Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{a}+\vec{b}|=15$ and

$$ \vec{a} \times(3 \hat{i}-4 \hat{j}+5 \hat{k})=(3 \hat{i}-4 \hat{j}+5 \hat{k}) \times \vec{b} $$

What is the value of $|(\vec{a}+\vec{b}) \cdot(2 \hat{i}+3 \hat{j}+\hat{k})|$ ?

What is the derivative of $\log \left(\sin ^2 x\right)$ with respect to $\sin x$ ?

Let $S_n$ denote the sum of the first $n$ terms of a sequence $a_1, a_2, a_3, \ldots$. If $S_{n+3}-S_n=13 n+7$ for all $n$, what is the value of $a_{13}-a_{10}$ ?

Five fair coins are tossed independently. What is the probability that at least two heads appear?