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AS Pure Maths
A Pure Maths
Indices and surds
Polynomials and Quadratics
Inequalities
Algebraic Fractions
Simultaneous Equations
Graphs and Transformations
Coordinate geometry
Binomial
Differentiation
Integration
Logarithms
Trigonometry
Polynomials and Quadratics - AS Level
Polynomials
Expand 2 brackets
Expand 3 brackets
Factorise (1): \( ax^2 + bx + c \)
Polynomial Division
Factorising (2) : cubic function
Verify a factor
Finding k given a factor
Finding k given a root
Finding k - mixture
Polynomials
Finding unknowns given 2 roots
Finding unknowns given 2 factors
Finding unknowns Mixture
Finding the remainder
Finding Unknowns
Factor and Remainder
Expressing : \( (Ax^2 + Bx + c)(x + D) + E \)
Quadratics
Expanding (1): \( (x \pm a)(x \pm b) \)
Expand \( (ax \pm b)(cx \pm d) \)
Factorising (1): \(x^2 + bx + c \)
Factorising (2): \(ax^2 + bx + c \)
Express as \( (x \pm a)^2 \pm b \)
Express as \(a(x \pm a)^2 \pm b \)
Solving (1): \( (x \pm a)^2 \pm b = 0 \)
Solving (2): a\( (x \pm b)^2 \pm c = 0 \)
Solving (3): Factorising (\ a = 1 \)
Solving (4): Factorising
Solving (5): Formula \( x^2 + bx + c = 0 \)
Solving (6):Formula \( ax^2 + bx + c = 0 \)
Discriminant (1): Determine the number of roots
Discriminant (2): Find the Discriminant
Discriminant (3): one repeated root Roots
Quadratics
Discriminant (4): 2 real roots
Discriminant (5): No real roots
Discriminant (6): Mixture
Find the equation (1): two roots Given
Find the equation (2): two points Given
Find the equation (3): Mixed Questions
Intersections (1): Quadratic & \( y = mx+c \)
Intersections (2): Quadratic & \( a + bx = c \)
Intersections (3): Mixed Questions
Solving (1): \( ax^4 +bx^2 + c = 0 \)
Solving (2): \( ax^ {2/n} + bx^{1/n} = c \)
Solving (3): \( ax^ {2/n} + bx^{1/n} = c \)
Solving (4): \( ax+\frac{c}{x} = b\)
Indices and surds - AS Level
Indices
Positive Unit Fractions
Negative Unit Fractions
Mixed Unit Fractions
Positive Fractions
Negative Fractions
Mixed Fractions fractions
Multiplying \( ax^n \times bx^m \)
Dividing \( \frac{ax^n y^m}{bx^p y^q} \)
Brackets \( (ax^n)^m \)
Mixed Questions
Indices
Simplifying (1):\( \frac{x^n\times x^m}{x^p} \)
Simplifying (2): \( \frac{ax^n\times bx^m}{cx^p} \)
Simplifying (3): \( nx^\frac{a}{b} \div mx^\frac{c}{b} \)
Simplifying (4): \( (nx^\frac{a}{b})^2 \div mx^\frac{c}{b} \)
Simplifying (5): \( x^\frac{a}{b} \div x^n \)
Simplifying (6): \( nx^\frac{a}{b} \div mx^c \)
Simplify (7): \( \frac{ax^n(bx^m + c)}{x^p} \)
Simplify (8): \( \frac{(bx^m + c)^2}{x^p} \)
Solving \( x ^\frac{a}{b} = n \)
Surds
Simplifying (1): \( \sqrt{a} \)
Simplifying (2): \(\sqrt{a} \pm b + \sqrt{a} \pm c\)
Simplifying (3): \( n\sqrt{a} \pm b + m\sqrt{a} \pm c\)
Simplifying (4): \( n\sqrt{a} + \sqrt{b} \)
Simplifying (5): \( (a\sqrt{b})^n \pm c\sqrt{b} \)
Simplifying Mixture (6): Sets 1 - 4
Simplifying Mixture (7) Sets 1 - 5
Multiplying (1): \( n\sqrt{a} \times m\sqrt{a} \)
Multiplying (2): \( \sqrt{a} \times \sqrt{b} \)
Multiplying (3):\( m\sqrt{a} \times n\sqrt{b} \)
Multiplying (4): Mixture
Expanding (1): \( \sqrt{a} (\sqrt{a} \pm b )\)
Expanding (2): \( ( \sqrt{a} + b)(\sqrt{a} - b )\)
Expanding (3): \( ( \sqrt{a} \pm b)(\sqrt{a} \pm c )\)
Expanding (4): Mixture
Expanding (5): \( \sqrt{a} (n\sqrt{a} \pm b )\)
Expanding (6): \( ( c\sqrt{a} + b)(c\sqrt{a} - b )\)
Expanding (7): \( ( m\sqrt{a} \pm b)(n\sqrt{a} \pm c )\)
Expanding (8): Single bracket mixture
Expanding (9): Mixture
Surds
Rationalising (1): \( \frac{a}{\sqrt{b}} \)
Rationalising (2): \( \frac{a}{c \pm \sqrt{b}} \)
Rationalising (3): \( \frac{a \pm \sqrt{b}}{c \pm \sqrt{b}} \)
Rationalising (4): Mixture
Rationalising (5): \( \frac{a}{n\sqrt{b}} \)
Rationalising (6): \( \frac{a}{c \pm n\sqrt{b}} \)
Rationalising (7): \( \frac{a \pm n\sqrt{b}}{c \pm m\sqrt{b}} \)
Rationalising (8): \( \frac{a \pm n\sqrt{b}}{m\sqrt{b} \pm c} \)
Rationalising (9): Mixture(2)
Solving (1): \( m - x\sqrt{b} = \frac{ax}{\sqrt{b}} \)
Solving (2): \( m - x\sqrt{c} = \frac{ax}{\sqrt{b}} \)
Solving (3): \( \sqrt{a} \pm \frac{\sqrt{b}}{cx} = \frac{1}{x\sqrt{d}} \)
Solving (4): Mixed
Inequalities - AS Level
Inequalities
Linear Inequalities
Solving Quadratic (1) :\( x^2 \pm bx \pm c <> 0\)
Solving Quadratic (2) :\( ax^2 \pm bx \pm c <> 0\)
Simultaneous (1): \( x^2 \pm bx \pm c < d \) , \( x \pm m < n \)
Inequalities - using set notation
Solving Quadratic (3) :\( x^2 \pm bx \pm c < 0\)
Solving Quadratic (4) :\( ax^2 \pm bx \pm c < d\)
Simultaneous (2): \( x^2 \pm bx \pm c < d \) , \( x \pm m < n \)
Inequalitites -defining shaded regions
Representing (1):\( y \le a \) or \( y \lt a \)
Representing (2):\( y \ge a \) or \( y \gt a \)
Representing (3):\( x \le a \) or \( x \lt a \)
Representing (4):\( x \ge a \) or \( x \gt a \)
Representing (5):\( ax+by \le c \) or \( ax+by \lt c \)
Representing (6):\( ax+by \ge c \) or \( ax+by \gt c \)
Representing (7):\( ax+by \le c \) and axes bounds
Representing (8):\( ax+by \ge c \\\ \)axes bounds
Representing (9):\( ax+by \le c \\\ x \ge p \\\ , \\\ y \ge 0 \)
Representing (10):\( ax+by \ge c \\\ x \le p \\\ , \\\ y \le 0 \)
Representing (11):\( ax+by \le c \\\ x \ge 0 \\\ , \\\ y \ge p \)
Representing (12):\( ax+by \ge c \\\ x \le 0 \\\ , \\\ y \le p \)
Representing (13):\( ax+by \le c \\\ x \ge p \\\ , \\\ y \ge q \)
Representing (14):\( ax+by \ge c \\\ x \le p \\\ , \\\ y \le q \)
Inequalitites -defining unshaded regions
Representing (15):\( y \ge a \) or \( y \gt a \)
Representing (16):\( y \le a \) or \( y \lt a \)
Representing (17):\( x \ge a \) or \( x \gt a \)
Representing (18):\( x \le a \) or \( x \lt a \)
Representing (19):\( ax+by \ge c \) or \( ax+by \gt c \)
Representing (20):\( ax+by \le c \) or \( ax+by \lt c \)
Representing (21):\( ax+by \ge c \\\ \) and axes bounds
Representing (22):\( ax+by \le c \\\ \) and axes bounds
Representing (23):\( ax+by \ge c \\\ x \le p \\\ , \\\ y \le 0 \)
Representing (24):\( ax+by \le c \\\ x \ge p \\\ , \\\ y \ge 0 \)
Representing (25):\( ax+by \ge c \\\ x \le 0 \\\ , \\\ y \le p \)
Representing (26):\( ax+by \le c \\\ x \ge 0 \\\ , \\\ y \ge p \)
Representing (27):\( ax+by \ge c \\\ x \le p \\\ , \\\ y \le q \)
Representing (28):\( ax+by \le c \\\ x \ge p \\\ , \\\ y \ge q \)
Rational Expressions - AS Level
Simplifying - Rational Expressions
Simplifying (1)
Simplifying (2)
Simplifying (3)
Multiplication and division
Multiplication (1)
Multiplication (2)
Division (1)
Division (2)
Working with Algebraic Fractions - GCSE
Simplifying - Addition and Subtraction
Simplify (1):\( \frac{x}{a}+ \frac{x}{b} \)
Simplify (2):\( \frac{x}{a}- \frac{x}{b} \)
Simplify (3):\( \frac{ax}{b} + \frac{cx}{d} \)
Simplify (4):\( \frac{ax}{b} - \frac{cx}{d} \)
Simplify (5):\( \frac{x+a}{b} + \frac{x+c}{d} \)
Simplify (6):\( \frac{x+a}{b} - \frac{x+c}{d} \)
Simplify (7):\( \frac{x+a}{b} - \frac{x-c}{d} \)
Simplify (8):\( \frac{x-a}{b} - \frac{x+c}{d} \)
Simplify (9):\( \frac{x-a}{b} - \frac{x-c}{d} \)
Simplify (10):\( \frac{x-a}{b} + \frac{x-c}{d} \)
Simplify (11):\( \frac{x+a}{b} - \frac{x-c}{d} \)
Simplify Mixture
Simplifying - Multiplication and division
Multiply algebraic fractions
Divide algebraic fractions
Simultaneous Equations
Linear
(1): \( ax + by = r \) and \( cx + dy = s \)
(2): \( ax - by = r \) and \( cx - dy = s \)
(3): \( ax - by = r \) and \( cx + dy = s \)
(4): \( ax + by = r \) and \( cx - dy = s \)
Mixed questions (1 - 19)
Non-linear and Linear
(1)\(y= x^2 + bx + c \) and \( y = dx + e \)
(2) \( ax^2 \pm by^2 = c \) and \( x + y = d \)
(3) \(x^2 + y^2 \pm xy = c \) and \( x + y = d \)
(4) \(x^2 + y^2 \pm axy = c \) and \( x + y = d \)
(5) \(x^2 + y^2 \pm ax = c \) and \( x + y = d \)
(6) \(x^2 + y^2 \pm ay = c \) and \( x + y = d \)
(7) \(x^2 \pm ax y^2 \pm by = c \) and \( x + y = d \)
Circle and a line
(1) \(x^2 + y^2 = r^2 \) and \( y = x \pm c \)
(2) \(x^2 + y^2 = r^2 \) and \( y = ax \pm c \)
Mixture of (1 & 2)
Graphs and Transformations
Quadratic - vertex and symmetry
Complete the square and find the vertex (1)
Complete the square and find the vertex (2)
Symmetry (1) \( y= (x \pm a)(x\pm b) \)
Symmetry (2) \(y =x^2 \pm bx \pm c \)
Symmetry (3) \( y =ax^2 \pm bx \pm c \)
Roots (1) \(y =x^2 \pm bx \pm c \)
Roots (2) \( y =ax^2 \pm bx \pm c \)
Sketching Graphs - Other
Sketching \(y = x^2 \pm bx \pm c \)
Sketching \(y = -x^2 \pm bx \pm c \)
Sketching \(y = ax^3 \pm bx^2 \pm cx \pm d \)
Sketching \(y = - ax^3 \pm bx^2 \pm cx \pm d \)
Sketching \(y = asin \\\ x \)
Sketching \(y = acos x \)
Sketching \(y = sin \\\ ax \)
Sketching \(y = cos \\\ ax \)
Sketching \(y = sin \\\ (x \pm a) \)
Sketching \(y = cos \\\ (x \pm a) \)
Sketching \(y = sin \\\ x \\\ \pm a \)
Sketching \(y = cos \\\ x \\\ \pm a \)
Quadratic - transformations (function notation)
(1q)Finding \( f(x) + a \)
(2q)Finding \( f(x) - a \)
(3q)Finding \( f(x + a) \)
(4q)Finding \( f(x - a) \)
(5q)Finding \( -f(x) \)
(6q)Finding \( f(-x) \)
(7q)Finding \( af(x) \)
(8q)Finding \( f(ax) \)
(9q)Finding \( f(x + a) + b \)
(10q)Finding \( f(x + a) - b \)
(11q)Finding \( f(x - a) + b \)
(12q)Finding \( f(x - a) - b \)
Cubic - transformations - (function notation)
(1c)Finding \( f(x) + a \)
(2c)Finding \( f(x) - a \)
(3c)Finding \( f(x + a) \)
(4c)Finding \( f(x - a) \)
(5c)Finding \( -f(x) \)
(6c)Finding \( f(-x) \)
(7c)Finding \( af(x) \)
(8c)Finding \( f(ax) \)
(9c) Finding \( f(x + a) + b \)
(10c)Finding \( f(x + a) - b \)
(11c)Finding \( f(x - a) + b \)
(12c)Finding \( f(x - a) - b \)
(12c)Finding \( f(x - a) - b \)
Quadratic - transformations (descriptions)
(13q)Translation (+ve y)
(14q)Translation (-ve y)
(15q)Translation (-ve x)
(16q)Translation (+ve x)
(17q)Reflection (x-axis)
(18q)Reflection (y-axis)
(19q)Stretch (parallel to the y axis)
(20q)Strecth (parallel to the x axis)
(21q)Translation -ve x and +ve y
(22q)Translation -ve x and -ve y
(23q)Translation +ve x and +ve y
(24q)Translation +ve x and -ve y
Cubic - transformations (descriptions)
(13c)Translation (+ve y)
(14c)Translation (-ve y)
(15c)Translation (-ve x)
(16c)Translation (+ve x)
(17c)Reflection (x-axis)
(18c)Reflection (y-axis)
(19c)Stretch (parallel to the y axis)
(20c)Stretch (parallel to the x axis)
(21c)Translation -ve x and +ve y
(22c)Translation -ve x and -ve y
(23c)Translation +ve x and +ve y
(24c)Translation -ve x and -ve y
Coordinate geometry - AS Level
Gradient and intercept (1): \( y=mx + c \) or \(y = c - mx \)
Gradient and intercept (2): rearranging needed
Finding the equation (1): gradient and \( 0,a) \)
Finding the equation (2): gradient and \( (a,b) \)
Finding the equation (3): \( (a,b) \) and \( (0,c) \)
Finding the equation (4): \( (a,b) \) and \( (c,d) \)
Writing in the form: \( ax + by = c \)
Finding the equation a parallel line (1)
Finding the equation perpendicular line (1)
Finding the equation perpendicular line (2)
Finding the equation perpendicular bisector (2)
Finding the area of a triangle
Finding the centre of a circle
Finding the diameter of a circle
Finding the centre and radus
Finding the tangent to a circle
Binomial - AS Level
Expanding (a + x) and (x + a)
Expanding (1) : \( (1 + x)^n \)
Expanding (2) : \( (1 - x)^n \)
Expanding (3) : \( (a + x)^n \)
Expanding (4) : \( (a - x)^n \)
Expanding (5) : \( (x + a)^n \)
Expanding (6) : \( (x - a)^n \)
Expanding (a + bx) and (ax + b)
Expanding (7) :\( (1 + ax)^n \)
Expanding (8) :\( (1-ax)^n \)
Expanding (9) :\( (a + bx)^n \)
Expanding (10) :\( (a - bx)^n \)
Expanding (11) :\( (ax + b)^n \)
Expanding (12) :\( (ax - b)^n \)
Specific terms (a + x)
Specific terms (1): \( (1 + bx)^n \)
Specific terms (2): \( (1 - bx)^n \)
Finding \(b \) (1): \( (1 + bx)^n \)
Finding \(b \) (2): \( (1 - bx)^n \)
Specific terms (a + bx)
Specific terms (3): \( (a + bx)^n \)
Specific terms (4): \( (a - bx)^n \)
Finding \(b \) (3): \( (a + bx)^n \)
Finding \(b \) (4): \( (a - bx)^n \)
Trigonometry - AS Level
Exact values
Stating exact values in the range 0 to 360
Given that - find ratio for an acute angle
Given that - find ratio for an obtuse angle
Given that - find ratio for a reflex angle
Given that - mixture
Solving \( a sin \theta = \pm b \)
(1) : \(asin \\\ \theta = \pm b \\\ \\\ ( 0^o \le \theta \le 360^o ) \)
(2) : \(acos \\\ \theta = \pm b \\\ \\\ (0^o \le \theta \le 360^o ) \)
(3) : \(atan \\\ \theta = \pm b \\\ \\\ ( 0^o \le \theta \le 360^o ) \)
Mixture \( \\\ \\\ ( 0^o \le \theta \le 360^o ) \)
Mixture \( (1 - 6) \)
(1a) : \(asin \\\ \theta = \pm b \\\ \\\ ( -180^o \le \theta \le 180^o ) \)
(2a) : \(acos \\\ \theta = \pm b \\\ \\\ ( -180^o \le \theta \le 180^o ) \)
(3a) : \(atan \\\ \theta = \pm b \\\ \\\ ( -180^o \le \theta \le 180^o ) \)
Mixture \( (1a - 3a) \): Mixture \( \\\ \\\ ( -180^o \le \theta \le 180^o ) \)
Solving \( sin^2 \\\ \theta = b \)
(4) : \(sin^2 \\\ \theta = b \\\ \\\ ( 0^o \le \theta \le 360^o ) \)
(5) : \(cos^2 \\\ \theta = b \\\ \\\ (0^o \le \theta \le 360^o ) \)
(6) : \(tan^2 \\\ \theta = b \\\ \\\ ( 0^o \le \theta \le 360^o ) \)
Mixture \( 4-6 \)
Mixture (all) \)
(4a) : \(sin^2 \\\ \theta = b \\\ \\\ ( -180^o \le \theta \le 180^o ) \)
(5a) : \(cos^2 \\\ \theta = b \\\ \\\ ( -180^o \le \theta \le 180^o ) \)
(6a) : \(tan^2 \\\ \theta = b \\\ \\\ ( -180^o \le \theta \le 180^o ) \)
Mixture \( (4a - 6a) \)
Solving \( sin \\\ a\theta = \pm b \)
(7) : \( sin a \\\ \theta = b \\\ \\\ ( 0^o \le \theta \le 360^o ) \)
(8) : \(cos \\\ a \theta = b \\\ \\\ ( 0^o \le \theta \le 360^o ) \)
(9) : \(tan \\\ a \theta = b \\\ \\\ ( 0^o \le \theta \le 360^o ) \)
(7a) : \(sin \\\ a \theta = b \\\ \\\ ( -180^o \le \theta \le 180^o ) \)
(8a) : \(cos \\\ a \theta = b \\\ \\\ ( -180^o \le \theta \le 180^o ) \)
(9b) : \(tan \\\ a \theta = b \\\ \\\ ( -180^o \le \theta \le 180^o ) \)
Solving \( sin \\\ ( \theta \pm a) = \pm b \)
(10) : \(sin \\\ ( \theta \pm a) = b \\\ \\\ ( 0^o \le \theta \le 360^o ) \)
(11) : \(cos \\\ ( \theta \pm a)= b \\\ \\\ ( 0^o \le \theta \le 360^o ) \)
(12) : \(tan \\\ ( \theta \pm a )= b \\\ \\\ ( 0^o \le \theta \le 360^o ) \)
(10a): \(sin \\\ ( \theta \pm a) = b \\\ \\\ ( -180^o \le \theta \le 180^o ) \)
(11a): \(cos \\\ ( \theta \pm a )= b \\\ \\\ ( -180^o \le \theta \le 180^o ) \)
(12a): \(tan \\\ ( \theta \pm a )= b \\\ \\\ ( -180^o \le \theta \le 180^o ) \)
Solving - using identities
(13) : \( a sin \\\ \theta = bcos \\\ \theta \\\ ( 0^o \le \theta \le 360^o ) \)
(14) : \(asin^2 \\\ \theta \pm b sin \\\ \theta \pm c = 0 \\\ \\\ ( 0^o \le \theta \le 360^o ) \)
(15) : \(acos^2 \\\ \theta \pm b cos \\\ \theta \pm c = 0 \\\ \\\ ( 0^o \le \theta \le 360^o ) \)
(16) : \(atan^2 \\\ \theta \pm b tan \\\ \theta \pm c = 0 \\\ \\\ ( 0^o \le \theta \le 360^o ) \)
(17) : \(acos^2 \\\ \theta \pm b sin \\\ \theta \pm c = 0 \\\ \\\ ( 0^o \le \theta \le 360^o ) \)
(18): \(asin^2 \\\ \theta \pm b cos \\\ \theta \pm c = 0 \\\ \\\ ( 0^o \le \theta \le 360^o ) \)
(19) : \(acos^2 \\\ \theta = b sin \\\ \theta \pm c \\\ \\\ ( 0^o \le \theta \le 360^o ) \)
(20): \(acos^2 \\\ \theta = b sin \\\ \theta \pm c \\\ \\\ ( 0^o \le \theta \le 360^o ) \)
(21): \(asin^2 \\\ \theta \pm b sin \\\ \theta \pm c = dcos^2 \theta \\\ \\\ ( 0^o \le \theta \le 360^o ) \)
(22): \(acos^2 \\\ \theta \pm b cos \\\ \theta \pm c = dsin^2 \theta \\\ \\\ ( 0^o \le \theta \le 360^o ) \)
(13a) : \( a sin \\\ \theta = bcos \\\ \theta \\\ ( -180^o \le \theta \le 180^o ) \)
(14a): \(asin^2 \\\ \theta + b sin \\\ \theta + c = 0 \\\ \\\ ( -180^o \le \theta \le 180^o ) \)
(15a): \(acos^2 \\\ \theta + b cos \\\ \theta + c = 0 \\\ \\\ ( -180^o \le \theta \le 180^o ) \)
(16a): \(atan^2 \\\ \theta + b tan \\\ \theta + c = 0 \\\ \\\ ( -180^o \le \theta \le 180^o ) \)
(17a): \(acos^2 \\\ \theta - b sin \\\ \theta - c = 0 \\\ \\\ ( -180^o \le \theta \le 180^o ) \)
(18a): \(asin^2 \\\ \theta - b cos \\\ \theta - c = 0 \\\ \\\ ( -180^o \le \theta \le 180^o ) \)
(19a): \(acos^2 \\\ \theta = \pm b sin \\\ \theta \pm c \\\ \\\ ( -180^o \\\ to \\\ 180^o ) \)
(20a): \(acos^2 \\\ \theta = \pm b sin \\\ \theta \pm c \\\ \\\ ( -180^o \\\ to \\\ 180^o ) \)
(21a): \(asin^2 \\\ \theta \pm b sin \\\ \theta \pm c = dcos^2 \theta \\\ \\\ ( -180^o \\\ to \\\ 180^o ) \)
(22a): \(acos^2 \\\ \theta \pm b cos \\\ \theta \pm c = dsin^2 \theta \\\ \\\ ( -180^o \\\ to \\\ 180^o ) \)
Sine Rule
Sine rule - calculating an angle (1)
Sine rule - calculating an angle (2)
Sine rule - calculating an angle (mix)
Sine rule - calculating sides (1)
Sine rule - calculating sides (2)
Sine rule - calculating sides (mix)
Sine rule - calculating angles and sides
Trigonometry - Cosine Rule
Cosine rule - calculating an angle
Cosine rule - calculating sides
Cosine rule - calculating angles and sides
Sine and cosine rules calculating sides
Sine and cosine rules calculating an angle
Sine and cosine rules mixture
Sine and area
Calculating the area (1)
Calculating the area (2)
Calculating the area (1 & 2)
Calculating an angle
Calculating a side
Side and angle mixture
Mixture
Logarithms - AS Level
Using the laws of logarithms
Solving (1) : \( log_{a} \\\ b = x \)
Solving (2) : \( log_{a} \\\ \frac{1}{b} = x \)
Solving (3) : Mixture (1 & 2)
Solving (4) : \( log_{x} \\\ a = b \)
Expressing as \( log_{a} \\\ x \) (1) : \( log_{a} \\\ m + log_{a} \\\ n \)
Expressing as \( log_{a} \\\ x \) (2) : \( log_{a} \\\ m - log_{a} \\\ n \)
Expressing as \( log_{a} \\\ x \) (3) : \( nlog_{a} \\\ m \)
Expressing as \( log_{a} \\\ x \) (4) : Mix ( 1 to 3)
Expressing as \( log_{a} \\\ x \) (5) : \( \frac{1}{m}log_{a} \\\ n + log_{a} \\\ p \)
Expressing as \( log_{a} \\\ x \) (6) : \( \frac{1}{m}log_{a} \\\ n + plog_{a} \\\ q \)
Expressing as \( log_{a} \\\ x \) (7) : \( nlog_{a} \\\ m - \frac{1}{q}log_{a} \\\ r \)
Expressing as \( log_{a} \\\ x \) (8) : \( nlog_{a} \\\ m - \frac{p}{q}log_{a} \\\ r \)
Expressing as \( log_{a} \\\ x \) (9) : \( log_{a} \\\ \frac{1}{m} + \frac{1}{p}log_{a} \\\ \sqrt{r} \)
Expressing as \( log_{a} \\\ x \) (10) : \( log_{a} \\\ \frac{1}{m} - \frac{1}{p}log_{a} \\\ \sqrt{r} \)
Expressing as \( log_{a} \\\ x \) (11) : Mix (5-10)
Using the laws of logarithms - in terms of \( x \)
Expressing as \( nlog_{a} \\\ x \) (12): \( log_{a} \\\ x^n \)
Expressing as \( nlog_{a} \\\ x \) (13): \( \frac{1}{m}log_{a} \\\ x^n \)
Expressing as \( nlog_{a} \\\ x \) (14): \( \frac{1}{m}log_{a} \\\ \frac{1}{x^n} \)
Expressing as \( nlog_{a} \\\ x \) (15): \( nlog_{a} \\\ \frac{1}{ \sqrt[m]{x} } \)
Expressing as \( nlog_{a} \\\ x \) (16): \( log_{a} \\\ x^m + log_{a} \\\ x^n \)
Expressing as \( nlog_{a} \\\ x \) (17): \( log_{a} \\\ \frac{1}{x^m} + log_{a} \\\ \frac{1}{x^n} \)
Expressing as \( nlog_{a} \\\ x \) (18): \( log_{a} \\\ \frac{1}{x^m} - log_{a} \\\ \frac{1}{x^n} \)
Expressing as \( nlog_{a} \\\ x \) (19): Mixture (12 to 18)
Solving
Solving (1) : \( a^{bx} = c \)
Solving (2) : \( a^{x+b} = c \)
Solving (3) : \( a^{bx+c} = d \)
Solving (4) : \( e^{ax}=b \)
Solving (5) : \( e^{x \pm a} = b \)
Solving (6) : \( e^{ax \pm b} = c \)
Solving (7) : \( a^{x \pm b} = c^{x \pm d} \)
Solving
Solving (8) : \( \log_{m}(ax \pm b) = 1 \)
Solving (9) : \( \log_{m}(ax \pm b)-\log_{m}(cx \pm d) = 1 \)
Solving (10) :\( \log_{m}(ax \pm b)+log_{m}(x \pm d) = 1 \)
Differentiation - AS Level
Differentiating (1)
Differentiation (1) : \( ax \)
Differentiation (2) : \( x^n \)
Differentiation (3) : \( x^{-n} \)
Differentiation (4) : \( ax^{n} \)
Differentiation (5) : \( ax^{-n} \)
Differentiation (6) : \( x^{\frac{m}{n}} \)
Differentiation (7) : \(x^{-\frac{m}{n}} \)
Differentiation (8) : \(ax^{\frac{m}{n}} \)
Differentiation (9) : \(ax^{-\frac{m}{n}} \)
Differentiation (10) : \(y = mx \pm c \)
Differentiation (11) : \(y=mx^2 \pm c \)
Differentiation (12) : \(y=ax^2 \pm bx \)
Differentiation (13) : \(y=ax^2 \pm bx \pm c \)
Differentiation (14) : \(y=ax^3 \pm c \)
Differentiation (15) : \(y=ax^3 \pm bx \)
Differentiation (16) : \(y=ax^3 \pm bx \pm c \)
Differentiation (17) : \(y=ax^3 \pm bx^2 \)
Differentiation (18) : \(y=ax^3 \pm bx^2 \pm d \)
Differentiation (19) : \(y=ax^3 \pm bx^2 \pm cx \)
Differentiation (20) : \(y=ax^3 \pm bx^2 \pm cx \pm d \)
Differentiation (21) : \( y = (x\pm a)(x \pm b) \)
Differentiation (22) : \( y = (ax \pm c)(bx \pm d) \)
Finding the gradient
Find the gradient (1) : \(y=mx \pm c \)
Find the gradient (2) : \(y=ax^2 \pm c \)
Find the gradient (3) : \(y=ax^2 \pm bx \)
Find the gradient (4) : \(y=ax^2 \pm bx \pm c \)
Find the gradient (5) : \(y=ax^3 \pm c \)
Find the gradient (6) : \(y=ax^3 \pm bx \)
Find the gradient (7) : \(y=ax^3 \pm bx \pm c \)
Find the gradient (8) : \(y=ax^3 \pm bx^2 \)
Find the gradient (9) : \(y=ax^3 \pm bx^2 \pm c \)
Find the gradient (10) : \(y=ax^3 \pm bx^2 \pm cx \)
Find the gradient (11) : \(y=ax^3 \pm bx^2 \pm cx \pm d \)
Finding the points and stationary points
Find the point : \( y = ax^2 \pm bx \)
Find the point : \( y = ax^2 \pm bx \pm c \)
Stationary point : \(y = ax^3 \pm bx^2 \pm cx \pm d \)
Integration - AS Level
Indefinite Integration
(1) Integration \( ax \)
(2) Integration \( ax^2 \)
(3) Integration \( ax^{-n} \)
(4) Integration \( x^{\frac{m}{n}} \)
(5) Integration \( x^{- \frac{m}{n}} \)
(6) Integration \( ax^{\frac{m}{n}} \)
(7) Integration \( ax^{- \frac{m}{n}} \)
(8) Integration \( ax \pm b \)
(9) Integration \( ax^2 \pm b \)
Indefinite Integration
(10) Integration \( ax^2 \pm bx \)
(11) Integration \( ax^2 \pm bx \pm c \)
(12) Integration \( ax^3 \pm b \)
(13) Integration \( ax^3 \pm bx \)
(14) Integration \( ax^3 \pm bx \pm c \)
(15) Integration \( ax^3 \pm bx^2 \)
(16) Integration \( ax^3 \pm bx^2 \pm c \)
(17) Integration \( ax^3 \pm bx^2 \pm cx \)
(18) Integration \( ax^3 \pm bx^2 \pm cx \pm d \)
Mixed Integration
Find the function through a given point
Definite Integration
Definite Integration (1) \(ax \pm b \)
Definite Integration (2) \(ax^2 \pm bx \)
Definite Integration (3) \(ax^2 \pm bx \pm c \)
Definite Integration (4) \(ax^3 \pm bx^2 \pm cx \pm d \)
Definite Integration
Definite Integration (5)
Definite Integration (6)
Definite Integration (7)
Definite Integration (8)
Numerical methods - AS Level
Vectors - AS Level
Binomial
Differentiation (1)
Differentiation (2)
Modulus Functions
Partial Fractions
Sequences and Series
Trigonometry
Partial Fractions (Rational Functions)
Rational Functions
(1) Express as \( A + \frac{B}{ax \pm b}\)
(2) Express as \( A + \frac{B}{ax \pm b}\)
(3) Express as \( Ax + B + \frac{C}{ax \pm b}\)
Partial fractions
(1) Express as \( \frac{a}{bx \pm c} \pm \frac{d}{ex \pm f} \)
(2) Express as \( \frac{a}{(bx \pm c)^2} \pm \frac{d}{bx \pm c} \)
(3) Express as \( \frac{a}{bx + c} \pm \frac{d}{bx - c} \)
(4) Express as \( \frac{a}{(bx \pm c)^2} \pm \frac{d}{bx \pm c} \pm \frac{e}{gx \pm h}\)
Sequences and Series
Arithmetic
(1) : Express as a recurrance relation
(2) : Find a specific term (1)
(3) :Find the first term
(4) : Find the common difference
(5) : Find a specific term (2)
(6) : Find the sum - first n terms (1)
(7) : Find the first term - sum known
(8) : Find the last term - sum known
(9) : Find the first term
(10) : Find the number of terms - sum known
(11) : Find the sum - first n terms (2)
(12) : Find the common difference
(13) : Find the first term
(14) : Calculate the number of terms
(15) : Calculate the sum
Geometric
(1) : Sum of the first n terms
(2) : Find the first term - sum known
(3) : Sum of an infinite series (1)
(4) : Find the ratio - infinite series
(5) : Find first term - infinite series
(6) : Find sum - infinite series (2)
Summation notation - arithmetic
(1) : Arithmetic 1 to n
(2) : Arithmetic m to n
Binomial Series
First 3 terms
(1) : \( (1 \pm ax)^n \)
(2) : \( (1 \pm ax)^{-n} \)
(3) : \( (1 \pm x)^{\frac{n}{m}} \)
(4) : \( (1 \pm x)^{-\frac{n}{m}} \)
(5) : \( (1 \pm ax)^{\frac{n}{m}} \)
(6) : \( (1 \pm ax)^{-\frac{n}{m}} \)
(7) : \( (a \pm x)^n \)
(8) : \( (a \pm x)^{-n} \)
(9) : \( (a \pm x)^{\frac{n}{m}} \)
(10) : \( (a \pm x)^{-\frac{n}{m}} \)
(11) : \( (a \pm bx)^n \)
(12) : \( (a \pm bx)^{-n} \)
(13) : \( (a \pm bx)^{\frac{n}{m}} \)
(14) : \( (a \pm bx)^{-\frac{n}{m}} \)
Finding specific terms
(15) : \( (1 \pm ax)^{\pm n} \)
(16) : \( (1 \pm x)^{ \pm \frac{n}{m}} \)
(17) : \( (a \pm x)^{\pm n} \)
(18) :\( (a \pm x)^{ \pm \frac{n}{m}} \)
(19) : \( (a \pm bx)^{\pm n} \)
(20) : \( (a \pm bx)^{\ pm \frac{n}{m}} \)
Mixed Forms
(21) : \( \frac{ax}{(b \pm cx)^n} \)
(22) : \( \frac{ax}{(b \pm cx)^{\frac{n}{m}}} \)
Mixed Forms
(23) : \( \frac{a}{(1 \pm bx)^{n}} \pm \frac{c}{(1 \pm dx)^{m}} \)
Modulus Functions
Equations
(1) : \( |ax|=b \)
(2) : \( |x \pm a|=b \)
(3) : \( |a - x|=b \)
(4) : \( |ax \pm b|=c \)
(5) : \( |b - ax|=c \)
(6) : \( b \pm |ax \pm c = d \)
(7) : \( a|bx \pm c| \pm d = e \)
(8) : \( a|x| \pm b = c|x| \pm d \)
(9) : \( |ax \pm b| = |cx \pm d| \)
(10) : \( |ax \pm b| = | d - cx |\)
Inequalities
(1) : \( |ax|<>b \)
(2) : \( |x \pm a|<>b \)
(3) : \( |a - x|<>b \)
(4) : \( |ax \pm b|<>c \)
(5) : \( |b - ax|<>c \)
(6) : \( b \pm |ax \pm c <> d \)
(7) : \( a|bx \pm c| \pm d <> e \)
(8) : \( a|x| \pm b <> c|x| \pm d \)
(9) : \( |ax \pm b| <> |cx \pm d| \)
(10) : \( |ax \pm b| <> | d - cx |\)
Trigonometry
Radians
(1) Degress to radians (exact)
(2) Radians to degrees (exact)
(3) Degress to radians (2 d.p.)
(4) Radians to degrees (2 d.p.)
Sectors
(1) Calculate the arc length
(2) Calculate the perimeter
(3) Calculate the area
(4) Calculate the angle - area known
(5) Calculate the radius - area known
(6) Calculate the perimeter - area known
(7) Calculate the perimeter - area known
Segments
(1) Calculate the area
(2) Calculate the perimeter
Period and amplitude
Identify the period and amplitude
Solving equations
(1): \( sin \ cos \ tan \theta \ (range \ 0 \ to \ 2 \pi) \)
(2): \( sin \ cos \ tan \theta \ (range \ -\pi \ to \ \pi) \)
(3): \( sin \ a\theta \ (mixed \ range) \)
(4): \( cos \ a\theta \ (mixed \ range) \)
(5): \( tan \ a\theta \ (mixed \ range) \)
(6): \( sin \ (\theta \pm \frac{\pi}{n}) \)
(7): \( cos \ (\theta \pm \frac{\pi}{n}) \)
(8): \( tan \ (\theta \pm \frac{\pi}{n}) \)
(9): \( sin \ ( a\theta \pm \frac{\pi}{n}) \)
(10): \( cos \ ( a\theta \pm \frac{\pi}{n}) \)
(11): \( tan \ ( a\theta \pm \frac{\pi}{n}) \)
Small angle approximations
(1) \( acos \ \theta cos b \theta \)
(2) \( cos \ a \theta cos b \theta \)
(3) \( cos \ a \theta cos \frac{b}{ \theta} \)
(4) \( cos \ a \theta (1 \pm b sin \ \theta) \)
(5) \( cos \ a \theta (1 \pm sin \ b \theta) \)
(6) \( cos \ a \theta (1 \pm sin \ b \theta)^2 \)
(7) \( (1 \pm cos \ a \theta)(tan \ b \theta \pm 1)^2 \)
(8) \( \frac{1 - cos \ a \theta}{sin \ b \theta } \)
(9) \( \frac{\theta^2 \pm cos \ a \theta \pm 1}{sin \ b \theta} \)
(10) \( \frac{sin^2 \ a \theta + cos^2 \ b \theta}{tan c \theta} \)
(11) \( \frac{1 - cos \ a \theta}{ b \theta sin \ c \theta} \)
Addition Formula and (R)
(1): Express \( sin(x \pm \theta ) \) as \( a sinx \pm b cos x \)
(2): Express \( cos(x \pm \theta ) \) as \( a cosx \pm b sin x \)
(3): Max \( asinxcos\theta +acosxsin\theta \)
(4): Max \( asinxcos\theta -acosxsin\theta \)
(5): Max \( acosxcos\theta + asinxsin\theta \)
(6): Max \( acosxcos\theta - asinxsin\theta \)
(7): Min \( asinxcos\theta +acosxsin\theta \)
(8): Min \( asinxcos\theta -acosxsin\theta \)
(9): Min \( acosxcos\theta + asinxsin\theta \)
(10): Minx \( acosxcos\theta - asinxsin\theta \)
(11): Express \( asinx + b cos x \) in the form \( Rsin(x + \theta ) \)
(12): Express \( asinx - b cos x \) in the form \( Rsin(x - \theta ) \)
(13): Express \( acosx - b sin x \) in the form \( Rcos(x + \theta ) \)
(14): Express \( acosx + b sin x \) in the form \( Rcos(x - \theta ) \)
Solving Equations
(1): Solve \(asin \\\ 2x=bsin \\\ x \)
(2): Solve \(asin \\\ 2x + bsin \\\ x=0 \)
(3): Solve \(asin \\\ 2x = bcos \\\ x \)
(4): Solve \(asin \\\ 2x + bcos \\\ x =0 \)
(5): Solve \(acos \\\ 2x - b sin^2 \\\ x = c \)
(6): Solve \(acos \\\ 2x + b cos^2 \\\ x = c \)
(7): Solve \(asin \\\ 2x = btanx \\\ x \)
(8): Solve \(acos^2 \\\ x -bcos \\\ 2x = c \)
(9): Solve \(asin^2 \\\ x -bcos \\\ 2x = c \)
(10): Solve \(asin \\\ 2x \\\ cos \\\ x = bsin \\\ x \)
Reciprocals - Solving (1)
(1): Solve \( cosec \\\ x = a \)
(2): Solve \( sec \\\ x = a \)
(3): Solve \( cot \\\ x = a \)
Reciprocals - Exact values
(4): \( tan \\\ \theta \) known find \( sec \\\ \theta \)
(5): \( cosec \\\ \theta \) known find \( cot \\\ \theta \)
(6): \( cot \\\ \theta \) known find \( sec \\\ \theta \)
(7): \( cot \\\ \theta \) known find \( cosec \\\ \theta \)
Reciprocals - Solving (2)
(8): Solve \( acot^2 \\\ \theta + b = c cosec \\\ \theta \)
(9): Solve \( atan^2 \\\ \theta + bsec \\\ \theta = c \)
(10): Solve \( asec^2 \\\ \theta + btan \\\ \theta = c \)
Differentiation (1)
Simple
(1): \( y= ae^x \)
(2): \( y= ae^{x \pm b} \)
(3): \( y= asin \\\ x \)
(4): \( y= acos \\\ x \)
(5): \( y= atan \\\ x \)
(6): \( y= aln \\\ x \)
(7): \( y= aln \\\ x^n \)
(8): \( y= aln \\\ x \pm be^x\) find gradient
(9): \( y= a \pm be^x\) gradient known - find x
(10): \( y= x^2 - aln \\\ x\) gradient known - find x
(11): \( y= \frac{a}{x} +bln \\\ x\) gradient known - find x
Chain rule
(1): \( y= ae^{ \\\ bx} \)
(2): \( y= ae^{ \\\ bx \pm c} \)
(3): \( y= ae^{ \\\ c-bx } \)
(4): \( y= \frac{e^{ \\\ ax} \pm b }{e^{ \\\ cx} } \)
(5): \( y= (e^{ \\\ ax} \pm b)(e^{ \\\ cx} \pm d) \)
(6): \( y= (e^{ \\\ ax} \pm b)(e^{ \\\ -cx} \pm d) \)
(7): \( y= (ax \pm b)^n \)
(8): \( y= (ax^2 \pm b)^n \)
(9): \( y= (ax^2 \pm bx)^n \)
(10): \( y= (ax^2 \pm b)^{-n} \)
(11): \( y= (ax^2 \pm bx)^{-n} \)
Chain Rule
(12): \( y= (ae^{bx} \pm cx)^{-n} \)
(13): \( y= asin(bx \pm c) \)
(14): \( y= asin(bx^2 \pm c) \)
(15): \( y= acos(bx \pm c) \)
(16): \( y= acos(bx^2 \pm c) \)
(17): \( y= atan(bx \pm c) \)
(18): \( y= atan(bx^2 \pm c) \)
(19): \( y= \sqrt{ax \pm b} \)
(20): \( y= \sqrt{ax^2 \pm bx} \)
Chain Rule (use twice)
(21): \( y= sin^n \\\ ax \)
(22): \( y= cos^n \\\ ax \)
(23): \( y= sin (ln \\\ ax^n) \)
(24): \( y= cos (ln \\\ ax^n) \)
(25): \( y= ln( asin \\\ bx) \)
(26): \( y= ln( acos \\\ bx) \)
Chain Rule (use twice)
(27): \( y= (ln \ ax)^n \)
(28): \( y= e^{\sqrt{ax \\\ \pm b}} \)
(29): \( y= sin^n \\\ \sqrt{ax} \)
(30): \( y= cos^n \\\ \sqrt{ax} \)
Differentiation (2)
Product Rule
(1): \( y= ax^n \\\ (bx \pm c)^m \)
(2): \( y= ax^n \\\ \sqrt{bx \pm c} \)
(3): \( y= (ax \pm b)^n \\\ (cx \pm d)^m \)
(4): \( y= e^{ax} \\\ (cx \pm d)^m \)
(5): \( y= ax^n \\\ sin \\\ bx \)
(6): \( y= ax^n \\\ cos \\\ bx \)
(7): \( y= ax^n \\\ tan \\\ bx \)
(8): \( y= ax^n \\\ e^{ \\\ bx} \)
Product Rule
(9): \( y= ax^n \\\ ln \\\ bx \)
(10): \( y= ax^n \\\ ln \\\ (bx \pm c) \)
(11): \( y= e^{ \\\ ax} \\\ ln \\\ bx \)
(12): \( y= e^{ \\\ ax} \\\ sin \\\ bx \)
(13): \( y= e^{ \\\ ax} \\\ cos \\\ bx \)
(14): \( y= e^{ \\\ ax} \\\ tan \\\ bx \)
(15): \( y= e^{ \\\ ax} \\\ tan \\\ bx \)
Quotient Rule
(1): \( y= \frac{ax}{bx \pm c} \)
(2): \( y= \frac{ax \pm b}{cx \pm d} \)
(3): \( y= \frac{ (ax \pm b)^n }{cx \pm d} \)
(4): \( y= \frac{\sqrt{ax \pm b} }{cx} \)
(5): \( y= \frac{\sqrt{ax \pm b} }{cx \pm d} \)
(6): \( y= \frac{ax^n}{\sqrt{cx \pm d}} \)
(7): \( y= \frac{ax^2 \pm b}{cx \pm d} \)
(8): \( y= \frac{ax \pm b}{cx^2 \pm d} \)
(9): \( y= \frac{ln \\\ x}{bx \pm c} \)
(10): \( y= \frac{ln \\\ ax}{bx^2 \pm c} \)
(11): \( y= \frac{e^{ax}}{bx \pm c} \)
(12): \( y= \frac{e^{ax}}{ \sqrt{bx \pm c}} \)
Implicit Differentiation
(1): \( ax^2 \pm by^2 = c \)
(2): \( ax^n \pm by^m = c \)
(3): \( ay^n \pm bxy = c \)
(4): \( ax^ny \pm bxy^m = c \)
(5): \( \frac{ ax \pm y}{bx \pm y} = cy \)
(6): \( \frac{ ax \pm y}{bx \pm y }= cx \)
(7): \( axe^y \pm yln \\\ y = cx^n \)
(8): \( axe^y \pm byln \\\ y = cx^n \)
(9): \( \frac{ay^n}{bxy \pm c} = d \)
(10): \( \frac{ax^n}{bxy \pm c} = d \)
(11): \( y = a^x \)
(12): \( y = a^{bx \pm c} \)
Random Order
Yes
No