IIT JEE , AIEEE Forum - Mathematics , Algebra

shajikoshy

Q 1 How many different values are taken by the expression [x] + [2x] + [5x/3] + [3x]+ [4x] for real x in the range 0 < x < 100? [ ] denotes the greatest integer function.

Q2. Let a and b be positive real numbers for which

60^a = 3 and 60^b = 5. Without the use of a calculator or

of logarithms, determine the value of

12^{[(1-a-b)/{2(1-b)}]}

divalli_oct07

can u make ur ques a bit clr???

Greatdreams

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Edited......thanks for pointing my mistake but please provide a solution..

elastiboysai

Great dreams

your working out is clearly wrong!!

chec urself by plugging in values.

[2x]=2[x]????

plug in x= 0.5

its clearly wrong.

simlarly for all others

vivsarda

yup u r 100 % rite mr elstiboysai
It can be only taken wen x is an integer
even if x is an integer u cnt take (5/3x)=5/3 x
(X) denotes GIF of x
HOPE HELPS
RATE IF USEFUL!!

elastiboysai

heres da soln 4 da 2nd ques

a ln60=ln3

bln60=ln5

a=ln3/ln12+ln5

b=ln5/ln12+ln5

plug in the values in da expression

1-a-b/2(1-b)

[(1-ln15)/(ln12+ln5)]/[2ln12/(ln12+ln5)]

simplif.

ln12-ln3/2ln12

0.5-log₁₂_³

^{_{so 12 power the abuv is sqrt(12)/3}}

elastiboysai

well

shajikoshy ji,

for the 1st question its not clear if 0 and 100 are included or not. i've done the problem including them.

[x]+[2x]+[5x/3]+[3x]+[4x]

write it as 35x/3 -{x}-{2x}-{5x/3}-{3x}-{4x}

now the fractional part terms become zero at all integral multiples of 1/coeff.of x

i.e

{x} = 0 at integers

{2x}=0 at integral multiples of1/2

{5x/3}=0 at integral multiples of 3/5

{3x}=0 at integral mult of 1/3

{4x}=0 at integr. multiples of 1/4

and importantly*** all the fractional parts are zero at

lcm of 3/5,1/2,1,1/3,1/4 wich is 3.

so total different values =

100[2+5/3+1+3+4]=1166

but we are addin integral points 4times so-(100*3)

subtr. integr pts for 5/3.= -33

u can skip 2 as well so-(100*2) [mult of 1/4 will take care of dem]

and - subtr pts were all change simultaneously = -33

well the ans turns abt to be 600

shajikoshy

Q 1 How many different values are taken by the expression [x] + [2x] + [5x/3] + [3x]+ [4x] for real x in the range 0 < x <100 ? [ ] denotes the greatest integer function.

Sorry for typing error.........

karthik_18049200

anmswer for 2nd Q is 2

try by taking logarithms

cheers

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