You uploaded an exam paper for "Applied Mathematics Class – XI" with various questions covering topics such as set theory, functions, probability, derivatives, sequences, and more. I'll assist with solving these problems. Here are a few worked-out examples from the paper:

Q1. If A = {1, 2, 3, 4} and B = {1, 2, 3}, find A × B.

The Cartesian product $A \times B$ represents all ordered pairs where the first element is from $A$ and the second from $B$.

• $A \times B = \{(1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3), (4,1), (4,2), (4,3)\}$.

Q2. If the ordered pair (x – 1, y + 3) = (2x + 4), find x and y.

We can equate the components of the ordered pairs:

1. $x - 1 = 2x + 4$
   Solving for $x$: $x - 1 = 2x + 4$ $-1 - 4 = 2x - x \implies x = -5$

2. $y + 3 = 7$
   Solving for $y$: $y + 3 = 7 \implies y = 4$ Thus, $x = -5$ and $y = 4$.

Q4. Let f be a function defined as $f(x) = 5x^2 + 2$, find $x$ such that $f(x) = 22$.

We need to solve $f(x) = 22$, i.e., $5x^2 + 2 = 22$ $5x^2 = 20 \implies x^2 = 4$ $x = \pm 2$ Thus, $x = 2$ or $x = -2$.

Would you like to proceed with more questions or clarify any specific ones from this exam?

5 Related Questions:

1. What is the Cartesian product, and how is it used in set theory?
2. How can ordered pairs be used to solve linear equations?
3. What other methods exist to solve quadratic equations like in Q4?
4. How do you find the domain and range of a function?
5. What are mutually exclusive events in probability?

Tip:

When solving equations involving ordered pairs, always separate and equate the corresponding components for clarity.