Full Practice Exams — Exam 13

Questions in this quiz (45)

1. A water tank can hold $$24000$$ gallons of water when full. It is being filled by $$8$$ pipes, each of which pumps $$5$$ gallons of water per minute. How many hours will it take to completely fill the tank?

   • 40
   • 60
   • 10
   • 600

2. A jar contains $$r$$ red marbles and $$g$$ green marbles. If the probability of randomly drawing a red marble from the jar is $$\frac{4}{9}$$, what is the value of $$\frac{g}{r}$$?

   • $$\frac{9}{4}$$
   • $$\frac{4}{7}$$
   • $$\frac{5}{4}$$
   • $$\frac{4}{9}$$

3. A hotel has only standard rooms and deluxe rooms. A renovation project is planned for all the rooms to modernize their interiors. The renovation cost is $$2500$$ for each standard room and $$4800$$ for each deluxe room. If the ratio of standard rooms to deluxe rooms in the hotel is $$4:1$$, what is the average renovation cost per room?

   • $$5000$$
   • $$14800$$
   • $$2960$$
   • $$2500$$

4. If $$(x-4)$$ and $$(x+2)$$ are factors of the quadratic expression below, what is the value of $$a+b$$?$$x^2+(a-6)x+a-b$$

   • 24
   • 10
   • 16
   • 8

5. The ratio of the volume of a cube (in cubic units) to its surface area (in square units) is $$3:2$$. What is the surface area of the cube?

   • 243
   • 486
   • 94
   • 181

6. If $$\log_b \left(\frac{1}{9}\right) = -\frac{4}{3}$$, what is $$b$$?

   • $$\sqrt{27}$$
   • $$\sqrt{3}$$
   • $$\sqrt{7}$$
   • $$9^ {\frac{3}{2}}$$

7. For positive real numbers $$a, b,$$ and $$c$$, consider the function:$$g(x) = |-(x-a)^2 + b| - c$$What is the minimum value of $$g(x)$$ for all $$x$$?

   • $$b-c$$
   • $$-c$$
   • $$-a^2-c$$
   • $$a^2-c$$

8. Determine the type of conic represented by the equation: $$x^2 = -3y^2 + 4y - 8$$

   • Hyperbola
   • Circle
   • Parabola
   • Ellipse

9. If the ratio of the perimeters of two similar rectangles is $$3 : 4$$, what is the ratio of their areas?

   • $$9 : 16$$
   • $$7 : 6$$
   • $$12 : 16$$
   • $$2 : 3$$

10. The interior angles of a triangle are in the ratio $$1 : 2 : 3$$. What is the ratio of the longest side to the shortest side?

    • $$3 : 1$$
    • $$2 : 3$$
    • $$2 : 1$$
    • $$4 : 5$$

11. Mary is playing a game that involves rolling a standard six-sided dice 6 times. If she rolls each number exactly once, she will win a prize. She has already rolled a "3" and a "6." What is the probability that she will win a prize? (Note: The faces of a standard six-sided dice are numbered from 1 to 6.)

    • $$\frac{1}{24}$$
    • $$\frac{1}{54}$$
    • $$\frac{3}{32}$$
    • $$\frac{1}{1296}$$

12. Which of these lines will intersect the vertex of a parabola with the equation $$y = 4x^2 + 3$$?

    • y = -2
    • y = -1
    • y = 3
    • y = 2

13. In a group of 40 students, 25 applied to Columbia and 30 applied to Cornell. If 3 students applied to neither Columbia nor Cornell, how many students applied to both schools?

    • 18
    • 37
    • 25
    • 13

14. The coordinates of a midpoint and one endpoint are $$M(6,2)$$ and $$A(0,-4)$$, respectively. What are the coordinates of the other endpoint B?

    • $$(3,-1)$$
    • $$(12,8)$$
    • $$(-3,1)$$
    • $$(6,-2)$$

15. The coordinates of a midpoint and one endpoint are $$M(6,2)$$ and $$A(0,-4)$$, respectively. What are the coordinates of the other endpoint B?

    • $$(3,-1)$$
    • $$(12,8)$$
    • $$(-3,1)$$
    • $$(6,-2)$$

16. The graphs of $$h(x)$$ and $$k(x)$$ are shown below. Which of the following must be true?

    • $$k(x)<0$$ for all values of $$x$$
    • $$h(x)$$ > $$k(x)$$ for all values of $$x$$
    • $$k(x)$$ > $$h(x)$$ for all values of $$x$$
    • $$h(x)=k(x)$$ at two points

17. The value of $$k$$ for which the system of equations$$kx - y = 2$$$$6x - 2y = 3$$has a unique solution is:

    • $$k = 0$$
    • $$k = 3$$
    • $$k \ne 3$$
    • $$k \ne 0$$

18. If the radii of two concentric circles are $$10\text{ cm}$$ and $$26\text{ cm}$$, find the length of a chord of the larger circle that is tangent to the smaller circle.

    • 24
    • 20
    • 52
    • 48

19. A man on the top of a vertical observation tower observes a car moving at a uniform speed coming directly towards it. If it takes $$8$$ minutes for the angle of depression to change from $$30^\circ$$ to $$45^\circ$$, how soon after this will the car reach the observation tower?

    • $$7.63$$
    • $$6.53$$
    • $$8.33$$
    • $$10.93$$

20. Tickets numbered 1 to 30 are mixed up and one ticket is selected at random. What is the probability that the selected ticket has a number that is a multiple of 4 or 6?

    • $$\frac{1}{5}$$
    • $$\frac{1}{15}$$
    • $$\frac{1}{3}$$
    • $$\frac{1}{4}$$

21. The price of a new laptop can be expressed as the original price $$P$$, which does not include any fees. A store is offering a promotion in which each laptop has a 25 percent discount off the original price. Additionally, a recycling fee of 5 percent is added to the discounted price. If the final price of the laptop (after the discount and fee) can be expressed as $$A\times P$$, with $$A$$ as a constant, what is the value of $$A$$?

    • 8
    • 7
    • 0.75
    • 0.79

22. If $$f(x)=x^2+2x$$ and $$f(x+n)=x^2+8x+12$$, what is the value of the constant $$n$$?

    • 2
    • 4
    • 6
    • 3

23. In the xy-plane, find the equation of the line passing through point $$B(-2,5)$$ and perpendicular to the line given by the equation $$4x+y=7$$.

    • $$y=4x+\frac{7}{2}$$
    • $$y=\frac{1}{4}x+\frac{11}{2}$$
    • $$y=\frac{1}{2}x+\frac{11}{3}$$
    • $$y=-4x+\frac{7}{2}$$

24. Jack and John are meeting at a restaurant. The graph shows the distance-time relationship for each person's drive from their office to the restaurant. Which of the following statements is true?

    • It took John longer to arrive to the pub since his office is farther away.
    • It took Jack longer to arrive to the pub since his office is farther away.
    • Jack drove to the pub at a faster speed than John.
    • John drove to the pub at a faster speed than Jack.

25. We survey 125 employees at random from each of two companies labeled A and B, and separated into groups based on how many siblings they have. The results are shown in the table below.What is the median number of siblings of the sample of employees in company B?

    • 1
    • 2
    • 3
    • 4

26. The graph of the function $$f(x)=-3(x-1)(x+7)$$ is a parabola. If the line $$x=k$$ is the axis of symmetry of the parabola, what is the value of $$k$$?

    • 1
    • -7
    • -3
    • 4

27. If the values in the table below represent a linear relationship, what is the missing value?

    • 9
    • -2
    • 4
    • 5

28. The shaded region in the graph represents the solution of the system$$\begin{cases}4x+y$$ < $$2\\y\ge-2\end{cases}$$For $$x=0$$, what is the highest integer value of $$y$$?

    • 1
    • -2
    • 0
    • 6

29. If the expression $$(3i^2+4i)\div(2+i)$$ is written in the form $$a+bi$$ where $$a$$ and $$b$$ are real numbers and $$i=\sqrt{-1}$$, what is the value of $$a$$?

    • $$\frac{-1}{2}$$
    • $$\frac{3}{4}$$
    • $$\frac{-2}{5}$$
    • $$\frac{-3}{4}$$

30. What is the area of the sector defined by a central angle of 60° in the circle of equation $$x^2+y^2+4x-6y-12=0$$?

    • $$\frac{5\pi}{6}$$
    • $$\frac{4\pi}{5}$$
    • $$\frac{3\pi}{4}$$
    • $$\frac{25\pi}{6}$$

31. The table below summarizes the results of a survey about favorite extracurricular activities for a group of 800 university students from 4 different majors. What fraction of people who prefer Sports or Music come from Biology or Math majors?

    • $$\frac{3}{4}$$
    • $$\frac{25}{35}$$
    • $$\frac{29}{56}$$
    • $$\frac{10}{19}$$

32. The selling price of an apartment is directly proportional to its area. If an apartment of 127 sqm is sold for 168275 dollars, what would be the price of a 156 sqm apartment?

    • $$257,500$$
    • $$175,600$$
    • $$198,700$$
    • $$206,600$$

33. The right triangles ADE and ABC are similar. If $$AD = \frac{4}{3}ED$$, how long is $$DB$$?

    • 4
    • 5
    • 3
    • 6

34. If the system of linear equations below admits no solutions, and m is an integer, what is the value of m?$$\frac{2}{5}x - \frac{1}{3}y = 7 \\-\frac{m}{10}x + \frac{5}{6}y = 3$$

    • 10
    • 12
    • 5
    • 6

35. Which of the following is equivalent to the expression $$\sqrt{50x^4y^6 - 32x^2y^3}$$? Assume $$x$$ > $$0$$ and $$y$$ > $$0$$.

    • $$3xy^2\sqrt{3xy} - \sqrt{2}$$
    • $$9xy^2\sqrt{3xy} - \sqrt{2}$$
    • $$3xy^2\sqrt{3xy - 2}$$
    • $$9xy^2\sqrt{3xy - 2}$$

36. A liquid fills 15 cm of the height of a cylindrical container with a diameter of 10 cm. The liquid is then poured into a conical container with a diameter of 15 cm. What height will the liquid reach in the cone?

    • 17
    • 20
    • 15
    • 14

37. In the xy-plane below, ABC is an equilateral triangle with sides of length 2. If point A has coordinates $$(-2,0)$$, what are the coordinates of point B?

    • $$(-\sqrt{3},-3)$$
    • $$(-2,-\sqrt{3})$$
    • $$(-3,\sqrt{2})$$
    • $$(-3,-\sqrt{3})$$

38. Which of the following best approximates the range in monthly rainfall over the course of a year?

    • 16
    • 26
    • 10
    • 11

39. In triangle PQR, QS and SR are angle bisectors, and the measure of angle P is 80 degrees. How many degrees are there in angle QSR?

    • $$40^\circ$$
    • $$80^\circ$$
    • $$100^\circ$$
    • $$130^\circ$$

40. In a certain electric circuit, the generator has an adjustable feature called the frequency. For different frequencies between 60 and 80 Hertz, one obtains different intensities of current flowing in the circuit measured in Amperes. The graph given below represents the variation of the effective value of the current as a function of the frequency. Which of the following values of the current can be attained by only one value of the frequency?

    • 0.3
    • 32
    • 0.05
    • 80

41. Which graph best represents a quadratic function that has only one zero?

    • None of the above

42. If $$a$$ is a solution of the equation $$|2x - 4| = 5$$, what is the distance between $$a$$ and the point of coordinate $$2$$ on the number line?

    • $$\frac{5}{2}$$
    • $$\frac{3}{2}$$
    • $$\frac{2}{3}$$
    • $$\frac{9}{2}$$

43. In trapezoid ABCD below, $$AB||CD$$. If $$AE=5.2$$, $$AC=11.7$$, and $$CD=10.5$$, what is the length of $$AB$$ to the nearest tenth?

    • 6.5
    • 4.7
    • 8.5
    • 14.5

44. What is the area of the figure below if the all the segments are congruent?

    • 16
    • 20
    • 24
    • 32

45. The graphs of linear function $$f$$ and $$g$$ are shown on the grid.Which function is best represented by the graph of $$g$$?

    • $$g(x) = f(x) = 3$$
    • $$g(x) = 3f(x)$$
    • $$g(x) = f(x) = 2$$
    • $$g(x) = \frac{1}{3}f(x)$$