What is the value of ${ }^{23} C_0+{ }^{23} C_2+{ }^{23} C_4+\cdots+{ }^{23} C_{22}$ ?

Let $f: Q \rightarrow Q$ be a function such that $f(x+y)=f(x)+f(y)$ for all $x, y \in \mathbf{Q}$, and $f(1)=$ 10. Which one of the following statements is Correct?

Let $I=\int_{e^{-\pi / 2}}^{e^{\pi / 2}}\left(\sin ^2(\log (x))+\sin \left(\log \left(x^2\right)\right)\right) d x$. What is the value of $I$ ?

Consider the following subset of the $X Y$-plane.

$$ S=\left\{\left(|z-\mathrm{i} z|,|z|^2\right): \quad z \text { is a complex number }\right\} $$

Which one of the following statements is correct?