Here are the solutions to the given questions:

Question 12

Solve the equation and check for extraneous solutions:

$\frac{e}{e+5} + \frac{5}{e-5} = \frac{2e + 24}{(e+5)(e-5)}$

Step 1: Find the LCD

The least common denominator (LCD) is $(e+5)(e-5)$, so we multiply each term by it:

$(e(e-5) + 5(e+5) = 2e + 24)$

Expanding:

$e^2 - 5e + 5e + 25 = 2e + 24$

$e^2 + 25 = 2e + 24$

Step 2: Solve for $e$

Rearrange:

$e^2 - 2e + 1 = 0$

Factor:

$(e-1)(e-1) = 0$

$e = 1$

Step 3: Check for Extraneous Solutions

$e = \pm5$ would be extraneous because they make denominators zero. Since $e = 1$ is not extraneous, it is the final answer.

Answer: (A) 1

Question 13

Simplify:

[ \frac{4x}{x+3} \div \frac{x^2