KVPY SX SB Exam 20141 Question Paper

(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]
$\mathit{x}=2;\mathit{y}=\frac{1}{2}$
$\left[2.\frac{1}{2}\right]\le \left[2\right]\left[\frac{1}{2}\right]$
⇒ 1 ≤ 0 wrong
(C) $\left[2^{\mathit{x}}\right]\le 2^{\left[\mathit{x}\right]}$
$\mathit{x}=0.99\left[2^{0.99}\right]\le 2^{\left[0.99\right]}$
$\left[2^{0.99}\right]\le 2\mathit{º}=1$ wrong
(D) $\left[\frac{\mathit{x}}{\mathit{y}}\right]\le \frac{\left[\mathit{x}\right]}{\left[\mathit{y}\right]}$
given x, y ≥ 1
if true
if $\mathit{x}\ge \mathit{y}\left[\frac{\mathit{x}}{\mathit{y}}\right]\le \frac{\left[\mathit{x}\right]}{\left[\mathit{y}\right]}$ always true