Unit 1 Flashcards
What is the discriminant?
b² - 4ac
Two distinct real roots?
b² - 4ac > 0
Two real roots?
b² - 4ac => 0
One real root?
b² - 4ac = 0
No roots?
b² - 4ac < 0
What is remainder theorum?
When f(x) is divided by the polynomial x - a, the remainder of the calculation is f(a)
What is factor theorum?
If f(a) = 0 then x - a is a factor of the f(x)
What are the types of proof?
- proof by deduction
- proof by exhaustion
- disproof by counter example
What is the length of a line?
√(x2 - x1)² + (y2 - y1)²
What are the two equations for lines?
y = mx + c
y - y1 = m(x - x1)
What is the midpoint of a line?
(y1 + y2 / 2, x1 + x2 / 2)
What is the gradient of a line?
y2 - y1 / x2 - x1
What is the sine rule?
a/sinA = b/sinB = c/sinC or vice versa
What is the cosine rule?
a² = b² + c² - 2bccosA
or cosA = b² + c² - a² / 2bc
What is the area of a triangle?
1/2 ab sinC
What are the circle theorems?
- the angle in a semi circle is a right angle
- the perpendicular from the centre to a chord bisects the chord
- the radius of a circle at a given point on its circumference is perpendicular to the tangent to the circle at that point
What are the trig identities?
sin²θ + cos²θ = 1
sin²θ = 1 - cos²θ
tanθ = sinθ/cosθ
What is the symbol for “is a member of”?
A big curved E
What are the symbols for the set of all integers and the set of all positive integers?
A big Z and a big Z with a +
What is the symbol for the set of all real numbers?
A big R
What does it mean when numbers are in curly brackets?
That all the numbers within it are included e.g. {1,2,3,4,5} 1, 2, 3, 4 and 5 are all included.
What does it mean when numbers are in square brackets?
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.
What does it mean when numbers are in regular brackets?
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.
What are the log laws?
- log(xy) = log(x) + log(y)
- log(x/y) = log(x) - log(y)
- nlog(x) = log(x^n)
What are concentric circles?
Circles that have the same centre point.
How can you prove that 2 circles don’t touch?
d > r1 + r2
How can you prove that 2 circles touch externally?
d = r1 + r2
How can you prove that 2 circles meet at 2 distinct points?
d < r1 + r2
How can you prove that two circles touch internally?
d = r1 - r2
How can you prove that 2 circles don’t touch, but one is within the other?
d < r1 - r2
How do you know when to use the sine rule?
- when there are 2 angles and 1 side
- when there are 2 sides and a non included angle
How do you know when to use the cosine rule?
- when there are 3 sides
- when there are 2 sides and an included angle
What transformation is f(x+a)?
Horizontal translation of f(x) (negative is right, positive is left)
What transformation is f(x) + a?
Vertical translation of f(x) (positive up, negative down)
What transformation is f(ax)?
Horizontal stretch of a scale factor of 1/a, so you multiply the x co-ords by 1/a
What transformation is af(x)?
A vertical stretch of a scale factor of a, so you multiply the y co-ords by a
What transformation is f(-x)?
Change the x co-ords, reflection in the y axis
What transformation is -f(x)?
Change in y co-ords, reflection in the x axis
How do you differentiate using first principles?
f(x+h) - f(x) / h
How do you differentiate?
- multiply power by coefficient of x
- decrease the power by 1
How do you integrate?
- increase the power by 1
- divide by the new power
- add + c (if needed)
What are differentiation and integration used for?
Differentiation: finding the gradient of a curve
Integration: finding the area under a curve
What is a normal?
A line perpendicular to a tangent
How do you determine whether a stationary point is minimum or maximum?
If the second differential > 0, it is a minimum.
If the second differential < 0, it is a maximum.
