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IMPORTANT FORMULAE

ATIO PROPORTION VARIATION

Ifa : b : : c : d, then ad= bc

Ifa : b : : c : d, then a + b : b : : c + d: d

Ifa : b : : c : d, then ab : b : : cd: d

Ifa : b : : c : d, then a + b : ab : : c + d: cd

If then k=

UMBERSa3 + b3 + c3 3abc = (a + b + c) (a2 + b2 + c2 ab bc ca)

The product ofn consecutive integers is always divisible by n! (n factorial)

The sum of any number of even numbers is always even

The sum of even number of odd numbers is always even

The sum of odd number of odd numbers is always odd

If N is a composite number such that N = ap. bq. cr .... where a, b, c are prime factors of N andp, q, r.... are positive integers,then

a) the number of factors of N is given by the expression (p + 1) (q + 1) (r+ 1) ...b) it can be expressed as the product of two factors in 1/2 {(p + 1) (q + 1) (r+ 1).....} waysc) if N is a perfect square, it can be expressed

(i) as a product of two DIFFERENT factors in 1/2 {(p + 1) (q + 1) (r+ 1) ... 1 } ways(ii) as a product of two factors in 1/2 {(p + 1) (q + 1) (r+ 1) ... +1} ways

d) sum of all factors of N =

e) it can be expressed as a product of two co-primes in 2n1 ways, where n is the number of different prime factors of the givennumber N

f) the number of co-primes of N (< N), (N) =

g) sum of the numbers in (e) =

MPLE INTEREST AND COMPOUND INTEREST

= Interest, P is Principle, A = Amount, n = number of years, ris rate of interestInterest under

a) Simple interest, I =

,kf

e

d

c

b

a..........===

.........

.........

fdb

eca

.............1c

1c.

1b

1b.

1a

1a 1r1q1p

+++

............c

11

b

11

a

11N

)N(.2

N

100

Pnr

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b) Compound interest, I = P

Amount under

a) Simple interest, A =

b) Compound interest, A = P

Effective rate of interest when compounding is done ktimes a year

re

=

IXTURES AND ALLIGATIONIfp1,p2 andp are the respective concentrations of the first mixture, second mixture and the final mixture respectively, and q1 and

q2

are the quantities of the first and the second mixtures respectively, then Weighted Average (p)

p =

If C is the concentration after a dilutions, V is the original volume andx is the volume of liquid. Replaced each time then

C =

UADRATIC EQUATIONS

Ifa, b and c are all rational andx + is an irrational root ofax2 + bx + c = 0, thenx is the other root

Ifand are the roots ofax2 + bx + c = 0, then += and =

When a > 0, ax + bx + c has a minimum value equal to , atx =

When a < 0, ax + bx + c has a maximum value equal to , atx =

ROGRESSIONSithmetic Progression (A.P)

s the first term, dis the last term and n is the number of terms

+ 1

100

r1

n

+

1001P

nr

n

100

r1

+

1100

1

+

k

k

r

)qq(

)qpqp(

21

2211

+

+

nx

V

V

y y

a

b

a

c

a

bac

4

4

a

b

2

a

bac

4

4

a

b

2

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Tn

= a + (n 1)d

Sn

= =

Tn

= Sn

Sn1

Sn

= A.M n

ometric Progression (G.P)s the first term, ris the common ratio and n is the number of terms

Tn

= arn1

Sn = =

armonic Progression (H.P)

H.M ofa and b =

A.M > G.M > H.M

(G.M)2 = (A.M) (H.M)

Sum of first n natural numbers n =

Sum of squares of first n natural numbers n2 =

Sum of cubes of first n natural numbers n3 = = (n)2

EOMETRY

In a triangle ABC, if AD is the angular bisector, then

In a triangle ABC, if E and F are the points of AB and AC respectively and EF is parallel to BC, then

In a triangle ABC, if AD is the median, then AB2 + AC2 = 2(AD2 + BD2)

In parallelogram, rectangle, rhombus and square, the diagonals bisect each other

n+

2

last term)m(First tern

dna

+

2

))1(2(

1

)termfirst()termlast(

r

r

1

)1(

r

ran

ba

ab

+

2

2

)1( +nn

6

)12()1( ++ nnn

2

2

)1(

+nn

DC

BD

AC

AB=

ACAF

ABAE =

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a

b

hc

h

b

a a

r

b

a22

ba +

a2

3a a

a

r

r

Sum of all the angles in a polygon is (2n 4)90

Exterior angle of a polygon is

Interior angle of a polygon is

Number of diagonals of a polygon is

The angle subtended by an arc at the centre is double the angle subtended by the arc in the remaining part of the circle

Angles in the same segment are equal

The angle subtended by the diameter of the circle is 90

ENSURATIONPlane figures

Figure Perimeter Area

Triangle = a + b + c

Right angledtriangle

a + b +

Equilateraltriangle

3a

Isocelestriangle

2a + b

Circle 2r r2

Sector of acircle

+ 2r

( is indegrees)

n

360

n

360180

2

)3n(n

bh2

1

)cs()bs()as(s

22 ba + ab2

1

2a4

3

22 ba44

b

r

2360

2

360r

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l

b

h

a

r

lh

h

r

a

aa

a

lb

b

a

h dc

h

b

aa

Square 4a a2

Rectangle 2(l + b) lb

Trapezium a + b + c + d(a + b)h

Parallelogram 2(a + b) bh or absin

2. Solids

FigureLateral Surface

AreaTotal Surface

AreaVolume

Cube 4a2 6a2 a3

Cuboid 2h(l + b) 2(lb + bh + lh) lbh

Cylinder 2rh 2r(r + h) r2h

Cone rl r(l + r) r2h

Sphere

2

1

3

1

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r

r

ha

h

a

R

h lr

r

a

a ah

h

A1

A2

r

a

GHER MATHS I

ERMUTATIONS & COMBINATIONS, PROBABILITY)n (A B) = n (A) + n (B) n (A B)

If A and B are two tasks that must be performed such that A can be performed in 'p' ways and for each possible way of

performing A, say there are 'q' ways of performing B, then the two tasks A and B can be performed inpq ways

The number of ways of dividing (p + q) items into two groups containingp and q items respectively is

4r2

r3

Hemisphere 2r2 3r2 r3

Right prism(i)Equilateral

triangularprism

(ii) Squareprism

(iii) Hexagonal

Prism

3ah

4ah

6ah

3ah +

2a(2h + a)

+ haa 2

2

33

a2h

2

332 ha

Frustum of acone

l(R + l)

l = (R2 + r2

+ Rl +

rl)

h

(R2 + Rr+ r2)

Frustum of aPyramid

Perimeter

of

base Slantheight

L.S.A + A1

+ A2

Torus 42ra 22r2a

3

4

3

2

2a2

3ha

4

3 2

22 h)rR( +

3

1

2

1

)AAAA(

2

1

2121 ++

h

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The number of ways of dividing 2p items into two equal groups ofp each is , when the two groups have distinct

identity and , when the two groups do not have distinct identity

nCr

= nCn r

The total number of ways in which a selection can be made by taking some or all out of (p + q + r+ .....) items wherep are alikeof one kind, q alike of a second kind, ralike of a third kind and so on is {(p + 1) (q + 1) (r+ 1) ....} 1

P(Event) = and 0 P(Event) 1

P(A B) = P(A) P(B), if A and B are independent events

P(A B) = 1, if A and B are exhaustive events

Expected Value = [Probability (Ei)] [Monetary value associated with event E

i]

GHER MATHS II

TATISTICS, NUMBER SYSTEMS, INEQUALITIES & MODULUS, SPECIAL EQUATIONS)

G.M. = (x1x

2 ...... .x

n)1/n

For any two positive numbers a, b(i) A.M. G.M. H.M. (ii) (G.M.)2 = (A.M.) (H.M.)

Range = Maximum value Minimum value

Q.D. = (i.e., one-half the range of quartiles)

Ifa > b, , for any two positive numbers a and b

|x +y| |x| + |y|, for any two real numbersx andy

If for two positive values a and b; a + b = constant (k), then the maximum value of the product ab is obtained for

a = b =

( )2

)!p(

!p2

( )2

)!p(!2

!p2

casesofnumberTotal

casesfavourableofNumber

i

n21 x

1..........

x

1

x

1H.M.

+++

=n

2

|QQ| 13

ba

11

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