ACT--Sample Mathematics Questions: Set 5

DIRECTIONS: Solve each problem, choose the correct answer, and then fill in the corresponding oval on your answer document.

Do not linger over problems that take too much time. Solve as many as you can; then return to the others in the time you have left for this test.

You are permitted to use a calculator on this test. You may use your calculator for any problems you choose, but some of the problems may best be done without using a calculator.

1. Ms. Hernandez began her math class by saying:
   
   I'm thinking of 5 numbers such that their mean is equal to their median. If 4 of the numbers are 14, 8, 16, and 14, what is the 5th number?
   What is the 5th number Ms. Hernandez is thinking of?
   1. A. 13
   2. B. 14
   3. C. 15
   4. D. 16
   5. E. 18
2. The graph of a certain hyperbola, y = h(x), is shown in the standard (x,y) coordinate plane below.
   
   Among the following graphs, which best represents y = –h(x) ?
   1. F. 
   2. G. 
   3. H. 
   4. J. 
   5. K. 
3. In the figure below, H  F; E, G, and I are collinear; and G is the midpoint of FH.
   
   To prove that HI  FE given the conditions stated above, which of the following is a logical order for the 5 steps in the table below?

   Statement	Reason
   1. HG  FG	The midpoint of a line segment divides the segment into 2 congruent segments
   2. EGF  IGH	Vertical angles are congruent
   3. GHI  GFE	Angle-side-angle congruence theorem
   4. EGF and IGH are vertical angles	Definition of vertical angles
   5. HI  FE	Corresponding parts of congruent triangles are congruent

   1. A. 1, 2, 3, 4, 5
   2. B. 1, 2, 3, 5, 4
   3. C. 1, 2, 4, 3, 5
   4. D. 1, 4, 2, 3, 5
   5. E. 1, 5, 4, 2, 3
4. Each of the variables t, w, x, y, and z represents a differentpositive real number. Given the equations below, which of the 4 variables w, x, y, and z necessarily has the greatest value?
   1.23w = t
   1.01x = t
   0.99y = t
   0.23z = t
   1. F. w
   2. G. x
   3. H. y
   4. J. z
   5. K. Cannot be determined from the given information
5. Which of the following is equivalent to  ?
   1. A. 
   2. B. 
   3. C. 
   4. D.
   5. E. 
6. In the 2 × 2 matrix below, b1 and b2 are the costs per pound of bok choy (Chinese greens) at Market 1 and Market 2, respectively; r1 and r2 are the costs per pound of rice flour at these 2 markets, respectively. In the following matrix product, what does q represent?
   
   1. F. The cost of r1 pounds of rice flour at $0.50 per pound
   2. G. The cost of a half-pound of rice flour at Market 1
   3. H. The total cost of a half-pound of bok choy and a half-pound of rice flour at Market 1
   4. J. The total cost of a half-pound of bok choy and a half-pound of rice flour at Market 2
   5. K. The total cost of a half-pound of rice flour at Market 1 and a half-pound of rice flour at Market 2
7. The 2 diagrams below show a circle of radius 1 inch with shaded sectors of angle x°, for 2 different values of x.
   
   One of the following is the graph in the standard (x,y) coordinate plane of the area, y, of a shaded sector with angle x°, for all values of x between 0 and 360. Which is that graph?
   1. A. 
   2. B. 
   3. C. 
   4. D.
   5. E. 
8. If h(x) = x³ + x and g(x) = 2x + 3, then g(h(2)) = ?
   1. F.   7
   2. G. 10
   3. H. 17
   4. J. 19
   5. K. 23
9. In the figure below, points A and B are on opposite banks of a small stream. Point C is on the same bank of the stream as point Band approximately 18 meters from B. The measure of CBA is 45°, and the measure of BCA is 60°.
   
   Which of the following expressions gives the approximate distance, in meters, between point A and point B ?
   (Note: For PQR, where p, q, and r are the lengths of the sides opposite P, Q, and R, respectively, .)
   1. A. 
   2. B. 
   3. C. 
   4. D.
   5. E. 
10. Each side of the smaller square in the figure below is x inches long, and each side of the larger square is c inches longer than a side of the smaller square. The area of the larger square is how many square inches greater than the area of the smaller square?
    
    1. F. c2
    2. G. xc
    3. H. 4c
    4. J. (x + c)2
    5. K. 2xc + c2
11. A cube with edges  inch long is shown below. What is the length, in inches, of a diagonal that runs from one corner of the cube to the opposite corner?
    
    1. A. 
    2. B. 
    3. C. 
    4. D.
    5. E. 
12. Which of the following is equivalent to sin csc(–) wherever sin csc(–) is defined?
    1. F. –1
    2. G.   1
    3. H. –tan 
    4. J.   tan 
    5. K. –sin2