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SOME IMPORTANT MATHEMATICAL FORMULAE

Circle
: Area = π r2; Circumference = 2 π r
...

Rectangle: Area = xy ; Perimeter = 2(x+y)
...

2
3 2
Area of equilateral triangle =
a
...

3
2
3
Cube
: Surface Area = 6a ; Volume = a
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Cuboid : Total surface area = 2 (ab + bh + lh); Volume = lbh
...

SOME BASIC ALGEBRAIC FORMULAE:

1
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2
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3
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4
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2
2
2
2
5
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6
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7
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8
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9
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10
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11
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12
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INDICES AND SURDS
m n
mn (ab)m = a m b m
am
1
...

= a m − n
...
(a ) = a

...


...
  =

...
a 0 = 1, a ≠ 0
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a

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a x = a y ⇒ x = y
m
am
b
b
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a ± 2 b = x ± y , where x + y = a and xy = b
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loga mn = logm + logn
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loga   = logm – logn
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loga mn = n logm
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logba =

...
logaa = 1
...
loga1 = 0
...
logba =

...
loga1= 0
...
log (m +n) ≠ logm +logn
...
e logx = x
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logaax = x
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P
...

n
Sum to n terms, Sn = [ 2a + (n − 1)d ]
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P, then 2b = a + c
...
P
...

1− r
r −1
a
Sum to infinite terms of G
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1− r
If a, b, c are in A
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HARMONIC PROGRESSION
Reciprocals of the terms of A
...
P
1
1
1
,
,
, ----------------- are in H
...
P, then b =

...

2
n(n + 1)(2n + 1)
2
12+22 +32 + -----------------+n2 = ∑ n =

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PERMUTATIONS AND COMBINATION
n!
n Pr =
( n − r) !
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r!( n − r ) !
n!= 1
...
--------n
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nCr + nCr-1 = (n + 1) Cr
...

m!n!
BINOMIAL THEOREM
(x +a)n = xn + nC1 xn-1 a + nC2 xn-2 a2 + nC3 xn-3 a3 +------------+ nCn an
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PARTIAL FRACTIONS
f (x)
is a proper fraction if the deg (g(x)) > deg (f(x))
...

g(x)
1
...

(ax + b)(cx + d) ax + b (cx + d)
2
...

2
(ax + b)(cx + d)
ax + b (cx + d) (cx + d) 2
3
...

2
2
(ax + b)(cx + d) ax + b (cx 2 + d)
ANALYTICAL GEOMETRY
1
...


2
...

m−n 
 m−n
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3
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2 
 2
4
...

3
3


5
...

Slope of a X- axis = 0
Slope of a line parallel to X-axis = 0
Slope of a Y- axis = ∞
Slope of a line parallel to Y-axis = ∞
y 2 − y1
Slope of a line joining (x1, x2) and (y1, y2) =

...
y = mx + c (slope-intercept form)
y - y1 = m(x-x1) (point-slope form)
y −y
y − y1 = 2 1 (x − x1 ) (two point form)
x 2 − x1
x y
+ = 1 (intercept form)
a b
x cosα +y sinα = P (normal form)
Equation of a straight line in the general form is ax2 + bx + c = 0
a
Slope of ax2 + bx + c = 0 is –  
b
m1 − m 2
2
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Sc
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Two lines meeting a point are called intersecting lines
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Equation of bisector of angle between the lines a1x + b1y+ c1 = 0 and
a1x + b1 y + c1
a x + b 2 y 2 + c2
=± 2
a2x + b2y + c2 = 0 is
a12 + b12
a 22 + b22
PAIR OF STRAIGHT LINES
1
...

a+b
ax2 +2hxy +by2 = 0, represents a pair of coincident lines, if h2 = ab and the same
represents a pair of perpendicular lines, if a + b = 0
...

b
2
...

The angle between the lines ax2 +2hxy +by2+2gx +2fy +c = 0 is given by
tanθ =

2 h 2 − ab

...

a(a + b)
ax2 +2hxy +by2+2gx +2fy +c = 0, represents a pair of perpendicular lines
,if a + b = 0
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Arc length, S = r θ
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hyp
hyp
adj
opp
adj
opp
1
1
1
1
Sinθ =
or cosecθ =
, cosθ =
or secθ =
,
cos ecθ
sin θ
sec θ
cos θ
1
1
sin θ
cos θ
tanθ =
or cotθ =
, tanθ =
, cotθ =

...

STANDARD ANGLES
π
π
0 or
or
450 or
00 0 30
6
4
Sin
Cos
Tan
Cot
Sec
Cosec

0

1
0

∞
1
∞

1
2

1

3
2
1

1

3
3

2

3
2
1
2

1

3

2

1

2
3
2

600 or

2

1
3
1
2

2

3

π
3

900 or
1
0

∞
0

∞
1

π
π
150 or
2
12
3 −1
2 2
3 +1
2 2

750 or

5π
12

3 +1
2 2
3 −1
2 2

3 −1

3 +1

3 +1

3 −1

3 +1

3 −1

3 −1

3 +1

2 2
3 +1
2 2
3 −1

2 2
3 −1
2 2
3 +1

ALLIED ANGLES
Trigonometric functions of angles which are in the 2nd, 3rd and 4th quadrants can be
obtained as follows :
If the transformation begins at 900 or 2700, the trigonometric functions changes as
sin ↔ cos
tan ↔ cot
sec ↔ cosec
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where as the transformation begins at 1800 or 3600, the same trigonometric functions
will be retained, however the signs (+ or -) of the functions decides ASTC rule
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Sin(A-B)= sinAcosB-cosAsinB
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Cos(A-B)=cosAcosB+sinAsinB
...
sin 2A=2 sinA cosA
...
sin 2A=

2 tan A

...
cos 2A = cos 2 A − sin 2 A
=1-2 sin 2 A
...
tan 2A=
, 5
...
cos 2 A = (1 + cos 2A)
...
1-cos 2A= 2sin 2 A , 8
...
1+sin 2A= (sin A + cos A) 2 ,
2
2
10
...
cos 3A= 4 cos3 A − 3cos A ,
12
...
tan 3A=

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3 tan A − tan 3 A

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2) sin θ =

...

θ
2
2
2
2
1 + tan 2  
2
θ
1 − tan 2  
2
2 θ
2 θ
4) cos θ = 1 − 2sin
...
6) cos θ =

...
8) 1 + cos θ = 2 cos
...

θ
2
2
1 − tan 2  
2
PRODUCT TO SUM
2 sinA cosB = sin(A+B) + sin(A-B)
...

2 cosA cosB = cos(A+B) + cos(A-B)
...

SUM TO PRODUCT
C+D
C−D
Sin C + sin D = 2sin 
 cos 

...

 2   2 
C+D
C−D
Cos C + cos D = 2 cos 
 cos 

...

b 2 + c2 − a 2
Cosine Rule: a2 = b2 + c2 -2bc cosA or cosA =
,
2bc
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a 2 + c2 − b2
,
2ac
a 2 + b2 − c2
c2 = a2 + b2 -2ab cosC or cosC =

...

=
 2  a+b
2
Half angle formula:
(s − b)(s − c)
(s − b)(s − c)
s(s − a)
A
A
A
sin   =
, cos   =
, tan   =

...

s(s − b)
ac
ac
2
2
2
(s − a)(s − b)
(s − a)(s − b)
s(s − c)
C
C
C
sin   =
, cos   =
, tan   =

...

2
2
2
LIMITS

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If f ( − x ) = − f ( x ) , then f ( x ) is called Odd Function

3
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4
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Sc
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sin x 0
tan x 0
π
= Lt
=
x →0
x →0
x
x
180

8
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lim

sin −1 x
tan −1 x
= 1 = lim
x →0
x
x

10
...

x−a

11
...


1

lim  1 +  = e ,
x →∞ 
n

13
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lim  f ( x ) ± g ( x )  = lim f ( x ) ± lim g ( x )



15
...
g ( x ) = lim f ( x )
...


1

lim ( 1 + n ) n = e
x →0

x→a

x→a

x→a

x→a

x →a

x→a

 lim f ( x )
x→a
provided lim g( x ) ≠ 0
=
x→a
lim g ( x )
 x→a

A function f ( x ) is said to be continuous at the point x = a if
(i) lim f ( x ) exists
x→a

17
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If two functions f ( x ) and g ( x ) are continuous then f ( x ) + g ( x ) is continuous

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