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7/29/2019 Www.time4education.com CAT Important Formulaes CAT Impor
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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