Important formulae

INDICES AND SURDS

a^m x aⁿ = a^m+n

a^m/aⁿ = a^m-n

(a^m)ⁿ = a^mn

a^–m = a^1/am

m√a = a^1/m

(ab)^m = a^m . b^m

a⁰ = 1

a¹ = a

√a+√a+√a + ----∞ = a

√a+√a+√a + ----∞ = (1 ± √1 + 4a )/2

RATIO – PROPORTION – VARIATION

Ratio

If a/b = c/d = e/f , then each of these ratios is equal to (a+c+e)/(b+d+f)

Proportion

If a : b = c : d, then ad = bc = b : a = d : c a : c = b : d
If a : b = c : d, then a + b : a – b = c + d : c – d and This is called as COMPONENDO-DIVIDENDO.
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

Direct variation: If A ∝ B, then A₁/A₂ = B₁/B₂
Inverse variation: If A ∝ 1/B, then A₁B₁ = A₂B₂

PERCENTAGES

x as a percent of y = X/Y x 100
y as a percent of x = Y/x x 100
Percentage Increase = (Final initial/initial) x 100 =(Difference/smaller) x 100.
Percentage More (or) Percentage Exceeding = (Difference/smaller) x 100.
Percentage Less (or) Percentage Decrease =(Difference/Greater) x 100.
If the price of a commodity increases by r%, then the reduction in consumption so that the expenditure remains the same, is ( (r /100 + r) x 100)%

(

)

If the value of a variable is first increased x% and then decreased by y%, then there is ( (X- Y)- XY/100)% increase or decrease, according to the +ve, or –ve sign respectively.
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 X²/100%

PROFIT AND LOSS

Profit = Selling Price (S.P) – Cost Price (C.P)
When S.P. > C.P.,
1. Profit = S.P. – C.P.
2. S.P. = C.P. + Profit
3. C.P. = S.P. – Profit
Profit % = (Profit/C.P.) X 100
S.P. = C.P. X (100 + Profit%)/100)
When S.P. < C.P., Loss = Cost Price (C.P) – Selling Price (S.P)
Loss % = ( Loss /C.P) X 100
S.P. = C.P. X ( 100 - Loss%)/ X 100
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.
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(X/10)² %
Discount = M.P. – S. P.
Discount % = ( Discount/M.P.) x 100
When M.P and Discount % are given,
S.P. = (M.P. X (100 - Diccount&))/100
When an article is sold after two successive discounts of p% and q%, then the final selling price = M.P.((100-p)(100-q))/100 x 100.
If cost price of x articles is equal to the selling price of y articles, then profit or loss percentage =(X - Y/Y) x 100 %
If the shopkeeper sells goods at cost price but gives lesser weight than true weight, then
Gain percentage =(True weight - False weight)/(False weight) x 100
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 [100 - a] c/b - 100 is as the signis +ve or –ve.

SIMPLE INTEREST – COMPOUND INTEREST

Simple Interest = PTR/100.
Amount (A) = P(1 + TR/100)
Compound interest = P[(1 + R/100) - 1]
Amount under C.I = P(1 + R/100)
If a sum becomes x times in y years at CI then it will be (x)ⁿ times in ny years.
If a sum of money becomes ‘m’ times in ‘t’ years at SI, the rate of interest is given by
(a) (100 (m - 1))/t %
(b) Also to become n times, time taken = (n - 1) x t/(m - 1)
If amount under Compound Interest for n years and (n + 1) years is known, then the rate of interest is given by
(Difference of amount after n yrs and (n+1)/rs x 100)/Amount after n yrs
Difference between the compound interest and simple interest on a certain sum of money for 2years at r% rate is given by
Sum(r/100)²
Difference between CI and SI on a certain sum for 3 years at r% is given by
P(r/100)² (r/100 + 3)

AVERAGES – MIXTURES – ALLIGATION

Average = Sum of all items in the group/Number of items in the group.
Quantity of Cheaper/Quantity of Dearer = Rate of deared - Average Rate/Average Rate - Rate of Cheaper

TIME AND WORK – PIPES AND CISTERNS

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
M₁D₁H₁/W₁ = M₂D₂H₂/W₂
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 XY/(X + Y) days
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 = XY/(X - Y)

TIME AND DISTANCE – CIRCULAR TRACKS

Speed

Distance covered per unit time is called speed.

i.e., Speed = Distance/Time
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 ∝ 1/Speed (Inverse Variation)

Joint variation of these is Speed ∝ Distance/Time
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).
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.
Average Speed = Total distance travelled/Total time taken
(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 2XY/X + Y km/hr

(b) Instead if distances are in the ratio m : n, then average speed = ((m + n)xy)/my + nx
Speed of the boat against stream (or) Upstream Speed = Speed of the boat in still water – Speed of the stream.
Speed of the boat with the stream (or) Downstream Speed = Speed of the boat in still water + Speed of the stream.
If two persons cover a certain distance at different speeds reaching at different times, then that distance = (Product of two speeds)/(Difference of two speeds) x Difference between arrival times
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 = √b : √a

Trains

Time taken by a train of length ℓt travelling at a speed of s_t to cross a

(a) pole (or) a tree = ℓ_t/s_t

(b) platform of length ℓ_p = (ℓ_t + ℓ_p)/s_t
Time taken by two trains of lengths ℓ₁and ℓ₂ travelling at speeds of s1 and s2 to cross each other is given by

= (ℓ₁ + ℓ₂)/(s₁ + s₂) (opposite Direction) = (ℓ₁ + ℓ₂)/(s₁ ∼ s₂)(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.

SituationTime 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	Length of the track/Relative Speed = L/(a + b)	Length of the track/Relative Speed = L/(a + b)	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 =( (2n - 4) /n )x 90°

Each exterior angle = 360^o/n

No. of diagonals = n(n - 3)/2

Area of a regular polygon = 1/2 x Perimeter x Perpendicular distance from the centre to any side.

MENSURATION

Triangles

For any triangle

(a) When the measurements of sides a, b, c are given, Area = √s(s - a)(s - b)(s - c)
where s = ^{(a + b + c)}/₂

(b) When height and base are given, Area = 1/2 x Base x Altitude = 1/2 a.h

(c) Area = 1/2 ab . sinC = 1/2 bc.sinA =1/2 ca.sinB

(d) Area =^(abc)/_4R where R is the circumradius.

(e) Area = s.r where r is the inradius.
(a) For a right angled triangle, area = 1/2 x Product of the sides containing the right angle.
(b) In a right angled triangle, circumradius = ^{hypothenuse/₂}
For an equilateral triangle, area =^{√3 . a
²}/₄ where "a" is the side of the triangle. =^{√3 . a}/₄
For an isosceles triangle where "a" is length of two sides which are equal and b is the third side.

Area = b/4 √4a² - b²

Triangles

For any quadrilateral

(a) Area = 1/2 x one diagonal x sum of offsets (or perpendiculars) drawn to this diagonal from the two opposite vertices

(b)Area = 1/2 x Product of diagonals x Sine of the angle between them
= 1/2 . AD.BC.sinq
For a cyclic quadrilateral, Area
=√(s - a)(s - b)(s - c)(s - d) where s is the semi-perimeter, i.e., s = (a + b + c + d)/2
For a trapezium, area = 1/2 x Sum of parallel sides x Height
For a Parallelogram
(a) Area = Base x height
(b) Area = Product of two sides x Sine of included angle
Rhombus, Area = 1/2 x Product of the diagonals Side = 1/2 √sum of the squares of diagonals
For a Rectangle, Area = Length x Breadth
For a Square
(a) Area = side²
(b) Area = 1/2 x Diagonal² [Diagonal =√2 x Side]
For a Polygon

(a) Area of a regular Polygon = 1/2 x height x 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 X - Y - XY/100 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 (x(x + 200))/100 % or [2x + x²/100]%

Circle

Area = πr² where r is the radius of the circle. Circumference = 2πr
Sector of a circle: Area =Θ/360 x πr² Where θ is the angle of the sector and r is the radius of the circle. Length of arc = Θ/360 x2πr
Circular ring : Area = πR² – πr² = π(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 x Area of base

Volume = area of Base x 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 = √ℓ² + b² + h²

Cube

Volume = a³ where "a" is the side of the cube.

Lateral surface Area = 4a² and Total Surface Area = 6a²

Cylinder

Volume = Πr²h, where "r" is the radius of the base and "h" is the height

Curved surface area = 2Πrh and Total Surface Area = 2 Πrh + 2Πr² (or) 2Πr(r + h)

A hollow cylinder has a cross-section of a circular-ring.

Volume of the material contained in a hollow cylindrical ring = Π(R² – r²)h where R is the outer radius, r the inner radius and h1 the height.

Cone

A cone is equivalent to a right pyramid whose base is a circle. Volume = 1/3 Πr²h ; 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 r² = Π r(ℓ + r)

Sphere

Surface area of a sphere = 4Πr², where r is radius.

Volume of a sphere = (4/3)Πr³

SETS AND RELATIONS

Union of Sets:
1. A ∪ B = {x │x∈A or x∈B}
2. If A ⊂ B, then A ∪ B = B
3. A ∪ φ = A
4. A ∪ μ = μ
Intersection of Sets:
1. A ∩ B = {x | x ∈ A and x ∈ B}.
2. A ⊂ B, then A ∩ B = A.
3. A ∩ φ = φ.
4. A ∩ μ = A.
Difference of Sets:
1. A - B = {x | x ∈ A and x ∈ B}
  Similarly, B - A = {x ∈ B│ x ∉ A}
A Δ B = (A ∪ B) – (A ∩ B) = (A – B) ∪ (B – A)
Some Results:
1. A ∪ A = A; A ∩ A = A
2. A ∪ B = B ∪ A
3. A ∩ B = B ∩ A
4. A ∪ (B ∪ C) = (A ∪ B) ∪ C
5. A ∩ (B ∩ C) = (A ∩ B) ∩ C
6. A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
7. A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
8. C - (C - A) = C ∩ A
9. C - (A ∩ B) = (C - A) ∪ (C - B)
10. C - (A ∪ B) = (C - A) ∩ (C - B)
11. (A^c)^c = A
12. (A ∪ B)^c = A^c ∩ B^c
13. (A ∩ B)^c = A^c ∪ B^c
14. A Δ B = (A – B) ∪ (B – A) = (A ∪ B) – (A ∩ B)
15. n(A ∪ B) = n(A) + n(B) –n(A ∩ B)
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 ∈ A and b ∈ B.

The product is denoted by A ´ B

A x B = { (a, b)│ a ∈ A, b ∈ B}
1. A x B ≠ B x A
2. n(A x B) = n(B x A) = n(A) x n(B)
3. The number of relations defined from A to B = 2^{n(A) x n(B)}
Reflexive Relation:
1. A relation R on a set A is said to be reflexive, if for every x ∈ A, (x, x) ∈ R.
2. If 'A' has n elements then a reflexive relation must have at least 'n' ordered pairs.
3. The number of reflexive relations defined from A to A = 2^{r² - 1}.
Symmetric Relation:
A relation R on a set is said to be symmetric if (x, y) ∈ R ⇒ (y, x) ∈ R
1. The necessary and sufficient condition for a relation R to be symmetric is R = R^-1.
Transitive Relation:
A relation R on a set is said to be symmetric if (x, y) ∈ R ⇒ (x, z) ∈ R
Equivalence Relation:
1. A relation which is reflexive, symmetric and transitive is called an equivalence relation.
2. The smallest equivalence relation on a set A is the identity relation.
3. The largest equivalence relation on a set A is the cartesian product A x A.
Antisymmetric Relation:

R is antisymmetric if (a, b) ∈ R, (b, a) ∈ R ⇒ a = b
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

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):

f: A → B is one-one if x1 ≠ x2 ⇒ f(x1) ≠ f(x2), Equivalently f(x1) = f(x2) ⇒ x1 = x2.
The number of one-one functions defined from set A to set B is ^n(B)p₁(A).
Onto Function (Surjection):
1. If f is onto, range of f = co-domain of f.
2. 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.

If f: A → B and g: B → C are two bijective functions, then (g o f)^{- 1} = f^{- 1} o g^{- 1}.
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.
(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) φ/2 \| n ∈ Z }	R
cot x	R - {nφ \| n ∈ Z}	R
sec x	R-{(2n+1) φ/2 \| n ∈ Z}	(- ∞, -1] ∪ [1, ∞)
cosec x	R-{nφ \| n ∈ Z}	(- ∞, -1] ∪ [1, ∞)

MATHEMATICAL LOGIC

If p = T and q = T then p ∧ q = T
If p = F, q = F then p ∨ q = F
If p = T, q = F then p ⇒ q = F
(p ⇒ q) = ∼ p ∨ q
Converse of p ⇒ q = q ⇒ p
Inverse of p ⇒ q = ∼p ⇒ ∼q
Contropositive of p ⇒ q = ∼q ⇒ ∼p
p ⇔ q = (p ⇒ q) ∧(q ⇒ p)
∼(p ∨ q) = ∼p ∧ ∼q
∼(p ∧ q) = ∼p ∨ ∼q
∼(p ⇒ q) =p ∧ ∼q
∼(p ⇔ q) = ∼p ⇔ q
= p ⇔ ∼q

Some Logical Equivalences:

Commutative Properties:
p ∨ q ≡ q ∨ p ; p ∧ q ≡ q ∧ p
Associative Properties:
p ∨ (q ∨ r) ≡ (p ∨ q) ∨ r ; p ∧ (q ∧ r) ≡ (p ∧ q) ∧ r
Distributive Properties:
p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r) ;
p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)
Idempotent Properties:
p ∨ p = p ; p ∧ p = p
Absorption :
p ∨ (p ∧ q) = p
p ∧ (p ∨ q) = q

List of Tautologies:

p ∧ q ⇒ p or p ∧ q ⇒ q
p ⇒ p ∨ q or q ⇒ p ∨ q
∼p ⇒ (p ⇒ q) or ∼(p ⇒ q) ⇒ p
(p ∧(p ⇒ q)) ⇒ q
(∼p ∧ (p ∨ q)) ⇒ q
(∼q ∧ (p ⇒ q)) ⇒ ∼p
((p ⇒ q) ∧ (q ⇒ r)) ⇒ (p ⇒ r)
hypothetical syllogism or transitive rule

Operations with F and T:
(T – Tautology and F – Contradiction)

p ∨ ∼p = T
p ∧ ∼p = F
T ∨ p = T
F ∧ p = F
F ∨ p = p
T ∧ p = p
∼F =
∼T = F

INEQUALITIES AND MODULUS

If a > b, then b < a
If a > b and b > c, then a > c
If a < b and b < c, then a < c
If a > b and c > 0 then a ± c > b ± c
If a > b and c > 0, then ac > bc
If a < b and c > 0, then ac < bc
If a > b and c < 0, then ac < bc
If a < b and c < 0, then ac > bc
If a > 0, then -a < 0 and if a > b, then -a < -b
If a and b are positive numbers and a > b, then 1/a < 1/b,
If A, G and H are the Arithmetic mean, Geometric mean and Harmonic mean of n positive real numbers, then A ≥ G ≥ H, the equality occurring only when the numbers are all equal.

Properties of modulus:

x = 0 ⇔ |x| = 0
|x| ≥ 0 and – |x| ≤ 0
|x + y| ≤ |x| + |y|
| |x| - |y| | ≤ |x - y|
-|x| ≤ x ≤ |x|
|xy| = |x| . |y|
|x/y| =|y/x| , y ≠ 0
|x|² = x²

MATRICES AND DETERMINANTS

A = , then A² –(a + d) A = [bc – ad]I.
If A = , then Aⁿ =
If A = , then det A = ad – bc
If A is a n ´ n matrix, then det(KA) = Kⁿ det A.
det(A^-1) =
If A is a n ´ n matrix, then det(adjA) = (det A)^{n - 1}.
If A = , then adj A =
If A = , then A^–1 =
det(AB) = detA.detB

BINOMIAL THEOREM AND REMAINDER THEOREM

The nP_r and nC_r representation:
1. For 0 ≤ r ≤ n,
  nP_r = n!/(n - r)! and nC_r = n!/r!(n - r)!
2. nC_r = nC_n - r
3. If nC_r = nC_s, then r = s or r + s = n
Binomial Theorem for a Positive Integral Index:
1. If 'n' is a positive integer, then
  (x + y)ⁿ = nC₀xⁿ + nC₁x^{n - 1}y¹ + nC₂x^{n - 2}y²+..... + nC_rx^{n - r}y^r +…..+nC_nyⁿ,
2. Number of terms in the expansion is n + 1
3. General Term in the Expansion of (x + y)ⁿ is
  T_{r + 1} = nC_r x^{n - r} y^r,
4. The general term in the expansion of (x - y)^{n is T_{r + 1} = (-1)^r nC_r x^{n - r} . y^r}
5. If 'n' is even, there exists only one middle term, which is (n/2 + 1)ⁿ term
6. If 'n' is odd, there will be two middle terms i.e.(n + 1)/2 ⁿand(n + 3)/2ⁿ terms.
7. 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 = (n + 1)/2 or n_c (n - 1)/2 (if n is odd)
8. The coefficient of xk in the expansion(ax^p + b/x^q)ⁿ is T_{r + 1}, then r =( n_p - k )/p + q.
9. The constant term in(ax^p + b/x^q)ⁿ is T_{r + 1}, then r =( n_p )/p + q. .
10. The number of terms in (a + b + c)n is (n + 1)(n +2)/2
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 ((n + 1)|x|)/|x| + 1:
1. If ((n + 1)|x|)/|x| + 1 = an integer, say 'k', then k^th and (k + 1)^th terms are numerically greatest terms
2. If ((n + 1)|x|)/|x| + 1 is not an integer, say k + a; where 0 < a < 1, (k + 1)^th term is numerically greatest term.
Properties of Binomial Coefficients:
1. C₀ + C₁ + C₂ + ....... + C_n = 2ⁿ
  i.e., sum of all the binomial coefficients in the expansion of (1 + x)ⁿ is 2ⁿ
2. C₁ + C₃ + C₅ + . . . . =2ⁿ/2 = 2^{n - 1}
  C₀ + CC₂ + C₄ . . . . = 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}
3. Let f(x) = (ax +b)ⁿ
  The sum of all coefficients = f(1)
  The sum of even coefficients =(f(1) + f(-1))/2
  The sum of odd coefficients = (f(1) - f(-1))/2
4. The ratio of the two successive terms in (ax + by)ⁿ is T_r/T_{r + 1} = nr/n- r + 1 (by/ax)
Expansions:
1. (1 - x)^-1= 1 + x + x² + x³ + ........ + x^r + .......
2. (1 + x)^-1 = 1 - x + x² - x³ + ....... + (-1)^r x^r +......
3. (1 - x)^-2 = 1 + 2x + 3x² + 4x³ + ........ + (r + 1)x^r + .......
4. (1 + x)^-2 = 1 - 2x + 3x² - 4x³ + ....... + (-1)^r(r + 1)x^r + ......
Remainder Theorem:
1. If a polynomial, say f(x), is divided by (x - a), then the remainder is f(a).
2. If f(x) is a polynomial such that f(a) = 0, then (x - a) is a factor of f(x).
3. 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).
4. 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.
5. If f(x) is divided by ax + b, then remainder is f (- b/a).
6. 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 x Quotient] + Remainder we have, f(x) = p(x) x q(x) + r(x)

STATISTICS

Arithmetic mean (A.M. or ):
1. Individual series: =
2. Discrete series: = where x₁, x₂, .… x _n are n distinct values with frequencies f₁, f₂, f₃, …., f _n respectively.
3. Continuous series: = = where f₁, f₂, f₃ ,…. fn are the frequencies of the classes whose mid-values are m₁, m₂, … m_n respectively.
4. The algebraic sum of deviations taken about mean is zero.
5. Mean of first n natural numbers is (n+1)/2
6. Arithmetic mean of two numbers a and b is (a + b)/2
7. Combined Mean: If x1 and x2 are the arithmetic means of two series with n1 and n2 observations respectively, the combined mean,
Geometric Mean (G.M.) for Individual series
1. G.M. = (x₁. x₂. x₃ …. x_n)^{1/ n} ,
2. Geometric mean of two numbers a and b is√ab
3. If b is G.M. of a and c then a, b and c are in a geometric progression.
Harmonic Mean (H.M.) for Individual series:
1. H =
  where x₁, x₂, …. x_n are n observations.
2. Harmonic mean of two numbers a and b is .
3. (GM)² = AM x HM.
Median
1. Individual series:
  If x₁, x₂, …. x_n are arranged in ascending order of magnitude then the median is the size of item.
2. Discrete series:
  Median is the value of the variable x for which the cumulative frequency just exceeds or equals , N being the total frequency.
3. Continuous series:
  Median = l +
  where l = lower boundary of the median class (or the class in which N/2th 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 = ∑fi
4. The sum of absolute deviations taken about median is least.
Mode for continuous series:
Mode =
where l₁ = lower boundary of the modal class (class, where the frequency is maximum)

c = width of the class Δ1 = f – f₁ Δ2 = f – f₂
f = frequency of the modal class f₁ = frequency of the class which immediately precedes modal class

f₂ = frequency of the class which immediately succeeds modal class
Empirical Formula:
1. For moderately symmetrical distribution,
  Mode = 3 median – 2 mean
2. For a symmetric distribution,
  Mode = Mean = Median.
Measures of Dispersion:
Quartile Deviation (Q.D.): Q.D. =
where Q1 ® size of (n+1)/4th item
Q3 ® size of 3(n+1)/4th item
Mean Deviation (M.D.) for Individual series:
1. M.D. =
  where x₁, x₂ …. x_n are the n observations and A is the mean or median or mode.
2. Mean deviation about the median is the least.
3. Mean deviation of two numbers a and b is .
Standard Deviation (S.D.):
Individual series:
1. S.D. (σ ) = where x₁, x₂, …. x_n are n observations with mean as x
2. σ =
3. 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.
Coefficient of variation (CV) is defined as, CV =
Correlation: We confine our study of correlation to Spearman's rank correlation co-efficient.
ρ = 1 – [di = (xi - yi); i = 1, 2, 3, . . . . , n]
Co-efficient of correlation:
1. Limits of correlation coefficient –1 ≤ r ≤ 1.
2. ρ = + 1, perfect positive correlation.
3. ρ = – 1, perfect negative correlation.
4. ρ = 0, we infer that there is no linear correlation.
5. ρ > 0, positive correlation.
6. ρ < 0, negative correlation.
7. Coefficient of skewness:
  Sk =

PROBABILITY

P(E) =
P = 1 – P(E).
P(A ∩ B) = P(A).P(B) when A, B are independent events.
P(A ∪ B) = P(A) + P(B) –P(A ∩ B)
When n coins are tossed the probability of getting r heads =
The odd infavour of E is a : b, then P(E) = ,P = ,

LIMITS

k f(x) = k f(x)
[f(x) ± g(x)] = f(x) ± g(x)
[f(x) . g(x)] = f(x) . g(x)
=
= 1
= 1
= a
= a/b
= a/b
=
If k and ‘n’ are constants, |x| > 0 and n > 0 then and
= 1
(1 + x)1/x = e
(1 + 1/x)x = e
= 1
= loge a
L’ Hospitals Rule
(i) If is of the indeterminate form , then .
(ii) If is also of indeterminate form and so on