Set 2

Given two real numbers with the following conditions:

1. Twice the first number minus three times the second number equals $14$.
2. If the first number is multiplied by $k$ plus twice the second number, the result is $4$, where $k$ is a real number.

If $k > 0$ and $k$ is equal to the sum of the first and second numbers, the value of the first number is ....

Given two real numbers with the following conditions:

1. Twice the first number minus three times the second number equals $14$.
2. If the first number is multiplied by $k$ plus twice the second number, the result is $4$, where $k$ is a real number.

If the value of the first number is $2$ more than $k$, the largest possible value of the second number is ....

Function $f$ is defined as $f(x) = 2x + 5$ and $(f \circ g)(x) = 2x^2 - 6x + 9$.

How many of the following four statements are true based on the information above?

1. $g(1) = 0$.
2. Function $g$ intersects the x-axis at exactly one point.
3. $f^{-1}(x) = \frac{1}{2}x + \frac{5}{2}$.
4. $f(x) + g(x) = x^2 - x + 7$.

The following table presents the grade data of students who took the Math and Science exams in a tutoring class. A student is considered to pass if their average exam score exceeds $85$ and their Science score is higher than the median Science score of all students.

Based on the information provided, which of the following relationships between quantity $P$ and $Q$ is correct?

The following table presents data on the number of students related to their favorite subjects.

The total number of students from all classes is $150$.

Based on the information provided, which of the following relationships between quantity $P$ and $Q$ is correct?

From the digits $1, 2, 3, 4, 5,$ and $6$, a four-digit odd number with distinct digits will be formed. The number of such numbers that can be formed is....

If $a$ satisfies $5a - 7 \leq 3a + 9$ and $a$ is an integer greater than $7$, then the value of $a$ is ....

Given the function $f(x) = \frac{2x + 1}{3x - 5}$.

Based on the information above, determine whether each of the following statements is TRUE or FALSE.

1. $f^{-1}(x) = \frac{5x + 1}{3x - 2}$.
2. The value $y = \frac{5}{3}$ is not a member of the range of $y = f^{-1}(x)$.
3. The value $x = \frac{1}{2}$ satisfies $f(x) = f^{-1}(x)$.

The correct sequence of truth values for statements 1, 2, and 3 is ....

Matrix $M$ has an inverse $\begin{pmatrix} a & 1 \\ 3 & 2 \end{pmatrix}$ and satisfies the equation:

Based on the information above, determine whether each of the following statements is TRUE or FALSE.

1. $a = 2$.
2. $a + b = 15$.
3. The determinant of the inverse matrix $M$ is $3$.

The correct sequence of truth values for statements 1, 2, and 3 is ....

Three distinct whole numbers less than $10$ are selected simultaneously at random.

How many of the following four statements are true based on the information above?

1. The probability that all selected numbers are even is greater than $\frac{1}{20}$.
2. The probability that the number $3$ is one of the selected numbers is $\frac{3}{10}$.
3. The probability that at least one selected number is a factor of $9$ is less than $\frac{1}{2}$.
4. The probability that the selected numbers are not odd prime numbers is less than $\frac{1}{3}$.

Given the step function $f(x)$ as follows.

The interval of $x$ values that satisfies the inequality $f(x) \geq 5$ is ....

Region $S$ on the Cartesian plane is the set of all solutions to the following system of linear inequalities in two variables.

The area of region $S$ is ... square units.

The sequence of real numbers $a_1, a_2, a_3, \dots$ satisfies $a_{n+1} = 4a_n + 1$ for natural number $n$ and $a_4 = 85$. The value of $a_3 = \dots$

The sequence of real numbers $a_1, a_2, a_3, \dots$ satisfies $a_{n+1} = 4a_n + 1$ for natural number $n$ and $a_4 = 85$. The value of $n$ such that $a_n$ is divisible by $5$ is ....

The sequence of real numbers $a_1, a_2, a_3, \dots$ satisfies $a_{n+1} = 4a_n + 1$ for natural number $n$ and $a_4 = 85$. The number that divides $a_{2025} - a_{2024}$ is ....

Data from two groups is presented in the following table.

The average of Group II is two more than the mode of Group I. The value of $a$ is ...

Data from two groups is presented in the following table.

The average of Group II is two more than the mode of Group I.

Two numbers will be used to replace the lowest data in both groups so that the averages of the groups become equal. The difference between the two replacement numbers is ...

Function $f$ is defined as $f(x) = \frac{1}{\sqrt{x^2 - x + n}}$ with $f(-2) = \frac{1}{2}$.

How many of the following four statements are true based on the information above?

1. $n$ is an odd number.
2. $0$ is a member of the range of function $f(x)$.
3. The function $f$ has a value of $\frac{1}{4}$ when $x = -4$.
4. The domain of function $f$ is $x < -1 \cup x > 2$.

Functions $f$ and $g$ are defined as follows.

Based on the information provided, which of the following relationships between quantity $P$ and $Q$ is correct?

If the value of $\int_{1}^{3} (3x + 5) dx + \int_{2}^{b} (2x - 3) dx = 28$ with $b > 0$, then the value of $b = \dots$