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![[x] denotes to be the least integer no less than x. Is [2d]](/forum/styles/gmatclub_light/imageset/icon_post_target_unread.svg "[x] denotes to be the least integer no less than x. Is [2d]"){kind=icon}
17 Apr 2012, 19:53
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B
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[x] denotes to be the least integer no less than x. Is [2d] = 0?
(1) [d] = 0
(2) [3d] = 0
(1) [d] = 0
(2) [3d] = 0
17 Apr 2012, 21:17
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[x] denotes to be the least integer no less than x. Is [2d] = 0?
[x] denotes to be the least integer no less than x, means that some function [] rounds UP a number to the nearest integer, for example:
\([1.5] = 2\) since 2 is the least integer which no less than 1.5;
\([-0.5]=0\) since 0 is the least integer which no less than -0.5;
\([1] = 1\) since 1 itself is the least integer which no less than 1;
...
(1) [d] = 0 --> \(-1<d\leq{0}\). Now, if \(d=0\) then \([2d]=[0]=0\) but if \(d=-0.5\) then \([2d]=[-1]=-1\). Not sufficient.
Or: \(-1<d\leq{0}\) --> \(-2<d\leq{0}\) --> \([2d]=0\) (if \(-1<2d\leq{0}\)) or \([2d]=-1\) (if \(-2<2d\leq{-1}\)). Not sufficient.
(2) [3d] = 0 --> \(-1<3d\leq{0}\) --> \(-\frac{1}{3}<d\leq{0}\). Even if \(d=-\frac{1}{3}\) then \([2d]=[-\frac{2}{3}]=0\) (again since 0 is the least integer which no less than -2/3). Sufficient.
Or: \(-1<3d\leq{0}\) --> \(-\frac{2}{3}<2d\leq{0}\) --> \([2d]=0\). Sufficient.
Answer: B.
Similar questions to practice:
if-denotes-the-greatest-integer-less-than-or-equal-to-x-94687.html
for-all-z-denotes-the-least-integer-greater-than-or-110561.html
Hope it helps.
[x] denotes to be the least integer no less than x, means that some function [] rounds UP a number to the nearest integer, for example:
\([1.5] = 2\) since 2 is the least integer which no less than 1.5;
\([-0.5]=0\) since 0 is the least integer which no less than -0.5;
\([1] = 1\) since 1 itself is the least integer which no less than 1;
...
(1) [d] = 0 --> \(-1<d\leq{0}\). Now, if \(d=0\) then \([2d]=[0]=0\) but if \(d=-0.5\) then \([2d]=[-1]=-1\). Not sufficient.
Or: \(-1<d\leq{0}\) --> \(-2<d\leq{0}\) --> \([2d]=0\) (if \(-1<2d\leq{0}\)) or \([2d]=-1\) (if \(-2<2d\leq{-1}\)). Not sufficient.
(2) [3d] = 0 --> \(-1<3d\leq{0}\) --> \(-\frac{1}{3}<d\leq{0}\). Even if \(d=-\frac{1}{3}\) then \([2d]=[-\frac{2}{3}]=0\) (again since 0 is the least integer which no less than -2/3). Sufficient.
Or: \(-1<3d\leq{0}\) --> \(-\frac{2}{3}<2d\leq{0}\) --> \([2d]=0\). Sufficient.
Answer: B.
Similar questions to practice:
if-denotes-the-greatest-integer-less-than-or-equal-to-x-94687.html
for-all-z-denotes-the-least-integer-greater-than-or-110561.html
Hope it helps.
Smita04
'no less than' indicated [x]\(>=\) x....with the equality holding when x is an integer
st1: [d]=0 indicates -1<d<0 (eg. -0.5) => -2<2d<0(2x -0.5=-1) =>[2d]=-1
but if d= -0.3......2d= -0.6.....[2d]= 0 (not -1 as -1< -0.6)
NOT Sufficient
st2: -1<3d<0 => -1/3<d<0
d can be -0.65...then [2d]=0..... d can be [-0.01]....[d]=0....
(taking extreme values )
Suffecient
Answer:B
