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maths as > Unit 1 > Flashcards

Unit 1 Flashcards

1
Q

What is the discriminant?

A

b² - 4ac

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1
Q

Two distinct real roots?

A

b² - 4ac > 0

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2
Q

Two real roots?

A

b² - 4ac => 0

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3
Q

One real root?

A

b² - 4ac = 0

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4
Q

No roots?

A

b² - 4ac < 0

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5
Q

What is remainder theorum?

A

When f(x) is divided by the polynomial x - a, the remainder of the calculation is f(a)

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6
Q

What is factor theorum?

A

If f(a) = 0 then x - a is a factor of the f(x)

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7
Q

What are the types of proof?

A
  • proof by deduction
  • proof by exhaustion
  • disproof by counter example
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8
Q

What is the length of a line?

A

√(x2 - x1)² + (y2 - y1)²

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9
Q

What are the two equations for lines?

A

y = mx + c
y - y1 = m(x - x1)

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10
Q

What is the midpoint of a line?

A

(y1 + y2 / 2, x1 + x2 / 2)

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11
Q

What is the gradient of a line?

A

y2 - y1 / x2 - x1

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12
Q

What is the sine rule?

A

a/sinA = b/sinB = c/sinC or vice versa

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13
Q

What is the cosine rule?

A

a² = b² + c² - 2bccosA
or cosA = b² + c² - a² / 2bc

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14
Q

What is the area of a triangle?

A

1/2 ab sinC

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15
Q

What are the circle theorems?

A
  1. the angle in a semi circle is a right angle
  2. the perpendicular from the centre to a chord bisects the chord
  3. the radius of a circle at a given point on its circumference is perpendicular to the tangent to the circle at that point
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16
Q

What are the trig identities?

A

sin²θ + cos²θ = 1
sin²θ = 1 - cos²θ
tanθ = sinθ/cosθ

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17
Q

What is the symbol for “is a member of”?

A

A big curved E

18
Q

What are the symbols for the set of all integers and the set of all positive integers?

A

A big Z and a big Z with a +

19
Q

What is the symbol for the set of all real numbers?

A

A big R

20
Q

What does it mean when numbers are in curly brackets?

A

That all the numbers within it are included e.g. {1,2,3,4,5} 1, 2, 3, 4 and 5 are all included.

21
Q

What does it mean when numbers are in square brackets?

A

That the numbers from the first number to the second number are included, including the first and second number e.g. [1,5] numbers 1-5 are included as well as 1 and 5.

22
Q

What does it mean when numbers are in regular brackets?

A

That the numbers from the first number to the second number are included, NOT including the first and second number e.g. (1,5) numbers 1-5 are included MINUS 1 and 5.

23
Q

What are the log laws?

A
  1. log(xy) = log(x) + log(y)
  2. log(x/y) = log(x) - log(y)
  3. nlog(x) = log(x^n)
24
Q

What are concentric circles?

A

Circles that have the same centre point.

25
Q

How can you prove that 2 circles don’t touch?

A

d > r1 + r2

26
Q

How can you prove that 2 circles touch externally?

A

d = r1 + r2

27
Q

How can you prove that 2 circles meet at 2 distinct points?

A

d < r1 + r2

28
Q

How can you prove that two circles touch internally?

A

d = r1 - r2

29
Q

How can you prove that 2 circles don’t touch, but one is within the other?

A

d < r1 - r2

30
Q

How do you know when to use the sine rule?

A
  • when there are 2 angles and 1 side
  • when there are 2 sides and a non included angle
31
Q

How do you know when to use the cosine rule?

A
  • when there are 3 sides
  • when there are 2 sides and an included angle
32
Q

What transformation is f(x+a)?

A

Horizontal translation of f(x) (negative is right, positive is left)

33
Q

What transformation is f(x) + a?

A

Vertical translation of f(x) (positive up, negative down)

34
Q

What transformation is f(ax)?

A

Horizontal stretch of a scale factor of 1/a, so you multiply the x co-ords by 1/a

35
Q

What transformation is af(x)?

A

A vertical stretch of a scale factor of a, so you multiply the y co-ords by a

36
Q

What transformation is f(-x)?

A

Change the x co-ords, reflection in the y axis

37
Q

What transformation is -f(x)?

A

Change in y co-ords, reflection in the x axis

38
Q

How do you differentiate using first principles?

A

f(x+h) - f(x) / h

39
Q

How do you differentiate?

A
  • multiply power by coefficient of x
  • decrease the power by 1
40
Q

How do you integrate?

A
  • increase the power by 1
  • divide by the new power
  • add + c (if needed)
41
Q

What are differentiation and integration used for?

A

Differentiation: finding the gradient of a curve
Integration: finding the area under a curve

42
Q

What is a normal?

A

A line perpendicular to a tangent

43
Q

How do you determine whether a stationary point is minimum or maximum?

A

If the second differential > 0, it is a minimum.
If the second differential < 0, it is a maximum.

maths as (2 decks)

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