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KVPY SX SB Exam 20141 Question Paper
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© examsnet.com
Question : 2
Total: 120
For a real number r we denote by [r] the largest integer less than or equal to r . If x, y are real numbers with x ,y ≥ 1 then which of the following statements is always true?
[
x
+
y
]
≤
[
x
]
+
[
y
]
[
x
y
]
≤
[
x
]
[
y
]
[
2
x
]
≤
2
[
x
]
[
x
y
]
≤
[
x
]
[
y
]
Validate
Solution:
(A) [x + y] ≤ [x] + [y]
let x = 0.1
y = 0.9
[0.1 + 0.9] ≤ [0.1] + [0.9]
1 ≤ 0 + 0 wrong
(B) [xy] ≤ [x ] [y]
x
=
2
;
y
=
1
2
[
2.
1
2
]
≤
[
2
]
[
1
2
]
⇒ 1 ≤ 0 wrong
(C)
[
2
x
]
≤
2
[
x
]
x
=
0.99
[
2
0.99
]
≤
2
[
0.99
]
[
2
0.99
]
≤
2
º
=
1
wrong
(D)
[
x
y
]
≤
[
x
]
[
y
]
given x, y ≥ 1
if
x
<
y
[
x
y
]
=
0
0
≤
[
x
]
[
y
]
true
if
x
≥
y
[
x
y
]
≤
[
x
]
[
y
]
always true
© examsnet.com
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