Core 1 Mathematics 4.0 / 5 based on 2 ratings ? MathematicsCore 1 MathsASAQA Created by: crabbdanie09Created on: 07-01-16 09:30 Surds Rules of Surds √ab = √a√b √(a/b) = √a/√b a√c ± b√c = (a±b)√c Rationalising the denominator Multiply the top and bottom of the fraction by the reverse of the denominator 1 of 20 Sketching Quadratic Graphs ① UP/DOWN – Look at the coefficient of χ² - positive u shaped/negative n shaped ② AXES – Put χ=0 for the y-axis and solve Υ=0 for the x-axis (can use discriminant to see if it crosses x-axis) ③MAX/MIN – find the vertex by completing the square or finding the value of x at the line of symmetry (halfway between roots) 2 of 20 Completing the Square aχ²±bχ±c=0 ① Halve the coefficient of χ ② Square the halved value ③ Put the halved value in the (χ±b) and take the squared value away from c 3 of 20 The Quadratic Formula aχ²±bχ±c=0 4 of 20 The Discriminant b²-4ac<0 – NO ROOTS b²-4ac=0 – ONE REPEATED ROOT b²-4ac>0 – TWO DISTINCT ROOTS 5 of 20 Simulatenous Equations Elimination ① Match the coefficients ② Eliminate and then solve for one variable ③ Find the other variable ④ Check the answer works 6 of 20 Simultaneous Equations Cont. One Quadratic and One Linear ① Isolate variable in linear equation ② Substitute into quadratic equation ③ Solve to get values for one variable ④ Put values into linear equation 7 of 20 Geometric Interpretation Two Solutions – Two points of intersection One Solution – One point of intersection (line is a tangent to curve) No Solutions – The graphs never meet 8 of 20 Factorising Cubics Ƒ(χ)= χ³±χ²±χ±c ① Find one factor ② Use grid method to find a quadratic ③ Factorise quadratic 9 of 20 The Remainder Theorem Ƒ(χ)= χ³±χ²±χ±c ① Substitute (χ±a) into ƒ(χ) ② Solve for remainder 10 of 20 Coordinate Geometry Finding the equation of a line Υ-γ₁=m(χ-χ₁) m - gradient Finding the midpoint of a line ① (χ₁+χ₂) , (Υ₁+Υ₂) ② Then divide both values by 2 Distance between two points D=√(χ₂-χ₁)+(Υ₂-Υ₁) 11 of 20 Parallel & Perpendicular Lines Parallel lines have the same gradient Perpendicular lines product of gradient = -1 so m₂= -1/m₁ 12 of 20 Curve Sketching Υ=kχⁿ n positive and even – u-shape or n-shape n positive and odd – ‘corner-to-corner’ shape 13 of 20 Curve Sketching Cont. Υ=kχⁿ n negative and even – two bits next to each other n negative and odd – two bits opposite each other 14 of 20 Circles Equation of a circle (χ-a)²+(Υ-b)²=r² Centre = (a,b) Radius = r Complete the square to find the equation of a circle 15 of 20 Cicles Continued Tangents/Normals Tangents are perpendicular to the radius Normals have the same gradient Circle Properties The angle in a semicircle is a right-angle The perpendicular from the centre to a chord bisects the chord A radius and tangent to the same point will meet at right angles 16 of 20 Differentiation dy/dx=nxⁿˉ¹ Finding Tangents/Normals ① Differentiate the function ② Find gradient (Tan= Same, Norm= -1/m) ③ Υ-Υ₁=m(χ-χ₁) 17 of 20 Differentiation Cont. Stationary Points dy/dx=0 d²y/dx²<0 maximum d²y/dx²>0 minimum Increasing/Decreasing Positive gradient - increasing Negative gradient - decreasing 18 of 20 Integration ʃχⁿ dx = χⁿ⁺¹/n+1 +c Area under curve ²ʃ₁ χ³ dx = [x⁴/4]²₁ = [2⁴/4]-[1⁴/4] sometimes you have to find the limits by factorising the quadratic function 19 of 20 Integration Cont. Adding integrals Diagrams Diagrams Diagrams Subtracting integrals Diagrams Diagrams Diagrams 20 of 20
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