Advanced Quizzes — Quiz 17

Questions in this quiz (10)

1. The system of equations $$ax+5y=11$$ $$6x+10y=k$$ has no solution, where $$a$$ and $$k$$ are constants. Which of the following must be true?

   • $$a=3$$ and $$k\ne22$$
   • $$a=3$$ and $$k=22$$
   • $$a=-3$$ and $$k\ne22$$
   • $$a=-3$$ and $$k=22$$

2. A state with a population of $$1,200,000$$ residents conducted a random survey of $$4,000$$ residents to estimate internet access rates. The survey found that $$72\%$$ of the surveyed residents have home internet access, with an associated margin of error of $$3\%$$. Based on these results, which of the following is a plausible value for the total number of residents in the state who have home internet access?

   • $$780,000$$
   • $$828,000$$
   • $$900,000$$
   • $$930,000$$

3. A behavioral scientist investigated the relationship between sleep duration and academic performance among college students. The researcher recruited $$300$$ volunteers and allowed each participant to decide whether to follow a structured sleep schedule for $$8$$ weeks or continue their normal routines. At the end of the study, participants completed a standardized academic assessment. The students who followed the structured sleep schedule had significantly higher average assessment scores than those who did not. Based on this study, which of the following conclusions is most appropriate?

   • Following a structured sleep schedule causes improved academic performance.
   • There is a strong causal relationship between sleep duration and academic performance for all students.
   • The study provides evidence of an association between structured sleep schedules and higher academic performance, but it cannot establish causation.
   • The results can be generalized to all college students because the sample size was large.

4. The graph of the function $$f(x)=3x^2+bx+c$$ is a parabola in the $$(x,y)$$-plane, where $$b$$ and $$c$$ are constants. The vertex of the parabola is at $$(h,k)$$, where $$(h,k)$$ is the solution to the system $$y=5x-22$$ $$y=-x+14$$ What is the value of $$b+c$$?

   • $$-168$$
   • $$-84$$
   • $$80$$
   • $$144$$

5. A scientist is observing the growth of a mold culture. At the beginning of the experiment, there are $$M_0$$ mold cells. Every hour, the number of new mold cells produced is equal to $$20\%$$ of the current number of mold cells already present. Which of the following functions best models the total number of mold cells, $$M(t)$$, after $$t$$ hours?

   • Increasing linear
   • Decreasing exponential
   • Increasing exponential
   • Increasing quadratic

6. In $$\triangle PQR$$, point $$S$$ is on $$\overline{PQ}$$ and point $$T$$ is on $$\overline{PR}$$ such that $$\overline{ST}\parallel\overline{QR}$$. The lengths of the segments are given by $$PS=x+1$$, $$SQ=3$$, $$PT=2x-1$$, and $$TR=5$$. What is the value of $$x$$?

   • 8

7. A circle is tangent to the $$x$$-axis. Its center lies on the line given by the equation $$y=3x-8$$ If the circle passes through the point $$(4,7)$$, what is the sum of all possible values for the radius of the circle?

   • $$14$$
   • $$18$$
   • $$28$$
   • $$134$$

8. A data set $$S$$ consists of $$N$$ distinct positive integers, where $$N$$ is an odd number greater than $$3$$. The mean of $$S$$ is $$m_S$$ and the median is $$M_S$$. A new data set $$S'$$ is formed by removing the smallest value from $$S$$. Which of the following statements must be true?I. The mean of $$S'$$ is greater than $$m_S$$.II. The median of $$S'$$ is greater than or equal to $$M_S$$.III. The standard deviation of $$S'$$ is less than the standard deviation of $$S$$.

   • I only
   • I and II only
   • II and III only
   • I, II, and III

9. In the $$(x,y)$$-plane, a line passes through the points $$(7,4)$$ and $$(1,-2)$$. This line can be represented by the equation $$ax+by=1$$ where $$a$$ and $$b$$ are constants. What is the value of $$a-b$$?

   • $$-\frac{1}{3}$$
   • $$\frac{1}{3}$$
   • $$\frac{5}{3}$$
   • $$\frac{2}{3}$$

10. A data set $$R$$ consists of $$2n$$ distinct positive integers with mean $$a$$ and median $$b$$. The smallest value in the data set is increased by $$100$$, while all other values remain unchanged. The new data set has mean $$a'$$ and median $$b'$$. Which of the following must be true?

    • $$a'=a+\frac{50}{n}$$ and $$b'=b$$
    • $$a'$$ > $$a$$ and $$b'$$ > $$b$$
    • $$a'$$ > $$a$$ and $$b'=b$$
    • $$a'=a+100$$ and $$b'=b$$