Edexcel A-Level Mathematics - Pure Maths

1. Gradient y2-y1/x2-x1 2. Length of a line sqrt((x2-x1)^2+(y2-y1)^2) 3. Midpoint (x1+x2/2, y1+y2/2) 4. How do you know if two lines are perpendic- ular? When you multiply the gra- dients together they should equal -1. The general rule is perpendicular = -1/m. Or you could use Pythagoras to show there is a right angle 5. How do you know if two lines are parallel? The gradients should be the same 6. What should you always remember when finding the square roots? The answer could be posi- tive or negative 7. What is the quadratic formula? x = (-b ± b^2 - 4ac)/2a (will get 2 answers) 8. Describe the proof for the quadratic formula 1) Divide by a on both sides 2) Complete the square for ax2/a + bx2/a 3) Expand the second bracket, (b/2a)2 4) Multiply by c/a by 4 and add the fractions 5) Factorise into brackets 6) Add the sec- ond bracket to the right side 7) Square root both sides 8) Subtract the b/2a on both sides 9) Simplify the answer 9. What's the alternative formula you can use for completing the sqauare? 10. Describe y=(x+a) ax2+bx+c = a(x + b/2a)2 + (c-b2/2a) You then simpli- fy the second bracket and solve the equation Horizontal translation, +a move left, -a move right, use vector (-a,0) 11. Describe y=f(x)+a Vertical translation, +a move up, -a move down, use vector (0/a) 12. Describe y=f(ax) Horizontal stretch by scale factor 1/a 13. Describe y=af(x) Vertical stretch by scale fac- tor a 14. Describe y=-f(x) Reflection of the graph in the x axis 15. Describe y=f(-x) The graph is reflected in the y-axis 16. Equation for real roots b2-4ac>0 Quadratic graph passes through x axis 17. Equation for equal roots b2-4ac=0 Quadratic graph touches the x axis 18. Equation for no real roots b2-4ac<0 Quadratic graph doesn't touch x axis 19. What is a linear model? Shows the relationship be- tween two variables, x and y. The graph has a straight line and the variables are related by y = mx + c 20. What are the qualities of direct proportion? The two quantities increase at the same rate and the straight line passes through the origin (0,0). Therefore y is directly proportional to x 21. How do you decide whether a linear model is appropriate? so y = k x where k is the real constant - Points should be on the line or close to the straight line, the further away the points then the less appro- priate it is - A mathematical model is an attempt to represent a real life situation, however assumptions are needed to sketch it 22. What does k represent? K represents the constant and the gradient, so on a ex- tension graph then k repre- sents the increase in exten- sion in cm when the mass is increased by 1 gram. 23. What is the equation of a circle? (x-a)^2 + (y-b)^2 = r^2 cen- ter (a,b) radius r 24. What is a perpendicular bisector, tangent and a chord? - The perpendicular bisec- tor is the shortest distance between the center and a chord is a right angle and is also passes through the center. - The tangent is a straight line that intersects the circle at one point. - A chord is a line segment that joins two points on the circumference. 25. What are the rules for a triangle in a circle? A triangle consists of three vertices and you can draw a circle around those three 26. What are the rules for a right-angled triangle in a circle? 27. How do you find the center of a circle given any three points on the circumference? vertices. The circle is called the circumcircle and the center is called the circum- center, this is also the point where all the perpendicular bisectors intersect. For a right angled triangle, the hypotenuse of the trian- gle is a diameter of the cir- cumcircle. If angle PRQ is 90o then R lies on the circle with the diameter PQ. The angle in the semi circle is always a right angle. Find the equations of the perpendicular bisectors of two different chords. Find the coordinates of the point of intersection of the per- pendicular bisectors. 28. What is the graph for real roots? The regions outside the graph are shaded, above the x-axis if the graph is pos- itive, below if it's negative. -2 > x & 2 < x 29. What is the graph for no real roots? The regions inside the graph are shaded, below the x-axis if the graph is posi- tive, above if it's negative. -2 < x < 2 30. Formula to rationalise 1/ square root (a) Multiply the numerator and denominator by square root (a) 31. Formula to rationalise 1/a + square root (b) Multiply the numerator and denominator by a - square root (a) 32. Formula to rationalise 1/a - square root (b) Multiply the numerator and denominator by b + square root (a) 33. Coordinates of the turning point Completing the square for- mula: a(x+p)^2 + q, turning points = (-p,q) 34. Colour code for linear inequality number lines 35. Inequality for when the graph of y = f(x) is below the graph of g(x) 36. Inequality for when the graph y = f(x) is above the curve y = g(x) Black circles: less/greater than or equal to. White cir- cles: less/greater than f(x) < g(x) f(x) > g(x) 37. Lines to represent inequalities on graphs Dotted: greater/less. Solid line: greater/less than or equal to 38. Equation of a cubic graph y = (ax)^3 + (bx)^2 + cx + d 39. Equation of a quartic graph y = (ax)^4 + (bx)^3 + (cx)^2 + dx + e (looks like a w if it is positive and an m if it is negative) 40. What is a polynomial? A polynomial is a finite ex- pression with positive whole number indices and you can use long division to divide them 41. What is the factor theorem? If f(x) is a polynomial then if f(p) = 0, then (x-p) is a factor of f(x) or if (x+p) is a factor if f(x), then f(-p) = 0 42. What is a theorem and what is a conjecture? A theorem is a statement that has been proven and a conjecture has not been proven 43. How do you prove by deduction? You start from know facts and definitions, then use logical steps to reach a conclusion. You use num- bers for a demonstration and then use algebra to prince that statement and you then write a statement of proof 44. How do you prove by exhaustion? You break the statement into smaller cases and break each case separately, but this is better for statements with small cases. 45. How do you disprove a mathematical state- ment? You can use counter-ex- amples which is an exam- ple that does not work for the statement and you only need to give one example. 46. Pascal's Triangle A pattern for finding the co- efficients of the terms of a binomial expansion and it is formed by adding the adjacent pairs of numbers to find the numbers on the next row. The first row ex- pands brackets/numbers to the power of 0. 47. What is factorial notion? 3! = 3 x 2 x 1 or 2! = 2 x 1 48. What is the formula using factorial notion to find entries on Pascal's triangle? n C r = (n/r) = n!/r!(n-r)!, the rth entry would be n-1 C r-1 = (n-1 / r-1) 49. What is the binomial expansion? (a+b)^n = a^n + (n C 1)(a)^n-1(b) + (n C 2)(a)^n-2(b)^2 ..... (n C r)(a)^n-r(b)^n 50. Cosine rule a² = b² + c² - 2bcCosA 51. Sine rule a/sinA = b/sinB = c/sinC 52. Sine rule (angle) sinA/a = sinB/b = sinC/c 53. Two outcomes of sine rule for a missing an- gle sin(X) = sin (180 - X) 54. Area of a triangle Area= 1/2absinC 55. Cosine Rule for Angles CosA=(b2+c2-a2)/2bc 56. Sine graph 57. Cosine graph 58. Tan graph 59. Trig Quadrants 60. Trig Values 61. Trigonometric ratio relating sin and cos sin^2(X) + cos^2(X) = 1 1 - sin^2(X) = cos^2(X) 1 - cos^2(X) = sin^2(X) 62. Trigonometric rations relating sin, cos and tan tan(X) = sin(X) / cos (X) 63. What is the principle value? The angle you get when you use the inverse trigonomet- ric functions on your calcu- lator 64. What is a vector? A quantity that has both magnitude and direction 65. What is the resultant vector? The sum of two or more vec- tors 66. What are the two forms of writing vectors? Column vector and compo- nent vector (using i for east and west and j for north and south). Using pi + qj is also a two dimensional vector 67. How do you find the magnitude of the vector? [a] = square root (x^2 + y^2) 68. What is a unit vector? A vector with the magnitude of 1, a unit vector in the di- rection of a is a/[a] 69. How do you calculate vector AB? AB = OB -OA where O is the origin 70. How do you calculate speed and distance from vectors? Speed is the magnitude of the velocity vector and dis- tance between A and B is the magnitude of AB 71. What is the formula for the derivative to dif- ferentiate from first principles? 72. Equation of the tangent to the curve y - f(a) = f'(a)(x-a) 73. Equation to the normal of the curve y - f(a) = -1/f'(a) (x-a) 74. How do you know if a function is increasing on a interval? [a,b] f' > 0 for a < x < b 75. How do you know if a function is decreasing on a interval? [a,b] f' < 0 for a < x < b 76. Types of stationary points: Local minimum f'(x-h) - Positive f'(0) - 0 f'(x+h) - Negative f''(a) > 0 or = 0 77. Types of stationary points: Local maximum f'(x-h) - Negative f'(0) - 0 f'(x+h) - Positive f''(a) < 0 or = 0 78. Types of stationary points: Points of inflec- tion f'(x-h) - Negative / Positive f'(0) - 0 / 0 f'(x+h) - Positive / Negative f''(a) = 0 79. Sketching gradient functions Min/max = cuts x-axis Inflection = touches x axis Positive gradient = above Negative gradient = below Vertical asymptote = same Horizontal asymptote = hor- izontal asymptote at x-axis 80. How do you use differentiation in modelling? f'(x) represents the rate of change, so you can use it to find the rate in volume for example