Core 1 Key points 1.0 / 5 MathematicsCore 1ASAQA Created by: LShan261Created on: 24-04-15 21:37 1: Algebra and functions Simplifying expressions --> Like terms & rules of indices A^m x A^n = A^m+n A^m / A^n = A^m-n A^-m = 1/A^m A^1/m = m√A A^n/m = m√A^n (A^m)^n = A^mn A^0 = 1 Expanding expressions --> Each term inside X each term outside --> Factorising = opposite Quadratic expression --> ax^2 + bx + c Difference of squares --> x^2 - y^2 = (x+y)(x-y) Surd = √prime number Manipulation surds √ab = √a x √b √a/b = √a / √b Rationalise surds 1/√a --> multiply top and bottom by √a 1/(a+√b) --> multiply top and bottom by (a-√b) --> opposite for 1/(a-√b) 1 of 8 2: Quadratic functions General form --> y=ax^2 + bx + c Solving quadratic equations factorisation Completing the square --> x^2 +bx = (x + b/2)^2 - (b/2)^2 Formula --> x= -b+-√(b^2 - 4ac) / 2a QE has 2 solutions - may be = To sketch a Quadratic graph Shape: a > 0 = Happy , a < 0 = Sad X-axis and y-axis crossing points discriminant b^2 - 4ac --> general shape 2 of 8 3: Equations and Inequalities Linear simultaneous equations --> elimination or substitution 1. linear and 1. quadratic (simultaneous) --> substitutiom X or / inequality by -ve no. --> </> = opposite To solve a quadratic inequality Solve corresponding quadratic equation sketch graph of quadratic function use sketch to find required set of values. 3 of 8 4: Sketching Curves Shapes of the basic curves y = x^2 y = x^3 y = (x-a)(x-b)(x-c) y=1/x Transformation f(x+a) --> -a in the x-direction f(x) + a --> +a in the y-direction f(ax) --> stretch of 1/a in x-direction (X x-coordinates by 1/a) af(x) --> stretch of a in y-direction (X y-coordinate by a) 4 of 8 5: Coordinate geometry in the (x,y) plane General form - y = mx + c --> m = gradient & c = y intercept Gradient of line joining two points together m = y2-y1 / x2 - x1 equation of line w/ m and passes through (x1, y1) y - y1 = m(x -x1) equation of line passing through two points y-y1/y2-y1 = x-x1/x2-x1 perpendicular has gradient -1/m two lines perpindicular gradient = -1. 5 of 8 6: Sequences and series sequence = series of numbers with a set rule each number = term nth term = general term expressed as a formular for the nth term e.g. Un = 4n + 1 Arithmetic sequence Uk+1 = Uk + n a + (a+d) + (a + 2d) + (a + 3d) + (a + 4d) nth term of an arithmetic seqence a + (n-1)d --> a = 1st term and d = common difference Sum of an arithmetic sequence Sn = n/2(2a + (n-1) d) Sn = n/2(a+L) --> L = last term Use ∑ 10∑r=1 (5 + 2r) = 7 + 9 + ... + 25 6 of 8 7: Differentiation m of y = f(x) @ specific point = m of tangent to curve @ that point m of tangent @ any point = dy/dx f'(x) = derived function y = f(x) --> dy/dx = y = f'(x) --> dy/dx = nx^n-1 2nd order derivative = d^2y/dx^2 Equation of tangent to curve @ point A y - f(a) = f'(a)(x-a) Equation of normal to curve @ point A y - f(a) = -1/f'(a) (x-a) 7 of 8 8: Integration If dy/dx = x^n --> y = (1/n+1)(x^n+1)+c If dy/dx = kx^n --> y = (kx^n+1/n+1) + c ∫x^ndx = (x^n+1/n+1) +c 8 of 8

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