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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.

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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