1. The problem asks to find the intersection $P \cap Q$ where: - $P$ is the set of prime factors of 210. - $Q$ is the set of prime numbers less than 10. 2. First, determine the prime factors of 210: 210 can be factorized as: $$210 = 2 \times 3 \times 5 \times 7$$ So, $P = \{2, 3, 5, 7\}$. 3. The prime numbers less than 10 are: $$Q = \{2, 3, 5, 7\}$$ 4. Now, find the intersection $P \cap Q$ which means the elements common to both sets. Since both $P$ and $Q$ are $\{2, 3, 5, 7\}$, their intersection is: $$P \cap Q = \{2, 3, 5, 7\}$$ **Final answer:** $\boxed{\{2, 3, 5, 7\}}$ --- **Second question:** Find for what value of $x$ the matrix $$\begin{pmatrix} 2x & -2 \\ 6 & 3 \end{pmatrix}$$ is singular. 1. A matrix is singular if its determinant is zero. 2. Compute the determinant: $$\det = (2x)(3) - (-2)(6) = 6x + 12$$ 3. Set determinant equal to zero: $$6x + 12 = 0$$ 4. Solve for $x$: $$6x = -12$$ $$x = -2$$ **Final answer:** $\boxed{-2}$