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SOME IMPORTANT MATHEMATICAL FORMULAE
Circle : Area =
π
r
2
;
Circumference = 2
π
r.
Square : Area = x
2
; Perimeter = 4x.
Rectangle: Area = xy ; Perimeter = 2(x+y).
Triangle : Area =
(base)(height) ; Perimeter = a+b+c.
Area of equilateral triangle =
a
2
.
Sphere : Surface Area = 4
π
r
2
; Volume =
π
r
3
.
Cube : Surface Area = 6a
2
; Volume = a
3
.
Cone : Curved Surface Area =
π
rl ; Volume =
π
r
2
h
Total surface area =.
π r l +
π r
2
Cuboid : Total surface area = 2 (ab + bh + lh); Volume = lbh.
Cylinder : Curved surface area = 2
π
rh; Volume =
π
r
2
h
Total surface area (open) = 2
π
rh;
Total surface area (closed) = 2
π
rh+
π
r
2
SOME BASIC ALGEBRAIC FORMULAE:
1.(a + b)
2
= a
2
2
. 2. (a - b)
2
= a
2
2
3.(a + b)
3
= a
3
3
3
= a
3
3
5.(a + b + c)
2
= a
2
2
2
+2ab+2bc +2ca.
6.(a + b + c)
3
= a
3
3
3
+3a
2
b+3a
2
c + 3b
2
c +3b
2
a +3c
2
a +3c
2
a+6abc.
7.a
2
2
= (a + b)(a – b ).
8.a
3
3
= (a – b) (a
2
2
9.a
3
3
= (a + b) (a
2
2
10.(a + b)
2
2
= 4ab.
11.(a + b)
2
2
= 2(a
2
2
12.If a + b +c =0, then a
3
3
3
= 3 abc.
INDICES AND SURDS
- a
m
a
n
=
a
m + n
m
a
m n
a
n
a
m n mn
(a ) = a
. 4.
m m m
(ab) = a b
.
m m
a a
m
b
b
a = 1, a ≠ 0
m
a
m
a
y x
a = a ⇒ x =y
x x
a = b ⇒ a = b
a ± 2 b = x ± y
, where x + y = a and xy = b.
S B SATHYANARAYANA
M. Sc., M.I.E ., M Phil.
LOGARITHMS
x
a m log m x
a
(a > 0 and a ≠ 1)
- log a
mn = logm + logn.
- log a
m
n
= logm – logn.
- log a
m
n
= n logm.
- log b
a =
log a
log b
- log a
a = 1.
- log a
- log b
a =
a
log b
- log a
- log (m +n) ≠ logm +logn.
- e
logx
= x.
- log a
a
x
= x.
PROGRESSIONS
ARITHMETIC PROGRESSION
a, a + d, a+2d,-----------------------------are in A.P.
n
th
term, T n
= a + (n-1)d.
Sum to n terms, S n
= [ ]
n
2a (n 1)d
If a, b, c are in A.P, then 2b = a + c.
GEOMETRIC PROGRESSION
a, ar, ar
2
,--------------------------- are in G.P.
Sum to n terms, S n
n
a(1 r )
1 r
if r < 1 and S n
n
a(r 1)
r 1
if r > 1.
Sum to infinite terms of G.P,
a
S
1 r
∞
If a, b, c are in A.P, then b
2
= ac.
HARMONIC PROGRESSION
Reciprocals of the terms of A.P are in H.P
a a + d a + 2d
----------------- are in H.P
If a, b, c are in H.P, then b =
2ac
a + c
MATHEMATICAL INDUCTION
1 + 2 + 3 + -----------------+n =
n(n 1)
n
2
2
2
2
2
n(n 1)(2n 1)
n
S B SATHYANARAYANA
M. Sc., M.I.E ., M Phil.
- Mid point formula
1 2 1 2
x x y y
- Centriod formula
1 2 3 1 2 3
x x x y y y
- Area of triangle when their vertices are given,
[ ]
1 2 3
1 2 3 2 3 1 3 1 2
x (y y )
x (y y ) x (y y ) x (y y )
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 = ∞
Slope of a line joining (x 1
, x 2
) and (y 1
, y 2
2 1
2 1
y y
x x
If two lines are parallel, then their slopes are equal (m 1
= m 2
If two lines are perpendicular, then their product of slopes is -1 (m 1
m 2
EQUATIONS OF STRAIGHT LINE
- y = mx + c (slope-intercept form)
y - y 1
= m(x-x 1
) (point-slope form)
2 1
1 1
2 1
y y
y y (x x )
x x
(two point form)
x y
a b
(intercept form)
x cosα +y sinα = P (normal form)
Equation of a straight line in the general form is ax
2
+ bx + c = 0
Slope of ax
2
a
b
- Angle between two straight lines is given by, tanθ =
1 2
1 2
m m
1 m m
Length of the perpendicular from a point (x 1
,x 2
) and the straight line ax
2
= 0 is
1 1
2 2
ax by c
a b
S B SATHYANARAYANA
M. Sc., M.I.E ., M Phil.
Equation of a straight line passing through intersection of two lines a 1
x
2
x + c 1
= 0 and a 2
x
2
x + c 2
= 0 is a 1
x
2
x + c 1
x
2
x + c 2
) = 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 a 1
x + b 1
y+ c 1
= 0 and
a 2
x + b 2
y + c 2
= 0 is
1 1 1 2 2 2 2
2 2 2 2
1 1 2 2
a x b y c a x b y c
a b a b
PAIR OF STRAIGHT LINES
- An equation ax
2
+2hxy +by
2
= 0, represents a pair of lines passing through origin
generally called as homogeneous equation of degree2 in x and y and
angle between these is given by tanθ =
2
2 h ab
a b
ax
2
+2hxy +by
2
= 0, represents a pair of coincident lines, if h
2
= ab and the same
represents a pair of perpendicular lines, if a + b = 0.
If m 1
and m 2
are the slopes of the lines ax
2
+2hxy +by
2
= 0,then m 1
2h
b
and m 1
m 2
a
b
- An equation ax
2
+2hxy +by
2
+2gx +2fy +c = 0 is called second general second
order equation represents a pair of lines if it satisfies the the condition
abc + 2fgh –af
2
2
2
The angle between the lines ax
2
+2hxy +by
2
+2gx +2fy +c = 0 is given by
tanθ =
2
2 h ab
a b
ax
2
+2hxy +by
2
+2gx +2fy +c = 0, represents a pair of parallel lines, if h
2
= ab and
af
2
= bg
2
and the distance between the parallel lines is
2
2 g ac
a(a b)
ax
2
+2hxy +by
2
+2gx +2fy +c = 0, represents a pair of perpendicular lines
,if a + b = 0.
S B SATHYANARAYANA
M. Sc., M.I.E ., M Phil.
where as the transformation begins at 180
0
or 360
0
, 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+B)=
tan A tan B
1 tan A tan B
tan(A-B)=
tan A tan B
1 tan A tan B
tan A
π
1 tan A
1 tan A
tan A
π
1 tan A
1 tan A
tan(A+B+C)=
tan A tan B tan C tan A tan B tan C
1 (tan A tan B tan B tan C tan C tan A)
sin(A+B) sin(A-B)=
2 2 2 2
sin A − sin B = cos B −cos A
cos(A+B) cos(A-B)=
2 2
cos A −sin B
MULTIPLE ANGLES
1.sin 2A=2 sinA cosA. 2. sin 2A=
2
2 tan A
1 + tan A
3.cos 2A =
2 2
cos A −sin A
2
sin A
2
cos A − 1
2
2
1 tan A
1 tan A
- tan 2A=
2
2 tan A
1 − tan A
, 5. 1+cos 2A=
2
2 cos A , 6.
2
cos A =
(1 cos 2A)
- 1-cos 2A=
2
2sin A , 8.
2
sin A =
(1 cos 2A)
, 9.1+sin 2A=
2
(sin A + cos A) ,
- 1-sin 2A=
2
(cos A − sin A) =
2
(sin A − cos A), 11.cos 3A=
3
4 cos A − 3cos A
- sin 3A=
3
3sin A − 4sin A, 13.tan 3A=
3
2
3 tan A tan A
1 3 tan A
S B SATHYANARAYANA
M. Sc., M.I.E ., M Phil.
HALF ANGLE FORMULAE
- sin θ =
2sin cos
θ θ
. 2) sin θ =
2
2 tan
1 tan
θ
θ
. 3) cos θ =
2 2
cos sin
θ θ
2
cos 1 2sin
θ
θ = −
. 5)
2
cos 2 cos 1
θ
θ = −
. 6)
2
2
1 tan
cos
1 tan
θ
θ =
θ
2
2 tan
tan
1 tan
θ
θ =
θ
2
1 cos 2 cos
θ
. 9)
2
1 cos 2sin
θ
− θ = .
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
Sin C + sin D =
C D
2sin
C D
cos
Sin C –sin D =
C D C D
2 cos sin
Cos C + cos D =
C D C D
2 cos cos
Cos C- cos D =
C D C D
2sin sin
OR
Cos C- cos D =
D C D C
2sin sin
PROPERTIES AND SOLUTIONS OF TRIANGLE
Sine Rule:
a b c
2R
sin A sin B sin C
= = = , where R is the circum radius of the
triangle.
Cosine Rule: a
2
= b
2
2
-2bc cosA or cosA =
2 2 2
b c a
2bc
S B SATHYANARAYANA
M. Sc., M.I.E ., M Phil.
1
0
p
n
Lt
n
→∞
=
, if
p > 0
and
p
n
Lt n
→∞
= ∞
if
p > 0
0 0 0 0
sin tan
in radians 1
sin tan
x x x x
x x x x
Lt Lt x Lt Lt
x x x x
→ → → →
= = = =
π
→ →
= =
0 0
0 0
sin tan
180
x x
x x
Lt Lt
x x
π
π →
=
2
sin 2
x
x
Lt
x
1 1
0 0
sin tan
lim 1 lim
x x
x x
x x
− −
→ →
= =
−
→
−
=
−
1
lim
n n
n
x a
x a
na
x a
, where n is an integer or a fraction.
0 0
1 1
lim log , lim log 1
x x
x x
a e
a e
x x
→ →
− −
= = =
1
0
1
lim 1 , lim 1
n
n
x x
e n e
n
→∞ →
→ →
=
lim lim
x a x a
kf x k f x
lim lim lim
x a x a x a
f x g x f x g x
→ → →
± = ±
lim ( ). ( ) lim ( ) .lim ( )
x a x a x a
f x g x f x g x
→ → →
=
→
→ →
→
= ≠
lim
lim lim ( ) 0
lim
x a
x a x a
x a
f x
f x
provided g x
g x g x
- A function
f ( x ) is said to be continuous at the point
x = a
if
(i)
lim ( )
x a
f x
→
exists (ii) ( )
f a is defined (iii)
lim ( ) ( )
x a
f x f 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 ( )
x a
f x
→
does not exist at
x = a
(iii)
0 0
lim lim
x a x a
f x f x f a
→ + → −
≠ ≠
- If two functions
f x
and
g x
are continuous then
f x + g x
is continuous
S B SATHYANARAYANA
M. Sc., M.I.E ., M Phil.
