wiz-icon
MyQuestionIcon
MyQuestionIcon
1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

If x0[x]dx=[x]0xdx, then which of the following is true?

(where [.] and {.} denote the greatest integer and fractional parts respectively.)

A
x is an integer
No worries! We‘ve got your back. Try BYJU‘S free classes today!
B
x is purely fractional
Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
C
{x}=12
Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
D
[x]>1
No worries! We‘ve got your back. Try BYJU‘S free classes today!
Open in App
Solution

The correct options are
A x is purely fractional
B {x}=12
Let x=I+f[x=I] (i)
Now, [x]0xdx=I0xdx=I22
and x0[x]dx=I+f0[x]dx=100dx+211dx++II1(I1)dx+I+fIIdx
={1+2+3++(I1)}+I(I+fI)
=I(I1)2+I(f)=I(I1)2+I(xI) [using equation (i)]
Given, x0[x]dx=[x]0xdx
I22=I(I1)2+I(xI)
I(I+12x+2I1)=0
I(2I2x+1)=0
I=0, 2I2x+1=0
ie, [x]=0 or x=2I+12=I+12
0x<1 or x=[x]+12
0x<1 or {x}=12
Ans: B,C

flag
Suggest Corrections
thumbs-up
0
Join BYJU'S Learning Program
similar_icon
Related Videos
thumbnail
lock
Geometric Progression
MATHEMATICS
Watch in App
Join BYJU'S Learning Program
CrossIcon