Maths Unit 3 revision

7/31/2019 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

Maths Unit 3 revision

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