Tüm alıştırma soruları

A bookshelf holds $4$ distinct fiction novels and $3$ distinct non-fiction books. If a reader chooses exactly $1$ fiction novel and $1$ non-fiction book to take on a trip, how many different pairs of books can the reader select?

In the $xy$-plane, line $k$ has a slope of $\frac{3}{4}$ and passes through the point $(2, 1)$. Line $k$ intersects the line $x = 10$ at point $P$. What is the distance between point $P$ and the point $(2, 1)$?

For all real numbers $x$, the function $g$ is defined by $g(x) = cx + 5$, where $c$ is a constant. The custom operation $\diamond$ is defined for all real numbers $a$ and $b$ by $a \diamond b = g(a+b) - g(a-b)$. If $3 \diamond 1 = 12$, what is the value of $g(4)$?

In the $xy$-plane, line $m$ has an $x$-intercept of $-6$ and a $y$-intercept of $3$. Line $k$ is perpendicular to line $m$ and passes through the point $(4, 9)$. What is the $x$-intercept of line $k$?

A commercial bakery uses two automated ovens, Oven X and Oven Y, to bake identical orders of bread. Working alone at its constant rate, Oven X bakes a full order of bread in $8$ hours, while Oven Y, working alone at its constant rate, bakes the same order in $12$ hours. Both ovens begin baking a full order together at 8:00 a.m. At 10:00 a.m., Oven X shuts down due to a maintenance alert, and Oven Y continues working alone at its constant rate until the order is completed. How many total hours, from 8:00 a.m. until completion, does it take to finish the order?

In the $xy$-coordinate plane, point $A$ has coordinates $(-9, 0)$ and point $C$ has coordinates $(0, 12)$. Point $B$ lies on the positive $x$-axis such that line segment $BD$ is perpendicular to segment $AC$, with point $D$ lying on segment $AC$. If the area of right triangle $ABD$ is $54$, what is the length of segment $OB$, where $O$ is the origin $(0,0)$?

A reliability study recorded the operating lifespan, $t$ (in thousands of hours), for a sample of $250$ semiconductor laser diodes. The results are summarized in the grouped frequency table below.

Laser diodes with an operating lifespan of at least $6{}000$ hours ($t \ge 6$) but less than $14{}000$ hours ($t < 14$) are designated as high-efficiency units. Assuming that the values within each class interval are uniformly distributed, what percentage of the $250$ laser diodes in the sample are designated as high-efficiency units?

A community health center conducted a study of $250$ adults regarding their participation in three wellness programs: Nutrition Counseling ($N$), Exercise Coaching ($E$), and Stress Management ($S$). The survey revealed the following data:

• $125$ adults participate in Nutrition Counseling.
• $105$ adults participate in Exercise Coaching.
• $85$ adults participate in Stress Management.
• $15$ adults participate in all three programs.
• $30$ adults participate in none of the three programs.

How many adults participate in exactly two of the three programs?

A committee of $8$ people consists of $4$ men and $4$ women. A subcommittee of $4$ people is to be selected from this group such that the subcommittee contains at least one man and at least one woman. If two specific members, one man and one woman, refuse to serve together on the same subcommittee, how many different valid subcommittees of $4$ people can be formed?

A machine operates using two independent components, Component $A$ and Component $B$. The probability that Component $A$ functions properly on a given day is $0.90$, and the probability that Component $B$ functions properly on that same day is $0.80$. What is the probability that at least one of the components functions properly on a given day?

A commercial print shop uses two high-speed printing presses, Press Alpha and Press Beta. Press Alpha operates at a constant rate of $120$ pages per minute, while Press Beta operates at a constant rate of $180$ pages per minute. Press Alpha begins printing a job of $15,000$ pages at 9:00 AM. At 9:15 AM, Press Beta is turned on to assist Press Alpha, and both presses continue printing simultaneously at their respective constant rates until the job is completed. How many total minutes after 9:00 AM will the entire $15,000$-page job be finished?

For all non-zero real numbers $a$ and $b$, the custom binary operation $\diamondsuit$ is defined by $a \diamondsuit b = \frac{a^2 - b^2}{ab}$. If the function $f$ is defined for all $x \neq 0$ by $f(x) = x \diamondsuit 2$, what is the value of $f(4) - f(1)$?

A research committee must select a delegation of $6$ members from a pool of $4$ senior fellows and $4$ junior analysts to sit around a circular conference table with $6$ evenly spaced seats. The delegation must consist of exactly $3$ senior fellows and $3$ junior analysts. If no two senior fellows may sit in adjacent seats around the table, how many distinct delegation seating arrangements are possible? (Two seating arrangements are considered identical if one can be rotated to match the other.)

In the $xy$-plane, triangle $ABC$ has vertices $A(1, 2)$, $B(4, 2)$, and $C(1, 6)$. The triangle is reflected across the line $y = x$, and then translated $2$ units to the left and $3$ units downward. What is the $y$-coordinate of the image of vertex $C$?

A dataset $S$ consists of 12 numbers listed in increasing order: $x_1, x_2, \dots, x_{12}$. The median of dataset $S$ is 40. The arithmetic mean of the 6 smallest numbers in $S$ is 28, and the arithmetic mean of the 6 largest numbers in $S$ is 56. A new dataset $T$ is formed by subtracting 4 from each of the 6 smallest numbers in $S$ and adding 8 to each of the 6 largest numbers in $S$. What is the positive difference between the arithmetic mean of dataset $T$ and the median of dataset $T$?

In the $xy$-plane, line $k$ is defined by the equation $3x - 4y = 12$. Line $m$ is parallel to line $k$, and the perpendicular distance between line $k$ and line $m$ is $5$ units. If the $y$-intercept of line $m$ is greater than the $y$-intercept of line $k$, what is the $y$-intercept of line $m$?