1

SOME IMPORTANT MATHEMATICAL FORMULAE

Circle
: Area = π r2; Circumference = 2 π r.
Square : Area = x2 ; Perimeter = 4x.
Rectangle: Area = xy ; Perimeter = 2(x+y).
1
Triangle : Area = (base)(height) ; Perimeter = a+b+c.
2
3 2
Area of equilateral triangle = a .
4
4
Sphere : Surface Area = 4 π r2 ; Volume = π r3.
3
2
3
Cube
: Surface Area = 6a ; Volume = a .
1
Cone
: Curved Surface Area = π rl ; Volume = π r2 h
3
π r l + π r2
Total surface area = .
Cuboid : Total surface area = 2 (ab + bh + lh); Volume = lbh.
Cylinder : Curved surface area = 2 π rh; Volume = π r2 h
Total surface area (open) = 2 π rh;
Total surface area (closed) = 2 π rh+2 π r2 .
SOME BASIC ALGEBRAIC FORMULAE:

1.(a + b)2 = a2 + 2ab+ b2 .
2. (a - b)2 = a2 - 2ab+ b2 .
3.(a + b)3 = a3 + b3 + 3ab(a + b).
4. (a - b)3 = a3 - b3 - 3ab(a - b).
2
2
2
2
5.(a + b + c) = a + b + c +2ab+2bc +2ca.
6.(a + b + c)3 = a3 + b3 + c3+3a2b+3a2c + 3b2c +3b2a +3c2a +3c2a+6abc.
7.a2 - b2 = (a + b)(a – b ) .
8.a3 – b3 = (a – b) (a2 + ab + b2 ).
9.a3 + b3 = (a + b) (a2 - ab + b2 ).
10.(a + b)2 + (a - b)2 = 4ab.
11.(a + b)2 - (a - b)2 = 2(a2 + b2 ).
12.If a + b +c =0, then a3 + b3 + c3 = 3 abc .
INDICES AND SURDS m n mn (ab)m = a m b m am 1. am an = am + n 2.
= a m − n . 3. (a ) = a
. 4.
.
an m am
−m = 1
a
5.   =
.
6. a 0 = 1, a ≠ 0 .
7. a
. 8. a x = a y ⇒ x = y m am
b
b
9. a x = b x ⇒ a = b 10. a ± 2 b = x ± y , where x + y = a and xy = b.

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LOGARITHMS
a x = m ⇒ log m = x (a > 0 and a ≠ 1) a 1. loga mn = logm + logn.
m
2. loga   = logm – logn.
n
3. loga mn = n logm. log a
4. logba =
.
log b
5. logaa = 1.
6. loga1 = 0.
1
7. logba =
.
log a b
8. loga1= 0.
9. log (m +n) ≠ logm +logn.
10. e logx = x.
11. logaax = x.
PROGRESSIONS
ARITHMETIC PROGRESSION a, a + d, a+2d,-----------------------------are in A.P. nth term, Tn = a + (n-1)d. n Sum to n terms, Sn = [ 2a + (n − 1)d ] .
2
If a, b, c are in A.P, then 2b = a + c.
GEOMETRIC PROGRESSION a, ar, ar2 ,--------------------------- are in G.P. a(1 − r n ) a(r n − 1)
Sum to n terms, Sn = if r < 1 and Sn = if r > 1.
1− r r −1 a Sum to infinite terms of G.P, S∞ =
.
1− r
If a, b, c are in A.P, then b2 = ac.
HARMONIC PROGRESSION
Reciprocals of the terms of A.P are in H.P
1
1
1
,
,
, ----------------- are in H.P a a + d a + 2d
2ac
If a, b, c are in H.P, then b =
.
a+c
MATHEMATICAL INDUCTION n(n + 1)
1 + 2 + 3 + -----------------+n = ∑ n =
.
2 n(n + 1)(2n + 1)
2
12+22 +32 + -----------------+n2 = ∑ n =
.
6
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13+23 +33 + ----------------+ n3 =

∑n

3

=

n 2 (n + 1) 2
.
4

PERMUTATIONS AND COMBINATION n! n Pr =
( n − r) ! . n! nCr =
.
r!( n − r ) ! n!= 1.2 3.--------n. nCr = nCn-r. nCr + nCr-1 = (n + 1) Cr.
(m + n)!
(m + n)Cr =
.
m!n!
BINOMIAL THEOREM
(x +a)n = xn + nC1 xn-1 a + nC2 xn-2 a2 + nC3 xn-3 a3 +------------+ nCn an. nth term, Tr+1 = nCr xn-r ar .
PARTIAL FRACTIONS f (x) is a proper fraction if the deg (g(x)) > deg (f(x)). g(x) f (x) is a improper fraction if the deg (g(x)) ≤ deg (f(x)). g(x) 1. Linear non- repeated factors f (x)
A
B
=
+
.
(ax + b)(cx + d) ax + b (cx + d)
2. Linear repeated factors f (x)
A
B
C
=
+
+
.
2
(ax + b)(cx + d) ax + b (cx + d) (cx + d) 2
3. Non-linear(quadratic which can not be factorized) f (x)
Ax + B Cx + D
= 2
+
.
2
2
(ax + b)(cx + d) ax + b (cx 2 + d)
ANALYTICAL GEOMETRY
1. Distance between the two points (x1, y1) and (x2, y2) in the plane is
(x 2 − x1 ) 2 + (y 2 − y1 ) 2 OR

(x1 − x 2 ) 2 + (y1 − y 2 ) 2 .

2. Section formula
 mx 2 + nx1 my 2 + ny1 
,

 (for internal division), m+n 
 m+n
 mx 2 − nx1 my 2 − ny1 
,

 (for external division). m−n 
 m−n
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3. Mid point formula
 x1 + x 2 y1 + y 2 
,

.
2 
 2
4. Centriod formula
 x1 + x 2 + x 3 y1 + y 2 + y3 
,

.
3
3


5. Area of triangle when their vertices are given,
1
∑ x1 (y2 − y3 )
2
1
= [ x1 (y 2 − y3 ) + x 2 (y3 − y1 ) + x 3 (y1 − y 2 ) ]
2
STRAIGHT LINE
Slope (or Gradient) of a line = tangent of an inclination = tanθ.
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) =
.
x 2 − x1
If two lines are parallel, then their slopes are equal (m1= m2)
If two lines are perpendicular, then their product of slopes is -1 (m1 m2 = -1)
EQUATIONS OF STRAIGHT LINE
1. 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. Angle between two straight lines is given by, tanθ =
1 + m1m 2
Length of the perpendicular from a point (x1,x2) and the straight line ax2 + bx + c ax1 + by1 + c
= 0 is a 2 + b2

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Equation of a straight line passing through intersection of two lines a 1x2 + b1x + c1
= 0 and a2x2 + b2x + c2 = 0 is a1x2 + b1x + c1 + K(a2x2 + b2x + c2 ) = 0, where K is any constant.
Two lines meeting a point are called intersecting lines.
More than two lines meeting a point are called concurrent lines.
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. An equation ax2 +2hxy +by2 = 0, represents a pair of lines passing through origin generally called as homogeneous equation of degree2 in x and y and
2 h 2 − ab angle between these is given by tanθ =
.
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.
2h
If m1 and m2 are the slopes of the lines ax2 +2hxy +by2 = 0,then m1 + m2 = − b a and m1 m2 = . b 2. An equation ax2 +2hxy +by2+2gx +2fy +c = 0 is called second general second order equation represents a pair of lines if it satisfies the the condition abc + 2fgh –af2 – bg2 – ch2 = 0.
The angle between the lines ax2 +2hxy +by2+2gx +2fy +c = 0 is given by tanθ =

2 h 2 − ab
.
a+b

ax2 +2hxy +by2+2gx +2fy +c = 0, represents a pair of parallel lines, if h2 = ab and af2= bg2 and the distance between the parallel lines is
2 g 2 − ac
.
a(a + b) ax2 +2hxy +by2+2gx +2fy +c = 0, represents a pair of perpendicular lines
,if a + b = 0.

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TRIGNOMETRY
1 2
Area of a sector of a circle = r θ .
2
Arc length, S = r θ. opp adj opp adj hyp hyp sinθ =
,cosθ =
,tanθ =
,cotθ =
, secθ =
, cosecθ =
.
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 =
.
cot θ tan θ cos θ sin θ sin2θ + cos2θ = 1; ⇒ sin2θ = 1- cos2θ; cos2θ = 1- sin2θ; sec2θ - tan2θ = 1; ⇒ sec2θ = 1+ tan2θ; tan2θ = sec2θ – 1; cosec2θ - cot2θ = 1; ⇒ cosec2θ = 1+ cot2θ; cot2θ = cosec2θ – 1.
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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7 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.
COMPOUND ANGLES
Sin(A+B)=sinAcosB+cosAsinB.
Sin(A-B)= sinAcosB-cosAsinB.
Cos(A+B)=cosAcosB-sinAsinB.
Cos(A-B)=cosAcosB+sinAsinB. tan A + tan B tan(A+B)= 1 − tan A tan B tan A − tan B tan(A-B)= 1 + tan A tan B
π
 1 + tan A tan  + A  =
4
 1 − tan A
π
 1 − tan A tan  − A  =
4
 1 + tan A tan A + tan B + tan C − tan A tan B tan C tan(A+B+C)= 1 − (tan A tan B + tan B tan C + tan C tan A) sin(A+B) sin(A-B)= sin 2 A − sin 2 B = cos 2 B − cos 2 A cos(A+B) cos(A-B)= cos 2 A − sin 2 B
MULTIPLE ANGLES
1.sin 2A=2 sinA cosA.

2. sin 2A=

2 tan A
.
1 + tan 2 A

3.cos 2A = cos 2 A − sin 2 A
=1-2 sin 2 A .
= 2 cos 2 A − 1
1 − tan 2 A
=
1 + tan 2 A
2 tan A
1
4. tan 2A=
, 5. 1+cos 2A= 2 cos 2 A , 6. cos 2 A = (1 + cos 2A) .
2
2
1 − tan A
1
7. 1-cos 2A= 2sin 2 A , 8. sin 2 A = (1 − cos 2A) , 9.1+sin 2A= (sin A + cos A) 2 ,
2
2
10. 1-sin 2A= (cos A − sin A) = (sin A − cos A) 2 , 11.cos 3A= 4 cos3 A − 3cos A ,
12. sin 3A= 3sin A − 4sin 3 A , 13.tan 3A=

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3 tan A − tan 3 A
.
1 − 3 tan 2 A

8
HALF ANGLE FORMULAE
θ
2 tan   θ θ
2
2 θ
2 θ
1) sin θ = 2sin cos . 2) sin θ =
. 3) cos θ = cos − sin .
θ
2
2
2
2
1 + tan 2  
2
θ
1 − tan 2  
2
2 θ
2 θ
4) cos θ = 1 − 2sin . 5) cos θ = 2 cos − 1 . 6) cos θ =
.
θ
2
2
1 + tan 2  
2
θ
2 tan  
2
2 θ
2 θ
7) tan θ =
. 8) 1 + cos θ = 2 cos . 9) 1 − cos θ = 2sin .
θ
2
2
1 − tan 2  
2
PRODUCT TO SUM
2 sinA cosB = sin(A+B) + sin(A-B).
2 cosA sinB = sin(A+B) – sin(A-B).
2 cosA cosB = cos(A+B) + cos(A-B).
2 sinA sinB = 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
Sin C –sin D = 2 cos 
 sin 
.
 2   2 
C+D
C−D
Cos C + cos D = 2 cos 
 cos 
.
 2 
 2 
 C+D C−D
Cos C- cos D = −2sin 
 sin 

 2   2 
OR
 D+C  D−C
Cos C- cos D = 2sin 
 sin 

 2   2 
PROPERTIES AND SOLUTIONS OF TRIANGLE a b c =
=
= 2R , where R is the circum radius of the
Sine Rule: sin A sin B sin C triangle. b 2 + c2 − a 2
Cosine Rule: a2 = b2 + c2 -2bc cosA or cosA =
,
2bc
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9 a 2 + c2 − b2
,
2ac a 2 + b2 − c2 c2 = a2 + b2 -2ab cosC or cosC =
.
2ab
Projection Rule: a = b cosC +c cosB b = c cosA +a cosC c = a cosB +b cosA
Tangents Rule:
 B−C b−c
A
tan  cot   ,
=
 2  b+c
2
C−A c−a
B
tan  cot   ,
=
 2  c+a
2
A−B a−b
C
tan  cot   .
=
 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 − a) bc bc
2
2
2
b2 = a2 + c2 -2ac cosB or cosB =

(s − a)(s − c)
(s − a)(s − c) s(s − b)
B
B
B
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   =
.
s(s − c) ab ab
2
2
2
Area of triangle ABC = s(s − a)(s − b)(s − c) ,
1
1
1
Area of triangle ABC = bcsin A = ac sin B = ab sin C .
2
2
2
LIMITS

1.

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

2.

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

3.

If P is the smallest + ve real number such that if f ( x + P ) = f ( x ) , then f ( x ) is called a periodic function with period P.

4.

lim
Right Hand Limit (RHL) = x → a + ( f ( x ) ) = lim ( f ( a + h ) ) h →0 lim Left Hand Limit (LHL) = x → a − ( f ( x ) ) = lim ( f ( a − h ) ) h →0
If RHL=LHL then lim ( f ( x ) ) exists and x→a lim ( f ( x ) ) = RHL=LHL x→a S B SATHYANARAYANA
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10
5.
6.

1 p = 0 , if p > 0 and nLt n = ∞ if p > 0
→∞
np

Lt

n →∞

Lt

x →0

sin x tan x x x
= Lt
= Lt
=1
( x in radians ) = xLt0 x →0
→ sin x x → 0 tan x x x

7.

sin x 0 tan x 0 π = Lt
=
x →0 x →0 x x
180

8.

Ltπ

Lt

x→

2

sin x 2
=
x π 9.

lim

sin −1 x tan −1 x
= 1 = lim x →0 x x

10.

lim

x n − an
= nan − 1 , where n is an integer or a fraction. x−a 11.

lim

ax − 1
= log a , x 12.

1

lim  1 +  = e , x →∞  n 13.

lim kf ( x )  = k lim f ( x )

x→a  x→a 14.

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

15.

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

x →0

x→a

x →0

lim x →0

n

 f ( x) lim  x→a  g ( x)

16.

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.

ex − 1
= log e = 1 x (ii) f ( a ) is defined

(iii) lim f ( x ) = f ( a ) x→a A function f ( x ) is said to be discontinuous or not continuous at x = a if
(i) f ( x ) is not defined at x = a

(ii) lim f ( x ) does not exist at x = a x→a (iii) xlim0 f ( x ) ≠ xlim0 f ( x ) ≠ f ( a )
→a+
→a−

18.

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

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