Full Practice Exams — Exam 19

Questions in this quiz (45)

1. The square below is divided into 3 rows of equal area. In the top row, the region labeled A has the same area as the region labeled B. In the middle row, the 3 regions have equal areas. In the bottom row, the 4 regions have equal areas. What fraction of the square's area is in a region labeled A?

   • $$\frac{13}{36}$$
   • $$\frac{1}{3}$$
   • $$\frac{1}{6}$$
   • $$\frac{2}{3}$$

2. As part of a probability game, Sarah has to answer 5 multiple-choice questions. For each question, there are 4 possible answers, only 1 of which is correct. If Sarah randomly and independently answers each question, what is the probability that she will answer all 5 questions correctly?

   • $$\frac{1}{4}$$
   • $$\frac{4}{5}$$
   • $$\frac{5}{4}$$
   • $$\frac{1}{1024}$$

3. The sides of an acute triangle measure 14 cm, 18 cm, and 20 cm, respectively. Which of the following equations, when solved for θ, gives the measure of the smallest angle of the triangle?

   • $$\frac{\sin\theta}{18}=\frac{1}{14}$$
   • $$14^2=18^2+20^2-2(18)(20)\cos\theta$$
   • $$\frac{\sin\theta}{14}=\frac{1}{20}$$
   • $$20^2=14^2+18^2-2(18)(14)\cos\theta$$

4. The sum of 2 positive numbers is 151. The lesser number is 19 more than the square root of the greater number. What is the value of the greater number minus the lesser number?

   • 91
   • 85
   • 65
   • 125

5. The 3rd and 4th terms of an arithmetic sequence are 13 and 18, respectively. What is the 50th term of the sequence?

   • 250
   • 255
   • 236
   • 248

6. The points E(6,4) and F(14,12) lie in the standard (x,y) coordinate plane. Point D lies on EF between E and F such that the length of EF is 4 times the length of DE. What are the coordinates of D?

   • $$(7,5)$$
   • $$(10,12)$$
   • $$(8,6)$$
   • $$(12,10)$$

7. If $$a,b,c$$ are different prime numbers and $$abc^2=140$$, what is the product of $$a,b,c$$?

   • 140
   • 14
   • 70
   • 28

8. Given that its diagonal is 10 and its height is 6, what is the perimeter of the rectangle?

   • 22
   • 24
   • 26
   • 28

9. The perimeter of an equilateral triangle is 72 centimeters. The three vertices of the triangle lie on a circle. The radius of the circle is $$w\sqrt{3}$$ centimeters. What is the value of $$w$$?

   • 24
   • $$8\sqrt{3}$$
   • 8
   • $$24\sqrt{3}$$

10. The lengths of the corresponding sides of two similar right triangles are in the ratio of $$2:5$$. If the hypotenuse of the smaller triangle is $$5$$ inches long, how many inches long is the hypotenuse of the larger triangle?

    • 2.5
    • 2
    • 10
    • 12.5

11. In the equation $$\frac{x}{3}+\frac{y}{12}=z$$, where $$x$$, $$y$$, and $$z$$ are positive integers, all of the following could be a possible value of $$y$$ except:

    • 4
    • 6
    • 8
    • 12

12. The function $$f$$ is graphed in its entirety. If the function $$g$$ is defined so that $$g(x)=f(-x)$$, then for what value of $$x$$ does $$g$$ attain its maximum value?

    • 2
    • -2
    • 3
    • -3

13. In the figure, two line segments in the x-y plane form a right triangle with the x-axis. What is the value of $$a$$?

    • $$4$$
    • $$4\sqrt{2}$$
    • $$5$$
    • $$5\sqrt{2}$$

14. When a baseball is hit by a batter, the height of the ball, at a time $$t$$, is determined by the equation $$h(t) = -16t^2 + 64t + 4$$ where $$t \geq 0$$. For which interval of time, in seconds, is the height of the ball at least 52 feet above the playing field?

    • $$2 \leq t \leq 4$$
    • $$\frac{3}{2} \leq t \leq \frac{7}{2}$$
    • $$\frac{1}{2} \leq t \leq \frac{5}{2}$$
    • $$1 \leq t \leq 3$$

15. The set of numbers is $$\{8, 12, 9, 15, x\}$$.Which of the following cannot be the median of the five numbers listed above?

    • 9
    • 11
    • 12
    • 13

16. Given $$y=\frac{x^3-3x^2+2x-1}{x-3}$$Which of the following expressions is equivalent to $$y$$?

    • $$y=x^2-3+\frac{8}{x-3}$$
    • $$y=x^2+2x+2+\frac{3}{x-3}$$
    • $$y=x^2+2+\frac{5}{x-3}$$
    • $$y=x^2-2x+2+\frac{3}{x-3}$$

17. When designing a stairway, an architect can use the riser-tread formula $$2h + d = 25$$, where $$h$$ is the riser height in inches, and $$d$$ is the tread depth in inches. For any given stairway, the riser heights are the same and the tread depths are the same for all steps in that stairway. The number of steps in a stairway is the number of its risers. For example, there are 5 steps in the stairway in the figure. The total rise of a stairway is the sum of the riser heights. An architect wants to use the riser-tread formula to design a stairway with a total rise of 9 feet, a riser height between 7 and 8 inches, and an odd number of steps. With the architect’s constraints, which of the following must be the tread depth, in inches, of the stairway? (1 foot = 12 inches)

    • 10.6
    • 8.2
    • 9.7
    • 14.5

18. The glass pictured above has volume given by the formula $$V = \frac{7\pi k^3}{48}$$ where $$k$$ is the height of the glass and also the diameter of the top. Water pours into the glass slowly and at a constant rate. Which of the following graphs best illustrates the height of the water level in the glass as it fills?

19. Given the function $$f(x) = 75(0.3)^x$$The function $$f(x)$$ is a decreasing exponential function. By what percent is $$f(n)$$ less than $$f(n-1)$$?

    • $$30\%$$
    • $$75\%$$
    • $$70\%$$
    • $$63\%$$

20. Given $$f(x) = 2x^2 - 3$$ and $$g(x)$$ is linear with $$g(f(2)) = 7$$ and $$g(f(3)) = 13$$, what is $$g(x)$$?

    • $$g(x) = -3x + 5$$
    • $$g(x) = -3x^2 + 5$$
    • $$g(x) = 3x + 2$$
    • $$g(x) = \frac{3}{5}x + 4$$

21. In △PQR, points A,B, and C are the midpoints of its sides. In △ABC, points M,N, and O are the midpoints of its sides. The shaded regions are △PAB, △AQC, △BCR, and △MNO. What portion of the area of △PQR is shaded?

    • $$\frac{13}{16}$$
    • $$\frac{3}{16}$$
    • $$\frac{3}{4}$$
    • $$\frac{10}{16}$$

22. In the standard (x,y) coordinate plane, only one parabola of the form $$y=(x-h)^2+k$$ has x-intercepts of 2 and 8. Which of the following equations represents the axis of symmetry of this parabola?

    • $$y=x-5$$
    • $$x=5$$
    • $$3x+15y=0$$
    • $$x=-2$$

23. In the figure, WXYZ is a rectangle and MNOP is a square inside it. Segment ZY is the diameter of a semicircle, and R is the midpoint of ZY. Point Q is the midpoint of WX and also the midpoint of MN. The given lengths are in meters: WX = 16, XY = 10, OP = 4.The figure is placed on the standard (x,y) coordinate plane so that R is at the origin, WX is parallel to the x-axis, and 1 meter equals 1 coordinate unit. Which of the following values could be the y-coordinate of point M?

    • $$-2$$
    • $$4$$
    • $$10$$
    • $$2$$

24. If $$\log_b x=2$$ and $$\log_b z=5$$, what is $$\log_b\left(\sqrt{x^4z^2}\right)$$?

    • $$4$$
    • $$8$$
    • $$9$$
    • $$12$$

25. Over 10 basketball games, a team averaged 9 points per game. If the team averaged 7 points per game for the first 6 games, and scored the same number of points in each of the last 4 games, how many points did the team score during each of the last 4 games?

    • $$9$$
    • $$7$$
    • $$12$$
    • $$10$$

26. Given the equation $$|x^2-20|-5=0$$, which of the following is a solution but not a rational number?

    • $$2\sqrt{5}$$
    • $$\sqrt{15}$$
    • $$\sqrt{20}$$
    • $$5$$

27. In the fall semester of his science class, Daniel's test scores were 92, 88, 76, 95, 84, and 90. What was his average test score in the fall semester?

    • $$87.5$$
    • $$88$$
    • $$90$$
    • $$85.5$$

28. If $$0^\circ\le x\le90^\circ$$ and $$4\cos^2x=1$$, find $$x$$.

    • $$30^\circ$$
    • $$45^\circ$$
    • $$60^\circ$$
    • $$120^\circ$$

29. In a class of 34 students, the mathematics test grades are 15, 13, 8, 9, 11, 5, 19, 3, 7, 15, 11, 7, 3, 8, 4, 9, 7, 17, 9, 15, 11, 9, 8, 5, 9, 10, 12, 9, 11, 13, 11, 9, 8, 14. Determine the mode of this statistical series.

    • $$9$$
    • $$11$$
    • $$8$$
    • $$15$$

30. The function is defined by $$f(x)=(4x-1)(x+2)-4\left(x+\frac{\sqrt2}{2}\right)\left(x-\frac{\sqrt2}{2}\right)$$. Find all real numbers x such that the image of x by the function f is -1/2.

    • $$\frac{1}{7}$$
    • $$-\frac{1}{14}$$
    • $$-\frac{1}{2}$$
    • $$-\frac{1}{7}$$

31. In the figure, two lines intersect at point O. The angle POM measures $$115^\circ$$. Find the measure of z.

    • $$22.5^\circ$$
    • $$25^\circ$$
    • $$35^\circ$$
    • $$30^\circ$$

32. In right triangle PQR, points X,Y lie on PQ. The areas of triangles RPY and RYQ are 90 and 45, respectively. If $$PY=XQ=10$$, find the length of $$XY$$.

    • 7.5
    • 5
    • 6
    • 7

33. In circle C, the radii CP and CQ lie on a straight line (a diameter), and the radii CR and CS lie on another straight line (a diameter). The shaded regions are the two central sectors ∠PCR and ∠QCS. If ∠PCS=120^\circ, what fraction of circle C is shaded?

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

34. The operation ⊙ is defined on real numbers by a ⊙ b = 3a + 2b + ab. Find the value of (4 ⊙ 3) ⊙ 2.

    • $$154$$
    • $$96$$
    • $$86$$
    • $$126$$

35. The number of elements in the union of sets A and B is 160, and the number of elements in the intersection of sets A and B is 20. If the number of elements in set A is 50, what is the number of elements in set B?

    • 130
    • 140
    • 150
    • 155

36. The midnight temperatures over a week are shown in the table below. What is the range of these data?

    • 13°F
    • 35°F
    • 6°F
    • 20°F

37. $$3^x+3^x+3^x=?$$

    • $$3^{x+1}$$
    • $$3^{2x}$$
    • $$3^{3x}$$
    • $$9^x$$

38. Two trains starting from cities 300 miles apart head toward each other at rates of 70 mph and 50 mph, respectively. How long does it take the trains to cross paths?

    • $$2.5 \text{ hours}$$
    • $$3\text{ hours}$$
    • $$1.5 \text{ hours}$$
    • $$3.5 \text{ hours}$$

39. Which of the following complex numbers is equal to $$\frac{1+i}{3-2i}?$$

    • $$\frac{1}{5}+i$$
    • $$\frac{3}{13}+\frac{3}{13}i$$
    • $$\frac{1}{13}+\frac{5}{13}i$$
    • $$-2-i$$

40. In the standard (x,y) coordinate plane, the graph of$$h(x)=\frac{5x-2}{x+7}$$ has a horizontal asymptote at:

    • $$y=7$$
    • $$y=-7$$
    • $$y=\frac{2}{5}$$
    • $$y=5$$

41. In the standard (x,y) coordinate plane, the graph of$$y=x^2+mx+12$$ crosses the x-axis at only one point. What does m equal?

    • $$2$$
    • $$2\sqrt{3}$$
    • $$4$$
    • $$4\sqrt{3}$$

42. What is the units digit of $$3^{402}$$?

    • 1
    • 3
    • 7
    • 9

43. Which of the following expressions is equivalent to $$\frac{x^2-3x+7}{x-4}$$?

    • $$x+1+\frac{11}{x-4}$$
    • $$x-7+\frac{28}{x-4}$$
    • $$x+1+\frac{3}{x-4}$$
    • $$x+7+\frac{35}{x-4}$$

44. Which equation represents the circle in the xy-plane with center (-4,7) and a radius whose endpoint is (2,-1)?

    • $$(x-4)^2+(y-7)^2=10$$
    • $$(x+4)^2+(y-7)^2=100$$
    • $$(x+4)^2+(y+7)^2=10$$
    • $$(x-4)^2+(y-7)^2=100$$

45. A student wants to know the surface area of a right circular cylinder. The cylinder has radius 5 units and height 6 units. Find the surface area of the cylinder in square units.

    • $$110\pi$$
    • $$150\pi$$
    • $$60\pi$$
    • $$120\pi$$