2019-20 N. American Planner_DP Sample
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USEFUL INFORMATION<br />
MATHEMATICAL LAWS, FORMULAE, SYMBOLS<br />
MENSURATION<br />
Cone<br />
Surface area<br />
of cone =<br />
x r 2 + rs<br />
Volume of<br />
s<br />
cone =<br />
r 2 h h<br />
3<br />
r<br />
Box<br />
Volume = l x w x h<br />
Parallelogram<br />
D<br />
a h<br />
A<br />
A<br />
b<br />
l<br />
b<br />
b<br />
B<br />
B<br />
C<br />
a<br />
Area of ABCD = bh<br />
Triangle<br />
C<br />
Area of<br />
∆ABC = bh 2<br />
h<br />
h<br />
w<br />
h<br />
Cylinder<br />
Surface area<br />
of cylinder =<br />
2 rh+2 r 2<br />
Volume of<br />
cylinder =<br />
r 2 h<br />
Circle Sphere<br />
Circumference Surface area<br />
of circle = of sphere =<br />
2 r<br />
4 r 2<br />
r<br />
r<br />
h<br />
Area of circle Volume of<br />
r<br />
= r 2 sphere = 4 r 3<br />
3<br />
Rectangle<br />
Perimeter = 2(l+w)<br />
Area = lxw<br />
Trapezium<br />
B a c<br />
h<br />
A b<br />
D<br />
Area of Trapezium<br />
ABCD = ½ h (a+b)<br />
A<br />
c<br />
l<br />
b<br />
w<br />
C<br />
B<br />
a<br />
Area of<br />
∆ABC = ab 2<br />
Pyramid<br />
Volume of<br />
pyramid =<br />
Bh<br />
h<br />
3<br />
(B = B<br />
area of base)<br />
Arc<br />
r r<br />
L<br />
Arc length L = r<br />
Sector Area<br />
A = ½ r 2<br />
QUADRATICS<br />
Solution of equation ax 2 + bx + c = 0 is given by the<br />
formula x = -b± b 2 -4ac<br />
2a<br />
If b 2 - 4ac > 0, 2 solutions<br />
If b 2 - 4ac = 0, 1 solution<br />
If b 2 - 4ac < 0, no solutions<br />
b 2 - 4ac is called the discriminant<br />
CALCULUS<br />
d (x n ) = nx n-1 ∫x n dx = ( l ) x n+1 + c, n ≠ -1<br />
dx n+1<br />
d (e ax ) = ae ax ∫e ax dx = ( l ) e ax + c<br />
dx<br />
a<br />
d (log e<br />
x) = l ∫( l )dx = log e<br />
x + c, for x >0<br />
dx x x<br />
d (sin ax) = a cos ax ∫sin ax dx = - l cos ax + c<br />
dx<br />
a<br />
d (cos ax) = -a sin ax ∫cos ax dx = l sin ax + c<br />
dx<br />
a<br />
d (tan ax) = +asec 2 ax ∫sec 2 ax dx = l tan ax + c<br />
dx<br />
a<br />
product rule:<br />
d (uv) = u dv + v<br />
dx dx du<br />
dx<br />
v dv - u dv<br />
quotient rule: d ( u ) = dx dx<br />
dx v v 2<br />
chain rule:<br />
dy =<br />
dy du<br />
dx du dx<br />
PYTHAGORAS THEOREM<br />
c 2 = a 2 + b 2<br />
Cosine Rule: c 2 = a 2 + b 2 - 2ab cos<br />
Sine Rule: a = b = c<br />
sinA sinB sinC<br />
Area of Triangle = ½ b.c.sin<br />
COMPLEX NUMBERS<br />
z = x + yi = r (cos + isin ) = rcis<br />
lzl = x 2 + y 2 = r - < Arg z<br />
z 1<br />
z 2<br />
= r 1<br />
r 2<br />
cis( 1<br />
+ 2<br />
) z1 = r1 cis ( - ) 1 2<br />
z 2 r 1<br />
z n = r n cis n (de Moivre’s theorem)<br />
FACTORISING<br />
a 2 + 2ab + b 2 = (a + b) 2<br />
a 2 - 2ab + b 2 = (a - b) 2<br />
a 2 - b 2 = (a + b) (a - b)<br />
a 3 + b 3 = (a + b) (a 2 - ab + b 2 )<br />
a 3 - b 3 = (a - b) (a 2 + ab + b 2 )<br />
ALGEBRA – EXPANDING<br />
a (b + c) = ab + ac<br />
(a - b) 2 = a 2 - 2ab + b 2<br />
(a + b) 2 = a 2 + 2ab + b 2<br />
(a + b)(a - b) = a 2 - b 2 (difference of two squares)<br />
(a + b) 3 = a 3 + 3a 2 b + 3ab 2 + b 3<br />
(a - b) 3 = a 3 - 3a 2 b + 3ab 2 - b 3<br />
(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<br />
1 2 n-2 n-1<br />
TRIGONOMETRY<br />
Sohcahtoa<br />
sin = opposite cos = adjacent<br />
hypotenuse hypotenuse<br />
tan = opposite = sin<br />
adjacent cos<br />
cos =<br />
1<br />
sec = 1 cot = 1<br />
sin cos tan<br />
cos 2 + sin 2 = 1, 1 + tan 2 = sec 2<br />
cos 2 + 1 = cosec 2<br />
sin (a + b) = sin a cos b + sin b cos a<br />
sin (a - b) = sin a cos b - sin b cos a<br />
cos (a + b) = cos a cos b - sin a sin b<br />
cos (a - b) = cos a cos b + sin a sin b<br />
tan (a + b) = tan a + tan b<br />
1 - tan a tan b<br />
tan (a - b) = tan a - tan b<br />
1 + tan a tan b<br />
sin 2 = 2 sin cos tan 2 = 2 tan<br />
1 - tan 2<br />
cos 2 = cos 2 = sin 2 = 2 cos 2 - 1 = 1 - 2 sin 2<br />
sin a + sin b = 2 sin ½ (a + b) cos ½ (a - b)<br />
sin a - sin b = 2 cos ½ (a + b) sin ½ (a - b)<br />
cos a + cos b = 2 cos ½ (a + b) cos ½ (a - b)<br />
cos a - cos b = -2 sin ½ (a + b) sin ½ (a - b)<br />
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