INDICES AND SURDS
(1) a^{m} x a^{n} = a^{m+n}
(2) _{}^{}= a^{m-n}
(3) (a^{m})^{n} = a^{mn}
(4) a^{–m}
= _{}^{}
(5) _{}^{}= a^{1/m}
(6) (ab)^{m} = a^{m} . b^{m}
(7) a^{0}
= 1
(8) a^{1} = a
(9) _{}
(10) _{} = _{}^{ }
RATIO – PROPORTION – VARIATION
Ratio
(1) If
_{}, then each of these ratios is equal to _{}
Proportion
(2) If
a : b = c : d, then ad = bc = b : a
= d : c a : c = b : d
(3) If
a : b = c : d, then a + b : a – b = c + d : c – d and This is called as COMPONENDO-DIVIDENDO.
(4) If
three quantities a, b and c are such that a : b :: b : c, then we say that they are in CONTINUED PROPORTION.
We also get, b² = ac.
Variation
(5) Direct
variation: If A a B, then _{} = _{}
(6) Inverse variation: If A a _{}, then A_{1}B_{1} = A_{2}B_{2}
PERCENTAGES
(1) x as a
percent of y = _{}´ 100
(2) y as a
percent of x = _{}´ 100
(3) Percentage
Increase = _{}= _{}.
(4) Percentage
More (or) Percentage Exceeding = _{}
(5) Percentage
Less (or) Percentage Decrease = _{}
(6) If
the price of a commodity increases by r%, then the reduction in consumption so
that the expenditure remains the same, is
_{}
Instead
if it decreases, increase in consumption = _{}
(7) If
the value of a variable is first increased x% and then decreased by y%, then
there is _{}% increase or decrease, according to the +ve, or –ve sign
respectively.
(8) If
the value of a number is first increased by x% and later decreased by x%, the
net change is always a decrease which is equal to _{}%
PROFIT AND LOSS
(1) Profit
= Selling Price (S.P) – Cost Price (C.P)
When
S.P. > C.P., (i) Profit
= S.P. – C.P.
(ii) S.P.
= C.P. + Profit
(iii) C.P.
= S.P. – Profit
(2) Profit
% = _{}´ 100
(3) S.P.
= C.P. ´ _{}
(4) When
S.P. < C.P., Loss
= Cost Price (C.P) – Selling Price (S.P)
(5) Loss
% = _{} ´ 100
(6) S.P.
= C.P. ´ _{}
(7) When there are two articles having the
same cost price and if one article is sold at a% profit and the other is sold
at a% loss, profit % or loss % is zero.
(8) If there are two articles having the same
selling price and one is sold at x% profit and the other is sold at x% loss,
effectively, a loss will always be made and the loss percent is_{}%
(9) Discount
= M.P. – S. P.
(10) Discount
% = _{} ´ 100
(11) When
M.P and Discount % are given,
S.P.
= _{}
(12) When
an article is sold after two successive discounts of p% and q%, then the final
selling price =_{}.
(13) If
cost price of x articles is equal to the selling price of y articles, then
profit or loss percentage
= _{}%
(14) If
the shopkeeper sells goods at cost price but gives lesser weight than true
weight, then
Gain
percentage = _{}
(15) If
the shopkeeper sells his goods at a% loss on cost price but uses b gm instead
of c gm, then his % profit or loss is _{}as the sign is +ve or –ve.
SIMPLE INTEREST – COMPOUND INTEREST
(1) Simple
Interest = _{}
(2) Amount
(A) = P_{}
(3) Compound
interest = P _{}
(4) Amount under C.I = P_{}
(5) If
a sum becomes x times in y years at CI then it will be (x)^{n} times in
ny years.
(6) If
a sum of money becomes ‘m’ times in ‘t’ years at SI, the rate of interest is
given by
(a) _{}
(b) Also to become n times, time taken = _{}
(7) If
amount under Compound Interest for n years and (n + 1) years is known, then the
rate of interest is given by
_{}
(8) Difference
between the compound interest and simple interest on a certain sum of money for
2years at r% rate is given by
Sum_{}
(9) Difference
between CI and SI on a certain sum for 3 years at r% is given by
_{}
AVERAGES –
MIXTURES – ALLIGATION
(1) Average
= _{}
(2) _{}
=
_{}
TIME AND WORK –
PIPES AND CISTERNS
(1) Work
and men are directly proportional to each other, i.e.,
Men and days are inversely
proportional
Men
and hours are inversely proportional
Joint
variation of the above can be written as _{}
(2) If
A can do a piece of work in x days and B can do it in y days, then A and B
working together will do the same work in _{}days.
(3) If
a pipe can fill a tank in x hours , and
another pipe can empty the full tank in y hours, then the time taken to fill the
tank, when both the pipes are opened = _{}
TIME AND DISTANCE – CIRCULAR
TRACKS
Speed
(1) Distance
covered per unit time is called speed.
i.e.,
Speed = Distance/Time
(2) If
two bodies travel with the same speed, Distance covered µ Time (Direct
Variation).
If
two bodies travel for the same period of time,
Distance covered µ Speed (Direct Variation).
If
two bodies travel in the same distance, Time µ _{} (Inverse Variation)
Joint
variation of these is _{}
(3) To
convert speed in kmph to m/sec, multiply it with (5/18).
To
convert speed in m/sec to kmph, multiply it with (18/5).
(4) Relative
Speed:
(i) In the same direction, difference of the
speeds of the two bodies.
(ii) In opposite direction = Sum of the speeds of
the two bodies.
(5) Average Speed = _{}
(6) (a) If a person travels a distance at a speed
of x km/hr and the same distance either in the same direction or
the
other way at a speed of y km/hr, then the average speed during the whole
journey is given by
_{}km/hr
(b) Instead if distances are in the ratio m : n,
then average speed = _{}
(7) Speed
of the boat against stream (or) Upstream Speed = Speed of the boat in still
water – Speed of the stream.
(8) Speed
of the boat with the stream (or) Downstream Speed = Speed of the boat in still
water + Speed of the stream.
(9) If
two persons cover a certain distance at different speeds reaching at different
times, then that distance
=
_{}
(10) If
two persons A and B start at the same time in opposite directions form two points and after
passing each other they complete the journey in ‘a’ and ‘b’ hrs respectively,
then A’s speed : B’s speed = _{}
Trains:
(1) Time
taken by a train of length ℓ_{t} travelling at a speed of s_{t}
to cross a
(a) pole (or) a tree = _{}
(b) platform of length ℓ_{p} = _{}
(2) Time
taken by two trains of lengths ℓ_{ 1} and ℓ_{ 2}
travelling at speeds of s_{1} and s_{2} to cross each other is
given by = _{}(opposite Direction)
=
_{}(Same Direction)
Circular Tracks
When
two people are running around a circular track starting at the same time from
the same point, we would be interested in finding out the time taken by them to
meet for the first time anywhere on the track or to meet for the first time at
the starting point.
They
start running from the same point at the same time. Then the following table
gives complete details for finding out the time required for various meetings
as discussed above.
Situation Time taken |
Two people A and B running in opposite directions |
Two
people A and B running in the same
direction |
Three
people A, B and C running in the same direction |
To meet
for the first time anywhere
on the track |
_{} |
_{} |
LCM of {L/(a –
b), L/(b –
c)} |
To meet
for the first time at the starting point |
LCM of
{L/a, L/b} |
LCM of
{L/a, L/b} |
LCM of
{L/a, L/b, L/c} |
GEOMETRY & MENSURATION
In
a regular polygon of n sides, each of the interior angles.
d
= _{}´ 90°
Each
exterior angle = _{}
No.
of diagonals = _{}
Area
of a regular polygon = 1/2 ´ Perimeter ´ Perpendicular
distance from the centre to any side.
MENSURATION
Triangles
(i) For
any triangle
(a) When the measurements of sides a, b, c are
given, Area
= _{}
where s = _{}
(b) When height and base are given,
Area
= 1/2 ´ Base x
Altitude = 1/2 a.h
(c) Area = 1/2 ab . sinC = 1/2 bc.sinA =1/2 ca.sinB
(d) Area = _{}where R is the circumradius.
(e) Area = s.r where r is the inradius.
(ii) (a) For a right angled triangle, area = 1/2 ´ Product of the sides
containing the right angle.
(b) In a right angled triangle, circumradius = _{}
(iii) For
an equilateral triangle, area = _{}where "a" is the side of the triangle.
_{}
(iv) For
an isosceles triangle where "a" is length of two sides which are
equal and b is the third side.
Area
= _{}
Quadrilaterals
(i) For
any quadrilateral
(a) Area = 1/2 ´ one diagonal ´ sum of
offsets (or perpendiculars) drawn to this diagonal from the two
opposite
vertices
(b) Area = 1/2 x Product of diagonals ´ Sine of the angle
between them
= 1/2 . AD.BC.sinq
(ii) For
a cyclic quadrilateral, Area
=
_{} where s is the
semi-perimeter, i.e., s = (a + b + c + d)/2
(iii) For
a trapezium, area = 1/2 ´ Sum of
parallel sides ´ Height
(iv) For
a Parallelogram
(a) Area = Base ´ height
(b) Area = Product of two sides ´ Sine of included
angle
(v) Rhombus,
Area = 1/2 ´ Product of
the diagonals
Side
= _{}
(vi) For
a Rectangle, Area = Length ´ Breadth
(vii) For
a Square
(a) Area = side^{2}
(b) Area = 1/2 ´ Diagonal^{2}
[Diagonal
= _{}´ Side]
(viii) For
a Polygon
(a) Area of
a regular Polygon = 1/2 ´ height ´ perimeter of the polygon
(where height is the perpendicular distance from the centre to any side. Please
note that the centre of a regular polygon is equidistant from all its sides)
(b) For
a polygon which is not regular, the area has to be found out by dividing the
polygon into suitable number of quadrilaterals and triangles and adding up the
areas of all such figures present in the polygon.
In
measuring the sides of a rectangle, one side is taken x% in excess and the
other y% in deficit. The error percent in area calculated from the measurement
is _{}in excess or deficit, according to the +ve or –ve sign.
If
the side of a triangle, rectangle, square, circle, rhombus (or any 2 -
dimensional figure) are increased by x% its area is increased by _{}% or _{}%
Circle
(i) Area
= pr^{2} where r
is the radius of the circle.
Circumference
= 2pr
(ii) Sector
of a circle:
Area
= _{}´ pr^{2} Where
θ is the angle of the sector and r is the radius of the circle.
Length
of arc = _{}´ 2pr
(iii) Circular ring : Area = pR^{2} – pr^{2} = p(R + r)(R - r)
Solids
Solids
are three-dimensional objects which, in addition to area, have volume also. For
solids, two different types of areas are defined
(a) Lateral
surface area or curved surface area and
(b) Total
surface area.
For any solid, whose faces are regular
polygons, No. of faces + No. of vertices = No. of edges + 2
Prism
Lateral surface area = perimeter of base x
height
Total surface area = Lateral Surface Area + 2
´ Area of base
Volume = area of Base ´ height
Cuboid
Lateral surface area = 2(ℓ + b).h, where
Total surface area = 2(ℓ + b)h + 2 ℓb = 2(ℓb + ℓh + bh)
Volume = ℓbh
Longest diagonal of the cuboid = _{}
Cube
Volume = a^{3} where "a" is
the side of the cube.
Lateral surface Area = 4a^{2} and Total Surface Area = 6a^{2}
Cylinder
Volume
= pr^{2}h, where
"r" is the radius of the base and "h" is the height
Curved
surface area = 2prh and Total
Surface Area = 2 prh + 2pr^{2} (or) 2pr(r + h)
A
hollow cylinder has a cross-section of a circular-ring.
Volume
of the material contained in a hollow cylindrical ring = p (R^{2} – r^{2})h where R is the outer radius, r the inner
radius and h_{1} the height.
Cone
A
cone is equivalent to a right pyramid whose base is a circle. Volume = 1/3
πr^{2}h ; Where r is the radius of the base, h is height of the
cone and ℓ is the slant height.
Curved surface area = π r.
ℓ;
Total
surface area = π r
ℓ
+ π r^{2} = π r(ℓ + r)
Sphere
Surface area of a sphere = 4πr^{2}, where r is radius.^{ }
Volume of a sphere = (4/3)πr^{3}
SETS AND RELATIONS
(1) Union of
Sets:
(i) A È B = {x │xÎA or
xÎB}
(ii) If A Ì B, then A È B = B
(iii) A È f = A
(iv) A È m = m
(2) Intersection of Sets:
(i) A Ç B = {x | x
Î A and x Î B}.
(ii) A Ì B, then A ÇB = A.
(iii) A Ç f = f.
(iv) A Ç m = A.
(3) Difference of Sets:
A
- B = {x | x Î A and x Ï B}
Similarly,
B - A = {x Î B│ x Ï A}
(4) A D B = (A È B) – (A Ç B) = (A – B) È (B – A)
(5) Some Results:
(1) A È A = A; A Ç A = A
(2) A È B = B È A
(3) A Ç B = B Ç A
(4) A È (B È C) = (A È B) È C
(5) A Ç (B Ç C) = (A Ç B) Ç C
(6) A È (B Ç C) = (A È B) Ç (A È C)
(7) A Ç (B È C) = (A Ç B) È (A Ç C)
(8) C - (C - A) = C Ç A
(9) C - (A Ç B) = (C - A) È (C - B)
(10) C - (A È B) = (C - A) Ç (C - B)
(11) (A^{c})^{c} = A
(12) (A È B)^{c}
= A^{c} Ç B^{c}
(13) (A Ç B)^{c} = A^{c} È B^{c}^{}
(14) A D B = (A –
B) È (B – A) =
(A È B) – (A Ç B)
(15) n(A È B) = n(A)
+ n(B) –n(A Ç B)
(6) Cartesian Product of Two Sets:
Let
A and B be any two sets. Then the cartesian product of A and B is the set of
all ordered pairs of the form (a, b), where a Î A and b Î B.
The
product is denoted by A ´ B
A
´ B = { (a, b)│ a
Î A, b Î B}
(i) A ´ B ¹ B ´ A
(ii) n(A ´ B) = n(B ´ A) = n(A) ´ n(B)
(iii) The number of relations defined from A to B = 2^{n(A) }^{´}^{ n(B)}
(7) Reflexive Relation:
(i) A relation R on a set A is said to be
reflexive, if for every x Î A, (x, x) Î R.
(ii) If 'A' has n elements then a reflexive
relation must have at least 'n' ordered pairs.
(iii) The number of reflexive relations defined from
A to A = _{}.
(8) Symmetric Relation:
A
relation R on a set is said to be symmetric if
(x, y) Î R Þ (y, x) Î R
(1) The necessary and sufficient condition for a
relation R to be symmetric is R = R^{-}^{1}.
(9) Transitive Relation:
A relation R on a set A is said to be transitive (x,
y) Î R, (y, z) Î R Þ (x, z) Î R.
(10) Equivalence Relation:
(i) A relation which is reflexive, symmetric and
transitive is called an equivalence relation.
(ii) The smallest equivalence relation on a set A
is the identity relation.
(iii) The largest equivalence relation on a set A is
the cartesian product A ´ A.
(11) Antisymmetric Relation:
R
is antisymmetric if (a, b) Î R, (b, a) Î R Þ a = b
(12) Partially Ordered Relation:
Let
R be a relation on a set A, then R is said to be a partial order relation if it
is reflexive, antisymmetric and transitive.
FUNCTIONS
(1) Number of functions from a set A
containing m elements to another set B containing n elements is n^{m}.
One-to-One Function (Injection):
(2) f:
A ® B is one-one if x_{1}
¹ x_{2} Þ f(x_{1}) ¹ f(x_{2}),
Equivalently f(x_{1}) = f(x_{2}) Þ x_{1} = x_{2}.
(3) The number of one-one functions defined from set A to set B is _{}._{}
(4) Onto
Function (Surjection):
(i) If f is onto, range of f = co-domain of f.
(ii) If n(A) = m; n(B) = 2, then the number of
onto functions defined from A to B is 2^{m}
– 2.
Bijection:
If a function is both one-to-one and onto,
then it is called a bijective function or bijection.
The number of bijective
functions defined from A to B is m! where n(A) = n(B) = m
Composite Function or
Product Function:
If
f: A ® B and g :
B ® C are two functions,
then g o f is a function from A to
C, such that g o f(a) = g[f(a)], for every a Î A and is called the composite mapping of f and g.
(1) If f: A ®
B and g: B ® C are two bijective functions, then (g o f)^{-}^{ 1} = f^{-}^{1 }o g^{-}^{1}.
(2) If
h: A ® B, g: B ® C and f: C ® D be any three
functions, then f o (g o h) = (f o g) o h.
(3) (f
o f^{ }^{-}^{1}) (x) = x
or f o f^{-}^{1} = I.
Real Function:
Given below are the domain and range of the
various trigonometric functions.
Function |
Domain |
Range |
sin x |
R |
[-1, 1] |
cos x |
R |
[-1, 1] |
tan x |
R-{(2n+1) ^{p}/_{2 }|
nÎZ } |
R |
cot x |
R - {np | nÎZ} |
R |
sec x |
R-{(2n+1) ^{p}/_{2}
| nÎZ} |
(- ¥, -1] È [1, ¥) |
cosec x |
R-{np | n Î Z} |
(- ¥, -1] È [1, ¥) |
MATHEMATICAL LOGIC
(1) If
p = T and q = T then p Ù q = T
(2) If
p = F, q = F then p Ú q = F
(3) If
p = T, q = F then p Þ q = F
(4) (p
Þ q) = ~ p Ú q
(5) Converse
of p Þ q = q Þ p
(6) Inverse
of p Þ q = ~p Þ ~q
(7) Contropositive
of p Þ q = ~q Þ ~p
(8) p
Û q = (p Þ q) Ù(q Þ p)
(9) ~(p Ú q) = ~p Ù ~q
(10) ~(p Ù q) = ~p Ú ~q
(11) ~(p Þ q) =p Ù ~q
(12) ~(p Û q) = ~p Û q
= p Û ~q
Some Logical Equivalences:
(1) Commutative
Properties:
p
Ú q º q Ú p ; p Ù q º q Ù p
(2) Associative
Properties:
p Ú (q Ú
r) º (p Ú q) Ú r ; p Ù (q Ù r) º (p Ù q) Ù r
(3) Distributive
Properties:
p
Ú (q Ù r) º (p Ú q) Ù (p Ú r) ;
p
Ù (q Ú r) º (p Ù q) Ú (p Ù r)
(4) Idempotent
Properties:
p
Ú p = p ; p Ù p = p
(5) Absorption
:
p
Ú (p Ù q) = p
p
Ù (p Ú q) = q
List of Tautologies:
(a) p
Ù q Þ p or p Ù q Þ q
(b) p
Þ p Ú q or q Þ p Ú q
(c) ~p Þ (p Þ q) or ~(p Þ q) Þ p
(d) (p
Ù(p Þ q)) Þ q
(e) (~p Ù (p Ú q)) Þ q
(f) (~q Ù (p Þ q)) Þ ~p
(g) ((p
Þ q) Ù (q Þ r)) Þ (p Þ r)
hypothetical
syllogism or transitive rule
Operations
with F and T:
(T –
Tautology and F – Contradiction)
(a) p
Ú ~p = T
(b) p
Ù ~p = F
(c) T
Ú p = T
(d) F
Ù p = F
(e) F
Ú p = p
(f) T
Ù p = p
(g) ~F =
(h) ~T = F
INEQUALITIES AND MODULUS
(i) If
a > b, then b < a
(ii) If
a > b and b > c, then a > c
(iii) If
a < b and b < c, then a < c
(iv) If
a > b and c > 0 then a ± c > b ± c
(v) If
a > b and c > 0, then ac > bc
(vi) If
a < b and c > 0, then ac < bc
(vii) If
a > b and c < 0, then ac < bc
(viii) If
a < b and c < 0, then ac > bc
(ix) If
a > 0, then -a < 0
and if a > b, then -a < -b
(x) If
a and b are positive numbers and a > b, then 1/a < 1/b,
(xi) If
A, G and H are the Arithmetic mean, Geometric mean and Harmonic mean of n
positive real numbers, then A ³ G ³ H, the equality
occurring only when the numbers are all equal.
Properties of modulus:
(1) x
= 0 Û |x| = 0
(2) |x| ³ 0 and – |x| £ 0
(3) |x + y| £ |x| + |y|
(4) | |x| - |y| | £ |x - y|
(5) -|x| £ x £ |x|
(6) |xy| = |x| . |y|
(7) _{} = _{}, y ¹ 0
(8) |x|^{2} = x^{2}
MATRICES AND DETERMINANTS
(1) A
= _{}, then A^{2} –(a + d) A = [bc – ad]I.
(2) If
A = _{}, then A^{n} = _{}
(3) If
A = _{}, then det A = ad – bc
(4) If
A is a n ´ n matrix,
then
det(KA)
= K^{n} det A.
(5) det(A^{–1})
= _{}
(6) If
A is a n ´ n matrix,
then
det(adjA)
= (det A)^{n – 1}.
(7) If
A = _{}, then adj A = _{}
(8) If
A = _{}, then A^{–1 }= _{}_{}
(9) det(AB)
= detA.detB
BINOMIAL
THEOREM AND REMAINDER THEOREM
(1) The ^{n}P_{r} and ^{n}C_{r}
representation:
(i) For 0 £ r £ n,
_{}and _{}
(ii) ^{n}C_{r} = ^{n}C_{n
}_{-}_{ r}
(iii) If ^{n}C_{r} = ^{n}C_{s},
then r = s or r + s = n
(2) Binomial Theorem for a Positive Integral
Index:
(i) If 'n' is a positive integer, then
(x
+ y)^{n} = ^{n}C_{0}x^{n} + ^{n}C_{1}x^{n
}^{-}^{ 1}y^{1}
+ ^{n}C_{2}x^{n }^{-}^{ 2}y^{2}+..... + ^{n}C_{r}x^{n
}^{-}^{ r}y^{r }+…..+^{n}C_{n}y^{n},
(ii) Number of terms in the expansion is n + 1
(iii) General
Term in the Expansion of (x + y)^{n} is
\ T_{r + 1} = ^{n}C_{r} x^{n }^{-}^{ r} y^{r},
(iv) The
general term in the expansion of (x -
y)^{n} is T_{r+1 }= (-1)^{r} ^{n}C_{r} x^{n }^{-}^{ r} . y^{r}
(v) If 'n' is
even, there exists only one middle term, which is _{}term
(vi) If 'n' is odd, there will be two middle terms
i.e. _{}and _{}terms.
(vii) The Greatest Coefficient in the Expansion of (1
+ x)^{n} (where n is a positive integer): is _{}
=
n/2 (if n is even)
n_{c
}= _{} or n_{c}
_{} (if n is odd)
(viii) The coefficient of x^{k} in the expansion _{}is T_{r + 1}, then r = _{}.
(ix) The constant term in _{}is T_{r + 1}, then r = _{}.
(x) The number of terms in (a + b + c)^{n}
is _{}
(3) Numerically Greatest term in the Expansion
of (1 + x)^{n}:
The
numerically greatest term in the expansion of (1 + x)^{n} is found out
using the following process. We calculate the value of _{}:
(i) If _{}= an integer, say 'k', then k^{th} and (k + 1)^{th}
terms are numerically greatest terms
(ii) If _{}is not an integer, say k + a; where 0 < a < 1, (k + 1)^{th}
term is numerically greatest term.
(4) Properties of Binomial Coefficients:
(i) C_{0} + C_{1} + C_{2}
+ ....... + C_{n} = 2^{n}
i.e., sum of all the binomial coefficients in the
expansion of (1 + x)^{n} is 2^{n}
(ii) C_{1} + C_{3} + C_{5}
+ . . . . = _{}= 2^{n }^{-}^{ 1}
C_{0} + C_{2} + C_{4}
. . . . = 2^{n }^{-}^{ 1}
i.e. sum of coefficients of odd terms is
equal to the sum of coefficients of even terms and each sum is 2^{n }^{-}^{ 1}
(iii) Let f(x) = (ax
+b)^{n}
The sum of all
coefficients = f(1)
The sum of
even coefficients = _{}
The sum of odd coefficients = _{}
(iv) The ratio of the two successive terms in (ax +
by)^{n} is _{} _{}
(5) Expansions:
(i) (1 - x)^{-}^{1} = 1 + x +
x^{2} + x^{3} + ........ + x^{r} + .......
(ii) (1 + x)^{-}^{1} = 1 - x + x^{2} - x^{3} +
....... + (-1)^{r}
x^{r} +......
(iii) (1 - x)^{-}^{2} = 1 + 2x +
3x^{2} + 4x^{3} + ........ + (r + 1)x^{r} + .......
(iv) (1 + x)^{-}^{2} = 1 - 2x + 3x^{2} - 4x^{3} +
....... + (-1)^{r}(r
+ 1)x^{r} + ......
(6) Remainder Theorem:
(i) If a polynomial, say f(x), is divided by (x - a), then the
remainder is f(a).
(ii) If f(x) is a polynomial such that f(a) = 0,
then (x - a) is a
factor of f(x).
(iii) If the sum
of the coefficients of all the terms in the polynomial f(x) is zero, then (x - 1) is a factor of f(x).
(iv) If the
sum of the coefficients of odd powers of x is equal to the sum of the
coefficients of even powers of x, then (x+1) is a factor.
(v) If f(x) is divided by ax + b, then remainder
is f (- b/a).
(vi) When a polynomial, say f(x) is divided by a polynomial p(x) to give
a quotient q(x) and a remainder r(x), by division algorithm
Dividend
= [Divisor ´ Quotient] + Remainder we have, f(x) = p(x) ´
q(x) + r(x)
STATISTICS
(1) Arithmetic mean (A.M. or _{}):
(i) Individual
series:
_{} = _{}
(ii) Discrete
series:
_{}= _{}
where
x_{1}, x_{2}, .… x _{n} are n distinct values with
frequencies f_{1}, f_{2}, f_{3}, …., f _{n }respectively.
(iii) Continuous
series:
_{}= _{}= _{}
where
f_{1}, f_{2}, f_{3 ,}…. f_{n} are the
frequencies of the classes whose
mid-values are m_{1}, m_{2}, … m_{n}
respectively.
(iv) The algebraic sum of deviations taken about
mean is zero.
(v) Mean of first n natural numbers is (n+1)/2
(vi) Arithmetic mean of two numbers a and b is (a +
b)/2
(vii) Combined Mean: If x_{1 }and x_{2}
are the arithmetic means of two series with n_{1 }and n_{2 }observations
respectively, the combined mean, _{}
(2) Geometric Mean (G.M.) for Individual
series
(i) G.M. = (x_{1}. x_{2}. x_{3}
…. x_{n})^{1/ n },
(ii) Geometric mean of two numbers a and b is _{}
(iii) If b is G.M. of a and c then a, b and c are in
a geometric progression.
(3) Harmonic Mean (H.M.) for Individual
series:
(i) H = _{}
where
x_{1}, x_{2}, …. x_{n}
are n observations.
(ii) Harmonic mean of two numbers a and b is _{}.
(iii) (GM)^{2} = AM ´ HM.
(4) Median:
(i) Individual series:
If
x_{1}, x_{2}, …. x_{n} are arranged in ascending order
of magnitude then the median is the size
of _{} item.
(ii) Discrete series:
Median
is the value of the variable x for which the cumulative frequency just exceeds
or equals _{}, N being the total frequency.
(iii) Continuous series:
Median = l + _{}
where l = lower
boundary of the median class (or the class in which N/2^{th} item lies)
m = cumulative frequency upto the median.
c = width of the class interval
f = frequency of median class
N = Total frequency i.e., N = åf_{i}
(iv) The sum of absolute deviations taken about
median is least.
(5) Mode for continuous series:
Mode = _{}
where l_{1} = lower boundary of the modal class
(class, where the frequency is maximum)
c = width of the
class
_{ }_{D}_{1 = f – f1}
_{ }_{D}_{2 = f – f2}
f = frequency of the modal class
f_{1 }= frequency of the class which immediately
precedes modal class
f_{2 }= frequency
of the class which immediately succeeds modal class
(6) Empirical Formula:
(i) For moderately
symmetrical distribution,
Mode = 3 median – 2 mean
(ii) For a
symmetric distribution,
Mode = Mean = Median.
(7) Measures of Dispersion:
(1) Quartile Deviation (Q.D.): Q.D. = _{}
where Q_{1} ® size of (n+1)/4^{th} item
Q_{3}
® size of 3(n+1)/4^{th} item
(2) Mean Deviation (M.D.) for Individual series:
(i) M.D. = _{}
where x_{1}, x_{2} …. x_{n}
are the n observations and A is the mean or median or mode.
(ii) Mean deviation about the median is the least.
(iii)Mean
deviation of two numbers a and b is _{}.
(3) Standard Deviation (S.D.):
Individual series:
(i) S.D. (s ) =_{} where x_{1},
x_{2}, …. x_{n} are n observations with mean as x
(ii) _{}
(iii) For a discrete series in the form a, a + d, a
+ 2d, ……(A.P.), the standard deviation is given by
S.D. = d_{}, where n is number of terms in the series.
(4) Coefficient of
variation (CV) is defined as, CV = _{}
(5) Correlation: We
confine our study of correlation to Spearman's rank correlation co-efficient.
r = 1 – _{}[d_{i} = (x_{i} - y_{i}); i = 1, 2,
3, . . . . , n]
Co-efficient of correlation:
(1) Limits of correlation coefficient –1 £ r £ 1.
(2) r = + 1, perfect positive
correlation.
(3) r = – 1, perfect negative
correlation.
(4) r = 0, we infer that there is no
linear correlation.
(5) r > 0, positive correlation.
(6) r < 0, negative correlation.
(7) Coefficient of skewness:
S_{k} = _{}
PROBABILITY
(1) P(E)
= _{}
(2) P_{} = 1 – P(E).
(3) P(A
Ç B) = P(A).P(B) when
A, B are independent events.
(4) P(A
È B) = P(A) + P(B) –P(A
Ç B)
(5) When
n coins are tossed the probability of getting r heads = _{}
(6) The odd infavour of E is a : b, then P(E) = _{},P_{} = _{},
LIMITS
(1) _{} k f(x) = k _{}f(x)
(2) _{} [f(x) ± g(x)] = _{}f(x) ± _{}g(x)
(3) _{} [f(x) . g(x)] = _{} f(x) . _{} g(x)
(4) _{} = _{}
(5) _{}= 1
(6) _{}= 1
(7) _{}= a
(8) _{}= a/b
(9) _{}= a/b
(10) _{} = _{}
(11) _{}
(12) If k and
‘n’ are constants, |x| > 0 and n > 0 then
_{} and _{}
(13) _{}
(14) _{}
(15) _{}
(16) _{}= 1
(17) _{} (1 + x)^{1/x} = e
(18) _{}(1 + 1/x)^{x} = e
(19) _{}= 1
(20) _{}= log_{e} a
(21) _{}
(22) _{}
(23) L’
Hospitals Rule
(i) If _{}is of the indeterminate form _{}, then _{}.
(ii) If _{}is also of indeterminate form _{}and so on
DIFFERENTIATION
(1) _{}= f '(x)
(2) _{}(k) = 0
(3) _{}k.f(x) = k _{}f(x)
(4) _{}[f(x) ± g(x)] = _{}f(x) ± _{}g(x)
(5) _{} {f(x).g(x)}= f '(x).g(x)+f(x).g ' (x)
(Product
rule or 'uv' rule)
(6) _{}
(7) If
y = f(u) and u = g(x) be two functions, then dy/dx = (dy/du) ´ (du/dx)
(8) (i) _{} (x^{n})= n.x^{n–1}
(ii) _{}= _{}
(9) _{}[ax + b]^{n} = n.a(ax + b)^{n–1}
(10) _{} [e^{ax}] = a.e^{ax}
(11) _{} [logx] = _{}
(12) _{} [a^{x}] = a^{x}.loga
(13) _{} [sinx] = cosx
(14) _{} [cosx] = – sinx
(15) _{} [tanx] = sec²x
(16) _{} [cotx] = – cosec²x
(17) _{} [secx] = secx.tanx
(18) _{} [cosecx] = – cosecx.cotx
(19) If
y = _{} then
then
_{}