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IMPORTANT FORMULAE FOR COMPETITIVE EXAMS
1. Important formulae used in simplification:
(1) (a + b)
2 = a
2
2
(2) (a – b)
2 = a
2
2
(3) (a + b)
2 = (a – b)
2
(4) a
2
2 = (a – b) (a + b)
(5) a
3
3 = (a + b) (a
2
2 )
(6) a
3
3 = (a – b) (a
2
2 )
2 2 2 2
a b [(a b) (a – b) ]
2. Rules of counting numbers 1. Sum of f irst n natural numbers
n n 1
- Sum of first n odd natural numbers
= n
2
- Sum of first n even natural numbers
= n(n + 1)
- Sum of the squares of first n natural
numbers =
n(n 1)(2n 1)
- Sum of the cubes of first n
natural numbers =
2
n(n 1)
PERCENTAGES
1. Two successive percentage changes of a% and
b% is an effective change of
ab
a+b+
2. If A is r% more/less than B,
B is
100 r
100 r
less/more than A.
INTEREST
1. P = Principal, A = Amount, I = Interest, n = no. of
years, r% = rate of interest
The Simple Interest (S.I.) =
P r n
2. If P is the principal kept at Compound Interest (C.I.)
@ r% p.a., amount after n years
n
r
P 1
3. Amount = Principal + Interest 4. Let P = Original Population, P = Population after
n years, r% = rate of anual growth
n
r
P' P 1
5. Difference between CI and SI for 2 and 3 years
respectively:
(CI)
2
– (SI)
2
= Pa
2
for two years
(CI)
3
– (SI)
3
= Pa
2
(a + 3) for three years
where, a =
r
6. A principal amounts to X times in T years at S.I. It
will become Y times in:
Y – 1
Years T
X – 1
7. A principal amounts to X times in T years at C.I. It
will become Y times in:
Years = T × n
where n is given by X
n = Y
PROFIT AND LOSS
1. Profit % =
Profit
CP
2. SP = CP + P% of CP = CP
P
3. Discount = Marked Price – Selling Price 4. Discount % =
Discount
Marked Price
5. The selling price of two articles is same.
If one is sold at X% profit and the other at loss of
X%, then there is always a loss of
2
X
IMPORTANT FORMULAE FOR COMPETITIVE EXAMS
IMPORTANT FORMULAE FOR COMPETITIVE EXAMS
RATIO & PROPORTION
1. It a : b : : c : d, then ad = bc 2. If a < b and x is a positive quantity, then
a a x
b b x
and
a a – x
b b – x
3. If a > b and x is a positive quantity, then
a a x
b b x
and
a a – x
b b – x
4. If
a c
b d
then:
(a)
a b c d
b d
(b)
a – b c – d
b d
(c)
a b c d
a – b c – d
Dividendo Law
(d)
a c a
b d b
5. If
a c e
b d f
= K, then:
(a)
a c e
K
b d f
(b)
pa qc re
pb qd rf
= K
(p, q and r are not all zero)
ALLIGATION, MIXTURES AND MEAN
1. Alligation is a method of calculating weighted
averages. The ratio of the weights of the two
items mixed will be inversely proportional to the
difference of each of these two items from the
average attribute of the resultant mixture.
x
x 1
x 2
x 2
w 1
w 2
x – x 1
2 1
2 1
w x – x
w x – x
2. Arithmetic Mean =
1 2 3 n
x x x ...... x
n
3. Geometric Mean =
n
1 2 3 n
x x x ...... x
4. Harmonic Mean =
n 1 2 3
n
x x x x
5. Let K o
be the initial concentration of a solution and
K is the final concentration after n dilutions.
V is the original volume and x is the volume of the
solution replaced each time, then
n
o
V x
K K
V
TIME, SPEED AND DISTANCE
1. 1 km/hr =
m/s and 1m/s =
km/hr
2. Average Speed =
Total Distance Travelled
Total Time Taken
3. When the distance is constant, the average speed
is the harmonic mean of the two speeds
1 2
avg
1 2
2S S
S
S S
4. When the time is constant, the average speed is
the arithmetic mean of the two speeds.
1 2
avg
S S
S
5. D – Speed of the boat downstream
U – Speed of the boat upstream
B – Speed of the boat in still water
R – Speed of the stream
D = B + R and
U = B – R.
Further, by adding and subt racti ng these
equations we get,
B =
D U
and R =
D – U
IMPORTANT FORMULAE FOR COMPETITIVE EXAMS
LINEAR EQUATION IN TWO VARIABLE
a b c
p q r
For the two simultaneous equations,
ax + by = c
px + qy = r
where a, b, c, p, q and r are constants
The same
equation/
Just one line/
Infinite Solutions
Inconsistent
Equations/
Two parallel
lines/
No Solutions
Two
intersecting
lines/
Unique
Solution
a b c
p q r
a b
p q
BINOMIAL THEOREM
1. (x+y)
n
= K 0
x
n
x
n–
.y
1
+K 2
x
n–
.y
2
x
0
.y
n
where K
0
, K
1
, K
2
, ... K
n
are constants
(called coefficients of binomial expansion)
2. Sum of exponents of x and y in any term = n 3. Any term is given by
n r r th
r 1 r
T K x .y (r 1) Term
4. K
r
= binomial coefficient of (r + 1)
th
term
n
r
n!
C
r! n r!
QUADRATIC EQUATIONS
1. General Form:
ax
2
Such an equation has two roots, usually denoted
by and .
2
–b + b – 4ac
2a
2
–b – b – 4ac
2a
2. Sum of roots: + =
b
a
3. Product of roots: × =
c
a
4. In ax
2
X
Y
y
x
The minimum value of ax
2
2
4ac – b
y
4a
at,
–b
x
2a 2
where,
are the roots of the equation
5. In ax
2
X
x
Y
y
The maximum value of ax
2
2
4ac – b
y =
4a
at,
–b
x
2a 2
where, , are the roots of the equation
6. If the roots of a quadratic equation are and ,
the equation can be re-constructed as
x
2
- (sum of roots) × x + (product of roots) = 0
IMPORTANT FORMULAE FOR COMPETITIVE EXAMS
CUBIC & HIGHER DEGREE EQUATIONS
Consider the cubic equation ax
3
2
The equation would have 3 roots (equal to the degree of
the equation). Some of them can be imaginary. If the roots
are denoted as , and , we have
b
a
c
a
d
a
4. The above can be extended for higher degree
equations as well. For an ‘n’ degree equation, Sum
of roots =
n–
n
co-efficient of x
co-efficient of x
5. Sum of roots taken two at a time
n–
n
co-efficient of x
co-efficient of x
6. Sum of roots taken three at a time
n–
n
co-efficient of x
co-efficient of x
7. And, sum of roots taken ‘r’ at a time
r
n–r
n
coefficient of x
coefficient of x
8. Product of roots
n
n
constant term
co - efficient of x
9. Remainder Theorem:
To identify whether a given expression is a factor
of another expression, we can take help of
Remainder Theorem.
According to the remainder theorem, when any
expression f(x) is divided by (x – a), the remainder
is f(a). (a is any constant in this example).
10. Factor Theorem:
An expression is said to be a factor of another
expression only when the remainder is 0 when the
latter is divided by the former.
(x – a) is a factor of f(x) if and only if f(a) = 0.
ARITHMETIC PROGRESSION (AP)
Let,
a = The first term,
d = Common difference,
T
n
= The n
th
term
= The last term,
S
n
= Sum of n terms,
1. The n
th term is given by,
T
n
= a + (n – 1)d
2. The sum of n terms is given by,
n
n
S = [2a + (n – 1)d]
or,
n
a +
S = × n
3. T
n
= S
n
– S
n – 1
GEOMETRIC PROGRESSION (GP)
Let,
a = The first term,
r = The common ratio
T
n
= The n
th
term and
S
n
= The sum of n terms we have the following
1. T
n
= ar
n – 1
n
n
(1– r )
S = a
(1– r)
, where r < 1
n
n
a(r – 1)
S =
(r – 1)
, where r > 1
4. Sum of infinite number of terms =
a
1– r
IMPORTANT FORMULAE FOR COMPETITIVE EXAMS
5. For obtuse triangle ABC
AC
2
= AB
2
2
A
B C D
6. The following are some properties of a triangle right
angled at A, where AD BC:
(i) AD
2
= BD × DC
(ii) AB
2 = BD × BC
(iii) AC
2
= CD × BC
A
B D C
Polygon
In a polygon of ‘n’ no. of sides,
1. Total number of diagonals =
n(n – 3)
2. Exterior angle of a regular polygon
n
3. Interior angle of a regular convex polygon
n
4. Sum of all the exterior angles of a regular convex
polygon = 360°
5. Sum of interior angles of a n sided polygon
= (n – 2) × 180°
Circles
1. If two chords, AB and CD intersect inside or outside
the circle at a point P,
A
B
C
D
P A
B
C
D
P
Then, PA × PB = PC × PD
2. If AB is any chord of a circle which is extended to
P, and PT is a tangent drawn from P on to the circle,
then
PA × PB = PT
2
P
A
B
T
3. Angle subtended by the chord at the center of a
circle i s twi ce of that subtended at the
circumference.
Reflex AOB
A
P
B
X
O
Thus AOB = 2 × AXB
4. An exterior angle of a cyclic quadrilateral is equal
to the angle opposite to its adjacent interior angle.
A
D C E
B
i.e. BCE DAB
IMPORTANT FORMULAE FOR COMPETITIVE EXAMS
5. This means that a parallelogram inscribed in a
circle is always a rectangle/square.
6. Also, when a square or rectangle is inscribed in a
circle, the diagonal of the square / rectangle is
equal to the diameter of the circle.
7. Common Tangents for a pair of circles:
For the two circles with centres O
1
and O
2
and
radius r
1
& r
2
P Q
R S
r 1
r 2
O
1
O
2
O
1
O
2
A
B
C
D
PQ, RS are Direct common tangents & AB, CD
are Transverse common tangents.
Length of PQ or RS
2 2
2 2
= (distance between centres) – (r – r )
Length of AB or CD
2 2
2 2
= (distance between centres) – (r r )
(a) When two circles touch externally
Distance between centres C
1
C
2
= r
1
2
and
2 direct common tangents and one transverse
common tangents are possible.
(b) When two circles touch internally
Only one common tangent is possible
(c) When two circles intersect.
Two direct common tangents are possible.
(d) When one circle is completely inside the other
without touching each other.
No common tangent is possible
(e) When two circles are apart i.e. not touching
each other
Two direct and two transverse tangents are
possible.
8. Alternate segment theorem:
Angle between any chord passing through the
tangent point and tangent is equal to the angle
subtended by the chord to any point on the other
side of circumference (alternate segment)
x
x
P B Q
A
C
9. Ptolmey’s theorem:
For a cyclic quadrilateral, the sum of products of
two pairs of opposite sides equals the product of
the diagonals
A
B
C
D
AB × CD + BC × DA = AC × BD
IMPORTANT FORMULAE FOR COMPETITIVE EXAMS
r
r r
r
R
w b
w
l
l
b
w
w
l
w
w
b
A
B C
r
Circle
Semicircle
Ring
(shaded region)
Sector of
a circle
Segment of
a circle
Pathways
running across
the middle of
a rectangle
Pathways
outside
Pathways
inside
S.No. Name Figure Perimeter Area Nomenclature
O
A B
r
C
Circumference
= 2 r
r + 2r
2 (R + r)
l + 2r where
2[ + b + 4w] l A = 2w( + b + 2w) l
2[ + b – 4w] l A = 2w( + b – 2w) l
A = w( + b – w) l
Area of
segment ACB
(Minor segment)
r = Radius of
the circle
or 3.
(approx.)
r = Radius of
the circle
R = Outer radius
r = Inner radius
=
r =
l =
r =
=
l =
b =
w =
Length
Breadth
Width of
the path
l = × 2 r
360°
× 2 r
360°
+ 2rsin
2
= r
2
360°
sin
2
× r
2
360°
(R – r )
2 2
r
2 1
2
=
22
7
r
2
IMPORTANT FORMULAE FOR COMPETITIVE EXAMS
Trigonometry
1. Angle Measures:
Angle are measured in many units viz. degree,
minute, seconds, radians. We have
1 degree = 60 minutes, 1 minute = 60 seconds,
radians = 180°
Trigonometrical Ratios:
In a right angled triangle ABP, if be the angle
between AP and AB we define
B A
P
(i)
Height PB
sin
Hypotenuse AP
(ii)
Base AB
cos
Hypotenuse AP
(iii)
Height PB
tan
Base AB
(iv)
1 Base AB
cot
tan Height PB
(v)
1 Hypotenuse AP
sec
cos Base AB
(vi)
1 Hypotenuse AP
cosec
sin Height PB
Solids
S.No. Name Figure Lateral/curved
surface area
Total surface
area
Volume Nomenclature
Cuboid
Cube
Right prism
Right circular
cylinder
Right pyramid
Right circular
cone
Sphere
Hemisphere
Spherical shell
2 rh
(Perimeter of
the base) ×
(Slant height)
1
2
r l
2 r
2
2( b+bh+ h) l l l bh
l =
b =
h =
Length
Breadth
Height
a = Edge a
3
6a
2
(Area of
base) ×
(Height)
2 r(r + h) r
2
h
r =
h =
(Area of
the base)
× Height
1
3
r( + r) l r
2
h
1
3
h =
r =
l =
=
Height
Radius
Slant height
r
2 2
4 r
2
3 r
2
4 + r )
2 2
(R
r = Radius
R =
r =
Outer radius
Inner radius
r = Radius r
4 3
3
r
2 3
3
4
3
IMPORTANT FORMULAE FOR COMPETITIVE EXAMS
7. If out of n things, p are exactly alike of one kind, q
exactly alike of second kind and r exactly alike of
third kind and the rest are different, then the
number of permutations of n things taken all at a
time =
n!
p!q!r!
8. Total number of ways in which a selection can be
made by taking some or all out of (p + q + r + ....)
items where p are of one type, q are of second
type and r are of another type and so on
= {(p + 1) (q +1) (r + 1) ...} – 1
9. The number of different relative arrangement for n
different things arranged on a circle = (n – 1)!
10. The number of ways in which (m + n) things can
be divided into two groups containing m and n
things respectively =
(m n)!
m! n!
11. If the numbers of things are equal, say
m = n, total ways of grouping = 2
(2m!)
2!(m! )
Probability
1. Probability of an event
Number of favourable outcomes
Number of all possible outcomes
2. The probability of E not occuring, denoted by
P (not E), is given by P (not E) or P ( E
= 1 – P (E)
3. Odds in favour
Number of favourable cases
Number of unfavourable cases
4. Odds against
Number of unfavourable cases
Number of favourable cases
5. If two events are said to be mutually exclusive
then if one happens, the other cannot happen and
vice versa. In other words, the events have no
simultaneous occurence.
In general P(A or B) = P(A) + P(B) – P (A B)
If A, B are mutually exclusive then
P (A B)= 0
If A, B are independent then
P(A B) = P(A) P(B)
6. Additional law of probability:
If E and F are two mutually exclusive events, then
the probability that either event E or event F will
occur in a single trial is given by:
P(E or F) = P(E) + P(F)
If the events are not mutually exclusive, then
P(E or F) = P(E) + P(F) – P(E and F together).
7. Multiplication law of probability:
If the events E and F are independent,
then P(E and F) = P (E) × P (F)
Coordinate Geometry
Some fundamental formulae:
1. Distance between the points (x 1
, y 1
) and (x 2
, y 2
) is
2 2
2 1 2 1
(x – x ) (y – y )
2. The area of a triangle whose vertices are
(x 1
, y 1
), (x 2
, y 2
) and (x 3
, y 3
[x
1
(y
2
3
) + x
2
(y
3
1
) + x
3
(y
1
2
)]
3. The point that divides the line joining two given
points (x
1
, y
1
) and (x
2
, y
2
) in the ratio m : n internally
and externally are
2 1 2 1
mx nx my ny
m n m n
Note: It would be '+' in the case of internal division
and '–' in the case of external division.
4. The coordinate of the mid-point of the line joining
the points (x
1
, y
1
) and (x
2
, y
2
1 2 1 2
x x y y
5. The centroid of a triangle whose vertices are
(x
1
, y
1
), (x
2
, y
2
) and (x
3
, y
3
1 2 3 1 2 3
x x x y y y
IMPORTANT FORMULAE FOR COMPETITIVE EXAMS
6. Slope of the line joining the points(x 1
, y 1
) and
(x 2
, y 2
) is
2 1
2 1
y – y
x – x
The slope is also indicated by m.
7. If the slopes of two lines be m 1
and m 2
, then the
lines will be
(i) parallel if m 1
= m 2
(ii) perpendicular if m 1
m 2
Standard forms:
1. All straight lines can be written as
y = mx + c,
where m is the slope of the straight line, c is the Y
intercept or the Y coordinate of the point at which
the straight line cuts the Y-axis.
2. The equation of a straight line passing through (x 1
y 1
) and having a slope m is
y – y 1
= m(x –x 1
3. The equation of a straight line passing through two
points (x 1
, y 1
) and (x 2
, y 2
) is
2 1
1 1
2 1
y – y
y – y (x – x )
x – x
4. The point of intersection of any two lines of the
form y = ax + b and
y = cx + d is same as the solution arrived at when
these two equations are solved.
5. The lenght of perpendicular from a given, point
(x
1
, y 1
) to a given line ax + by + c = 0 is
1 1
2 2
ax by c
p
(a b )
where p is the length of perpendicular.
In particular, the length of perpendicular from origin
(0,0) to the line ax + by + c = 0 is
2 2
c
a b
- Distance between two parallel lines
ax +by + c 1
ax + by + c 2
is
2 1
2 2
c c
a b
