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Let $x<0,\:0<y<1,\:z>1$. Which of the following may be false?

  1. $\left (x ^{2} -z^{2}\right )$ has to be positive.
  2. $yz$ can be less than one.
  3. $xy$ can never be zero.
  4. $\left (y ^{2} -z^{2}\right )$ is always negative.
in Quantitative Aptitude edited by
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Given that,

  • $x<0$
  • $0<y<1$
  • $z>1$

Let’s check all of the options.

$A. \; (x^{2} – z^{2})$ has to be positive.

Let’s take $x=-1, z=2$

Now, $(-1)^{2} – (2)^{2}>0$

$\Rightarrow 1-4>0$

$\Rightarrow \boxed{-3>0 \; (False)}$

$B. \; yz can be less than one.

$yz<1$

$\Rightarrow \frac{1}{4} \times 2<1$

$\Rightarrow \boxed{\frac{1}{2}<1 \; (True)}$

$C: xy$ can never be zero.

$\Rightarrow xy \ne 0$ 

$\Rightarrow \boxed{xy \ne 0} (True)$ 

$D. \; (y^{2} – z^{2})$ is always negative.

$\Rightarrow y^{2} – z^{2} <0$

Here $y<z$

$\Rightarrow y^{2} < z^{2}$

$\Rightarrow \boxed{y^{2} - z^{2} < 0 (True)Always}$

Correct Answer $:\text{A}$

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