math short techniques and formulas - bangla
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Math Short Techniques and Formulas - BanglaTRANSCRIPT
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Mrinal K RoyTypewritten textMathametics Short Techniques and Trics
Mrinal K RoyTypewritten textCollected ByMrinal Kanti [email protected]
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+ Plus / add [ Sum / Total ] Minus / Subtract /Deduct [ Difference ] Times / Into/ dot /of /Multiply
[Product / Multiplication ]
, /,
Division , Divisible , Remainder a(dividend-) b(Divisor-) = c(Quotient-)
Plus or Minus = Equal / Is Approximately Equal
Is not Equal < Is Less then {5 < 10} Is not less then { 10} Is less than or equal to { x X
}
> Is greater then {5> 2} Is not greater then {5 20} Is greater than or equal to ( / ) % Percentage( 100
)
Infinity / Varies as / is Proportional Implies/or If and only if
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Pi 1,2,3...n a,b,c Constant / x,y,z Variable /
2x x + x
=
a+bx+c Expression [a , bx,c Term]
|x| [x] Absolute Value ( ) Negative Positive
Square Root , =2, =3
Cube Root , =3, =4
n Root n
x 1/n
X y
=
Summation ,X n
,*( )+- Bracket
Logarithmsa x ( n= ) [ + , ] 10
n! n Factorial n n (n-1) (n-2)1 [ 5!=54321] npr Permutation [ n npr ,
npr= n(n-1) (n-2) (n-3) ..(n-r+1) Ex:- 6p3=654 ]
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ncr Combination / / ,ncr =
( )
A=*,, ,+ A , a,b,c,d
Intersection /cap [ / ] AB={x:xA xB}
Union/cup [ ]=> ( ) AB={x:xA xB}
Element of / Belongs to / [Ex: bA] Is not an Element of/not Belongs to / Subset of [A / B / ]
/ Superset Empty Set
A\B [B A , A / A ] A-B={x:xA xB}
A Complementary set [U\A=U-A={xU : xA }
P(A) Power Set [A n P(A) 2n ]
NZQR N( / +n), Z( +n,0,+n) Q(0,+n, ),R ( ), ( ) Q=R-Q
Straight line ( ) Ray ( ) Line Segment ( ) Similar to Almost Equal to Is Equivalent to / Congruent
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Angle / Right Angle
Measured Angle Perpendicular To Is Parallel to Therefore / Hence Since / Because / Triangle Rectangle/Square Circle
2 (Unit) () 3 (Digits) / 3 Ex:- 318 3+1+8=12 , 4 (Unit) (tens) 4 Ex:- 544 , 4 44 4
5 (Unit) () 5 6 3 2 8 (Unit) , (tens) (Hundredth) 8 Ex:- 77 264 , 264 8 , 77264 8
9 (Digits) / 9 Ex:- 288 2+8+8=18 , 9 , 288 9
(Unit) () 2, 5, 10 (oo ) , 4, 25, 100 (000) , 8, 125, 1000
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Digit () 0,1,2,3,4,5,6,7,8,9
1 100 0 11 2 9 (2,3,4,5,6,7,8,9) 20 1 21
Number () 87654321 8(Crores) 7(Nijut) 6(lacs) 5(Ajut) 4(Thousands) 3(Hundreds) 2(Tens) 1(Units) 4321=41000+3100+210+1
Integers ( n { -n.-5,-4,-3,-2,-1, 0, +1,+2,+3,+4,+..+n} ) Even Numbers ( 2 {0, 2, 4, 6, 8,.2n} ) Odd Number ( ) 2 {1, 3, 5, 7, 9.(2n+1)} Even Even= Even , Odd Odd=Odd , Even Odd=Even Even + Even=Even , Odd + Odd=Even , Even Odd=Odd Even / Even= Even , Even /Odd=Odd , Odd/Even= Odd Odd Odd =Odd , Odd Even =Odd , Even Even=Even , Even Odd =Even (2,4,6.) (3,5,7..)
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Consecutive Integers ( )
{ n,n+1,n+2,n+3,.} Prime Numbers ( ) ( even prime number) (Odd Prime number) 1 100 25 1060 1 200 46 1 500 95 1 1000 168 ( 2 ,3 3 , )
1 100 1-10 2,3,5,7 4 11-20 11,13,17,19 4 21-30 23,29 2 31-40 31,37 2 41-50 41,43,47 3 51-60 53,59 2 61-70 61,67 2 71-80 71,73,79 3 81-90 83,87 2 91-100 97 1
(4422322321) (1232232244) Composite Number ( ) /
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Fraction ()
Fraction ()= ( ) ()
( , x y )
Rules of fractions
+ =
- =
+ =
=
+
- =
=
-
=
= =
Continued Proportion ( ):-
=
a c = b2
Proper fraction ( ):-
> ( Ex:- ,
)
Improper fraction ( ):-
< ( Ex:- ,
)
Rational Number ( ):- - , - (0,1,2,3.n ), ()-16 , ( a/b - a b ) , -
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Irrational Number ( ): a/b - a b ) ,
( ) , ( ) ,
, ,
, , e Ex:- , , , , ,
Roman Numeral ( ):- I1, V5, X10, L50, C100, O500, M1000
Fators ( ):- ,
, 20 =5221 Multiples ():- ,
Ex:- 35/5=7 , 35/7=5 5 7 35 35 7 5
Common Factor ( ):- ,
Co-prime Number ( ) :- ,
Lowest Common Multiple L.C.M ( .. ): -
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Highest Common Factor H.C.F ( .. ):-
..
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2304 (Square Root )
74 28 1) 2) , 3) 74 64 , = 8 : 4) 8 74 64 5) 74 64 10 28 1028
( ) 6) 8 82 = 16 ( )
7) 16
1028 1028 8) 6 16 6 6 =996 7 ( 16 7 7 = 1169) 1028
9) 1028 996 32 862 = 172
32 10) , 86 11) 74 28 32 7396
86
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(Simplification) :- BODMAS [{(Braket)}]of / Division() Multiplication () Addition (+) Subtraction(-)
a b ?
=x , x-a =b-x => x=
Formula:-- =
1008 (Divisors) ?
1008 = 2222337=243271
= (4+1)(2+1)(1+1)=30
Formula:-- x =ambncl
=(m+1)(n+1)(l+1)
2\1008 2\504 2\252 2\126 3\63 3\21 7
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, n
Let the Numbers =x and (x+1) According to the Question, (x+1)2 x2 = n x
Formula:-- = , = +
n a , b , c d ? Formula:--
= ( a , b , c d) ..
n a , b , c d ? Formula:--
= ( a , b , c d) .. + ( ) a , b , c d n /
Formula:--
= ( a , b , c d) .. + / n .. L .. ? Formula:--
= .. .. a:b .. L .. ? Formula:--
.. = ..
.. .. ?
Formula:--
.. =
.. =
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(a) 2 (b) 3
(n) 10 ? Formula:--
(a) 2 (b) 3 = 6
1 = 6 10 =(610) Formula:-- = ..
=
=
M A N B , ? Formula:--
M A , M = M A [ A =
]
N B , N = N B
= M + N
= AM +BN
=
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Varies as / is Proportional
-
( , , , )
, = = , = a, b, c ,
a:b =b:c
s a:b:c
a+b+c
= S ()
a = S
a:b =x:y b:c =l:m a:b:c =?
a:b = x:y = (xl): (yl) = xl : yl b:c =l:m = (ly): (my)= yl : my a:b:c = xl : yl : my
n . a : b : c ?
Formula:-- = a + b + c
= n ( )
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7 : 3 8 ? Formula:- Required Amount: =
Required ratio
Water =
3 =6
30 7 : 3 3 : 7 (3 into 7) 7 : 5 Formula:-- = 7 + 3 = 10
30 = 30 ( ) ( )
= 21
30 = 30 ( ) ( )
= 9
, = 3 : 7 x 3 : 7 21 : (9 + x) = 3 : 7 x = 40 7 : 5 10 = 30 0/ 0 3
= 7 : (3+2) = 7 :5
1 = 3
6
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100 (100) x % =
N (90) P % (75% ) Find the number whose P % (75% ) is N (90) ? Number N (90) is P % (75% ) of What number?
Formula:-- Let the Number be = x
X P% = N
X 75% = 90
What Percentage % is R(90) of N(120)?
N X% =R
120 x% =90
: 100
100
, 100
,
, 100
X% (25%)
? Formula:--
25% = (100 + 25 ) = 125
125 = 100
,, 100 ,, ,, ,, =
= 80
= ( 00 80)% =20%
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Formula:-
[ % =
]
[ % =
]]
(Increase ) + (Decrease )
+ 25% 20%
+33.33% 25%
, % %
25% , %
25% % , 25%
10(O) 20(N)
?
, : 1(O) : 2(N) , ? Formula:-
% Change =
100%
Result + Increase
Result Decrease
=
100% =100% [ ]
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5% 1000 () ,
5 ?
Formula:-- 5% , 100 + 5 =105 100 = 105
,, 1000 ,, ,, ,, =
= 1050
5 = 1050 1000 = 50
1 ,, ,, ,, = 50/5 = 10
Formula:-
= 10
20% (x%) 10% (y%) , ? Solution: = 100x = 100y
= = 10000 xy
20% = (100+20)x =120x
10% = (100-10) = 90y
= 120x 90y = 10800xy
= 10800xy 10000xy = 800xy
10000xy = 800xy
,, ,, 100 ,, ,, ,, =
=8%
Formula:-
(C%) = x + y +
x First Value ( ) % y Second Value ( ) % , / (Increase) + / (Decrease) (Answer) + , (Answer) ,
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5% (x%) ,
?
Formula:-- +
C% = (+5) + (+5) + ( ) ( )
10 % (x%) , ?
Formula:-- , F = S = 10
C% = (10) + (10) + ( ) ( )
5% (x%) ?
Formula:-- , F = S = 5 , +
C% = (+5) + (+5) + ( ) ( )
25% , ( ) 5%
?
F= + 25%
S = 5%
C% = (+25) + (5) + ( ) ( )
20% , 20%
?
C% = (-20) + (+20) + ( ) ( )
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100 20%
2002 ?
C% = (+20) + (+20) + ( ) ( )
10% 20% %
C% = (+10) + (+20) + ( ) ( )
x% (voter , worker & Student) y % , %
voter , worker & Student ?
Formula:- Mix % =
f% , s% t%
n ? Formula:- - Rest (n) = 100x (f% + s% + t% )x n x%
a/b ?
Formula:- -pass = n x% (a/b) 25%
?
, = 100 = (100+25 ) 125 125 =100
,, ,, 100 ,, ,, ,, ,,
80
= (100 80 ) = 20
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64 25%
40% ? , 100 =25
,, 64 ,, ,, ,, ,, =
, = (6416) 48 , 40 = 100
,, 48 ,, ,, ,, ,, =
120
= (12048) 72 , = (72 16) 56
, ,
70% 80%
10% , 360 ? Formula:-- = (100 70 10)% = 20% = (100 80 10)% = 10% = (20 + 10 + 10)% = 40% = (100 40)% = 60%
60 , = 100
,, ,, 360 ,, ,, ,, ,, ,, ,, =
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200 70% ,60% 40%
1:- ?
2:- ?
Formula:-- = (60 40)% = 20% = (70 40)% = 30% = (40 + 30 + 20)% = 90% = (100 90)% = 10% [1 10% ] 100 = 10
,, 200 ,, ,, ,, ,, ,, ,, =
, , , , , () [ , ]
X = Y 1 = (X Y) [ , ] Z =( )
[ , ]
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9(M1) 3 (D1) 9(D2) ?
, 9(M1) 12(D1) , 3(M1)
?
Formula:- M1 D1 = M2 D2 1) M2 =
2) D2 =
500 20 5 100 ?
Solution : 100 =(500 100) = 400 5 , 500 = (20 5) 15
1 ,, ,, ,, ,, ,, = (15 500)
400 ,, ,, ,, ,, ,, = ( )
[ 500 , , ]
X m , Y n X Y ? Formula:-- X m = 1
1 ,, ,, =
Y 1 =
X Y 1 = ( + ) =
= 1
,, 1 ,, ,, ,, =
Formula:-
T =
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X m , X Y C Y
?
Formula:- T =
X , Y Z a , b c
?
Formula:-
T =
m n ( m n ) ?
Formula:-
T =
, m n
?
Formula:-
T =
[ ] X , Y Z 20 (a) , 24 (b) 30(c)
) ?
Solution
X 20 = 1
X 1 =
Y 1 =
Z 1 =
X , Y Z 1 = (
) =
1
,, 1 ,, ,, ,, = 8
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Formula:-- 1 = 1/8
,, 6 ,, ,, ,, = 6/8 = (1 ) =
X 1 = 20
,, ,, ,, = (20 ) =20/4 = 5
Formula:-- X 1 = 1/ 20
5 ,, ,, =
X 5
X , Y Z = (1 ) =
X , Y Z 1/8 = 1 ,, 1 ,, ,, ,, = 8
,, ,, ,, ,, = = 6
= X , Y Z + X 5 = 6 + 5 = 11
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3( M1) 5( W1) 20(T1) , 4( M2)
10( W2) ( T2) ?
Formula:-- 3 20 = 1
3 ,, ,, 1 ,, ,, =
1 ,, ,, 1 ,, ,, =
4 ,, ,, 1 ,, ,, =
=
[ ,
, ] , 5 20 = 1
1 ,, ,, 1 ,, ,, =
10 ,, ,, 1 ,, ,, =
=
4 10 , 1 = (
+
) =
= 1
1 ,, ,, ,, = 6
Formula :
T2 =
( ) ( )
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10(L) 1(t) 50(U)
25(D) , ? Formula:-
T = ( )
t
U = Up , D = Down
T = ( ) ( )
1= 77Min
,
,
n
?
Formula:-- ( ) n = 1 (
)
=
=
= +
=
, U V [ U > V ]
,
(Relative Speed) = U +V
,
(Relative Speed) = U V
/ / X T
T X
,, 1 ,, ,, ,, =
=
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Lp ( , , )
Lt T
T =( Lp + Lt)
,, 1 ,, ,, ,, = ( )
= ( )
45( ) 100( ) 60(T)
?
Formula:-- , 1 (60 60) = 45 = (45 1000)
60 ,, ,, =
750
60 ,
= +
750 = + 100 = 650 Formula:-
= ( )
45
= ( )
[ T C nver S eed in km/hr to m/sec multiply speed by
]
=650
121 ( x) 99 ( ) 40 ( ) 32 ( )
?
Formula:-
T =
, (Relative Speed) =
, (Relative Speed) =
T = ( )
= 11 Second [1 km/hr = meter/second ]
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150 ( ) 68( ) / , 8 ( ) /
?
Formula:-
T =
= 0
T = ( )
= 9 Second [1 km/hr = meter/second ]
20 ( ) 30 ( ) , ? Formula:-
T =
T = 0 0 0
P (Principal / Capital / Sum )[ Invested]
I / / ( Interest / Extra money for Interest) [ ]
N / [ Per-annum ]
R (Rate of Interest) [ , 100 1 ]
A /-/ (Amount/ Interest-principal/Increased Principal)
= +
P t = p t 1
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(Simple Interest)
P r % n ?
r , 100 1 r
1 1
P 1
P t n
I =
I = npr [r = r% = r/100]
=
[ ]
A = P + I
A = P + nPr = P (1 + nr)
= ( ) ( )
[ , , ]
C = p ( )
-
= ( )
( ) - = (
)
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(r=?) 700 (P) 5(t) 105(I) ? Solution-
700 5 = 105
100 ,, 1 ,, ,, =
Formula:-
I =
105 =
r =
5 - ? Formula:-- , X 5 - = 2X = - = 2X X = X
X 5 = X
100 ,, 1 ,, ,, =
= 20 = 20%
Formula:-
( ) =
=
r =20%
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8% (r1) [ ] 10% (r2) 5(t) 200 (I)
(p=?) ?
Formula:-- 100 1 = (10 8 ) = 2 100 ,, ,,, ,, 5 ,, ,, ,, = (5 2 ) = 10
10 = 100
,, 200 ,, ,, ,, =
Formula:- I =
200 =
p =2000
5 (r) 10 (t) 5000 (A)
(p=?)?
Formula:-- 5% , 100 10 = ( 5 10) = 50
100 10 = ( 100 + 50 ) = 150
150 = 100
,, 5000 ,, ,, ,, =
Formula:-
P = ( )
P =
( )
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300 ( ) 4 ( ) 500 ( ) 5 ( )
148 (I) , (r=?) ?
Formula:-- 300 4 = 1200 1 500 5 = 2500 1 (1200 + 2500 ) = 3700 1
3700 1 148
100 ,, ,, 1 ,, ,, =
= x % = x/100
, I1 =
, I2 =
, I1 + I2 =148
+
= 148
x = 4%
Formula:-
I = ( )
r =
r =
( 00 ) ( 00 )= 4%
10%(r% ) 600 (p) / 2 (n )
?
Formula:-
C = p ( )
C = 600 ( )
C = 726
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(Cost Price ) CP :- / (Selling Price )SP:-
(Profit) = (SP) - (CP) / (Profit /Gain) P:- (loss) = (CP) - (SP) / (Loss)L:-
, - X % x % x C x % ,
=
X % c = ( C +
)
X % C = ( C
)
C P ,
=
6 (n) 5 (n-1) ?
Furmula: 1
(Profit %) = ( )
Profit % =
%
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9 (n) 10 (n + 1) ?
Furmula: 1
(Loss %) =
( )
loss % =
9 (n) 11 (n+2) , 10 (n+1)
? Furmula:
(n) , (n+1) (n+2) ,
, ?
(loss) =
( , / -)
loss % =
( )
12(n) 25%(F%) 50%(S%)
?
Formula:- =n , First Value=F%, Second Value=S%
= n
[ + , ]
= 12
5(m) 2(n) 35(s) 40%(P%)
?
Formula:- = ( ) ( )
=
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150(S) 20(L/P) ? Solution = (150 + 20 ) =170 170 = 20
,, 100 ,, ,, ,, =
Formula:- = + [ = ]
=
Cost = 150 +20 =170 % =
900 (S) 20% (p%) ?
900 (S) 20% (p%) ?
Formula:- =
Cost =
900 (S) , 20% (L%) , ?
Formula:- = ( ) ( )
Cost =
1200(n) 15%(p%) ,
5%(L%) ? Solution: 100 , 100 P% , 100 = (100 + P )
,, n ,, ,, ,, = ( )
L% , 100 = (100 - L )
( )
,, ,, ,, =
( )
( )
[ P% , L% p% 5%(L%) ]
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8% (T%) 15%(p%)
280(C) , (Marked Price) ?
Formula:-- 15% , 100 = (100 +15 ) =115
,, 280 ,, ,, ,, =
= 322
8% ,
(100 - 8 ) 92 = 100
,, 322 ,, ,, ,, ,, = = 350
30(n) 55(C) 20%(n 20% ) 40(S) 12 ?
Formula : ( ) = 30 30 20% =24
(Profit) =(
)
Profit = (
0)
25%(R% -reduction) 120(T)
30(n) ? Formula:-
= R% (
)
Cost=
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10%(L%) 40(C) ,
20%(p%) ?
Solution %
10% = (100 10) = 90 20% = (100 + 20 ) = 120 (120 90) = 30 30 100
,, ,, 40 ,, ,, ,, ,,
Formula:-
=
40 10% 20% ,
10% + 20% =40
15% = 40
1% =
100% =
Formula:-
=
=
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10%(L%) 20%(C%)
50 , 40% (P%) ?
Solution: 3 % 20% = (100 20) = 80 40% = (100 + 40 ) = 140 40% 100 = 140
,, 80 ,, ,, ,,
= 112
10% , = (100 10 ) =90 = (112 -90) = 22
22 = 100
,, 50 ,, ,, ,, =
Formula:
10% , Sells1 = (100 10) = 90
20% = (100 20 ) = 80 ,
profit , 80 40% =32
,Sells2 = + = 80 + 32 = 112
=
[ Absolute Value Negative Positive
]
Cost =
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4 (X) 5 (n) , 5(Y) 4 (m) ?
Solution: 5 = 4 1 ,, ,, =
,
4 = 5 1 ,, ,, =
= - = =
[ ]
=
,, ,, 100 ,, ,, ,, =
Formula
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X Y ?
5 , 4 ? Solution: X = 1
1 ,, ,, =
Y = 1 1 ,, ,, =
= =
=
,, ,, 100 ,, ,, ,, = ( )
Formula
= ( )
= ( )
380 20 ( ) ? Solution: = (380+20) 400 400 =20
,, 100 ,, ,, ,,
10% 45 5% ? Solution: 10% = (100-10) 90
5% = (100+5) 105
= (105 90) 15
15 100
,, 45 ,, ,, ,, ,,
300
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% 10%
31 , 20% ? Solution: 20% = (100+20) 120
10% = (100-10) 90
20% ,
100 = 120
,, 90 ,, ,, ,, = ( ) =108
%
, = (100
)
= (100 )
312 =100
,, 31 ,, ,, ,, ,, =
200
n X % ( )
Y % ( ) ? Solution: x % , = 00
( 00 ) = 100
,, n ,, ,, ,, =
y % , = 00
100 = ( 00 )
,,
,, ,, ,,=
( )
( )
= n ( )( )
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36 20% , ?
Solution: 36 =12
100 ,, ,, ,,
20% , 100 = (100 + 20) 120
20% ,
= 120
20 ,, ,, ,,
1200 15% 5%
? Solution: 15% 1200 + 1200 15% (1200 15
100 )
1200+180 = 1380 5% 1380 1380 5%
1380 69 = 1311
5 2 35 40% ? Solution: 5 = 2
100 ,, ,, =
40
40% ,
100 (100+40) 140
140 =40
35 ,, ,, ,, ,, =
10
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20 12 2 ? Solution: 12 =20
1 ,, ,, ,, =
5
3
1 =
= 2 5
3 = 1
3
53 = 1
3
,, 100 ,, ,,
20
[ ]
3 2 ? Solution: 3 = 1
1 ,, ,, ,, =
2 = 1
1 ,, ,, ,, = 2
1 =
= 12 1
3 = 1
6
[ ]
13 = 1
6
,, 100 ,, ,,
50
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5 , 4
? Solution: , 5 = x 1 ,, ,, ,, =
4 = x
1 ,, ,, ,, =
1 = =
=
=
,, 100 ,, ,,
25
International Counting System ( ): -
1 Lac = 105
1 Million = 106 = 10 lac
1 Crore =107 =10 million
1 Billion = 109 (USA) =1012(Eu)
1 Trillion = 1012(USA) = 1018(Eu) Measurement And Unit ( )
S.I International Systems of Units 1960 CGS Centimeter . Gram .Second M.K.S Meter . Kilogram . Second F.P.S Foot .Pound .Second Local system( ) 12 Inch = 1 foot , 3 foot = 1 Yard , 1760 Yard = 1 mile
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Relationship Distance Area Volume 1 Inch = 2.54 Centimeter 1 Inchs2= 6.45 cm2 1 liter = 1000 Cubic Centimeter
1 Meter = 39.37 Inch 1 Hectare = 10000 meter2 1 Cubic meter = 1000 liters 1 mile = 1.61 kilometer 1 Katha = 720 feet2 1 Cubic foot = 28.31 liters 1 Nautical Mile = 1.853km 1 Acre = 4047 meter2 1 gallon = 3.78 liters(us) = 4.54 liters(uk) 1 km2 = 247 Acres 1 Cubic Centimeter= 1 gram
1 kilogram= 2.2 (lbs) pounds 1 Seer = .93 kg 1 mound = 40 seer = 37.2 kg 1 quintal = 100 kg 1 ton = 1000 kg 1 meter = 100 Centimeter = 39.37 Inchs = 3.28 feet = 1.0936 Y[rds = 0.00062 miles 1 kilogram = 1000 grams = 2.2 pounds = 35.27 Ounces = 0.157 Stone(uk) 1 meter 2 = 10.76 feet2 =1550 Inchs2 = 10000 centimeter2 1 Square kilometer = .3861 mile2 = 100 hectares = 247 acres 1 mile2 = 2.589 kilometers2 = 259 hectare = 640 acres 1 Celsius = 33.8 Fahrenheit = 274.15 kelvin
30(T) 20(F) 15(S) ? Formula : Total Number = First Individual group + Second Individual group (Common Group) F S F + S ( F S)
F S =
F S =
30 = 20 + 15 (F S )
(F S )
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1 12 12 = 3600
,, ,, ,, ,,
= 0
N = n 0
1 0 ( )
H M = ( ) ( )
HM =
[ () ]
0 / 0
Examples:
4 5 ( ) ( )
=
2 50 ( ) ( )
= [ > 0 ]
= 0
3 ( ) ( )
= 0
-
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ail:
( )( ) ( )
( )
(x)
( )
. /
. /
( )
(y
)
( )
. /
. /
( ) ( ) ( )
[
(x) +
(y)]
( ) ( )
( ) ( )
[ (x
) (y
)]
( ) ( )
( ) ( )
( ) ( )
-
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Em
ail:
( )
( ) ( )
( ) ( ) ( ) ( )
( )( )( ) ( ) ( )
( )
( ) ( )
( )
( ) ( )
( )( )
( ) *( ) ( ) ( ) +
( ) ( ) ( )
( )( )( )
( )( )
( )( )
( )( )
-
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ail:
, q
(a b
) x
q (
q= a+
b) r
(r=a
b)
( ) ( )( )
1)
q>
0, r>
0 ( )( )
2) q
0 ( )( )
3) q>
0, r0
1 r
,
n
r
< 0 1
r (.
1 -.9 )
,
n
-
1
-
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ail:
1,
4, 9,
16
. 5,
5 ]
1
, 9 , 2
5, 4
9, 8
1
,
5 7
9
]
9
36 8
1 144
.. 5, ( ) ( ) ]
81
, 27, . 3
, 1 9 ,
]
1,
2, 3,
5, 8
, 13 ,
21 , 34
. 55[
=
]
8,
11, 1
7, 29, 5
3,..
101
[8, 8
+(11-8
),11+(
17-11),
]
1,
3, 6,
10, 1
5, 21,.
28 [
1,(1+
2),(3+
3),(6+
4), (1
0+5)]
2,
8, 1
8, 32, 50
[ 2,
(2+6),
(8+6+4
),(18+6
+4+4), (3
2+6+4+
4+4)]
2,
4, 8,
14, 2
2, 32.
44 [
2,(2+2
),(4+2+
2),(8+
2+2+2)
, (14+2
+2+2+2
)]
0,
5, 1
2, 21, 3
2..
45 [ 0
,(0+5)
,(5+5+
2),(12
+5+2+2
), (21+
5+2+2+
2)]
13
, 17,
25, 4
1,..
73 [
13,(13+2
2 ),(17+2
3 ),(25+2
4 ), (41+
25 )]
5,
7, 11
, 19,
.. 5 [
5,(5+
21 )(7
+22 ),(1
1+23 ),(1
9+24 )]
8 [
4,(4+
21 )(6
+22 ),(1
0+23 ),(1
8+24 )]
3,
6, 4,
9, 5,
12, 6
,. 5 , t
1:t 3:t 5:
t 7 5 | t
2:t 4:t 6:
t 8 9 5-
8 9 7 , (t1
+4)
,(t2+
8)(t3
+4)
(t4+
8)]
7 5 5 8 9
, t 3
:t 5:t 7 ( 7-
2),(2
7-4)
,(27-
6),(2
7-8)
| t 4:t 6:
t 8 (5 ) (5 ) (5 9)-
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ail:
0,
9, 17, 2
435, 3
9, 42
.
5 8
9 5 8
7
7 5
7 9 5
9
11
, 17,
13,
13,
8 5 5 9 7
7 5 9 7 5
15,
13, 1
2, 11,
9, 9.
.
7 8 8 7
8 5 8 9
9 7 5 8
9 5 7 99
2,
11,
32,
65,
7
5 7 7
5 8 7 8
9 7 5 5 8
8 9 7
8 5 5 9
5 9
7 7 5 5
3,
5, 9
, 15,
23,
5
5 7
7 8 8 8 8
7 8 8 8 5
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Straight line ( ) Ray ( ) Line Segment ( ) Similar to Almost Equal to Is Equivalent to / Congruent Angle / Right Angle
Measured Angle Perpendicular To Is Parallel to Therefore / Hence Since / Because / Triangle Rectangle/Square Circle
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Euclids (): Elements 13 , 300
Straight line
Ray
Line Segment
( Parallel lines):
,
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AOD = COD
AOC = BOD
= 180
ADB = ADC = = 90
,
= 90
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0 < AOC < 90
90 < AOD < 180
180 < AOB < 360 AOB = 360 X
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90 ,
180 , DEF ACB
AB || CD EF (Transversal) , AGF = DHE DGF = CHE
AB || CD EF (Transversal) ,
AGF = CHF DGF = DHF
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ABC , BAC =B , ABC = B , ACB = C AB= c , BC = a AC = b
ACD = ACB =
,
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(Properties of Triangles): 180 ,
( ) ,
,
2:1 ( )
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(In-Centre): ( )
(Circumcentre): ( )
(Centroid):
(Orthocentre) :
ABC A , B c < 90
ABC A C < 90 B = (900 < < 1800)
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ABC B = = 900
A + C = 900 A = C = 900 , 900 = + = + 3:4:5 , 5 [ = + , 5 = 4 + 3 , 25 = 25 , ]
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ABC AB BC CA
ABC AB = AC BC b = c B = C , ,
ABC AB = BC = AC a= b = c A = B = C =600
, ,
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4 3600 A +B +C + D = 3600
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,
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n (Interior Angles) n = ( 2n 4) 900 = n 2 1800
=
180 )0
, n = 3600
= (
)0
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,
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(Arc):
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,
- ( ) ( ) 1 =
= 2
,
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, , ,
,
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= a = b , A = = a b S = 2 ( + ) = 2(a + b) d = +
= = a A = = a a =
S = 2(a + a) = 4 a d = + = 2 a
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=a h = = a h a , b
= a b n
d X
= (AC) (CG) = d X
a , b d
= 2 ( , ) [ , = S = = ] = 2 ( + ) BD = d1
AC(d2) = + = + + [ CF =DE] [DE AB DE = ] [BF BF = ]
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= = a AC= d1 BD= d2 A = (
) =
S = 4
AB =a DC = b CE = h
=
= (AB + CD ) CE =
(a +b) h
=
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BC= a AD = h
= (BC) (AD) =
a h
,
= =
(a) (b)
= a b n =
b c n =
c a n
ABC BC=a , AB =c AC=b (2S) = = a+b+c
S =
=
=
/ /
= b = a
= 4
/ ( = 60) / = a
=
= 3 a
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,
=
=3.1416() = 180
1 =
(S) (d) (r) = 2 = 2r
=
= = 2 r = 2 r
r S
360 = 2 r
,, ,, ,, ,, ,, =
=
S =
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r S ,
S = r [ 1 =
180 ]
= r , =
r S ,
=
OA= r1 OB = r2 ,
= = r1 r2
= 2
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=a , =b = c =d
(Area of the Whole Surface)=
= 2 (ab +bc +ca) (Volume) = = a b c
, d = + +
= = = a =d
(Area of the Whole Surface)=
= 6
(Volume) = =
, d = 3 a
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(Right Circular) BC = r , AB= h
AC = l
l = + [ = + ]
= (2r) (l)
= rl = r +
(Whole Surface) = ( l) + ( )
= r (l +r)
(Volume): = ( ) (h)
=
h
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(Right Circular) OB =r OC =h
= (2r) (h)
= 2 h
= (2rh) + (2 )
(Whole Surface) = 2r(h+r)
= ( ) (h) = h
= r
= 2 =4
=
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[MOP + OPM 90- = 900 PMO] ()
= + ()
(3:4:5), (5:12:13), (7:24:25) (8:15:17) ( 5 = 4 + 3 )
n =
[ ] =
=
[ ] =
= [ ] =
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n =
=
=
=
=
=
n =
=
n n
n + = 1
n = 1
= 1 n
= 1
= 1 +
= 1
= 1
= 1 +
= 1
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sin 0 12
1
2 3
2 1
cos 1 32
1
2
1
2 0
tan( ) 0 1
3 1 3
cot(
) 3 1 13
0
sec(
) 1 23
2 2
Cosec (
) 2 2 23
1
0, 1, 2, 3, 4, 4 , , , , n n 0 n 30 n 45 n 60 n 90
0,
1
sin cos sin cos tan
1 n +1
1 +1
1 -1
n
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sin / cos / tan {n 90 ) }
{n 90
}
(