All practice questions

A city planning board needs to form a $6$-member advisory task force selected from a pool of $5$ architects and $5$ civil engineers. The task force must include at least $2$ architects and at least $2$ civil engineers. However, two specific architects, Architect X and Architect Y, cannot both serve on the task force together. How many different $6$-member task forces can be formed satisfying these conditions?

A museum curator is arranging $6$ distinct paintings in a single row along a gallery wall. If $2$ specific paintings must not be placed adjacent to each other, how many different arrangements of the $6$ paintings are possible?

In rhombus $ABCD$, the side length is $10$ and the length of diagonal $BD$ is $12$. Line segment $AP$ is drawn perpendicular to side $BC$, with point $P$ lying on segment $BC$. What is the length of segment $AP$?

For all positive real numbers $a$ and $b$, the custom operation $\diamondsuit$ is defined by $a \diamondsuit b = \frac{a^2 + b^2}{ab}$. The function $f$ is defined for all $x > 0$ by $f(x) = x \diamondsuit 4$. If $f(x) = 2.5$, what is the value of $x$ that is greater than $4$?

A short-pulse laser emits a single pulse lasting $8.4 \times 10^{-9}$ seconds. A high-speed optical sensor completes one measurement cycle every $1.4 \times 10^{-11}$ seconds. How many measurement cycles does the sensor complete during the duration of a single laser pulse?

An automated risk-management system uses three independent algorithms—Algorithm X, Algorithm Y, and Algorithm Z—to detect fraudulent transactions. The probability that Algorithm X detects a given fraudulent transaction is $\frac{3}{5}$, the probability that Algorithm Y detects it is $\frac{2}{3}$, and the probability that Algorithm Z detects it is $\frac{3}{4}$. If a fraudulent transaction occurs, what is the probability that it will be detected by at least two of these three algorithms?

A university department tracked the number of research articles published by its 8 faculty members over a five-year period. The numbers of publications for 7 of the faculty members were $4, 7, 9, 12, 15, 18,$ and $23$. If the arithmetic mean of the number of publications for all 8 faculty members is equal to $1.25$ times their median, what is the number of publications for the 8th faculty member?

A laboratory technician has $12$ liters of a solution containing $15\%$ salt by weight. The technician wants to increase the salt concentration to $25\%$ by adding a second solution that contains $40\%$ salt by weight. How many liters of the $40\%$ solution must be added?

In the geometric configuration formed by adjacent triangles $ABC$ and $ACD$ sharing segment $AC$, $\angle ABC = 90^\circ$ and $\angle ACD = 90^\circ$. The lengths of the sides of triangle $ABC$ are $AB = 9$ and $BC = 12$. If $\angle ADC = 30^\circ$, what is the length of segment $AD$?

In the $xy$-plane, line $L_1$ is defined by the equation $y = 3x - 4$. Line $L_2$ is perpendicular to line $L_1$ and passes through the point $(6, 2)$. What is the $x$-intercept of line $L_2$?

A water reservoir initially contains $450$ liters of water. Water drains out of the reservoir at a constant rate of $r$ liters per hour, while an inlet pipe supplies water at a constant rate of $18$ liters per hour. If the reservoir contains $390$ liters of water after $6$ hours, what is the value of $r$?

If $N = 2^4 \times 3^3 \times 5^2$, how many positive integer factors of $N$ are divisible by $6$ but not divisible by $12$?

Let $n$ be an integer such that $-15 \le n \le 15$. How many integer values of $n$ satisfy both of the following conditions?

1. $(-1)^{n^2 + n + 1} < 0$
2. $(n - 3)(n + 4)^2 < 0$

A car travels a total distance of $120$ miles. For the first $60$ miles, the car travels at an average speed of $30$ miles per hour, and for the remaining $60$ miles, the car travels at an average speed of $60$ miles per hour. What is the average speed of the car, in miles per hour, for the entire $120$-mile trip?

A sequence of numbers begins with $t_1 = 4$. For all integers $n \geq 2$, each term is defined by $t_n = 3t_{n-1} - 2$. What is the value of $t_5$?

At the beginning of a quarter, the cost of raw material for a manufacturing plant was $x$ dollars per ton. During the first month, the cost per ton increased by $15\%$. During the second month, the cost per ton decreased by $20\%$ relative to the first month's price. If the cost per ton at the end of the second month was $\$552$, what was the initial cost $x$, in dollars?

A plumbing service charges a flat diagnostic fee of $\$45$ plus $\$65$ for each hour of repair work. If the total bill for a repair job was $\$305$, how many hours of repair work were performed?

On a real number line, point $A$ has coordinate $-5$ and point $B$ has coordinate $11$. Point $C$ has coordinate $x$ such that the distance between $C$ and the midpoint of line segment $AB$ is equal to $\frac{1}{3}$ of the distance between $C$ and point $B$. What is the maximum possible value of $x$?

For how many integer values of $x$ does the inequality $||2x - 7| - 5| \le 4$ hold true?