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xᵃ × xᵇ - Answer -xᵃ⁺ᵇ xᵃ ÷ xᵇ - Answer -xᵃ⁻ᵇ (xᵃ)ᵇ - Answer -xᵃ×ᵇ (xy)ᵇ - Answer -xᵇyᵇ When expanding two brackets - Answer -Middle number is them added together, End is them multiplied together When factorising two brackets - Answer -Multiply the 1st number by the end number, and find factors of that number that add to make the middle number and then simplify the brackets if they have any common factors x ⁰ - Answer - x⁻ᵃ - Answer -1/xᵃ x^ᵃ⁄ₖ - Answer -(ₖ√x)ᵃ √ab - Answer -√a√b √ᵃ⁄ₖ - Answer -√a ÷ √k Rationalise 1/(√a) - Answer -Multiply by (√a)/(√b) Rationalise 1/(a+√b) - Answer -Multiply by (a-√b)/(a-√b) Rationalise 1/(a-√b) - Answer -Multiply by (a+√b)/(a+√b)
Quadratic Formula - Answer - What does completing the square include - Answer -Turning ax²+bx+c=0 into a(x+p)²+q= What does the domain refer to - Answer -The x-axis What does the range refer to - Answer -y-axis What does f(x)∈ ℝ - Answer -Means that it is a real number so it can be anything What is a function - Answer -What takes an input value and turns it into an output value What would you do to find the minimum/maximum - Answer - Complete the square then make it equal to y, then make the x= so that when you sub it in you get the lowest value for y and that will be the coordinates What is the shape of the graph x² - Answer - What is the shape of the graph -x² - Answer -Upside down u How would you find something that crosses the x-axis - Answer - Make y= How would you find something that crosses the y-axis - Answer - Make x= What is the discrimnant - Answer -b²-4ac What does it mean if the discriminant is >0 - Answer -There is 2 real roots
What would the graph look like (x+f)³ - Answer -It crosses the x- axis at one place which is -f What does a quartic graph look like - Answer - What would you do if it asked to find the number of real solutions two equations have - Answer -Find how many points of intersection the two lines have and show that they are equal to each other and can be rearranged to the equation What are ways to work out an equation of a line - Answer - y=mx+c m=(y₂-y₁) ÷ (x₂-x₁) ax+by+c=0 but a,b,c have to be whole numbers What do parallel lines have - Answer -Same gradient What do perpendicular lines have - Answer -Reciprocal and opposite signs How would you find the distance between two points - Answer - Use pythagorus What is the area of a triangle - Answer -½ x base x height ½ absinC When would a linear model be appropiate - Answer -When the points lie on a straight line How would you find the midpoint - Answer -((x₂-x₁)÷2),((y₂-y₁) ÷2) What is an equation of a circle - Answer -(x-a)² + (y-b)² = r² where (a,b) is the centre
What would you do if it asks to show that a circle passes through a certain point - Answer -Put the x and y in the equation of the circle and show that it works Characteristics of a chord - Answer -2 points of intersection so b²- 4ac> Perpendicular bisector of a chord will go through the centre of a circle Characteristics of a tangent - Answer -1 point of intersection so b²-4ac= Perpendicular to radius of a circle Circumcircle - Answer -A circle that passes through all 3 vertices of a triangle and the perpendicular bisector of each side of the triangle passes through the centre of the circle What is the angle of a triangle in a semi-circle - Answer - 90degrees, where the line is the diameter and can be proven using pythagorus or that two chords are perpendicular What is the perpendicular bisector of any two chords of a circle like - Answer -Passes through the centre What would you write in factor theorum - Answer -if f(p)=0 then (x- p) is a factor What does 0! equal - Answer - ⁿCᵣ meaning - Answer -n = represent the power of the bracket r = represent the power of a n! ÷ r!(n-r)! Cosine rule - Answer -a² = b² + c² - 2bcCosA where the biggest angle will be opposite the longest side
0 then minimum <0 then maximum =0 then go to the first method Integration - Answer -Add one to the power and then divide by it and add a C What will happen when working out the area under the x-axis - Answer -It will come out as negative so you need to make it positive How would you find the area between two curves - Answer -∫ₐᵇ f(x) dx - ∫ₐᵇg(x) dx [integrate both functions then subtract] ∫ₐᵇ(f(x)-g(x)) dx [subtract the bottom function from the top then integrate] where f(x) is the top function and g(x) is the bottom function What does y=aˣ look like on a graoh - Answer -Crosses the y-axis at 1 What is special about y=eˣ graph - Answer -Gradient of the graph at x, is the same as its y value logₐn=x - Answer -aˣ=n logₐx + logₐy - Answer -logₐ(xy) logₐx - logₐy - Answer -logₐ(x/y) logₐ(xᵏ) - Answer -klogₐx What is the inverse of eˣ - Answer -lnx lnx - Answer -logₑx
How would you find the inverse of a graph - Answer -Reflect it in the line y=x When dealing with y=axⁿ or y=abˣ how would you turn into y=mx+c - Answer -You would log both sides then rearrange to get into that form where c is loga What do you do in a proof by contradiction question - Answer - Write an assumption, which is the opposite of the question, and try proving it wrong Even number - Answer -2n Odd number - Answer -2n+ Integer - Answer -n Irrational number - Answer -Cannot be written in the form a/b When do you use long division - Answer -When the numerator power is greater or equal to the denominator power What does a modulus do - Answer -Make any negative value positive What do you do when sketching y=|ax+b| - Answer -Sketch ax+b then reflect the section below the x-axis in the x-axis What do you do when solving a modulus - Answer -Replace the modulus with brackets and write it out twice and put a + infront of the first bracket and a - infront of the second and solve the two equations One-to-one function - Answer -When each input goes only to one output
secx - Answer -1/cosx cotx - Answer -1/tanx cosecx - Answer -1/sinx Year 2 trigonometric identities - Answer -1+tan²x = sec²x 1+cot²x = cosec²x (Worked out by sin²x+cos²x=1 and then dividing by cos or sin) arcsin - Answer -sin⁻¹ arccos - Answer -cos⁻¹ arctan - Answer -tan⁻¹ sin2A - Answer -2sinAcosA cos2A - Answer -cos²A-sin²A tan2A - Answer -2tanA/1-tan²A Parametric equation - Answer -When x and y are linked by a 3rd parameter Cartesian - Answer -When it links x and y How to turn a parametric equation into a cartesian equation - Answer -Get rid of t Differentiate sinkx - Answer -kcosx Differentiate coskx - Answer --ksinx
Differentiate eˣ - Answer -eˣ Differentiate lnx - Answer -1/x Differentiate e[ᶠ⁽ˣ⁾] - Answer -f'(x)eᶠ⁽ˣ⁾ Differentiate ln[f(x)] - Answer -f'(x)/f(x) Differentiate aᵏˣ - Answer -kaᵏˣlna Chain rule for differentiating - Answer -USED WHEN ITS ( )ˣ Differentiate the outside, then multiply by the inside differentiated Product rule for differentiating - Answer -USED WHEN TWO BRACKETS ARE MULTIPLIED WITH EACH OTHER Make one bracket = u and the other v then do uv' + u'v Quotient rule for differentiating - Answer -USED WHEN TWO THINGS ARE BEING DIVIDED ( vu' - uv' )/ v² Differentiate arcsinx - Answer -1/(√1-x²) Differentiate arccosx - Answer --1/(√1-x²) Differentiate arctanx - Answer -1/( 1+ x²) What to do in parametric differentiation - Answer -Y part differentiated / X part differentiated How do you know if a graph is concave - Answer -f''(x) < 0 or it the tangent of the curve is above the graph
where the limits will be put into to find out the new limits, then differentiate the u equation and rearrange to find dx and then substitute all these new values in the integration and then integrate as normal What does integration by parts involve - Answer -Making the polynomial (unless there's a ln|x|) equal to u and differentiate that, and make the other part of the equation equal dv/dx and integrate and then use the formula What to do when asked to find the exact area bounded by a parametric equation - Answer -Using ∫ydx = ∫y(dx/dt) and to find the new limits, sub in the x values into the x equation to find what t is equal to and use that as your new limits How would you work out the percentage error of a trapezium rule
- Answer -(Difference between the exact area and estimate/ exact area) x When would a trapezium rule be an overestimate - Answer -When the curve is U shaped and is because the top side of the trapezium will be above the curve When would a trapezium rule be an underestimate - Answer - When the curve is n shaped since the top side of the trapezium will be under the graph When would you do partial fractions in integration - Answer - When the denominator has two brackets What to do when solving differential equations - Answer -Times by dx, Get the x on one side and the y on the other and then integrate
If it says find the general equation don't find C, if it says find the particular equation find C which can be done by finding the general equation the subbing in values given in the question Position vector - Answer -Vector in reference to the origin Displacement vector - Answer -How to get to one point to another How to find an angle between a vector and positive coordinate axes - Answer -cosθₓ = x/|a| (length of vector using pythagorus) 3D vector represented as - Answer -(i,j,k) To find length/distance of a vector - Answer -Pythagorus To work out the nth term of an arithmetic sequence formula - Answer -a+(n+1)d a = first term n = term your trying to find d = difference between tow consecutive terms To find arithmetic series - Answer -n/2(2a + (n-1)d) To find nth term of a geometric sequence - Answer -ar^n- r = common ration a = first term To find a geometric series - Answer -a(1-r^n)/1-r Sum to infinity for a geometric series - Answer -a/1-r When is a geometric series convergent - Answer -|r|<
