Full Practice Exams — Exam 10

Questions in this quiz (45)

1. If two buildings are $$2.5\ \text{cm}$$ apart on the map, and $$1\ \text{cm}$$ on the map represents $$6\ \text{km}$$, what is the distance (in km) between the two buildings in real life?

   • $$6$$
   • $$15$$
   • $$25$$
   • $$12$$

2. A box of chocolate balls was shared among Alex, Tom, and Sarah. Alex got $$0.4$$ of them, and the remainder was shared between Tom and Sarah in the ratio $$3:2$$. Alex got $$2$$ chocolate balls more than Tom. What is the total number of chocolate balls in the box?

   • $$20$$
   • $$30$$
   • $$40$$
   • $$50$$

3. It takes 32 feet of fence to enclose a square garden in Jason's backyard. Jason wants to install grass everywhere except on a diagonal gravel path (shaded in the diagram below). The path is 2 feet wide and runs diagonally from one corner of the square to the opposite corner. What is the area of the gravel path?

   • 14
   • $$16\sqrt{2}$$
   • 16
   • $$32\sqrt{2}$$

4. Factorize $$x^3+64$$.

   • $$(x+4)(x^2+4x+8)$$
   • $$(x-4)(x^2-4x+8)$$
   • $$(x+4)(x^2-4x+16)$$
   • $$(x-4)(x^2+4x+8)$$

5. Solve the inequality: $$\frac{x+4}{3}-\frac{x-4}{5}$$ < $$2+\frac{3x-1}{15}$$

   • $$0$$ < $$x$$ < $$15$$
   • $$-3$$ < $$x$$ < $$5$$
   • $$x$$ > $$3$$
   • $$x$$ < $$5$$

6. ABC is an isosceles triangle such that the length of the side AB is 5 and the angle ∠A is 120 degrees. Let AH be the height of ABC. Determine the length of the side BC.

   • $$10$$
   • $$7$$
   • $$5$$
   • $$5\sqrt{3}$$

7. Solve for $$x$$:$$\log_3(7+\log_4(13+\log_2(x+4)))=2$$

   • $$2$$
   • $$13$$
   • $$9$$
   • $$4$$

8. Determine the value of $$\frac{2}{5}$$ of $$65\%$$ of $$10,000$$ dollars.

   • $$6,500$$
   • $$1,300$$
   • $$1,500$$
   • $$2,600$$

9. Sam traveled 300 miles at a constant speed, stopped for 30 minutes, then increased his speed by 20 mph to complete the remaining 200 miles and arrived on time. The total distance is 500 miles. Find Sam’s speed before the break.

   • $$40$$
   • $$20$$
   • $$50$$
   • $$30$$

10. The length of segment $$AB$$ is 5, and the coordinates of $$A$$ are $$(3,4)$$. The $$x$$-coordinate of $$B$$ is $$6$$. Determine the $$y$$-coordinate of the midpoint of $$AB$$.

    • $$4$$
    • $$6$$
    • $$5$$
    • $$16$$

11. The median of a set containing 15 items was found. Six data items were added to the set: three are greater than the original median, and three are smaller. Determine how the addition affects the median.

    • Not known
    • Decreases
    • Increases
    • Remains the same

12. Determine the value of $$\cos(A)$$ in the standard coordinate plane, where the terminal point of the angle $$A$$ is at $$(5,-3)$$.

    • $$-\frac{3\sqrt{34}}{34}$$
    • $$-\frac{5\sqrt{34}}{34}$$
    • $$\frac{5\sqrt{34}}{34}$$
    • $$\frac{3\sqrt{34}}{34}$$

13. Determine the coordinates $$(x, y)$$ at which the two lines $$-2x+y=6$$ and $$3x-y=-4$$ intersect.

    • $$(2, 10)$$
    • $$(-4, 2)$$
    • $$(2, -4)$$
    • $$(2, -6)$$

14. The swimming pool cross-section is shown. The shallow end (point A) has a depth of 5 feet, and the deep end (point D) has a depth of 10 feet. The horizontal distance between the extremities of the pool is 49 feet. Determine the slope of the segment connecting the shallow end to the deep end.

    • $$\frac{14}{30}$$
    • $$\frac{5}{14}$$
    • $$\frac{5}{44}$$
    • 1

15. The license plate numbers have exactly 6 characters:The first 3 characters are digits (0-9), giving 10 possible choices for each position. The last 3 characters are letters (A-Z), giving 26 possible choices for each position.Characters can be repeated on the license plate.

    • $$10^{26}$$
    • $$17,576,000$$
    • $$17,576,000$$
    • $$127,000$$

16. Given the equation $$3x - 2y = 4$$, determine the value of $$\frac{8^x}{4^y}$$.

    • $$16$$
    • $$4$$
    • $$2$$
    • $$8$$

17. A and B lie on a circle with center O, and C is a point outside the circle. Segments AC and BC are tangent to the circle at A and B, respectively. The radius of the circle is 10 inches, and the perimeter of quadrilateral AOBC is 200 inches. Determine the measure of segment OC.

    • $$95$$
    • $$10$$
    • $$82$$
    • $$10 \sqrt{82}$$

18. Solve the equation $$x^4 - 13x^2 + 36 = 0$$.

    • $${-3, -2, 2, 3}$$
    • $${-2, -3}$$
    • $${-3, -2, 0, 2}$$
    • $${-2 2}$$

19. The average of 12 test scores is $$a$$. After removing the lowest and highest scores, the new average becomes $$b$$. Define the average of the highest and lowest scores.

    • $$3a - 10b$$
    • $$5a - 10b$$
    • $$10a - 5b$$
    • $$6a - 5b$$

20. Determine in which set the difference between the mean and the median is the greatest.

    • $$100, 200, 300, 400$$
    • $$50, 60, 70, 80$$
    • $$500, 1000, 2000, 2500$$
    • $$1, 10, 100, 1000$$.

21. The ratio of the interior angles of a triangle is $$4:2:3$$. Determine the ratio of the shortest side to the longest side.

    • $$4:3$$
    • $$2:4$$
    • $$1.5:3$$
    • $$3:4$$

22. Determine the expression for $$\tan x \cdot \csc x$$ if $$0^\circ$$ < $$x$$ < $$90^\circ$$ and $$\cos x=t$$.

    • $$\frac{t}{2}$$
    • $$\frac{\sqrt{t}}{2}$$
    • $$\frac{1}{t}$$
    • $$\frac{1}{t^2}$$

23. What is the equivalent of $$\frac{9\pi}{12}$$ radians in degrees?

    • $$135$$
    • $$2.36$$
    • $$100$$
    • $$5$$

24. Determine the coefficient of the $$x$$ term of the third-degree polynomial with roots $$2i, -2i$$ and $$3$$.Note: $$i^2 = -1$$

    • $$4$$
    • $$-1$$
    • $$-12$$
    • $$1$$

25. Determine the equation of the tangent of the circle at the point $$(3,4)$$ if its equation is: $$x^2-4x+4=26-(y+1)^2$$

    • $$y=\frac{1}{4}x+\frac{26}{5}$$
    • $$y=-\frac{1}{4}x-\frac{26}{5}$$
    • $$y=\frac{1}{26}x+\frac{4}{5}$$
    • $$y=-\frac{1}{5}x+\frac{23}{5}$$

26. Alex and Matt are among 10 players in a video game tournament. If the pairing is random, what is the probability that they will play against each other in the first round?

    • $$\frac{1}{9}$$
    • $$\frac{8}{105}$$
    • $$\frac{11}{105}$$
    • $$\frac{13}{105}$$

27. Adel buys a truck for 120,000 dollars. If this truck depreciates at 10% per year, what is the price of the truck after 6 years?

    • $$21,258.73$$
    • $$63,772.92$$
    • $$57,395.63$$
    • $$63,772.92$$

28. Which of the following is an equation in which $$x$$ varies directly with the cube of $$z$$, inversely with the square of $$y$$, and inversely with the square root of $$t$$?

    • $$k \cdot \frac{z^3}{y^2 \sqrt{t}}$$
    • $$k \cdot \frac{y^2 \sqrt{t}}{z^2}$$
    • $$k \cdot \frac{z^2}{y^3 \sqrt{t}}$$
    • $$\frac{1}{k} \cdot \frac{y^2 \sqrt{t}}{z^2}$$

29. The mean price of a gallon of paint is $$30$$ dollars with a standard deviation of $$1.5$$ dollars. Which of the following is not within two standard deviations of the mean price?

    • 30
    • 28
    • 34
    • 32

30. A sample of 500 adults received either a vitamin C pill or a zinc pill each day for 21 days. After that, the adults reported if they contracted a cold during that period. The table below shows the numerical results of the study. Determine the percentage of adults who received a vitamin C pill and reported contracting a cold.

    • $$10.6\%$$
    • $$21.2\%$$
    • $$20.5\%$$
    • $$78.8\%$$

31. Determine the ratio of the least common multiple to the greatest common factor of 12 and 21.

    • $$21$$
    • $$84$$
    • $$24$$
    • $$28$$.

32. The coordinates of the points A, B, and C in the standard xy-plane are respectively $$(1,2)$$, $$(4,4)$$, and $$(6,1)$$. Determine the coordinates of point D in order for the quadrilateral ABCD to be a square.

    • $$(4, -2)$$
    • $$(7, -2)$$
    • $$(3, -1)$$
    • $$(6, 4)$$

33. Alex is making a poster of his favorite anime to hang it in the wall of his room. The initial poster he made was 5 feet long and 3 feet wide, but then he realized that it was too large, and so he decided to make a new one. The new poster’s area is half of the previous one, and its length is $$\frac{2}{5}$$ of the length of the first poster. Determine the width of the new poster.

    • $$7.5$$
    • $$2.5$$
    • $$3.75$$
    • $$5.0$$

34. Ali is using 50 feet of fencing to enclose a part of his backyard. What is the largest rectangular area that Ali can enclose?

    • $$156.25$$
    • $$250.50$$
    • $$125$$
    • $$210.25$$

35. A sequence is defined for all positive integers by: $$S_n=3S_{n-1}+n+2$$ and $$S_1=1$$What is the value of $$S_3$$?

    • $$7$$
    • $$21$$
    • $$5$$
    • $$26$$

36. The mean of the set of 7 numbers $$\{50,7,12,27,a,30,77\}$$ is 45 and the median of the set of 5 numbers $$\{64,7,31,93,b\}$$ is 38. Determine the value of $$a-b$$.

    • $$64$$
    • $$50$$
    • $$38$$
    • $$74$$

37. The dimensions of a rectangular prism are as follows: Length: 50 feet, Width: 30 feet, Height: 6 feet. If the length and width of the prism are increased by 10% and its height is decreased by 1%, determine the percentage of increase in the prism volume.

    • $$20.02\%$$
    • $$59.40\%$$
    • $$50.\%$$
    • $$9\%$$

38. Consider the graph of the function $$f$$ shown below. Determine which equation represents the vertical asymptote of the graph.$$f(x)=\frac{x+3}{2x-4}$$

    • $$x=3$$
    • $$x=-3$$
    • $$x=2$$
    • $$x=-4$$

39. Put the following fractions in decreasing order:$$\frac{6}{5}, \frac{4}{3}, \frac{3}{2}, \frac{8}{7}$$

    • $$\frac{3}{2}, \frac{4}{3}, \frac{6}{5}, \frac{8}{7}$$
    • $$\frac{4}{3}, \frac{8}{7}, \frac{6}{5}, \frac{3}{2}$$
    • $$\frac{8}{7}, \frac{6}{5}, \frac{4}{3}, \frac{3}{2}$$
    • $$\frac{4}{3}, \frac{6}{5}, \frac{3}{2}, \frac{4}{3}$$

40. In the standard xy-plane, determine the area of the region defined by the following:$$x^2+y^2=64$$$$x\geq0$$$$y\geq0$$

    • $$4\pi$$
    • $$8\pi$$
    • $$16\pi$$
    • $$64\pi$$

41. The figure below shows 2 circles and one ellipse in the standard xy-plane, all centered at (0,0). The equations of the two circles are also shown below. Determine the equation of the ellipse.$$x^2+y^2=64$$$$x^2+y^2=25$$

    • $$\frac{x^2}{8}+\frac{y^2}{5}=25$$
    • $$\frac{x^2}{8}+\frac{y^2}{5}=64$$
    • $$\frac{x^2}{8}+\frac{y^2}{5}=1$$
    • $$\frac{x^2}{64}+\frac{y^2}{25}=1$$

42. Jim ran at a rate of 8 mph for 20 minutes, then at a rate of 7 mph for 5 minutes, and finally at a rate of 6 mph for 5 minutes. Determine the average rate at which Jim ran over the whole time in mph.

    • $$5$$
    • $$6$$
    • $$7.5$$
    • $$12$$

43. If both $$x$$ and $$\left(\frac{x}{9}+\frac{x}{7}+\frac{x}{3}\right)$$ are positive integers, determine the least value of $$x$$.

    • $$63$$
    • $$37$$
    • $$39$$
    • $$21$$

44. In the figure shown below, the length of the segment AC is 10 inches. Determine which of the following values is the closest to the measure of BC.

    • $$5$$
    • $$\frac{10\sqrt{3}}{3}$$
    • $$\frac{5\sqrt{3}}{3}$$
    • $$10$$

45. Simplify the following expression:$$\frac{(n+1)!\cdot8!}{(n-1)!\cdot6!}$$

    • $$8n^2$$
    • $$6n(n-1)$$
    • $$56n^2$$
    • $$56n(n+1)$$