Download - Maths Unit 3 revision
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Reverse cosine rule: CosA = b + c - a/2bc
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Express x-6x-7=0 as a completed square and hence solve it
1) First, write out the first bracket (x + b/2)(x - 3)
2) Multiply out the brackets and compare to the original
(x - 3)(x - 3) = x - 6x + 9. You need to change the 9 into -7. So you have to minus 16
So now you have (x - 3) - 16 = o
3) Rearrange to make the squared bracket the subject
(x - 3) = 16
4) Square root both sides
x - 3 = 16
5) Rearrange to make x the subject and then solve
x = 3 16
x = 3 + 16 x = 7
x = 3 - 16 x = -1
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2x + 16x 3
1) Factorise out the 2
2(x + 8x) 3
2) Continue with the normal method
2((x + 4)-16) 3 (The negative 16 is cancel out the 2x4)
2(x + 4) - 35 (The negative 35 comes for the 2 x -16 = -32. -32 3 = -35
2(x + 4) = 35
2(x + 4) = 35
x + 4 = 35/2
x = -4 35/2
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Shade the region represented by: x + y < 5, y > x +2 and y >= 1
1) Convert each of the inequalities into an equation x + y < 5 becomes x + y = 5 and then y = -x + 5
y > x + 2 becomes y = x + 2
y >= 1 becomes y = 1 Not equal to
Equal to
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y-Stretch: y = kf(x)
Examples: y = 3f(x), y = 2x
You multiply the y coordinate by k
So if its y=3f(x) then (4,3) becomes (4,9)
y-Shift: y = f(x) + a
Examples: y = f(x) + 3, y = sinx + 2
You add a to each y coordinate
So if its y = f(x) + 4 then (4,3) becomes (4,7)
x-Sretch: y = f(kx)
Examples: y = f(3x), y = cos(4x)
You multiply the x coordinate by the reciprocal of k
So if its y = f(2x) then (4,3) becomes (2,3)
x-Shift: y = f(x
a)
Examples: y = f(x + 4), y = 1/3(x 2)
You add/subtract the opposite of a to the x coordinate
So if its y = f(x 2) then (4,3) becomes (6,3)
If its y = f(x + 1) then (4,3) becomes (3,3)
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If you have to find the distance between 2 points on a graph, use Pythagoras.
Example:
(-4,8)
(8,3)
12
5
5 + 12 = 169
169 = 13
13
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A
B
Angle of
Depression
Angle of
Elevation
The angle of Depression is the angle
downwards from the horizontal.
The angle of Elevation is the angle
upwards from the horizontal.
The angle of Depression is equal to
the angle of Elevation
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7m
83
53
Two angles + 1 side: Sine rule
53
8m7m
Two sides + an angle NOT enclosed by
them: Sine rule
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83
8m7m
Two sides + angle ENCLOSED by
them: Cosine rule
7m8m
10m
All 3 sides + no angles: Reverse Cosine rule
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a
a
b
bx y
Made 2 isosceles triangles.
2a + x = 180
2b + y = 180
x + y = 180
360 = x + y + 2(a + b)
360 = 180 + 2(a + b)
180 = 2a + 2b90 = a + b
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The line that cuts a chord directly inhalf (at 90 degrees) will go through the
centre of the circle and therefore be
the diameter
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a
a
b
b
All triangles drawn from a chord will
have the same angle where they touchthe circle. Also, two angles on the
opposite side of the chord = 180
a + b = 180
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An angle made at the centre from achord is always double the angle
made at the circumference.
Therefore the angle is double both of
the other angles. So the other angles
have to be equal
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a
2a
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x
a
a
b
by
z
180 = 2a + y
180 = 2b + z
360 = y + z + x
z + y = 360 - x
360 = 2(a + b) + 360 x
0 = 2(a + b) xx = 2( a + b)
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a b
c
d
a + c = 180
d + b = 180
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b
b
The angle between a tangent and
a chord that meet is equal to the
angle in the opposite segment
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Boat
10,000N
15,000N
20
40
Overall force from the two
tugs?
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Boat
10,000N
15,000N
20
40
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10,000N
15,000N
20
40120
x
x = 10,000 + 15,000 - (2 x 10,000x 15,000 x Cos120)
x = 21,794.49472N
a
15,000 x Sin120 / 21,794.49472
= 36.58677555.
a = 36.58677555b
40
40 36.58677555
= 3.413224451
B = 3.413224451
Boat3.4