1. **Problem 2:** Find the probability that a randomly drawn piece of paper from numbers 1 through 30 has a prime number. 2. **Step 1:** Identify all prime numbers between 1 and 30. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. The prime numbers between 1 and 30 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. There are 10 prime numbers. 3. **Step 2:** Calculate the probability. Probability = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{10}{30}. 4. **Step 3:** Simplify the fraction. $$\frac{\cancel{10}}{\cancel{30}} = \frac{1}{3}$$ So the probability is $\frac{1}{3}$, which is not exactly one of the options given, but closest to $\frac{10}{30}$ (option J). --- 5. **Problem 3:** Given $\sin \alpha = \frac{5}{13}$ and $\tan \alpha = \frac{5}{12}$, find $\cos \alpha$. 6. **Step 1:** Recall the definitions. $\sin \alpha = \frac{\text{opposite}}{\text{hypotenuse}}$, $\cos \alpha = \frac{\text{adjacent}}{\text{hypotenuse}}$, and $\tan \alpha = \frac{\text{opposite}}{\text{adjacent}}$. 7. **Step 2:** From $\sin \alpha = \frac{5}{13}$, opposite side = 5, hypotenuse = 13. From $\tan \alpha = \frac{5}{12}$, opposite side = 5, adjacent side = 12. 8. **Step 3:** Calculate $\cos \alpha$. $$\cos \alpha = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{12}{13}$$ **Final answers:** - Probability of prime number = $\frac{10}{30}$ (option J). - $\cos \alpha = \frac{12}{13}$.