2019-20 N. American Planner_DP Sample

USEFUL INFORMATION
MATHEMATICAL LAWS, FORMULAE, SYMBOLS
MENSURATION
Cone
Surface area
of cone =
x r 2 + rs
Volume of
s
cone =
r 2 h h
3
r
Box
Volume = l x w x h
Parallelogram
D
a h
A
A
b
l
b
b
B
B
C
a
Area of ABCD = bh
Triangle
C
Area of
∆ABC = bh 2
h
h
w
h
Cylinder
Surface area
of cylinder =
2 rh+2 r 2
Volume of
cylinder =
r 2 h
Circle Sphere
Circumference Surface area
of circle = of sphere =
2 r
4 r 2
r
r
h
Area of circle Volume of
r
= r 2 sphere = 4 r 3
3
Rectangle
Perimeter = 2(l+w)
Area = lxw
Trapezium
B a c
h
A b
D
Area of Trapezium
ABCD = ½ h (a+b)
A
c
l
b
w
C
B
a
Area of
∆ABC = ab 2
Pyramid
Volume of
pyramid =
Bh
h
3
(B = B
area of base)
Arc
r r
L
Arc length L = r
Sector Area
A = ½ r 2
QUADRATICS
Solution of equation ax 2 + bx + c = 0 is given by the
formula x = -b± b 2 -4ac
2a
If b 2 - 4ac > 0, 2 solutions
If b 2 - 4ac = 0, 1 solution
If b 2 - 4ac < 0, no solutions
b 2 - 4ac is called the discriminant
CALCULUS
d (x n ) = nx n-1 ∫x n dx = ( l ) x n+1 + c, n ≠ -1
dx n+1
d (e ax ) = ae ax ∫e ax dx = ( l ) e ax + c
dx
a
d (log e
x) = l ∫( l )dx = log e
x + c, for x >0
dx x x
d (sin ax) = a cos ax ∫sin ax dx = - l cos ax + c
dx
a
d (cos ax) = -a sin ax ∫cos ax dx = l sin ax + c
dx
a
d (tan ax) = +asec 2 ax ∫sec 2 ax dx = l tan ax + c
dx
a
product rule:
d (uv) = u dv + v
dx dx du
dx
v dv - u dv
quotient rule: d ( u ) = dx dx
dx v v 2
chain rule:
dy =
dy du
dx du dx
PYTHAGORAS THEOREM
c 2 = a 2 + b 2
Cosine Rule: c 2 = a 2 + b 2 - 2ab cos
Sine Rule: a = b = c
sinA sinB sinC
Area of Triangle = ½ b.c.sin
COMPLEX NUMBERS
z = x + yi = r (cos + isin ) = rcis
lzl = x 2 + y 2 = r - < Arg z
z 1
z 2
= r 1
r 2
cis( 1
+ 2
) z1 = r1 cis ( - ) 1 2
z 2 r 1
z n = r n cis n (de Moivre’s theorem)
FACTORISING
a 2 + 2ab + b 2 = (a + b) 2
a 2 - 2ab + b 2 = (a - b) 2
a 2 - b 2 = (a + b) (a - b)
a 3 + b 3 = (a + b) (a 2 - ab + b 2 )
a 3 - b 3 = (a - b) (a 2 + ab + b 2 )
ALGEBRA – EXPANDING
a (b + c) = ab + ac
(a - b) 2 = a 2 - 2ab + b 2
(a + b) 2 = a 2 + 2ab + b 2
(a + b)(a - b) = a 2 - b 2 (difference of two squares)
(a + b) 3 = a 3 + 3a 2 b + 3ab 2 + b 3
(a - b) 3 = a 3 - 3a 2 b + 3ab 2 - b 3
(x + a) n = x n + (n)x n-1 a + (n)x n-2 a 2 + ... ( n )x 2 a n-2 = ( n )xa n-1 + a n
1 2 n-2 n-1
TRIGONOMETRY
Sohcahtoa
sin = opposite cos = adjacent
hypotenuse hypotenuse
tan = opposite = sin
adjacent cos
cos =
1
sec = 1 cot = 1
sin cos tan
cos 2 + sin 2 = 1, 1 + tan 2 = sec 2
cos 2 + 1 = cosec 2
sin (a + b) = sin a cos b + sin b cos a
sin (a - b) = sin a cos b - sin b cos a
cos (a + b) = cos a cos b - sin a sin b
cos (a - b) = cos a cos b + sin a sin b
tan (a + b) = tan a + tan b
1 - tan a tan b
tan (a - b) = tan a - tan b
1 + tan a tan b
sin 2 = 2 sin cos tan 2 = 2 tan
1 - tan 2
cos 2 = cos 2 = sin 2 = 2 cos 2 - 1 = 1 - 2 sin 2
sin a + sin b = 2 sin ½ (a + b) cos ½ (a - b)
sin a - sin b = 2 cos ½ (a + b) sin ½ (a - b)
cos a + cos b = 2 cos ½ (a + b) cos ½ (a - b)
cos a - cos b = -2 sin ½ (a + b) sin ½ (a - b)
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2018/19 Intl Planner_DP_REAR.indd 156 25/5/18 3:15 pm