






Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Community
Ask the community for help and clear up your study doubts
Discover the best universities in your country according to Docsity users
Free resources
Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors
Formula sheet in which given mathematic formula, basic algebra formula, indices and surds, logarithms, progression, permutations and combinations.
Typology: Cheat Sheet
1 / 10
This page cannot be seen from the preview
Don't miss anything!
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
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.
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.
x
a m log m x
a
(a > 0 and a ≠ 1)
mn = logm + logn.
m
n
= logm – logn.
m
n
= n logm.
a =
log a
log b
a = 1.
a =
a
log b
logx
= x.
a
x
= x.
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.
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
1 r
∞
If a, b, c are in A.P, then b
2
= ac.
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
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.
1 2 1 2
x x y y
1 2 3 1 2 3
x x x y y y
1 2 3
1 2 3 2 3 1 3 1 2
x (y y )
x (y y ) x (y y ) x (y y )
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
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
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
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
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.
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
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
2
2 tan A
1 − tan A
, 5. 1+cos 2A=
2
2 cos A , 6.
2
cos A =
(1 cos 2A)
2
2sin A , 8.
2
sin A =
(1 cos 2A)
, 9.1+sin 2A=
2
(sin A + cos A) ,
2
(cos A − sin A) =
2
(sin A − cos A), 11.cos 3A=
3
4 cos A − 3cos A
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.
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 = .
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).
Sin C + sin D =
2sin
cos
Sin C –sin D =
2 cos sin
Cos C + cos D =
2 cos cos
Cos C- cos D =
2sin sin
Cos C- cos D =
2sin sin
Sine Rule:
a b c
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
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
→
=
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
→ + → −
≠ ≠
f x
and
g x
are continuous then
f x + g x
is continuous
S B SATHYANARAYANA
M. Sc., M.I.E ., M Phil.