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108 Cards in this Set
- Front
- Back
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The equation of the straight line
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y-y1=m(x-x1)
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What do parallel lines have in common?
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Equal gradients
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Gradient formula
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y2-y1/x2-x1
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Horizontal lines have equations that look like?
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y = a number
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Vertical lines have equations that look like?
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x = a number
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How do you find a perpendicular gradient?
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Turn the fraction upside down and change the symbol, but remember to say m1 x m2 = -1 first
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How do you find the gradient when told the angle the line makes with the x-axis?
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m = tan theta
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How do you find a midpoint?
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Add the two x co-ordinates together and half,
Add the two y co-ordinates together and half. |
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How do you find a perpendicular bisector?
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Find the gradient of the given line,
Get the perpendicular gradient, Find the midpoint of the given line, Use perpendicular gradient and midpoint in straight line formula |
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How do you find an altitude?
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Find the gradient of the opposite side from the point given,
Get the perpendicular gradient, Use the perpendicular gradient and the given point in the straight line formula |
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How do you find a median?
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Find the midpoint of the opposite side,
Use that and the given point to get the gradient, Sub the gradient and either point into the straight line formula |
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How do you prove a point lies on a line?
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Sub the x value and y value given into the equation and the left hand side should equal the right hand side
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What does collinear mean?
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Points that lie on the same straight line
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How would you prove that 3 points(A, B and C) are collinear?
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Find the gradient of AB, find the gradient of BC. Gradients should be equal and as they both go through B then they are collinear.
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How do you find where 2 lines cross?
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Simultaneous equations or substitution
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If given the equation 3x-2y+6=0, how do you find the gradient?
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Rearrange the equation to get y= .......... , and the gradient is the number before the x
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f(g(x))
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Use the circle, arrow technique.
Put the g(x) in the circle and arrow it over into the f(x), replacing the x with it. Remember to put it in a bracket when you substitute it in |
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f(f(x))
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Write out f(x) twice and do the circle, arrow technique
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State any restrictions on the domain
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This means your answer will be a fraction with x on the bottom line. You don't want the bottom line to equal zero, so say that x cannot equal that number.
Or it means you have a square root and you never want the bit under the root to be negative. |
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How would you sketch f(x) + 4?
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Moves the graph up 4
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How would you sketch f(x) - 2?
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Moves the graph down 2
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How would you sketch f(x+5)?
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Moves the graph 5 left
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How would you sketch f(x-1)?
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Moves the graph right 1
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How would you sketch f(-x)?
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Reflects the graph in the y axis
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How would you sketch
-f(x)? |
Reflects the graph in the x axis
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How would you sketch 3f(x)?
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Multiply the height only(the y coordinate) of all the points.
The curve will be strecthed |
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How would you sketch f(4x)?
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Divide the x coordinate of all the points by 4.
You'll see more curves |
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How would you sketch
-f(x+3)? |
Move the curve 3 places left then reflect in the x axis
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How do you sketch
2 - f(x)? |
Change to -f(x) + 2.
So reflect in the x axis and then move up 2 |
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Key points of an exponential curve
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Cross the y-axis at 1 and never crosses the x axis
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Key points of a logarithmic curve
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Cross the x axis at 1 and never crosses the y axis.
How far along it goes before reaching a height of 1,is the base number |
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What are radians?
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A different way of measuring angles.
They are linked to the Greek symbol pi |
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How do you change from degrees to radians?
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Put the degrees over 180 and simplify.
Put a pi symbol on the top line |
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How do you change from radians to degrees?
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Change the pi symbol to 180 and do the calculation
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What are the exact value triangles?
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Draw these in
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How would you find the exact value of cos 135?
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cos 135 = - cos 45 because of the symmetry in the cos curve. Use your exact value triangles to find cos 45 and make it negative
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How would you find the exact value of sin 120?
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Because of the symmetry of the curve sin 120 is the same as sin 60, so use your exact value triangles to find sin 60
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acos(bx) + c
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a is the amplitude - (maximum - minimum)/2
b - how many curves in 360 c - the movement up or down |
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How do you write a recurrence relation?
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You need to use the correct notation
To get the a value you need to add or take-away the percentage from 100% and change to a decimal. The b value is the extra add or take-away bit |
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Limit formula
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L = b/(1-a)
Remember the condition! A limit only exists if -1<a<1 |
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If you know a recurrence relation and u zero, how do you find u2?
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Sub u zero into the formula to get u1, then sub u1 into the formula to get u2
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If told u3, how would you find u1?
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Set recurrence relation equal to the u3 value, balance and solve to find u2.
Repeat this process with u2 to find u1 |
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How do you calculate a and b when given 3 consecutive answers to a recurrence relation?
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You set up 2 equations and solve them simultaneously
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How do you differentiate?
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Times by the power, take 1 off the power.
It must be in the correct format first, the x must be on the top line, so might require cancelling or indices work |
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When should you differentiate?
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When you see key words of phrases:
maximum, minimum, increasing, decreasing, stationary, rate of change, the derivative. |
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Differentiation and the gradient
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When you differentiate you get an expression for the gradient. If you sub in an x value you get the gradient at that point. This is used to find the gradient of a tangent to a curve
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Sketching the derived curve
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Mark the x-values of the stationary points on the x-axis of a new grid.
The gradient of the f(x) curve will tell you if you draw lines above or below the x-axis on your new grid |
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How do you find stationary points?
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1. Differentiate
2. Set equal to zero 3. Factorise(if necessary) and solve 4. Get y-values 5.Nature table and statement |
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How do you prove a function is increasing, decreasing or stationary at any given point?
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You differentiate and then sub in the x-value. If the answer is positive, the function is increasing.
If zero, the function is stationary. If negative, the function is decreasing |
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How do you find the maximum and minimum of a function in a given interval?
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1. Differentiate to check for stationary points.
2. Check the limits, so sub their value into the y = formula to see what they give you 3. Make a decision about what the maximum and minimum values are |
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How do you do synthetic division?
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To find the number outside,you put the factor equal to zero and solve.
The coefficients go at the top, but watch out for zeros and negatives! Remember your statement about why it's a factor |
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When do you use synthetic division?
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To factorise harder functions, usually cubics
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How do you find where a curve crosses the x and y axes?
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For the y axis, put x = 0 and solve
For the x axis, put y = 0 and solve, this might involve factorising/synthetic division |
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How do you work out the equation of a function when you know where it crosses the x and y axes?
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Write in this form y = k(x-a)(x-b)
Where a and b are where it crosses the x axis. Sub the y-intercept point into the formula, in place of and b, to find k |
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How do you solve a quadratic inequation?
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Factorise and solve. This allows us to draw the curve as these points are where it crosses the x axis and you decide if it's a happy or sad curve.
For <0, say when it's below the x-axis, for >0, say when it's above the x axis |
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What is the formula for the discriminant?
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b squared - 4ac
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What does the discriminant tell you?
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How many times a curve crosses/touches the x axis.
>0 real and unequal roots(2 times) =0 real and equal roots(once) <0 non-real roots(never) |
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Other uses of the discriminant
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If you put a curve and straight line equal to each other and balance so everything is on 1 side. The discriminant will tell you how many times the curve and line touch each other
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How do you complete the square?
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Half the number in front of the x, it goes in the bracket. Square the number and see what adjustment you have to make to get the number in the question
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How do you complete the square, when you have a number in front of the x squared?
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Still have the number in front of the x, but you then divide it by the number in front of the x squared, it then goes in the bracket. Once you square it, you have to multiply it by the number outside the bracket, before deciding on the adjustment
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Why would you complete the square?
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You'll be able to tell the maximum or minimum of the function very easily, and it helps you to sketch the curve
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How do you integrate?
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Add 1 to the power and divide by the power
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When do you put add C?
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When you are not calculating a shaded area, or when you are not evaluating the integral
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How do you find the area between a curve and a line?
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You do the top - the bottom, and then integrate and sub in the numbers
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What if the limits are not given?
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You put the 2 things equal to each other, balance and solve
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What if you are calculating the area under the x-axis?
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Do as normal, but the final answer will be negative, so change to positive and explain why
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In integration, are we interested in x-coordinates or y-coordinates?
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x-coordinates only
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What do you do if told what dy/dx is and asked for y in terms of x?
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This is differential equations, so integrate to get y =
but remember to write + c. They will give you some information about x and y, so that you can calculate c. |
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What do you do if told that sin x = 3/5?
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Sketch a right-angled triangle and mark in the opposite side is 3 and hypotenuse is 5(SOHCAHTOA).
Use Pythagoras to calculate the unknown side. |
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If told cos x = 5/13, how would you find sin 2x?
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Draw a right-angled triangle and mark in the 5 and 13 on the adjacent side and the hypotenuse. Use Pythagoras to calculate the missing side(12).
Expand sin 2x from the formula sheet(2sinxcosx) and then replace sinx and cosx with what they are. |
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In a trig equation, when do you need to expand?
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You are trying to get sin x = a number/fraction, or
sin 2x = number/fraction. If you can do this by balancing, then don't expand. If you can't do this, then expand the sin2x or cos2x |
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cos2x has 3 expansions, which one should you choose?
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In a trig equation, never choose the top 1!
If you already have a cosx in the question, then choose the expansion that has cosx in it. If you have sinx in the question, then go for the sinx expansion |
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Steps for solving trig equations
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1. Expand, if necessary
2. Factorise and solve 3. Use trig curves or CAST diagram to get more answers 4. Check the limits given in the question |
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What if the answer needs to be in radians but rounded to 1 or 2 decimal places?
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You need to change the calculator to radians and work through the question as normal. For 2nd answers the CAST diagram becomes:
sin 3.14 - answer tan 3.14 + answer cos 6.28 - answer |
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How do you work out the centre and radius from the short equation of the circle?
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For the centre, just change the symbols of the numbers. The radius is the square root of what the formula is equal to
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How do you work out the centre and radius from the expanded equation of the circle?
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Half the numbers in front of x and y and change the symbols.
Follow the radius formula from the formula sheet |
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How do you prove 2 circles touch?
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Calculate the distance between the 2 centre points, using Pythagoras, if the 2 radii add to make this, they touch once.
If the gap between centres is bigger than the 2 radii, then they don't touch. If the gap is smaller, they touch twice. |
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How do you prove something is not a circle?
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g squared + f squared - c must be less than or equal to zero
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How do you prove a line is a tangent to a circle?
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Sub the line into the circle and then either factorise and solve to reveal 1 solution, or use the discriminant and if it is zero, then you have a tangent
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How do you find the equation of the tangent to the circle?
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Remember you are finding the equation of a straight line, so need a gradient and point on the line.
For gradient, find the gradient of the radius, then the perpendicular gradient. Make sure you use a point that lies on the line, not the centre of the circle! |
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How do you find the gradient of a tangent to a curve?
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Differentiate and sub in the x-value to get the gradient
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What is a good strategy for finding out points you're not told?
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Stepping out, but remember to show working
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What do you need to remember when square rooting a number?
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The answer can be positive and negative
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How do you find the distance between 2 points?
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By using the distance formula/Pythagoras
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What is a tangent to a curve?
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A straight line that has an equal gradient with the curve, at a particular point
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What is a tangent to a circle?
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A straight line that touches a circle once
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How do you find the components of the vector AB?
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b - a
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How do you calculate the magnitude(size/length) of a vector?
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Use Pythagoras - square the 3 numbers, add together and square root
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How can you prove if 2 vectors are parallel?
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Show that one is a multiplication of the other
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How do you calculate a unit vector?
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Calculate the magnitude of the vector and multiply the vector by 1 over the magnitude
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What is the section formula?
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A way of finding the coordinates of a point, that lies a given ratio along a line
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How do you use the section formula?
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Add the ratio together, then take the number furthest away from each letter. Take out a common factor and complete the calculation
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What is the scalar product/dot product
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The result of 2 vectors being multiplied together.
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How many ways are there of calculating the scalar product?
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2
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What are the 2 ways of calculating the scalar product?
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a1b1 + a2b2 + a3b3 - Multiply the tops, middles and bottoms of the vectors together and add.
Or the size of vector a times the size of vector b, cos of the angle in between them |
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How do you calculate the angle in between 2 vectors (A VERY POPULAR EXAM QUESTION)
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Rearrange the dot product formula. Remember the vectors must point away from the angle you are calculating
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What must you remember about perpendicular vectors?
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Their dot product is zero
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How do you do this
a.(b+c) ? |
Multiply the bracket like normal
a.b + a.c Now find the dot product of a/b and a/c and add together |
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What is the wave function?
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A way of adding or subtracting a sine and a cos curve. You can combine them to make a new single cos or sine curve, using expansions
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How do you find k and a in
kcos(x-a) |
Expand the bracket and then identify what k cos a and k sin a are. For k, use Pythagoras and for a do tan a = sin a / cos a
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What can the wave function be used for?
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Identifying the maximum and minimum of 4 sinx + 3 cosx and when it occurs.
k will give the max and min |
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When do you use the chain rule?
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When you have a bracket to a power and you have to differentiate
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What is the rule for the quick method?
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Times by the power, take 1 off the power and multiply by the differentiated bracket
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How do you integrate a bracket to a power?
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Add 1 to the power and divide by the power, but you have to multiply the bottom of the fraction by the number in front of the x
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How do you differentiate and integrate sin and cos
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Use the formula sheet!
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What are the 3 laws of logs?
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An add becomes a multiply, a take-away becomes a divide and a number at the front becomes a power. These can also work in reverse.
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How do you simplify a logarithmic function? For example log 8 to the base 2
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You ask yourself 2 to the power of what makes 8 and the answer is 3
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How do you solve logarithmic equations
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You must have logs on both sides, sometimes you have to change the right hand side into a log. Use the laws and then cancel the logs out, before solving
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