Full Practice Exams — Exam 3

Questions in this quiz (45)

1. If $$abc-bc=2a$$ and all variables are greater than 1, what is $$a$$ in terms of the other variables?

   • $$\frac{bc}{2}$$
   • $$\frac{bc}{bc - 2}$$
   • $$\frac{b + c}{bc - 2}$$
   • $$\frac{bc + 2}{bc - 2}$$

2. If $$(x^b)^a = x^2$$, and $$a$$ and $$b$$ are both constants greater than zero, what must $$a$$ equal in terms of $$b$$?

   • $$\frac{b}{2}$$
   • $$\frac{2}{b}$$
   • $$\frac{b^2}{2}$$
   • $$b^2$$

3. $$y = (x - 4)(x + 10)$$What is the x-coordinate of the vertex of a parabola with the above equation?

   • -10
   • 4
   • -3
   • 4

4. What is the slope of the line provided by the equation $$18x + 4y - 10 = 0$$?

   • $$-\frac{2}{5}$$
   • $$\frac{2}{5}$$
   • $$-\frac{9}{2}$$
   • $$-\frac{9}{5}$$

5. The average salary of the 12 employees at Horizon Tech Solutions is $$48000$$. When a new employee, Maria, is hired, the average salary increases to $$48500$$. What is Maria’s salary?

   • 63050
   • 54500
   • 48500
   • 57600

6. During a school-wide vote on a new policy, $$320$$ students from the junior class and $$80$$ students from the senior class participated. A total of $$280$$ students voted "yes" on the policy. If the same percentage of students in both classes voted "yes," how many juniors voted "yes" on the policy?

   • 224
   • 320
   • 400
   • 280

7. Liam is laying tiles in a garden that is shaped like a parallelogram. The base of the garden is 10 feet and the height is 5 feet. One of the interior angles of the parallelogram is 60°. How many square feet of tiles will Liam need to completely cover the garden?

   • 50
   • 60
   • 25
   • $$25\sqrt{3}$$

8. A circle has its center at the origin and passes through the point with coordinates (3, 4). What is the area of this circle?

   • $$9\pi$$
   • $$25\pi$$
   • $$16\pi$$
   • $$12\pi$$

9. David received his allowance on Sunday. He spends $$\frac{1}{4}$$ of his allowance on Monday and $$\frac{2}{3}$$ of the remainder on Tuesday. What part of his allowance is left for the rest of the week?

   • $$\frac{3}{4}$$
   • $$\frac{1}{4}$$
   • $$\frac{1}{2}$$
   • $$\frac{1}{3}$$

10. The distance a rubber band stretches varies directly with the force applied to it. If a 5-lb weight stretches the band 40 inches, how many inches will the rubber band stretch if a 15-lb weight is applied?

    • 120
    • 40
    • 60
    • 90

11. If prices are increased by 30% and sales decrease by 15%, what is the net effect on gross receipts?

    • They decrease by 15%
    • They increase by 15%
    • They increase by 10%
    • They increase by 10.5%

12. A music streaming service charges a $$25$$ dollars activation fee and $$15$$ dollars per month for a subscription. The monthly fee is automatically withdrawn from your account. If your starting account balance is $$190$$ dollars, how many full months can you use the service before needing to add more money to your account?

    • 19
    • 16
    • 25
    • 11

13. The surface area of a cylinder is found using the formula $$SA = 2\pi rh + 2\pi r^2$$Which of the following shows the equation solved for $$h$$, the height of the cylinder, in simplest form?

    • $$\frac{SA - 2\pi r^2}{2\pi r}$$
    • $$\frac{SA + 2\pi r^2}{\pi r}$$
    • $$\frac{SA + 2\pi r^2}{2\pi}$$
    • $$\frac{SA - \pi r^2}{\pi r}$$

14. If a triangle with base 10 has the same area as a circle with radius 5, what is the altitude of the triangle?

    • $$25\pi$$
    • $$16\pi$$
    • $$5\pi$$
    • $$4\pi$$

15. If $$m(x) = 2x^2 + 7$$ and $$n(x) = \frac{4}{(x - 3)^2}$$, what is the value of $$m(4) - n(4)$$?

    • 32
    • 39
    • 35
    • 25

16. The number of textbooks in the math department is modeled by the equation $$m(x) = x^2 - 5x + 12$$ and the number of textbooks in the science department is modeled by the equation $$s(x) = 4x^2 + 2x + 18.$$ Which function models the total number of textbooks in stock?

    • $$4x^2 + 3x + 30$$
    • $$3x^2 - 7x + 6$$
    • $$-3x^2 - 7x - 6$$
    • $$5x^2 - 3x + 30$$

17. Lena practices both piano and violin each day. She plays her piano pieces correctly $$\frac{6}{7}$$ of the time and her violin pieces correctly $$\frac{5}{6}$$ of the time. The probability that she plays both piano and violin pieces correctly on the same day is $$\frac{1}{2}$$. What is the probability that she plays the piano pieces correctly given that she played the violin pieces correctly?

    • $$\frac{2}{5}$$
    • $$\frac{1}{2}$$
    • $$\frac{1}{42}$$
    • $$\frac{3}{5}$$

18. The scatterplot below depicts the relationship between two variables, $$x$$ and $$y$$. Which of the following best describes the line that best fits the data in the scatterplot?

    • The slope of the line of best fit is $$-3$$.
    • The slope of the line of best fit is $$-1$$.
    • The slope of the line of best fit is $$1$$.
    • The slope of the line of best fit is $$3$$.

19. In triangle OPQ, which is right-angled at point $$P$$, we are given:$$OP = 7 \ \text{cm}$$$$OQ - PQ = 1 \ \text{cm}$$Determine the values of $$\sin Q$$.

    • $$\frac{18}{25}$$
    • $$\frac{7}{25}$$
    • $$\frac{24}{25}$$
    • $$\frac{1}{25}$$

20. Suppose $$\frac{x}{4} = \frac{y}{6} = \frac{z}{9}$$ where $$y \ne 0.$$ Compute the value of $$\frac{2x - y + z}{y}.$$

    • $$\frac{11}{6}$$
    • $$\frac{3}{2}$$
    • $$\frac{2}{3}$$
    • $$\frac{9}{4}$$

21. Find the coordinates of the midpoint $$M$$ between two points $$A(4,-2)$$ and $$B(6,8)$$.

    • $$(1,5)$$
    • $$(2,10)$$
    • $$(5,3)$$
    • $$(10,6)$$

22. If the expression $$\frac{5+2i}{4+i}$$ is rewritten in the form $$a+bi$$, where $$a$$ and $$b$$ are real numbers, what is the value of $$a$$?

    • $$\frac{3}{17}$$
    • $$\frac{22}{17}$$
    • $$3$$
    • $$17$$

23. For all positive integers $$x$$, the function $$f$$ is defined by $$f(x)=\left(\frac{1}{2^a}\right)^x$$, where $$a$$ is a constant greater than 1. Which of the following is equivalent to $$f(4x)$$?

    • $$(\frac{1}{2^{4a}})^{2x}$$
    • $$(\frac{1}{4^{4a}})^{4x}$$
    • $$(\frac{1}{2^{4a}})^x$$
    • $$(\frac{1}{4^{2a}})^{2x}$$

24. For all real numbers $$a$$ and $$b$$, $$|a+b|$$ is equivalent to which of the following?

    • $$a+b$$
    • $$\sqrt{a+b}$$
    • $$(a+b)^2$$
    • $$\sqrt{(a+b)^2}$$

25. A lab stores a certain chemical in 3-hectogram containers. Based on the information given above, how many 1-milligram doses are there in one 3-hectogram container?Note:1 hectogram = 100 grams1 gram = 1000 milligrams

    • $$0.3$$
    • $$3$$
    • $$300$$
    • $$300000$$

26. Consider the system of equations:$$4x+5y=7$$$$6x-y=17$$What is the value of $$x+y$$?

    • $$\frac{13}{17}$$
    • $$\frac{46}{17}$$
    • $$\frac{85}{17}$$
    • $$\frac{33}{17}$$

27. A right circular cone has a height of $$12$$ inches as shown below. A $$60^\circ$$ angle is formed by the radius and the slant height of the cone. What is the volume of the cone in cubic inches?

    • $$16\pi$$
    • $$192\pi$$
    • $$4\pi$$
    • $$576\pi$$

28. Rectangle $$PQRS$$ shown below has diagonals $$PR$$ and $$QS$$. The measure of angle $$QRP$$ is $$45^\circ$$. If $$QS=10$$ meters in length, what is the area, in square meters, of the rectangle?

    • 100
    • 50
    • 25
    • 75

29. The pie chart below shows the results from a survey of the percentage of different ways people get to school each day. If $$60$$ people that were surveyed took a bus to school, how many people from the survey biked to school?

    • $$10$$
    • $$30$$
    • $$40$$
    • $$20$$

30. In triangle $$XYZ$$, the measure of $$\angle Y$$ is $$90^\circ$$, $$YZ=9$$, and $$XZ=15$$. Triangle $$MNO$$ is similar to triangle $$XYZ$$, where vertices $$M,N,O$$ correspond to vertices $$X,Y,Z$$ respectively, and each side of triangle $$MNO$$ is $$\frac{1}{2}$$ the length of the corresponding side of triangle $$XYZ$$. What is the value of $$\sin O$$?

    • $$\frac{9}{15}$$
    • $$\frac{5}{12}$$
    • $$\frac{4}{5}$$
    • $$\frac{3}{5}$$

31. The angles shown below are acute and $$\sin(m^\circ)=\cos(n^\circ)$$. If $$m=5k-10$$ and $$n=7k-25$$, what is the value of $$k$$?

    • $$\frac{115}{7}$$
    • $$\frac{115}{12}$$
    • $$\frac{115}{10}$$
    • $$\frac{125}{12}$$

32. A quality control manager at a factory selects $$5$$ batteries at random for inspection out of every $$250$$ batteries produced. At this rate, how many batteries will be inspected if the factory produces $$12000$$ batteries?

    • $$240$$
    • $$120$$
    • $$250$$
    • $$60$$

33. Which of the following equations has a graph in the $$xy$$-plane for which $$y$$ is always greater than or equal to $$2$$?

    • $$y=|x+1|+1$$
    • $$y=(x-1)^2+2$$
    • $$y=x^2+x+1$$
    • $$y=x^3+2$$

34. A line in the $$xy$$-plane passes through the origin and has a slope of $$\frac{3}{4}$$. Which of the following points lies on the line?

    • $$(5,3)$$
    • $$(3,4)$$
    • $$(8,6)$$
    • $$(12,8)$$

35. During an experiment, a ball was dropped and allowed to bounce multiple times until it came to a stop. The graph above shows how the height of the ball changed over time after it was dropped. Based on the graph, how many times did the ball reach a height of $$2$$ feet?

    • 1
    • 2
    • 3
    • 4

36. A grain silo is constructed from two right circular cones and a right circular cylinder, as shown in the figure below. The internal dimensions are as follows:- Height of each cone: $$12$$ feet- Height of the cylinder: $$6$$ feet- Radius of the base: $$4$$ feetOf the following, which is closest to the total volume of the grain silo, in cubic feet?

    • 703.36
    • 64.00
    • 192.00
    • 224.00

37. In a circle with center $$P$$, central angle $$QPR$$ has a measure of $$\frac{3\pi}{2}$$ radians. The area of the sector formed by central angle $$QPR$$ is what fraction of the area of the circle?

    • $$\frac{1}{4}$$
    • $$\frac{3}{4}$$
    • $$\frac{1}{2}$$
    • $$\frac{2}{3}$$

38. Points $$A$$, $$B$$, and $$C$$ lie in a plane. If the distance between $$A$$ and $$B$$ is $$15$$ and the distance between $$B$$ and $$C$$ is $$9$$, which of the following could be the distance between $$A$$ and $$C$$?I. $$5$$II. $$24$$III. $$25$$

    • I
    • II
    • I, III
    • III

39. The surface area of a cube is given by $$6\left(\frac{b}{3}\right)^2$$, where $$b$$ is a positive constant. Which of the following gives the length of one edge of the cube?

    • $$\frac{b}{3}$$
    • $$\frac{b}{\sqrt{6}}$$
    • $$\frac{b}{\sqrt{3}}$$
    • $$\frac{b}{6}$$

40. A shipping cart can carry packages that weigh either $$30$$ pounds or $$50$$ pounds. Let $$x$$ be the number of 30-pound packages and $$y$$ be the number of 50-pound packages. The cart has a limit of either 50 packages or a total weight of 2,000 pounds. Which of the following systems of inequalities represents this situation?

    • $$30x+50y\leq2000$$ $$x+y\leq50$$
    • $$\frac{x}{30}+\frac{y}{50}\leq2000$$ $$x+y\leq50$$
    • $$30x+50y\leq50$$ $$x+y\leq2000$$
    • $$x+y\leq2000$$ $$30x+50y\leq50$$

41. What is the range of values for $$y=2\sin x+\cos 2x$$ where $$x$$ is a real number?

    • $$[-3,\frac{3}{2}]$$
    • $$[-3,\infty)$$
    • $$(-\infty,\frac{3}{2}]$$
    • $$(-\infty,\infty)$$

42. What will be the value of $$c$$ < $$0$$ if the determinant of the matrix is equal to three times the value of $$c$$?

    • $$-3$$
    • $$\frac{-3}{2}$$
    • $$\frac{-3 - \sqrt{33}}{2}$$
    • $$2$$

43. The nth term $$(s_n)$$ of a certain sequence is defined as $$s_n=s_{n-1}+3$$. If $$s_1=5$$, then what is the value of $$s_{12}$$?

    • 16
    • 38
    • 33
    • 11

44. The expression $$\log_2(48)+\log_2(4)-\log_2(12)$$ equals which of the following?

    • $$4$$
    • $$12$$
    • $$3$$
    • $$16$$

45. Given $$f(x)=\sqrt[3]{x+1}+5$$, which of the following expressions is equal to $$f^{-1}(x)$$ for all real numbers $$x$$?

    • $$(x+1)^3-5$$
    • $$(x-1)^3+5$$
    • $$x^3-1$$
    • $$(x-5)^3-1$$