I faced a question in exam where we had to tick the correct negation of the following statement
$"\exists y\in \Bbb{Z}, \forall x\in \Bbb{R}, \text{such that}\ y^2<x"$.
I checked the option $"\forall\ y\in \Bbb{Z}, \exists\ x\in \Bbb{R}, \text{such that}\ y^2\ge x"$.
Does it seem correct to you? What is a general rule while negating statements involving $\exists$ and $\forall$ symbols?
Yes, that is correct. Here are the general rules involving quantifiers and negation:
Quantifier Negation
For any formula $\varphi$:
$\neg \forall x \varphi \Leftrightarrow \exists x \neg \varphi$
$\neg \exists x \varphi \Leftrightarrow \forall x \neg \varphi$
If we apply this to your statement, we get:
$\neg \exists y\in \Bbb{Z} \ \forall x\in \Bbb{R} \ y^2<x \Leftrightarrow$
$\forall y \in \Bbb{Z}\ \neg \forall x\in \Bbb{R}\ y^2<x \Leftrightarrow$
$\forall y \in \Bbb{Z}\ \exists x\in \Bbb{R}\ \neg y^2<x \Leftrightarrow$
$\forall y \in \Bbb{Z}\ \exists x\in \Bbb{R}\ y^2 \ge x $