du) Integration by parts ANS: ∫u (dv÷dx) = uv - ∫v (du÷dx) Trapezium Rule ANS: A = ½h (y₀ + 2(y₁ + y₂ ... yₙ-₁) + yₙ) How do you convert from degrees to radians? ANS: × (π÷180) How do you convert from radians to degrees? ANS: × (180÷π) Radians: Arc length ANS: rθ Radians: Sector Area ANS: ½r²θ Radians: Segment Area ANS: ½r²(θ - sinθ) Radians: Triangle Area ANS: ½r²sinθ ∫eⁿ ANS: eⁿ + c ∫1÷x ANS: ln|x| + c ∫cosx ANS: sinx + c ∫sinx ANS: -cosx + c ∫sec²x ANS: tanx + c ∫cosecxcotx ANS: -cosecx + c ∫secxtanx ANS: secx + c ∫cosec²x ANS: -cotx + c ∫tanx ANS: -ln|cosx| = ln|cosx|-¹ = ln|1÷cosx| = ln|secx| + c ∫cotx ANS: ln|sinx| + c ∫secx ANS: ln|secx + tanx| + c ∫cosecx ANS: -ln|cosecx + cotx| + c ∫sin²x ANS: ½x - ¼sin2x + c ∫cos²x ANS: ½x + ¼sin2x + c ∫sin³x ANS: -cosx + 1/3cos³x + c ∫cos³x ANS: sinx - 1/3sin³x + c Quadratic formula ANS: x = (-b ± √(b²-4ac)) ÷ 2a Discriminant rules ANS: b² - 4ac > 0 = Two distinct real roots b² - 4ac = 0 = One repeated root b² - 4ac < 0 = No real roots Chain Rule y = (f(x))ⁿ ANS: dy/dx = n(f(x))ⁿ-¹ × f'(x) Product Rule y = f(x)g(x) ANS: dy/dx = (g(x) × f'(x)) + (f(x) × g'(x)) Quotient Rule y = f(x)÷g(x) ANS: dy/dx = ((g(x) × f'(x)) - (f(x) × g'(x)) ÷ (g(x))² Implicit Differentiation ANS: f(y) --> f'(y) dy/dx yⁿ --> nyⁿ-¹ dy/dx xy --> x dy/dx + y Differential of eⁿ ANS: eⁿ Differential of lnx ANS: 1/x Differential of e^kx ANS: ke^kx Differential of sinx ANS: cosx Differential of cosx ANS: -sinx Differential of tanx ANS: sec²x Differential of cosecx ANS: -cosecxcotx Differential of secx ANS: secxtanx Differential of cotx ANS: -cosec²x Differential of arcsinx ANS: 1 ÷ (√1-x²) Differential of arccosx ANS: -1 ÷ (√1-x²) Differential of arctanx ANS: 1 ÷ (1+x²) Parametric Differentiation ANS: dy/dx = (dy/dt) ÷ (dx/dt) A function f(x) is increasing on the interval [a,b] if... ANS: f'(x) ≥ 0 for all values of x such that a < x < b A function f(x) is decreasing on the interval [a,b] if... ANS: f'(x) ≤ 0 for all values of x such that a < x < b 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 How do you know the nature of a stationary point? ANS: If d²y/dx² > 0 minimum If d²y/dx² = 0 classify points to find out If d²y/dx² < 0 maximum y = logₙx ANS: n^y = x 1:1 relationship ANS: One value of x is associated with on value of y e.g. x = y Many:1 relationship ANS: Many values of x go to the same value of y e.g. y = x² 1:Many relationship ANS: One value of x goes to many values of y e.g. y =√x Many:Many relationship ANS: Many values of x go to many values of y e.g. x² + y² = r² Function ANS: A relationship where each value of x is associated with a unique value of y. e.g. 1:1 and Many:1 relationships Modulus function: y = |f(x)| ANS: Reflection in x axis Anything below goes above Modulus function: y = f(|x|) ANS: RHS gets reflected to LHS, as soon as x becomes negative. Parametric Equation ANS: Links x and y separately Cartesian Equation ANS: Linking x and y together The function f(x) is concave on a given interval if... ANS: f''(x) ≤ 0 for every value of x in that interval The function f(x) is convex on the interval [a,b] if... ANS: f''(x) ≥ 0 for every value of x in that interval Integrating Parametric ANS: ∫y dx = ∫y (dx/dt) dt