AQA Maths Notes (complete)

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Maths Unit 3
Solving equations with graphs: The solution is always where the two graphs cross ­ use it to solve
simultaneous equations
Quadratic formula:
ax^2+bx+c = 0
X= (-b +/- sqrt (b^2 - 4ac)) / 2a
E.g 3x^2 +7x =1
3x^2+ 7x ­ 1 = 0
A= 3 b=7 c=-1
(-7 +/- sqrt (49 +12)) / 6
(-7 +/- 7.81) / 6
= -14.81 / 6
= 0.81 / 6
If it says proportional it means multiply
Inversely proportional to means divide
E.g y is proportional to the square of x = y = kx^2
d varies with the cube of t = d=kt^3
t is proportional to the square root of h = t= k sqrt h
v is inversely proportional to r^3 = v=k/r^3
The time (t) taken for a duck to fall down a chimney is proportional to the square of the diameter (d)
of the chimney
If she took 25secs to descend a chimney with a diameter of 0.3m ­ how long would it take her to fall
down a 0.2 diameter?
25 = k * 0.09
K = 25/0.09 = 278
T= 278 * 0.2^2 = 11.12secs

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Practice drawing graphs and quadratic graphs
Use the quadratic formula
Find the second formula to go on the graph
You have the graph y = 2x^2 + 3. Solve the equation 2x^2 ­ 3x = 4
1. 2x^2 -3x -4 = 0 (make the solution you are trying to solve equal 0)
2.…read more

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An equilateral triangle has 3 lines of symmetry and 3 rotational symmetry
Isosceles triangle has one line of symmetry and 1 rotational symmetry
Right angled triangle (unless it is a right angled isosceles triangle) has no lines of symmetry and 1
rotational symmetry
A square has 4 lines of symmetry and 4 rotational symmetry
A rectangle has 2 lines of symmetry and 2 rotational symmetry
A unequal parallelogram has 0 lines of symmetry and 2 rotational symmetry.…read more

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Front elevation ­ side elevation and plan (see pg 124)
Surface area of one end x length (for cylinders etc.)
Volume of sphere : 4/3 r3
Area of cone: 1/3 r2x h
Pyramid: 1/3 base area x h
Frustum of a cone: cone with top cut off (volume of whole cone- top part)
Angles of polygons ­ lines of symmetry and order of rotational symmetry is equal to number of sides.…read more

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The a locus of points that are equidistant from two different lines is an angle bisector
(a line)
o The locus of a point that are from two given points ­ a perpendicular bisector (a line)
o Constructing 60° angles ­ also needed for constructing equilateral triangles
Step one: Draw initial base line
Step two: Put your compass on the end of the line and draw two arcs, one on
the line and one at around 60°
Step three: Put your compass point on…read more

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Step two: From the point with a compass draw an arc on either side of the
line ­ through the line
Step three: From each of the arcs through the line draw an arc above the line
Step four: From the centre of the arcs to the point draw the perpendicular
Conversion factors
Converting between two measurements
Step one: Find the conversion factor e.g.…read more

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In this equation the vector v was added to the vector u. When v was added it
was `stuck' on the end of vector u. From the `start' point (indicated) to the end
point (indicated as `stop') another vector (u+v) is drawn.
E.g. In this vector equation I am adding vector u + vector v + vector w
o First I draw vector u as it is the first in the equation.…read more

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The second arrow down (2u) is where vector u is multiplied by 2 (a scalar) ­ this
means that it keeps its direction but it doubles in length (so becomes 2u).
The third arrow down is where vector u is multiplied by 1/2 (a scalar), this means
that it becomes half its length (1/2 u)
The fourth arrow down is when you multiply a vector by a negative scalar.…read more

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E.g. Enlarge wxyz with the scale factor of -2, centred around the origin
c. For this example you need to calculate the distance from a point to the centre of
enlargement (x to 0,0) then times it by the scale factor and, continuing the line,
measure that distance from the centre of enlargement and plot the point (and call it
x'). Do this with all the points and draw the enlargement.
3. Rotation
a.…read more

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A transformation, a reflection and a rotation would always give something congruent
An enlargement would always give something similar
Areas and volumes on enlargement
With a scale factor of n
o Sides are n times as long
o Areas are n^2 times as big
o Volumes are n^3 times as big
If two shapes are similar and you know the ratio of their area, volume or sides then you can work
out the ratio of anything else
o E.G.…read more


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