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# System prepended metadata

title: Sample midterm exam E

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**Problem 1.** Simplify the following expression.
$$
\frac{x^2 + 11x + 24}{x^2 + 8x + 15}
$$

**Problem 2.** Solve the following equation in the set of real numbers.
$$
7 - 4 \cdot \left(-\frac{x}{2} - 3\right) = 3 - 2x
$$

**Problem 3.** Determine the solution set of the following system of equations.
$$ 3y + 4 = 4x \ \ \ \ \ \ \ \ \ xy = 40 $$

**Problem 4.** Determine the largest possible domain of the function given by the following formula in interval notation.
$$
f(x) = \log_{4}(-3x^2 + 3x + 9)
$$

**Problem 5.** Simplify the following expression by rationalizing the denominator.
$$
\frac{1}{-\sqrt{5} + \sqrt{2}}
$$

**Problem 6.** Solve the following equation in the set of real numbers.
$$
\log_{4}(4x - 3) - 2 = 0
$$

**Problem 7.** Graph the function given by the following formula and determine its largest possible domain and range in interval notation.
$$
f(x) = 8 \cdot \left(\frac{1}{3}\right)^x - 6
$$

**Problem 8.** The price of a product is increased by 5%, then decreased 4 times, each time by 3%. By how much has the price of the product changed overall?

**Problem 9.** In triangle $BCD$, let $G$ be the trisection point of side $CD$ that is closer to point $D$.
From vertex $B$, let the side vectors be denoted by $\mathbf{p}$ and $\mathbf{q}$, where
$\mathbf{p} = \overrightarrow{BC}$ and $\mathbf{q} = \overrightarrow{BD}$.
Express the vector $\overrightarrow{BG}$ in terms of the vectors $\mathbf{p}$ and $\mathbf{q}$. Show your derivation.

**Problem 10.** In a right triangle, one leg is 15 cm and the other leg is 8 cm. Find the measure of the angle at the vertex opposite the longer leg.

