All pure formulas

Question 1

Rationalising surds (2)

Answer 1

x / √a = { x / √a } x { √a / √a }

a ± √b has a surd conjugate of a ∓ √b
a + √b has a conjugate of a - √b
a - √b has a conjugate of a + √b

Question 2

Difference of 2 squares

Answer 2

a² – b² = (a + b)(a - b)

Question 3

Perfect squares (2)

Answer 3

a² - 2ab + b² = (a - b)²

a² + 2ab + b² = (a + b)²

Question 4

Completing the square

Answer 4

ax² + bx + c = a(x ± h)² ± k

Question 5

Testing for number of solutions

Answer 5

b² - 4ac
> 0 , 2 real solutions
= 0 , 1 repeated solution
< 0 , no real solutions

Question 6

Sum of roots

Answer 6

-b / a

Question 7

Product of roots

Answer 7

c / a

Question 8

Horizontal asymptote •

Answer 8

y = a/c

Question 9

Vertical asymptote •

Answer 9

x = -d / c

Question 10

Vertex for graph of absolute value function
(in form y = mx + c)

Answer 10

x = -c / m

Question 11

Graphical translations by a translation vector (2)

Answer 11

f(x) ± b translates vertically
f(x ∓ a) translates horizontally

Question 12

Graphical reflections (2)

Answer 12

-f(x) reflects in x-axis
f(-x) reflects in y-axis

Question 13

Graphical stretches (2)

Answer 13

pf(x) stretches vertically by scale factor p
f(qx) stretches horizontally by scale factor 1/q

Question 14

Combined transformation rule

Answer 14

vertical transformations are normal

horizontal transformations (transformations within brackets) are either ∓, reciprocals, or follow reverse order of operations

Question 15

Distance between 2 points

Answer 15

d = √(x₂ - x₁)² + (y₂ - y₁)²

Question 16

Equation of line making angle θ with x-axis

Answer 16

tan(θ) = m

Question 17

Circle equation

Answer 17

(x - p)² + (y - q)² = r²

Question 18

Chain rule

Answer 18

dy/dx = dy/du • du/dx

Question 19

Second derivative

Answer 19

d²y/dx²

Question 20

Vertex in completed square form

Answer 20

a(x - h) + k

where (h, k) is the vertex

Question 21

Hyperbolic curve equation

Answer 21

(ax + b) / (cx + d)

Question 22

Square root function graph equation

Answer 22

y = √ax + b

Question 23

Properties of composite and inverse functions (4)

Answer 23

ff ⁻¹(x) = f ⁻¹ f(x) = x

(f ⁻¹) ⁻¹(x) = f(x)

(fg) ⁻¹(x) = g ⁻¹ f ⁻¹ (x)

f ⁻¹(x) ≠ 1 / f(x)

Question 24

Differentiations rules (2 - 2)

Answer 24

f(x) = xⁿ
f’(x) = n • xⁿ⁻¹

f(x) = axⁿ
f’(x) = n • axⁿ⁻¹

Question 25

Standard form of linear equation

Answer 25

y - y₁ = m(x - x₁)

Question 26

Stationary points

Answer 26

f’(x) = 0

Question 27

Increasing function

Answer 27

f’(x) > 0

Question 28

Decreasing function

Answer 28

f’(x) < 0

Question 29

Points of inflection

Answer 29

f’(x) = 0
f’‘(x) = 0

Question 30

Maximum stationary point

Answer 30

f’‘(x) < 0

Question 31

Minimum stationary point

Answer 31

f’‘(x) > 0

Question 32

Applications of rates of change, calculus (2)

Answer 32

v = s’ = ds/dt

a = v’ = dv/dt

Question 33

Sine rule (2)

Answer 33

sinA/a = sinB/b = sinC/c

a/sinA = b/sinB = c/sinC

Question 34

Cosine rule (2)

Answer 34

a² = b² + c² - 2bc x cosA

cosA = [a² - b² - c²] / -2bc

Question 35

Area of a non-right angle triangle

Answer 35

[bc sin(A)] / 2

Question 36

Radian/Degree conversions

Answer 36

Radian to degrees:
x radian • 180/π

Degrees to radian:
xº • π/180

Question 37

Arc length

Answer 37

S = rθ

Question 38

Area of a sector

Answer 38

A = 1/2r²θ

Question 39

Area of a segment

Answer 39

A = 1/2r²(θ - sinθ)

Question 40

General transformation notation (5)

Answer 40

y = A ___ (Bx - C) + D

where ___ is either; sin, cos, or tan

(4) A enlargement in y direction
(2) B enlargement in x direction
(1) C changes position along x-axis
(3) D changes position along y-axis

Question 41

Trigonometric quadrants

Answer 41

CAST:
Q1: All positive in A (0º - 90º) , [principle values]
Q2: Sin positive in S (90º - 180º) , [180 - principle value]
Q3: Tan positive in T (180º - 270º) , [180 + principle value]
Q4: Cos positive in C (270º - 360º) , [360 - principle value]

Question 42

Trigonometric identity rules (2)

Answer 42

sin²(x) + cos²(x) = 1
tanθ = sinθ/cosθ

Question 43

Integration (2 - 2)

Answer 43

∫ f(x) dx
∫ axⁿ dx = [axⁿ⁺¹ / n + 1] + c where n ≠ -1

∫ k(ax + b)ⁿ dx
{ [ k(ax + b)ⁿ⁺¹ ] / a(n + 1) } + c

Question 44

Properties of indefinite integrals (3)

Answer 44

∫[ f(x) + g(x)] dx = ∫ f(x) dx + ∫ g(x) dx

∫[ f(x) - g(x)] dx = ∫ f(x) dx - ∫ g(x) dx

∫ kf(x) dx = k x ∫ f(x) dx , where k is a constant

Question 45

Definite integral

Answer 45

∫ ᵇₐ f’ (x) dx = [f(x)ᵇₐ] = f(b) - f(a) where b > a

Question 46

Properties of definite integral

Answer 46

∫ᵇₐ f(x) dx = - ∫ᵃᵦ f(x) dx where b > a

∫ ᵃₐ f(x) dx = 0

∫βₐ f(x) dx + ∫ᶜᵦ f(x) dx = ∫ᶜₐ f(x) dx

Question 47

Area enclosed by a curve and the x-axis

Answer 47

∫ᵇₐ f(x) dx where b > a

Question 48

Area above and below the x-axis

Answer 48

∫ᵇₐ f(x) dx + | ∫ᶜᵦ f(x) dx | where c > b > a

Question 49

Area enclosed by a curve and the y-axis

Answer 49

∫ᵇₐ f(y) dy where b > a

Question 50

Area bounded by a curve and a line or by two curves

Answer 50

A = ∫ᵇₐ [top function] dx - ∫ᵇₐ [bottom function] dx
where b > a

Question 51

Volumes of solids around x-axis, y-axis

Answer 51

π ∫ᵇₐ f(x)² dx ,
π ∫ᵇₐ f(y)² dy

Question 52

Binomial expansion; term bʳ, term aʳ

Answer 52

(nCr)aⁿ⁻ʳbʳ

[nC(n-r)]aʳbⁿ⁻ʳ

Question 53

Geometric sequence

Answer 53

t₁ = a
t₂ = ar
t₃ = ar²
t₄ = ar³

r = tₙ₊₁ / tₙ