(Almost?) Everything a 2 Unit Student Needs For The HSC Examination (An Unofficial List of Things to Memorise) Probability
Probability of an event Real functions Even function
Odd function Circles Centre (0,0) Centre (h, k )
P(event)
number of favourable outcomes total number of outcomes
f ( x) f ( x) for all values of x in the domain f ( x) f ( x) fo for all values of x in the domain 2 2 2 x y r
( x h)2 ( y k )2 r 2
Trigonometry
Sine ratio
sin
opposite side
Cosine ratio
cos
adjacent side
Tangent ratio
tan
opposite side
hypotenuse hypotenuse adjacent side sin sin
30
Exact values
Sine rule
Cosine rule
45
60
a
1
6
2
2
1
1
4
2
3
3
2
1 3
1
1 2
2
b
tan tan
3
c
sin A sin B sin C c2 a 2 b2 2ab cos C or cos C tan
Identities
3
cos
a 2 b2 c 2
2ab sin
cos sin 2 cos2 1 1 cot 2 cosec2 tan 2 1 sec2
Linear functions and lines
x1 x2 y1 y2 , 2 2
Midpoint between two points
M
Distance between two points
2 2 d ( x2 x1 ) ( y2 y1 )
Perpendicular distance from a point to a line
d
ax1 by1 c a 2 b2 "#$% '
(Almost?) Everything a 2 Unit Student Needs For The HSC Examination (An Unofficial List of Things to Memorise)
m
Gradient of an interval
m
rise run y2 y1 x2 x1
or m tan
Gradient-intercept form of line
y mx b
General form of line
ax by c 0
Point-gradient formula
y y1 m( x x1 )
Two-point formula
y y1
x x1
y2 y1 x2 x1
The equation of a line passing through the point of intersection of two lines
a1 x b1 y c1 k (a2 x b2 y c2 ) 0
Parallel lines
m1 m2
Perpendicular lines
m1 m2 1
Series and applications
The nth term of an arithmetic series
Tn a (n 1)d
Sum to n terms of an arithmetic series
Sn
The nth term of a geometric series
n
2a (n 1)d or
2 n Tn ar 1
a(r 1)
Limiting sum of a geometric series Compound interest
S n
n
2
a l
a(1 r )
n
Sum to n terms of a geometric series
Sn
n
or S n r 1 a , if r 1 S 1 r r An P 1 100
1 r
n
The tangent to a curve and the derivative of a function
Differentiation using first principles
If y f x , then
Derivative of x n
If y x , then
Derivative of function of a function
n
dy
dy dx
f ' ( x) lim h 0
f x h f x h
nxn1
dx If y F (u ), where where F is any differentia differentiable ble function function
then
dy dx
F ' (u)
Derivative of product of functions
If y uv, then
Derivative of quotient of functions
u
If y
v
, then
du
dx dy
dx dy dx
u v
dv dx du dx
v u v
du dx dv dx
2
"#$% (
(Almost?) Everything a 2 Unit Student Needs For The HSC Examination (An Unofficial List of Things to Memorise) The quadratic polynomial and the parabola
The quadratic function
2 y ax bx c
If ax 2 bx c 0, 0, then
Solution of a quadratic equation
x
b b2 4ac 2a
b
Axis of symmetry of parabola
x
Discriminant
b2 4ac
Sum of roots of quadratic equation
Product of roots of quadratic equation
Parabola with vertex (h, k ) and axis of symmetry parallel to the y -axis Parabola with vertex (h, k ) and axis of symmetry parallel to the x -axis Integration
b a
c a
( y k )2 4a( x h)
A
Trapezoidal rule (multiple applications)
A
Simpson’s rule (one rule (one application)
A
Simpson’s rule (multiple applications)
A
ba
2 h
2
f (a) f (b)
y0 2 y1 2 y2 2 y3 2 y4 .......... yn
ba
ab f (b) f ( a) 4 f 6 2
h
3
y0 4 y1 2 y2 4 y3 2 y4 .......... yn x
n 1
x dx n 1 C n
n
Integral of (ax b)
2a
( x h)2 4a( y k )
Trapezoidal rule (one application)
Integral of x
, a0
n
(ax b) dx n
b
Area between two curves
A A
Volume of revolution around x - axis
A
Area between a curve and x - axis Area between a curve and y - axis
Volume of revolution around y around y - axis
A
a b
a b
a
(ax b)n 1 a(n 1)
C
y dx x dy
( f ( x) g ( x))dx
A
b
a b
a
2
y dx 2
x dy
"#$% )
(Almost?) Everything a 2 Unit Student Needs For The HSC Examination (An Unofficial List of Things to Memorise) Logarithmic and exponential functions
Exponents and logarithms
a x b x loga b
Change of base law
logb a
logc a logc b
log a xy loga x loga y log a
Identities
x
log a x loga y
y
log a xc c log a x Derivative of e x Derivative of e f ( x ) Integral of e x Integral of eaxb Derivative of loge x Derivative of loge f ( x) Integral of Integral of
1
x f ' ( x) f ( x)
If y e , then then x
dy
ex
dx dy f ( x ) f ' ( x)e f ( x) If y e , then dx
e dx e C 1 e d x e a x
x
ax b
ax b
C
If y log e x ln x, then
dy dx
1 x
If y loge f ( x) ln f ( x), then
1
x dx ln x C,
dy dx
f ' ( x) f ( x)
x0
'
f ( x)
f ( x) dx ln f ( x) C,
x0
Trigonometric functions
Derivative of sin x
If y sin sin x, the then
Derivative of sin ax
If y sin ax, then
Derivative of sin f ( x)
cos x Derivative of cos
cos ax Derivative of cos Derivative of cos f ( x) Derivative of tan x Derivative of tan ax Derivative of tan f ( x)
dy
cos x
dx dy
a cos ax dx dy f ' ( x) cos f ( x) If y sin f ( x), then dx dy sin x If y cos x, then dx dy a sin ax If y cos ax, then dx dy f ' ( x) sin f ( x) If y cos f ( x), then dx dy 2 sec If y tan tan x, the then sec x dx dy a sec2 ax If y tan ax, then dx dy f ' ( x) sec 2 f ( x) If y tan f ( x), then dx "#$% *
(Almost?) Everything a 2 Unit Student Needs For The HSC Examination (An Unofficial List of Things to Memorise)
Integral of sin x Integral of sin (ax b) Integral of cos x Integral of cos (ax b) Integral of sec2 x Integral of sec2 (ax b)
sin x dx cos x C 1 a x b d x s i n ( ) cos( ax b) C a cos x dx sin x C 1 a x b d x c o s ( ) sin( ax b) C a sec x dx tan x C 1 a x b d x s e c ( ) tan( ax b) C a 2
2
Applications of calculus to the physical world D S Average speed T
Exponential growth and decay
N Aekt can be used as the solution of
dN dt
kN
"#$% +
(Almost?) Everything a 2 Unit Student Needs For The HSC Examination (An Unofficial List of Things to Memorise)
Table 2: Number plane graphs that NSW Mathematics (2 Unit) students are required to draw by hand or recognise from memory horizontal lines y 1
square root
vertical lines
exponential
sine curve
y
x 1
x
y sin x
cosec curve y cosec x
y e x
oblique lines y x
logarithm y loge x or y ln x
cosine curve y cos x
absolute value y x
hyperbola
secant curve y sec x
parabola
y
1
x
circle y x
2
y x
3
cubic
2 2 x y 1
semicircle y 1 x
2
tangent curve y tan x
cotangent curve y cot x
"#$% ,
(Almost?) Everything a 2 Unit Student Needs For The HSC Examination (An Unofficial List of Things to Memorise)
Table 3: Plane geometry g eometry definitions, theorems, theorems, facts, properties and terminology Angles, including those associated with parallel lines and transversals
The sum of the angles in a right angle is 90o. The sum of the angles in a straight strai ght angle is 180o. Three points are collinear if they form a straight angle. The sum of angles about a point is 360 o. When two lines meet, vertically opposite angles are equal. Alternate angles on parallel lines are equal. Corresponding angles on parallel lines are equal. Co-interior angles on parallel lines are supplementary. Two lines are parallel if a pair of alternate angles are equal. Two lines are parallel if a pair of corresponding angles are equal. Two lines are parallel if a pair of co-interior angles are supplementary. supplementary. If a family of parallel lines cuts equal intercepts on one transversal, then it does so on all transversals. Parallel lines preserve ratios of intercepts on transversals. Triangles
The longest side in a triangle is opposite the largest angle and the shortest side is opposite the smallest angle. For a triangle to exist, the sum of the two shorter sides in a triangle tria ngle must be greater than the longest side. The interior angle sum of a triangle is 180 o. A scalene triangle is a triangle with no two sides equal in length. An isosceles triangle is a triangle with two sides equal in length. An equilateral triangle is a triangle with all three sides equal in length. The exterior angle of a triangle is equal to the sum of the opposite interior angles.
"#$% -
(Almost?) Everything a 2 Unit Student Needs For The HSC Examination (An Unofficial List of Things to Memorise)
The angles opposite equal sides of a triangle are equal. The sides opposite equal angles of a triangle ar e equal. All angles at the vertices of an equilateral triangle are 60o. Two triangles are congruent if three sides of one t riangle are equal to three sides s ides of the other triangle. (The SSS Test) Two triangles are congruent if two sides of one triangle are equal to two sides of the other triangle and the angles included by these sides are equal. (The SAS Test) Two triangles are congruent if two angles of one triangle ar e equal to two angles of the other triangle and one pair of corresponding sides is equal. (The AAS Test) Two right-angled triangles are congruent if their hypotenuses are equal and a pair of sides is also equal. (The RHS Test) Two triangles are similar if i f two angles of one triangle are equal to two angles of the other triangle. Two triangles are similar if i f the ratios of two pairs of corresponding corres ponding sides are equal and the angles included by these sides are equal. Two triangles are similar if the ratios of the three pairs of sides are equal. Two triangles are similar if i f the hypotenuse and a second side of a right-angled triangle a re proportional to the hypotenuse hypotenuse and a second side of another right-angled triangle. An interval parallel to a side of a triangle divides the other sides in the same ratio. An interval joining the midpoints of the sides of a tria ngle is parallel to the third side s ide and half its length. In a right angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides. A triangle is right-angled if the square on the longest side is equal to the sum of the squares on the other two sides. Quadrilaterals
The angle sum of a quadrilateral is 360 o. Definitions and properties for the special quadrilaterals
A trapezium is a quadrilateral with at least one pair of opposite sides parallel. A kite is a quadrilateral with two pairs of adjacent sides equal. Properties of a kite: One diagonal of a kite bisects the other diagonal One diagonal of a kite bisects the opposite angles The diagonals of a kite are perpendicular A kite has one axis of symmetry A parallelogram is a quadrilateral with both pairs of opposite sides parallel. Properties of a parallelogram: The opposite sides of a parallelogram are parallel The opposite sides of a parallelogram are equal
"#$% .
(Almost?) Everything a 2 Unit Student Needs For The HSC Examination (An Unofficial List of Things to Memorise) The opposite angles of a parallelogram are equal The diagonals of a parallelogram bisect each other A rhombus is a parallelogram with two adjacent sides equal in length. Properties of a rhombus: The opposite sides of a rhombus are parallel All sides of a rhombus are equal The opposite angles of a rhombus are equal The diagonals of a rhombus bisect the opposite angles The diagonals of a rhombus bisect each other The diagonals of a rhombus are perpendicular A rhombus has two axes of symmetry A rectangle is a parallelogram parallelo gram with one angle a right angle. Properties of a rectangle: The opposite sides of a rectangle are parallel The opposite sides of a rectangle are equal All angles at the vertices of a rectangle are 90o The diagonals of a rectangle are equal The diagonals of a rectangle bisect each other A rectangle has two axes of symmetry A square is a rectangle with a pair of adjacent sides equal. Properties of a square: Opposite sides of a square are parallel All sides of a square are equal All angles at the vertices of a square are 90o The diagonals of a square are equal The diagonals of a square bisect the opposite angles The diagonals of a square bisect each other The diagonals of a square are perpendicular A square has four axes of symmetry
Tests for special quadrilaterals
A quadrilateral is a trapezium if: It has one pair of parallel sides A quadrilateral is a kite if: Two pairs of adjacent sides are equal or The diagonals meet at right angles and one of them is bisected by the other A quadrilateral is a parallelogram if: both pairs of opposite sides are parallel or both pairs of opposite sides are equal or both pairs of opposite angles are equal or the diagonals bisect each other (i.e. the diagonals dia gonals have the same midpoint) or one pair of opposite sides are equal and parallel
"#$% &/
(Almost?) Everything a 2 Unit Student Needs For The HSC Examination (An Unofficial List of Things to Memorise) A quadrilateral is a rhombus if: all sides are equal or diagonals bisect each other at right angles or the diagonals bisect the angles at the vertices or a pair of adjacent sides are equal and opposite angles are equal A quadrilateral is a rectangle if: the diagonals are equal and they bisect each other or it has three right angles or it has two pairs of parallel sides and one right angle or it has two pairs of opposite sides equal and one right angle A quadrilateral is a square if: it has four equal sides and one right angle or the diagonals are equal, bisect each other and meet at right angles
Polygons
The angle sum of a n -sided polygon is (n 2) 180 . The sum of the exterior angles of a polygon pol ygon is 360o. A regular polygon has all sides equal and all interi or angles equal.
"#$% &&
