Pure Chapters 8-14 Flashcards
nCr = ?
n! / r!(n-r)!
What is the general term in the expansion of (a+b)^n
(n r) a^(n-r) b^r
Which row of Pascal’s triangle gives the coefficients in the expansion of (a+b)^n
(n+1)th row
What is the cosine rule?
a^2 = b^2 + c^2 -2bc cos(A)
What is the sine rule?
a / sin(A) = b / sin(B) = c / sin(C)
How do you find the two possible angles from the sine rule?
sinθ = sin(180-θ)
What is the area of a triangle?
1/2 ab sin(C)
What is a periodic graph?
One that repeats itself after a certain interval
Which trigonometric ratios are positive in the first quadrant?
sin θ - positive
cos θ - positive
tan θ - positive
Which trigonometric ratios are positive in the second quadrant?
sin θ - positive
cos θ - negative
tan θ - negative
Which trigonometric ratios are positive in the third quadrant?
sin θ - negative
cos θ - negative
tan θ - positive
Which trigonometric ratios are positive in the fourth quadrant?
sin θ - negative
cos θ - positive
tan θ - negative
Trigonometric Ratios - sin 30°
1/2
Trigonometric Ratios - sin 45°
1/✔️2
Or ✔️2 / 2
Trigonometric Ratios - sin 60°
✔️3 / 2
Trigonometric Ratios - cos 30°
✔️3 / 2
Trigonometric Ratios - cos 45°
1 / ✔️2
Or ✔️2 / 2
Trigonometric Ratios - cos 60°
1/2
Trigonometric Ratios - tan 30°
1 / ✔️3
Or ✔️3 / 3
Trigonometric Ratios - tan 45°
1
Trigonometric Ratios - tan 60°
✔️3
What are the two trigonometric identities? (2)
sin^2(θ) + cos^2(θ) = 1
tan θ = sin θ / cos θ
For what values of k do solutions for sin θ = k and cos θ = k exist?
-1 < k < 1
For what values of p do solutions for tan θ = p exist?
All values of p
What is the principal value?
The angle you get from your calculator when using inverse trigonometric functions
If PQ = RS, what can be said for the line segments PQ and RS? (2)
Equal in length
Parallel
Why does AB = -BA? (3)
AB is equal in length to BA
AB and BA are parallel
AB is in the opposite direction to BA
What is the triangle law for vector addition in terms of AB, AC and BC?
AB + BC = AC
What is the equivalent to subtracting a vector?
Add the negative vector
a - b = a + (-b)
PQ + QP = ?
0
How can a parallel vector be written?
Any vector parallel to a can be written as λa where λ is a non-zero scalar
How do you find the magnitude of a vector (x y)?
|a| = ✔️(x^2 + y^2)
How can a unit vector in the direction of a be written?
a / |a|
If a and b are two non-parallel vectors and pa + qb = ra + sb, What can be said for p, q, r and s? (2)
p = r q = s
What can the gradient function be used for?
Finding the gradient of the curve for any value of x
When is the function f(x) increasing?
On the interval [a,b] if f’(x) >/ 0 where a < x < b
When is the function f(x) decreasing?
On the interval [a,b] if f’(x) < 0 where a < x < b
What is a stationary point?
Any point on the curve y = f(x) where f’(x) = 0
How does f”(a) show the nature of a stationary point? (3)
If f”(a) > 0 - local minimum
If f”(a) < 0 - local maximum
If f”(a) = 0, the point is either a local minimum, maximum or point of inflection
Sketching gradient functions - maximum or minimum on y = f(x)
y = f’(x) - cuts the x-axis
Sketching gradient functions - point of inflection on y = f(x)
y = f’(x) - touches the x-axis
Sketching gradient functions - positive gradient on y = f(x)
y = f’(x) - above the x-axis
Sketching gradient functions - negative gradient on y = f(x)
y = f’(x) - below the x-axis
Sketching gradient functions - vertical asymptote on y = f(x)
y = f’(x) - vertical asymptote
Sketching gradient functions - horizontal asymptote on y = f(x)
y = f’(x) - horizontal asymptote at the x-axis
