Full Practice Exams — Exam 20

Questions in this quiz (45)

1. In the standard $$(x,y)$$ coordinate plane, the graph of the quadratic equation $$y = x^2 + mx + 12$$ touches the x-axis at only one point. What is the value of $$m$$?

   • $$\pm 4\sqrt{3}$$
   • $$\pm \sqrt{4}$$
   • $$\pm \frac{\sqrt{3}}{4}$$
   • $$\pm \frac{\sqrt{3}}{3}$$

2. If the minute hand of a clock has a length of $$\frac{12}{\pi} \, \text{cm}$$, how far in centimeters does the tip of the minute hand travel between $$3:15 \, \text{PM}$$ and $$7:30 \, \text{PM}$$?

   • $$102 cm$$
   • $$92 cm$$
   • $$67 cm$$
   • $$52 cm$$

3. Consider the equations:$$4x - 5y = 20$$$$20x - 25y = 100$$For any real number $$r$$, which of the following points lies on the graph of both equations?

   • $$(r, \frac{4r - 20}{5})$$
   • $$(\frac{r - 4}{5}, r)$$
   • $$(-\frac{4}{5}r + 20, \frac{4}{5}r - 100)$$
   • $$(\frac{4}{5}r + 20, -\frac{4}{5}r + 100)$$

4. A city's transportation department found that when the price of a bus ticket decreases, the number of tickets sold per hour increases. The equation below models this relationship: $$rx+s$$where $$x$$ is the ticket price in dollars, and $$r$$ and $$s$$ are constants.- When the price was $$4$$, the number of tickets sold per hour was $$60$$.- When the price was $$3$$, the number of tickets sold per hour was $$75$$.Find the values of $$r$$ and $$s$$.

   • $$r=-15, s=120$$
   • $$r=3, s=-40$$
   • $$r=-3, s=40$$
   • $$r=-4, s=60$$

5. A cylindrical tank has a radius of 3 meters and a height of 5 meters. If it is filled with water, and a cubic meter of water weighs 1,000 kilograms, about how many kilograms does the water in the tank weigh in thousands of kilograms?

   • $$45.0$$
   • $$1000.2$$
   • $$141.3$$
   • $$45\pi$$

6. In the standard $$(x,y)$$ coordinate plane, the region is bounded by the lines $$x = k$$, $$y = 0$$, and $$y = 3x$$. If the area of this region is $$54$$, what is the value of $$k$$?

   • 6
   • 2
   • $$\frac{1}{2}$$
   • $$\frac{1}{3}$$

7. Find the value of $$\log_3\left(\frac{27^{4x+2}}{9^{x+3}}\right).$$

   • $$9x$$
   • $$3x$$
   • $$10x$$
   • $$-3x$$

8. A line is described by the equation $$5y=-2x+20$$. Find the distance between the point where the line intersects the x-axis and the point where it intersects the y-axis.

   • $$\sqrt{29}$$
   • $$2\sqrt{29}$$
   • $$4$$
   • $$4\sqrt{29}$$

9. Determine the intersection point of the asymptotes for the function: $$y=\frac{(3x-7)(x+4)}{(2x+5)(x^2+3)}$$

   • $$(-\frac{3}{2}, 0)$$
   • $$(-\frac{5}{2}, 0)$$
   • $$(-\frac{5}{2}, \frac{3}{2})$$
   • $$(-\frac{5}{2}, \frac{5}{2})$$

10. If the point $$\left(6, \frac{5}{8}\right)$$ is on the graph of $$f(x) = \frac{x}{x + a}$$, what is $$f(10)$$?

    • $$\frac{5}{6}$$
    • $$\frac{25}{34}$$
    • $$\frac{5}{13}$$
    • $$\frac{20}{27}$$

11. Lines $$p$$ and $$q$$ are perpendicular. If the equation of line $$p$$ is $$y = mx + c$$ and the lines intersect at point $$(5,4)$$, which of the following points lies on line $$q$$?

    • $$(5+2m,2)$$
    • $$(3m,6)$$
    • $$(5-3m,7)$$
    • $$(5+m, 4)$$

12. If $$p$$ < $$\frac{57}{5y}$$ < $$q$$, and $$p = 6$$ and $$q = 12$$, which of the following could be true?

    • $$y$$ < $$q$$ and $$y$$ < $$p$$
    • $$y$$ > $$q$$ and $$y$$ < $$p$$
    • $$y$$ > $$p$$
    • $$y$$ > $$q$$ and $$y$$ > $$p$$

13. If the arc formed by a $$60^\circ$$ angle is $$8\pi$$, what is the area of the $$60^\circ$$ sector of the circle?

    • $$24\pi$$
    • $$60\pi$$
    • $$96\pi$$
    • $$80\pi$$

14. A sphere is inscribed in a cube with a diagonal of 4 meters. In meters, what is the diameter of the sphere?

    • $$\frac{7\sqrt{3}}{3}$$
    • $$\frac{2\sqrt{3}}{3}$$
    • $$\frac{4\sqrt{3}}{3}$$
    • $$\frac{\sqrt{3}}{12}$$

15. What are the coordinates of the point $$(4,-3)$$ after it is rotated $$90^\circ$$ clockwise about the origin?

    • $$(3,-4)$$
    • $$(-3,-4)$$
    • $$(-3,4)$$
    • $$(3,4)$$

16. How many 4-letter orderings with no letter repeated can be made from the letters $$a$$ to $$j$$?

    • 720
    • 5040
    • 2520
    • 560

17. Jason randomly selects one number from slips of paper labeled $${-3, -2, -1, 0, 1}$$ and randomly selects another number from slips of paper labeled $${-4, -3, -2, -1, 0, 1, 2}$$. What is the probability that the product of these two numbers will be positive?

    • 0.7
    • 0.4
    • 0.14
    • 0.35

18. What is the remainder when $$3x^2 - 5x + 11$$ is divided by $$x + 1$$?

    • 5
    • 11
    • 3
    • 19

19. How many real solutions does the following equation have?$$x + \sqrt{x} = 12$$

    • 1
    • 2
    • 0
    • Not known

20. If the shaded area in the following square is $$196 - 49\pi$$, what is the diagonal of the square?

    • 14
    • $$14\sqrt{2}$$
    • $$4-\pi$$
    • $$196\pi$$

21. Find the point $$Q$$ on the line $$l: 3x + y = 7$$ that is closest to the point $$P(2, -1)$$.

    • $$(1, 4)$$
    • $$(-1, 10)$$
    • $$(2, -1)$$.
    • $$(\frac{13}{5}, -\frac{4}{5})$$

22. A right triangle PQR is partially inscribed in a circle with center P and a radius of 7 units. The base QR measures 12 units. What is the approximate area of the shaded region inside the triangle but outside the circle?

    • $$10.30$$
    • $$25.24$$
    • $$16.76$$
    • $$59.04$$

23. Two zeros of the polynomial $$P(x) = x^3 - 4x^2 + x + 6$$ are $$2$$ and $$-1$$. Find the third zero of $$P(x)$$.

    • 3
    • 6
    • 9
    • -9

24. In figure, $$PB$$ and $$QA$$ are perpendicular to segment $$AB$$. If $$PO = 5\ \text{cm}$$, $$QO = 7\ \text{cm}$$ and area of $$\Delta POB = 150\ \text{cm}^2$$, find the area of $$\Delta QOA$$.

    • 150
    • 210
    • 300
    • 350

25. If $$\sin(A + B) = 1$$ and $$\cos(A - B) = \frac{\sqrt{3}}{2}$$, where $$0^\circ$$ < A + B \leq 90^\circ$$ and $$A$$ > $$B$$, then find the values of $$A$$ and $$B$$.

    • $$A = 60^\circ,\ B = 45^\circ$$
    • $$A = 45^\circ,\ B = 30^\circ$$
    • $$A = 45^\circ,\ B = 45^\circ$$
    • $$A = 60^\circ,\ B = 30^\circ$$

26. A simple pendulum of length $$40$$ cm subtends $$60^\circ$$ with respect to the vertical at the vertex in one full oscillation. What will be the shortest distance between the initial position and the final position of the bob?

    • $$35\sqrt{2}$$
    • $$25\sqrt{2}$$
    • $$82$$
    • $$40\sqrt{3}$$

27. Suppose you drop a coin at random on the rectangular region shown in the figure. What is the probability that it will land inside the square with side length $$1\text{ m}$$, which is inscribed inside the rectangle with dimensions $$3\text{ m}$$ by $$2\text{ m}$$?

    • $$\frac{1}{6}$$
    • $$\frac{1}{3}$$
    • $$\frac{1}{2}$$
    • $$\frac{1}{4}$$

28. In the figure shown, points U, V, X, and Z lie on the circle, and VX < VZ. Segment XZ is the diameter of the circle and has length 150. Segment XZ is perpendicular to segment UV at point Y, and $$VY = \sqrt{296}$$. If $$\frac{XY}{YZ} = k$$, what is the value of $$k$$?

    • $$\frac{1}{148}$$
    • $$148$$
    • $$\sqrt{74}$$
    • $$\frac{1}{74}$$

29. The function $$q$$ is defined by $$q(x)=a|x-9|^2-53|x-9|+b$$ where $$a$$ and $$b$$ are constants, and $$a$$ >b> $$53$$. If $$q(539)=h$$ and $$q(-521)=k$$, where $$h$$ and $$k$$ are constants, what is the value of $$32(-539)^{h-k}+347(9)^{k-h}?$$

    • $$539$$
    • $$529$$
    • $$379$$
    • $$530$$

30. A perfect square is an integer that is the square of an integer. Suppose that $$m$$ and $$n$$ are positive integers such that $$mn$$ > $$15$$. If $$15mn$$ is a perfect square, what is the least possible value of $$mn$$?

    • 45
    • 60
    • 90
    • 15

31. For a particular number $$a$$, the first term in the sequence $$a, 2a-1, 3a-2, 4a-3, \ldots$$ is equal to $$a$$, and each term thereafter is 7 greater than the previous term. What is the value of the 16th term in the sequence?

    • $$17a - 16$$
    • $$16a - 15$$
    • $$16a - 16$$
    • $$15a - 15$$

32. $$M$$ is a set consisting of a finite number of consecutive integers. If the median of the numbers in set $$M$$ is equal to one of the numbers in set $$M$$, which of the following must be true?I. The average (arithmetic mean) of the numbers in set $$M$$ equals the median.II. The number of numbers in set $$M$$ is odd.III. The sum of the smallest number and the largest number in set $$M$$ is even.

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

33. The figure on the left shows a wheel with a mark on its rim. The wheel is rolling on the ground at a constant rate along a level straight path from a starting point to an ending point. The graph of $$y = d(t)$$ on the right could represent which of the following as a function of time from when the wheel began to roll?

    • The speed at which the wheel is rolling
    • The distance of the wheel from its starting point
    • The distance of the mark on the rim from the center of the wheel
    • The distance of the mark on the rim from the ground

34. The figure shows a rectangular grass patch and an L-shaped footpath. The width of the footpath is 100 cm. The footpath is tiled using 38 circular tiles of diameter 50 cm, following the pattern shown. Each tile is in contact with those next to it. What is the area of the footpath not covered by the tiles?

    • $$45425$$
    • $$75425$$
    • $$35425$$
    • $$20425$$

35. Counting repetition, how many prime factors appear in any prime factorization of the integer $$(84)^4$$?

    • 4
    • 8
    • 12
    • 16

36. In the standard (x,y) coordinate plane the solution set of the system $$x+2y\le 6$$ and $$3x^2$$ > $$12-3y^2$$ is shown in one of the following graphs. Determine which of them is the correct one?

37. Carla is riding a bicycle with wheels 27 inches in diameter. The gear ratio is 4 pedal revolutions to 7 wheel revolutions. During a 5 minute ride Carla pedals at a steady rate of 60 pedal revolutions per minute. At what average speed in miles per hour was Carla riding during that ride? Use 1 mile equals 5280 feet and 12 inches equals 1 foot.

    • $$7.40$$
    • $$8.44$$
    • $$4.70$$
    • $$7.56$$

38. Data from a random sample of 290 customers are organized by age brackets as shown in the table below:Two customers are chosen at random without replacement. What is the probability that both chosen customers are from the same age bracket?

    • $$0.458$$
    • $$0.58$$
    • $$0.48$$
    • $$0.54$$

39. A pipe has radius 6 inches and sends water to three smaller pipes of equal size. Each smaller pipe allows exactly one-third as much water to flow as the larger pipe. What is the radius of one of the smaller pipes?

    • $$2\pi\sqrt{2}$$
    • $$2\sqrt{2}$$
    • $$2\sqrt{3}$$
    • $$\pi\sqrt{3}$$

40. From the figure below, find the value of angle x.

    • $$80^\circ$$
    • $$60^\circ$$
    • $$35^\circ$$
    • $$20^\circ$$

41. Quadrilateral ABCD has coordinates A(2,3), B(-2,3), C(-2,-3), D(2,-3). Determine the nature of ABCD.

    • Square
    • Rhombus
    • Rectangle
    • Trapezoid

42. In pentagon ABCDE, the interior angles are labeled as shown in the figure below. What is the measure of the largest angle?

    • $$150^\circ$$
    • $$170^\circ$$
    • $$210^\circ$$
    • $$110^\circ$$

43. In right triangle XYZ, the right angle is at Z and angle X equals 30 degrees. The side opposite angle X is YZ with length x, the side ZX has length y, and XY is the hypotenuse. Given $$x=4\sqrt{3}$$, find the value of y.

    • $$4\sqrt{3}$$
    • $$6\sqrt{3}$$
    • $$12$$
    • $$8\sqrt{3}$$

44. The bar graph shows the number of customers giving ratings 0 through 5. What is the median of the data?

    • 1
    • 2
    • 3
    • 4

45. In a survey of 200 students, 65 students said they like science and 40 students said they like math. If all the students who said they like math also said they like science, how many students said they like at least one of these subjects?

    • 40
    • 45
    • 65
    • 55