Edexcel A-Level Pure Maths ALL SOLUTION LATEST EDITION 2023 GUARANTEED GRADE A+

area of sector (where θ is in radians) ½r²θ angles in 3D vectors cosθₓ =x/|a| cosθᵧ =y/|a| cosθ₂ =z/|a| this is where vector a makes an angle θ with the x/y/z axis (because x/y/z would be the adjacent and a the hypotenuse) Expanding (a + bx)ⁿ (a + bx)ⁿ = aⁿ(1 + (b/a)x)ⁿ cosine rule a² = b² +c² -2bc cosA rule for parametric integration ∫y dx = ∫(y dx/dt)dt as (dx/dt) x dt would cancel out to give you ∫y dx unit vector (â) a / |a| vector / magnitude of vector rule for implicit differentiation of terms with xy d/dx (xy) = x dy/dx + y (basically use the product rule then times the x-term by dy/dx) e.g. d/dx (3xy) = 3x x 1(dy/dx) + 3y = 3x(dy/dx) + 3y arc length rθ (where θ is in radians) cosθ in terms of sin cosθ = sin(90° - θ) sinθ in terms of cos sinθ = cos(90° - θ) rules for domain and range of parametrics the domain of f(x) (i.e. the Cartesian equation) is the range of the x values, given the range of t values the range of f(x) (i.e. the Cartesian equation) is the range of the y values, given the range of t values quartic graphs set notation x a or x b = { x : x a } ∪ { x : x b } c x d = { x : c x } ∩ { x : x d } formula for nth term of arithmetic sequence uₙ = a + (n-1)d formula for nth term of geometric sequence uₙ = arⁿ⁻¹ symbol for 'is a subset of' ⊆ symbol for 'the set of integers' ℤ symbol for 'the set of positive integers' ℤ⁺ symbol for 'the set of non-negative integers' (includes 0) ℤ₀⁺ symbol for 'the set of real numbers' ℝ (not imaginary) [a, b] [a, b) (a, b] (a, b) [a, b] = the closed interval {x ∈ ℝ: a ≤ x ≤ b} [a, b) = the interval {x ∈ ℝ: a ≤ x b} (a, b] = the interval {x ∈ ℝ: a x ≤ b} (a, b) = the open interval {x ∈ ℝ: a x b} symbol for 'the set of rational numbers' ℚ a₁ + a₂ + ... + aₙ a₁ x a₂ x ... x aₙ symbol for 'an increment of x' ∆ₓ , δₓ symbol for 'the first, second, ... derivatives of x with respect to t' symbol for 'position vector' r sigma notation ∑ (4n) means you sum everything in the brackets. the limits of the n-values will be written above and below the ∑ (e.g if n=1 and 6 is written, you sum the equation in the brackets using values of n from 1 to 6) validity of fractional/negative binomial expansions (where n is a negative/fraction) for (1 + x)ⁿ, the expansion is valid where |x|1 for (1 + bx)ⁿ, the expansion is valid where |bx|1 for (a + bx)ⁿ, the expansion is valid where |(b/a)x|1 finding the length of a parametrically defined curve look at the bounds of t to work out how much of the circle is being included (e.g (-π/2) ≤ t ≤ π) find the difference between the upper and lower bound of t and divide by 2π (a full circle) (e.g (3π/2) / 2π) then use this to modify the equation of the circumference (2πr) (e.g [(3π/2) / 2π] x 2πr) and use the value of r from your Cartesian equation How to test for a point of inflection the point at which the curve changes from concave to convex (or vice versa) is the point of inflection 1) it is where f''(x) = 0 2) f''(x) must have a different sign either side of that point CONTINUED..