This is a revision of the basics and formulas that may be useful, enjoy.
Things that we may need to know that may or may not supplied (from what I've seen used in the VCAA exams) include:
PLEASE FEEL FREE TO ADD THINGS + CORRECT ANY ERRORS!Area of a segment(radians):
)
may be helpful when a line intersects a circle 'find the shaded area'
Area of a sector(radians):
Area of a sector(degrees):
(R = radius)
Supplementary angles (made this one up from some other formula):
where
Don't know what this is may be used for but:
where
Complex numbers:Typical complex number circle formulas: (good to know, as these questions pop up in multiple choice 'which one does not represent a circle in an Argand diagram')
, centre
)
radius
(\bar{z}-a+bi)=R^2)
, centre
radius
'Arg(z)' laws:
\leq\pi)
 = Arg(z_1) - Arg(z_2))
where
, Arg(z_2) \in Arg(z))
 = Arg(z_1) + Arg(z_2))
If a polynomial has all real coefficients, then roots contain even numbers of Imaginary (eg. if
) (Conjugate root theorem) else does not apply.
Linear:Midpoint
, (x_2, y_2): (\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}))
Bisector:
i.e. line perpendicular of the midpoint between points
and
)
from positive x-axis
Cubic factorisation:
(a^2-ab+b^2))
(a^2+ab+b^2))
Vectors:Scalar resolute of
Vector resolute of
\Hat{\tilde{b}})
Vector resolute of
\Hat{\tilde{b}})
Angle between two vectors:
Speed =
where r is the position vector
Tip: In questions that have 3-Dimensional vectors,
being the altitude, let the component of
to find when the particle 'hits' the ground
Linear dependence:
If one vector can be expressed by a scalar multiple of the others, then the vectors are linearly dependent.
ie.
and all 'k' values equal 0 then the vectors are linearly independent
Calculus:If
then
Newton's Law of Cooling:
, T_s)
= Temperature of surroundings
Partial fractions:
Type 1: Linear Factors
(cx+d)} = \frac{A}{ax+b} + \frac{B}{cx+d})
Type 2: Repeated Factor
^2} = \frac{A}{ax+b} + \frac{B}{(ax+b)^2})
Type 3: Irreducible quadratic factors
Inverse circular functions domain and range restricting:
eg. 'State the domain and range of
)+d)
Start with the 'original' domain and range
Domain:
Substitute x for whatever's inside the inverse function, i.e.
)
 \leq 1)
Simple rearrangement to find domain:
Range(slightly different approach):
, Let
= x for simplicity
 \leq \pi)
Observe the given equation; first multiply by
then add
 \leq \pix\frac{a}{\pi})
 \leq \pix\frac{a}{\pi}+d)
 \leq a+d)
Kinematics:(only for constant acceleration)
If the sign (i.e. + or -) of the acceleration = the sign of the velocity, the particle is speeding up
If the sign (i.e. + or -) of the acceleration
the sign of the velocity, the particle is slowing down up
Distance = speed x time
Average speed = total distance travelled / time (only positive)
Average velocity = displacement / time (can be positive or negative)
Instantaneous velocity:
, v)
In a velocity-time graph the DISTANCE = the area under the curve (if under x-axis then take the negative)
DISPLACEMENT = Add positive areas, subtract negative areas
eg.
velocity
|
|
|
\|
\|
\|
\ /ŻŻ| Distance = A1 + A2 + A3 (magnitude)
|
A1\ /A3| Displacement = A1 - A2 + A3
-----
\-------
/---------------time
|
\_A2_/|
|
MechanicsChanging momentum
, m is mass in kg, v is velocity
Friction: Friction is greatest when sliding is about to occur. When the body is about to slide down, its state is known as 'limiting equilibrium'.
Limiting friction force:
Friction opposes motion, however it cannot exceed
.
When motion is not about to occur
Recognising which method for integral calculus:
1) Divide first if possible
2) If the numerator is the derivative of the denominator -> log recognition
3) Can you factorise the denominator? Use partial fractions
4) Inverse circular functions
5) Using substitution (change of variable)
Slope fields:If
is in terms of x, the gradients follow the y-axis (i.e. on a point x, there is only one gradient)
If
is in terms of y, the gradients follow the x-axis (i.e. on a point x, there is more than one gradient)
Nature of a graph f(x)=y, some questions ask for 'show that point P is a maximum' etc. (imagination required)
increasing concave up increasing concave down positive P.O.I
| /ŻŻ /ŻŻ
__/ | __/ notice: P.O.I is has the steepest gradient
decreasing concave up decreasing concave down negative P.O.I
| ŻŻ\ ŻŻ\
\__ | \__
local minimum local maximum stationary point of inflection
\ / /ŻŻ\ __/ \__
\__/ / \ / or \
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