Math

7. In the figure shown, ∠ABC ≅ ∠ DFE, ∠ BAC ≅ ∠ FDE, D and F are on AB, AD ≅ FB, and distances in centimeters are as shown. What is the length of AD, in centimeters? A) 5 B) 4 C) 3 D) 2 E) 1

D

3. What is the value of x when 2x + 3 = 3x - 4? A) -7 B) -1/5 C) 1 D) 1/5 E) 7

E Explanation: You can solve this problem by first subtracting 2x from each side of the equation to get 3 = x - 4. Then add 4 to each side, so x = 7

12. If n = 8 and 16(2^m) = 4^(n-8), then m = ? F) -4 G) -2 H) 0 I) 1 J) 8

F

2. What is the probability that a number selected at random from the set {2, 3, 7, 12, 15, 22, 72, 108} will be divisible by both 2 and 3? F) 1/4 G) 3/8 H) 3/5 I) 5/8 J) 7/8

G

2. When x = 3 and y = 5, by how much does the value of 3x² - 2y exceed the value of 2x² - 3y? F) 4 G) 14 H) 16 I) 20 J) 50

G Explanation: 3x² - 2y = 3(3)² - 2(5) = 27 - 10 = 17 2x² - 3y = 2(3)² - 3(5) = 18 - 15 = 3 17 - 3 = 14

10. What is the slope of any line parallel to the line 9x + 4y = 7? F) -9 G) -9/4 H) 9/7 I) 7 J) 9

G Explanation: Since 4y = -9x + 7, y = -9/4x + 7/4.

2. If xy = 144, x + y = 30, and x > y, what is the value of x - y? F) 4 G) 6 H) 18 I) 22 J) 24

H

4. The measure of ∠ABC in the figure is x°. Which of the following is an expression for β° ? F) x° G) 2x° H) (90 + x)° I) (180 - x)° J) (180 - x/2)°

I Explanation: The angles of the rectangular pieces of lumber measure 90°, so the sum of the measure of the angles at β is 360°. β + 90 + x + 90 = 360, or β = 180 - x.

4. Ding's Diner advertised this daily lunch special: "Choose 1 item from each column—only $4.95!" Thus, each daily lunch special consists of a salad, a soup, a sandwich, and a drink. How many different lunch specials are possible? F) 4 G) 14 H) 30 I) 120 J) 180

I Explanation: 3(2)(5)(4)

10. How many irrational numbers are there between 1 and 6? F) 1 G) 3 H) 4 I) 10 J) infinitely many

J

4. What is the greatest common factor of 42, 126, and 210? F) 2 G) 6 H) 14 I) 21 J) 42

J

12. √-(-9)² *Note i = √-1 F) 9i G) 9 + i H) 9 - i I) 9 J) -9

F

8. The length, in inches, of a box is 3 inches less than twice its width, in inches. Which of the following gives the length, l inches, in terms of width, w inches, of the box? F) l = ½w + 3 G) l = w + 3 H) l = w - 3 I) l = 2w + 3 J) l = 2w -3

J

2. Which of the following is an equation of the circle with its center at (0,0) that passes through (3,4) in the standard (x,y) coordinate plane? F) x - y = 1 G) x - y = 25 H) x² + y = 25 I) x² + y² = 5 J) x² + y² = 25

J Explanation: The radius of the circle is the distance between (0,0) and (3,4), which is √((3-0)2+ (4-0)2) = 5. An equation of a circle where (h,k) is the center and r is the radius is (x - h)2 + (y - k)2 = r2. So (x - 0)2 + (y - 0)2 = 52 or x2 + y2 = 25.

11. An industrial cleaner is manufactured using only the 3 secret ingredients A, B, and C, which are mixed in the ratio of 2:3:5, respectively, by weight. How many pounds of secret ingredient B are in a 42 pound (net weight) bucket of this cleaner? A) 4.2 B) 12.6 C) 14.0 D) 18.0 E) 21.0

B Explanation: If you let 3x be amount of secret ingredient B, you can set up the equation 2x + 3x + 5x = 42. Since 10x = 42, x = 4.2, and B = 3x = 12.6.

1. What is the degree measure of the acute angle formed by the hands of a 12 hour clock that reads exactly 1 o'clock? A) 15° B) 30° C) 45° D) 60° E) 72°

B Explanation: One complete rotation of a clock hand is 360° and there are 12 hourly markings on a clock. When the hands read exactly 1 o'clock, the degree measure of the angle formed by the clock hands is 1/12 of a complete rotation. So (1/12)(360) = 30°

1. In the figure shown, A, B, C, and D are collinear, FC is parallel to ED, BE is perpendicular to ED, and the measures of ∠FAB and ∠EBA are as marked. What is the measure of ∠FCB ? A) 33° B) 57° C) 63° D) 84° E) Cannot be determined from the given information

B Explanation: Since FC and ED are two parallel line segments cut by transversal BE, ∠E and ∠BGC are corresponding angles. So, the measure of ∠BGC is 90°. Since ∠ABG ∠GBC are supplementary angles, the measure of ∠GBC = 180° - 147° = 33°. Looking at ΔBGC, the sum of the measures of angles ∠GCB, ∠BGC, and ∠GBC is 180°. So, the measure of ∠GCB + 90° + 33° = 180°, or 180° - 90° - 33° = 57°.

5. Sales for a business were 3 million dollars more than the second year than the first, and sales for the third year were double the sales for the second year. If sales for the third year were 38 million dollars, what were sales, in millions of dollars, for the first year? A) 16 B) 17.5 C) 20.5 D) 22 E) 35

A Explanation: If x = sales for the first year, then x + 3 = sales for the second year. Since sales for the third year were double the sales for the second year, sales for the third year = 2(x + 3). Sales for the third year were 38, so 2(x + 3) = 38. To solve this equation, you could first divide each side by 2 to get x + 3 = 19. Then, by subtracting 3 from both sides, x = 16.

5. In the standard (x,y) coordinate plane, what are the coordinates of the midpoint of a line segment whose endpoints are (-3,0) and (7,4)? A) (2,2) B) (2,4) C) (5,2) D) (5,4) E) (5,5)

A Explanation: To find the midpoint, you need to take the average of each of the coordinates ( (-3+7)/2) , (0+4)/2) ) = (2,2)

7. Which of the following statements must be true whenever n, a, b, and c are positive integers such that n < a, c > a, and b > c? A) a < n B) b - n > a - n C) b < n D) n + b = a + c E) 2n > a + b

B Explanation: Since b > a, subtracting n from each side, b - n > a - n, will not change the relationship between b and a.

5. The volume, V, of the right circular cone with radius r and height h, shown, can be found using the formula V = 1/3 πr²h. A cone-shaped paper cup has a volume of 142 cubic cm and a height of 8.5 cm. What is the radius, to the nearest cm, of the paper cup? A) 2 B) 4 C) 8 D) 12 E) 16

B Explanation: Solving for r = 3V/πh = r² So r = √3V/πh = √(3)(142)/8.5π ≈ 4

3. Which of the following is the sine of angle A in the right triangle? A) 5/13 B) 5/12 C) 12/13 D) 12/5 E) 13/5

C

9. What is x, the second term in the geometric series 1/4 + x + 1/36 + 1/108 + ...? A) 1/3 B) 1/9 C) 1/12 D) 1/16 E) 1/18

C

9. What is the difference between 1.8 and 1.08 (0.08 recurring)? A) 0.71 (1 recurring) B) 0.71 recurring C) 0.719 (.019 recurring) D) 0.72 (2 recurring) E) 0.72 recurring

C Explanation: 1.8 - 1.08080808080808 ≈ 0.7191919

11. A DVD player with a list price of $100 is marked down 30%. If John gets an employee discount of 20% off the sale price, how much does John pay for the DVD player? A) $86.00 B) $77.60 C) $56.00 D) $50.00 E) $44.00

C Explanation: 30% - 100(0.70) = 70 The discount of 20% means that the price will be 80% of the marked-down price. So, 70(0.80) = 56

3. A circle has a circumference of 16π feet. What is the radius of the circle in feet? A) √8 B) 4 C) 8 D) 16 E) 32

C Explanation: The formula for the circumference of a circle with radius r is 2πr. So 2πr= 16 , or r = 8.

1. A car averages 27 miles per gallon. If gas costs $4.04 per gallon, which of the following is closest to how much the gas would cost for this car to travel 2727 typical miles? A) $44.44 B) $109.08 C) $118.80 D) $408.04 E) $444.40

D Explanation: 2727/27 = 101 101(4.04) = 408.04

7. Abandoned mines frequently fill with water. Before an abandoned mine can be reopened, the water must be pumped out. The size of a pump required depends on the depth of the mine. If pumping out a mine that is D feet deep requires a pump a minimum of D²/25 + 4D - 250 gallons per minute, pumping out a mine that is 150 ft deep would require a pump that pumps a minimum of how many gallons per minute? A) 362 B) 500 C) 800 D) 1250 E) 1750

D Explanation: If you substitute D with 150 in the expression, you get (150²)/25 + 4(150) - 250 = 22500/25 + 600 - 250 = 1250

Use the following info to answer questions 3 - 5. Taher has decided to create a triangular flower bed border. He plans to use 3 pieces of rectangular lumber with lengths 4, 5, and 6ft, as shown in the figure. Points A, B, and C are located at the corners of the flower bed. 3. Taher plans to cut the 3 pieces of lumber for the flower bed border from a single piece of lumber. Each cut takes 1/8 inch of wood off the length of the piece of lumber. Among the following lengths, in inches, of pieces of lumber, which is the shortest piece that he can use to cut the pieces for the flower bed border? A) 178 B) 179 C) 180 D) 181 E) 182

D Explanation: The number of inches of wood needed if there were no cuts is 4 + 5 + 6 = 15 feet, or 180 inches. However, you need to add 2(<1/8) for 2 cuts that are needed so that you have lumber for each of the 3 sides. Since 180 + 2(1/8) = 180 + 1/4, the minimum piece needed to construct the flower bed border including the 2 cuts would be 181 inches.

9. In quadrilateral PQRS, sides PS and QR are parallel for what value of x? A) 158 B) 132 C) 120 D) 110 E) 70

D Explanation: The question states that PS and QR are parallel. If you treat PQ as a transversal, then ∠P and ∠Q are interior angles on the same side of a transversal, so their measures add up to 180°. Since the measure of ∠P is 70°, the measure of ∠Q is 180° - 70° = 110°.

1. The lead of a screw is the distance that the screw advances in a straight line when the screw is turned 1 complete turn. If a screw is 2½ inches long and has a lead of 1/8 inch, how many complete turns would get all the way into a piece of wood? A) 5 B) 10 C) 15 D) 20 E) 25

D Explanation: With every complete turn ⅛ inch of the screw goes into the wood. So after 8 complete turns, 1 inch of the screw would be in the wood. So, x(⅛) = 2½ . Multiplying by 8, x = 8(2½) = 8(5/2) = 20.

11. A typical high school student consumes 67.5 pounds of sugar per year. As part of a new nutrition plan, each member of a track team plans to lower the sugar he or she consumes by at least 20% for the coming year. Assuming each track member had consumed sugar at the level of a typical high school student and will adhere to this plan for this coming year, what is the maximum number of pounds of sugar to be consumed by each track member in the coming year? A) 14 B) 44 C) 48 D) 54 E) 66

D Explanation: For each member of the track team to consume 20% less sugar, the track member will consume 100% - 20% = 80% of the level of a typical high school student. 80% of 67.5 = 0.80(67.5) = 54.

6. Points A, B, C, and D are on a line such that B is between A and C, and C is between B and D. The distance from A and B is 6 units. The distance from B to C is twice the distance from A to B, and the distance from C to D is twice the distance from B to C. What is the distance, in units, from the midpoint of BC to the midpoint of CD? F) 18 G) 14 H) 12 I) 9 J) 6

F Explanation: BC = 2AB = 2(6) = 12 and CD = 2BC = 2(12) = 24. The distance between the midpoints of BC and CD is ½BC + ½ CD = ½(12) + ½(24) = 18

6. In the figure ray EF was constructed starting from rays ED and EG. By using a compass D and G were marked equidistant from E on rays ED and EG. The compass was then used to locate a point F, distinct from E, so that F is equidistant from D and G. For all constructions defined by the above steps, the measures of ∠DEF and ∠GEF ______. F) are equal G) are NOT equal H) sum to 30° I) sum to 45° J) sum to 60°

F Explanation: If you draw line segments DF and FG, you can show ΔDEF ≅ ΔGEF by SSS (side-side-side congruence). So, ∠DEF ≅ ∠GEF because corresponding parts of congruent triangles are congruent.

12. In the standard (x,y) coordinate plane below, 3 of the vertices of a rectangle are shown. Which of the following is the 4th vertex of the rectangle? F) (3, -7) G) (4, -8) H) (5, -1) I) (8, -3) J) (9, -3)

F Explanation: When moving from (2,1) to (-1,-1), you can go 3 units left and 2 units down. Since you want to form a rectangle, you will need to move in the same pattern from (6,-5) to the 4th vertex. Subtract 3 from the x-value, and subtract 2 from they-value, and you will find the point needed: (6 - 3, -5 - 2) = (3,-7).

10. Which of the following equations represents the linear relationship between time, t, and velocity, v, shown in the table? F) v = 32t G) v = 32t + 120 H) v = 120t I) v = 120t + 32 J) v = 120t + 120

G Explanation: A linear relationship means that the associated graph is a line. So (t,v). Since (0,120), (1,152), and (2,184) are points on a line, the slope is (152-120)/(1-0) = 32. Therefore, v = 32t + b, where b is the y-int. Since (0,120) is a point on the line, 120 = 32(0) + b or b = 120. Thus, an equation for the line is v = 32t + 120

4. A rectangle with a perimeter of 30 cm is twice as long as it is wide. What is the area of the rectangle in square cm? F) 15 G) 50 H) 200 I) 3^√15 J) 6^√15

G Explanation: If w = width, then 2w = length. So, the perimeter is 2(w + 2w) = 30, and w = 5. Since the width is 5, the length is 2(5) = 10. Then the area is 5(10) = 50.

6. A boat departs Port Isabelle, Texas, travelling to an oil rig. The oil rig is located 9 miles east and 12 miles north of the boat's departure point. About how many miles is the oil rig from the departure point? F) 3 G) √63 H) 15 I) 21 J) 225

H Explanation: Using pythagorean theorem 9² + 12² = c²

8. Which of the following is a factor of the polynomial 2x² - 3x - 5? F) x - 1 G) 2x - 3 H) 2x - 5 I) 2x + 5 J) 3x + 5

H Explanation: (x + 1)(2x - 5)

8. The distribution of Jamal's high school grades by percentage of course credits is given in the pie chart. What is Jamal's grade point average if each A is worth 4 points; each B, 3 points; and each C, 2 points? F) 3.0 G) 3.4 H) 3.6 I) 3.7 J) Cannot be determined from the given information

H Explanation: 4(0.7) + 3(0.2) + 2(0.1) = 3.6.