Let's start by solving some of the problems step by step from the image. I'll address the following:

Question 1: Domain and Range of the Relation

Problem: What will be the domain and range of the relation $R = \{(1, 2), (2, b), (3, c)\}?$

Solution:

• Domain is the set of all first elements in the ordered pairs of the relation $R$. $\text{Domain of } R = \{1, 2, 3\}$

• Range is the set of all second elements in the ordered pairs of the relation $R$. $\text{Range of } R = \{2, b, c\}$

Question 5: Range and Domain of the Function

Problem: If the range of function $f(x) = 3x + 5$ is 2, find its domain.

Solution: We are given that the range (output) is 2, so: $f(x) = 3x + 5 = 2$ Now solve for $x$: $3x = 2 - 5$ $3x = -3 \implies x = -1$ Thus, the domain for which the range is 2 is $x = -1$.

Question 8: Right-Angled Triangle, Find $\tan A$

From the given right-angled triangle (sides 3, 4, and 5, which follow the Pythagorean theorem), we can find $\tan A$.

Solution: $\tan A = \frac{\text{opposite}}{\text{adjacent}} = \frac{3}{4}$ So, $\tan A = \frac{3}{4}$.

Question 9b: Convert $\sec \theta$ in Terms of $\tan \theta$

Solution: We know the identity: $\sec^2 \theta = 1 + \tan^2 \theta$ Taking the square root of both sides: $\sec \theta = \sqrt{1 + \tan^2 \theta}$

Would you like me to continue with more problems or any particular question in detail?

Here are 5 related questions to explore further:

1. What is the formula to find the domain and range of more complex functions like quadratic ones?
2. How do you find the general term for an arithmetic sequence?
3. How do you determine if a relation is reflexive, symmetric, or transitive?
4. What is the difference between the domain and range in the context of relations and functions?
5. Can you prove $\sec \theta = \frac{1}{\cos \theta}$ using trigonometric identities?

Tip: When solving trigonometric problems, try drawing diagrams to visualize the relationships between angles and sides more easily!