General Mathematics 1 Mock Exam 1 Mock Exam 1

Calculate the value of $5 \cdot (3 + 2)$ using the distributive property and then verify the result by performing the addition first. Show both calculations.

If $\sin \theta = \frac{a}{b}$, find $\cos \theta$ in terms of $a$ and $b$. Assume $\theta$ is in the first quadrant.

What is the second term ($a_2$) of a geometric sequence if the sum of the three numbers is 13 and their product is 64?

The first term of an arithmetic series is 3, the common difference is 4, and the sum of all terms is 820. Find the number of terms ($n$) and the last term ($a_n$).

A student is trying to prove that $7^n - 1$ is divisible by 6 for all positive integers $n$. Base Case ($n=1$): $7^1 - 1 = 6$, which is divisible by 6. (True) Inductive Hypothesis: Assume $7^k - 1$ is divisible by 6 for some positive integer $k$. So, $7^k - 1 = 6m$ for some integer $m$. Inductive Step: Show that $7^{k+1} - 1$ is divisible by 6. Which of the following is the correct evaluation of $7^{k+1} - 1$ using the inductive hypothesis?

If set $M = \{x \mid x \text{ is a positive integer and } x < 10\}$ and set $N = \{x \mid x \text{ is a multiple of } 3 \text{ and } x \le 10\}$, find the set $M - N$.

Let $P(n)$ be the statement that $1+3+5+\dots+(2n-1) = n^2$. If $P(k)$ is true, what is $P(k+1)$?

Find the sum to which the geometric series converges: $1 + \frac{1}{5} + \frac{1}{25} + \frac{1}{125} + \dots$

Simplify the expression $\frac{\sin^2 x}{1 - \cos^2 x}$.

Consider the expression $(2x + 3y) + (4x - y)$. If we rearrange the terms to group like terms together, $(2x + 4x) + (3y - y)$, which algebraic property are we primarily using?