ACT Past Papers with Answers

Paper 75 · ACT January 2006 · Form 61C · Math Test

What is the value of the expression (x – y)2 when x = 5 and y = –1 ?

Paper 75 · ACT January 2006 · Form 61C · Math Test

On the first day of school, Mr. Vilani gave his third- grade students 5 new words to spell. On each day of school after that, he gave the students 3 new words to spell. In the first 20 days of school, how many new words had he given the students to spell?

Paper 75 · ACT January 2006 · Form 61C · Math Test

Which of the following is equivalent to (4x2 )3 ?

Paper 75 · ACT January 2006 · Form 61C · Math Test

Which of the following lists all the positive factors of 8 ?

Paper 75 · ACT January 2006 · Form 61C · Math Test

Which of the following is an equivalent simplified expression for 2(4x + 7) – 3(2x – 4) ?

Paper 75 · ACT January 2006 · Form 61C · Math Test

To determine a student’s overall test score for the semester, Ms. Lopez throws out the lowest test score and takes the average of the remaining test scores. Victor earned the following test scores in Ms. Lopez’s class this semester: 62, 78, 83, 84, and 93. What over- all test score did Victor earn in Ms. Lopez’s class this semester?

Paper 75 · ACT January 2006 · Form 61C · Math Test

Uptown Cable, a cable TV provider, charges each cus- tomer $120 for installation, plus $25 per month for cable programming. Uptown’s competitor, Downtown Cable, charges each customer $60 for installation, plus $35 per month for cable programming. A customer who signs up with Uptown will pay the same total amount for cable TV as a customer who signs up with Downtown if each pays for installation and cable pro- gramming for how many months? GO ON TO THE NEXT PAGE.

Paper 75 · ACT January 2006 · Form 61C · Math Test

In the 8-sided figure below, adjacent sides meet at right angles and the lengths given are in meters. What is the perimeter of the figure, in meters?

Paper 75 · ACT January 2006 · Form 61C · Math Test

The sum of the real numbers x and y is 11. Their dif- ference is 5. What is the value of xy ?

Paper 75 · ACT January 2006 · Form 61C · Math Test

For all x, (3x + 7)2 = ? + 14 + 49 + 21x + 49 + 42x + 49

Paper 75 · ACT January 2006 · Form 61C · Math Test

What is the slope of the line through (–5,2) and (6,7) in the standard (x,y) coordinate plane?

Paper 75 · ACT January 2006 · Form 61C · Math Test

When k + k = 1, what is the value of k ?

Paper 75 · ACT January 2006 · Form 61C · Math Test

What is the length, in feet, of the hypotenuse of a right triangle with legs that are 6 feet long and 7 feet long, respectively?

Paper 75 · ACT January 2006 · Form 61C · Math Test

Hexagon ABCDEF shown below was drawn on a grid with unit squares. Each vertex is at the intersection of 2 grid lines. What is the area of the hexagon, in square units?

Paper 75 · ACT January 2006 · Form 61C · Math Test

In the figure below, AD ___ is perpendicular to BD ___ , AC ___ is perpendicular to BC ___ , and AD ___ ≅ BC ___ . Which of the fol- lowing congruences is NOT necessarily true? ___ ≅ BD ___ ___ ≅ AE ___ ___ ≅ BE ___

Paper 75 · ACT January 2006 · Form 61C · Math Test

Leticia went into Discount Music to price CDs. All CDs were discounted 23% off the marked price. Leticia wanted to program her calculator so she could input the marked price and the discounted price would be the output. Which of the following is an expression for the discounted price on a marked price of p dollars?

Paper 75 · ACT January 2006 · Form 61C · Math Test

In the figure below, A, D, B, and G are collinear. If ∠CAD measures 76°, ∠BCD measures 47°, and ∠CBG measures 140°, what is the degree measure of ∠ACD ? C D C E A B F E A B C D 7 _ _ 2 12 __ 7 1 _ _ 7 1 _ _ 4 1 _ _ 3 5 __ 11 5 __ 11 20 8 12 GO ON TO THE NEXT PAGE. 2 2 ACT-61C-PRACTICE 27

Paper 75 · ACT January 2006 · Form 61C · Math Test

Ms. Lewis plans to drive 900 miles to her vacation destination, driving an average of 50 miles per hour. How many miles per hour faster must she average, while driving, to reduce her total driving time by 3 hours?

Paper 75 · ACT January 2006 · Form 61C · Math Test

For all positive integers x, what is the greatest common factor of the 2 numbers 216x and 180x ?

Paper 75 · ACT January 2006 · Form 61C · Math Test

The table below shows the price of different quantities of standard-sized lemons at Joe’s Fruit Stand. What is the least amount of money needed to purchase exactly 20 standard-sized lemons if the bags must be sold intact and there is no tax charged for lemons? Number of lemons: 1 bag of 6 bag of 12 Total price: $0.30 $1.20 $2.10

Paper 75 · ACT January 2006 · Form 61C · Math Test

The diameter, d centimeters, of the metal poles Goodpole Manufacturing produces must satisfy the inequality ⏐d – 3⏐ ≤ 0.001. What is the maximum diameter, in centimeters, such a metal pole may have?

Paper 75 · ACT January 2006 · Form 61C · Math Test

Which of the following is a factored form of the expression 5x2 – 13x – 6 ?

Paper 75 · ACT January 2006 · Form 61C · Math Test

A bag contains 6 red marbles, 5 yellow marbles, and 7 green marbles. How many additional red marbles must be added to the 18 marbles already in the bag so that the probability of randomly drawing a red marble is ?

Paper 75 · ACT January 2006 · Form 61C · Math Test

Which of the following trigonometric equations is valid for the side measurement x inches, diagonal mea- surement y inches, and angle measurement w° in the rectangle shown below?

Paper 75 · ACT January 2006 · Form 61C · Math Test

The slope of the line with equation y = ax + b is greater than the slope of the line with equation y = cx + b. Which of the following statements must be true about the relationship between a and c ?

Paper 75 · ACT January 2006 · Form 61C · Math Test

Minh cuts a board in the shape of a regular hexagon and pounds in a nail at an equal distance from each vertex, as shown in the figure below. How many rubber bands will she need in order to stretch a differ- ent rubber band across every possible pair of nails?

Paper 75 · ACT January 2006 · Form 61C · Math Test

There are 280 runners registered for a race, and the runners are divided into 4 age categories, as shown in the table below. under over Age category: 16 16–25 26–35 35 Number of runners: 40 76 112 52 The prize committee has 60 prizes to award and wants the prizes to be awarded in proportion to the number of runners registered in each category. How many prizes should be designated for the 26–35 age category? x _ _ y x _ _ y x _ _ y x _ _ y x _ _ y w° y x 3 _ _ 5 GO ON TO THE NEXT PAGE. 2 2 ACT-61C-PRACTICE 28 The youth center has installed a swimming pool on level ground. The pool is a right circular cylinder with a diameter of 24 feet and a height of 6 feet. A diagram of the pool and its entry ladder is shown below.

Paper 75 · ACT January 2006 · Form 61C · Math Test

To the nearest cubic foot, what is the volume of water that will be in the pool when it is filled with water to a depth of 5 feet? (Note: The volume of a cylinder is given by πr2 h, where r is the radius and h is the height.)

Paper 75 · ACT January 2006 · Form 61C · Math Test

A plastic cover is made for the pool. The cover will rest on the top of the pool and will include a wedge-shaped flap that forms a 45° angle at the center of the cover, as shown in the figure below. A zipper will go along 1 side of the wedge-shaped flap and around the arc. Which of the following is closest to the length, in feet, of the zipper?

Paper 75 · ACT January 2006 · Form 61C · Math Test

Two hoses are used to fill the pool. Twice as many gal- lons of water per minute flow through one of the hoses as through the other. Both hoses had been on for 12 hours and had filled the pool to the 4-foot mark when the hose with the faster flow stopped working. The hose with the slower flow then finished filling the pool to the 5-foot mark. Which of the following graphs shows the relationship between the time spent filling the pool and the height of the water in the pool?

Paper 75 · ACT January 2006 · Form 61C · Math Test

The directions for assembling the pool state that the ladder should be placed at an angle of 75° relative to level ground. Which of the following expressions involving tangent gives the distance, in feet, that the bottom of the ladder should be placed away from the bottom edge of the pool in order to comply with the directions? 1 _ _____ _ __ 6 tan 75° tan 75° _ ____ __ 6 6 _ ____ __ tan 75° height of water (in feet) time (in hours) height of water (in feet) time (in hours) time (in hours) height of water (in feet) time (in hours) height of water (in feet) time (in hours) height of water (in feet) 45° 75° 6 feet 24 feet ladder GO ON TO THE NEXT PAGE. 2 2 ACT-61C-PRACTICE Use the following information to answer questions 29–32. 29

Paper 75 · ACT January 2006 · Form 61C · Math Test

For a population that grows at a constant rate of r% per year, the formula P(t) = po冢1 + 冣 t models the popu- lation t years after an initial population of po people is counted. The population of the city of San Jose was 782,000 in 1990. Assume the population grows at a constant rate of 5% per year. According to this formula, which of the following is an expression for the population of San Jose in the year 2000 ?

Paper 75 · ACT January 2006 · Form 61C · Math Test

Tom’s long-distance service charges $0.10 per minute from 7:00 P.M. to 7:00 A.M. on weekdays, all day on Saturdays, and all day on holidays; $0.05 per minute all day on Sundays; and $0.25 per minute at all other times. The table below gives his long-distance calls for 1 week, including the date and day of each call, the time it was placed, and the number of minutes it lasted. Number of Date and day Time minutes 11/22 Tuesday 5:00 P.M. 8 11/23 Wednesday 10:30 A.M. 10 11/24 Thursday Thanksgiving holiday 11:30 A.M. 15 11/26 Saturday 9:30 A.M. 17 11/27 Sunday 12:15 P.M. 22 What did Tom’s long-distance service charge him for the calls in the table?

Paper 75 · ACT January 2006 · Form 61C · Math Test

The parallel sides of the isosceles trapezoid shown below are 10 feet long and 16 feet long, respectively. What is the distance, in feet, between these 2 sides?

Paper 75 · ACT January 2006 · Form 61C · Math Test

The inequality 3(x + 2) > 4(x – 3) is equivalent to which of the following inequalities?

Paper 75 · ACT January 2006 · Form 61C · Math Test

In the standard (x,y) coordinate plane, the midpoint of AB ___ is (4,–3) and A is located at (1,–5). If (x,y) are the coordinates of B, what is the value of x + y ?

Paper 75 · ACT January 2006 · Form 61C · Math Test

For all x in the domain of the function , this func- tion is equivalent to:

Paper 75 · ACT January 2006 · Form 61C · Math Test

In the figure below, line l is parallel to line m. Trans- versals t and u intersect at point A on l and intersect m at points C and B, respectively. Point X is on m, the measure of ∠ACX is 130°, and the measure of ∠BAC is 80°. How many of the angles formed by rays of l, m, t, and u have measure 50° ? A B C t u l m 80° 130° X 1 __ x3 1 _____ x2 – x 1 _____ x2 – 1 1 _ _ x 1 __ x3 1 __ x3 1 __ x2 x + 1 _____ x3 – x 16 ft 10 ft 5 ft 5 ft r _ ___ 100 GO ON TO THE NEXT PAGE. 2 2 ACT-61C-PRACTICE 30

Paper 75 · ACT January 2006 · Form 61C · Math Test

Tickets for the Senior Talent Show at George Washing- ton Carver High School are $3 for adults and $2 for students. To cover expenses, a total of $600 must be collected from ticket sales for the show. One of the fol- lowing graphs in the standard (x,y) coordinate plane, where x is the number of adult tickets sold and y is the number of student tickets sold, represents all the poss- ible combinations of ticket sales that cover at least $600 in expenses. Which graph is it?

Paper 75 · ACT January 2006 · Form 61C · Math Test

What is the median of the following 7 scores? 42, 67, 33, 79, 33, 89, 21

Paper 75 · ACT January 2006 · Form 61C · Math Test

What are the real solutions to the equation ⎪x⎪ 2 + 2⎪x⎪ – 3 = 0 ?

Paper 75 · ACT January 2006 · Form 61C · Math Test

The point (2,5) is shown in the standard (x,y) coordi- nate plane below. Which of the following is another point on the line through the point (2,5) with a slope of – ?

Paper 75 · ACT January 2006 · Form 61C · Math Test

For the triangles in the figure below, which of the fol- lowing ratios of side lengths is equivalent to the ratio of the perimeter of 䉭ABC to the perimeter of 䉭DAB ?

Paper 75 · ACT January 2006 · Form 61C · Math Test

In the figure below, 2 nonadjacent sides of a regular pentagon (5 congruent sides and 5 congruent interior angles) are extended until they meet at point X. What is the measure of ∠X ?

Paper 75 · ACT January 2006 · Form 61C · Math Test

The edges of a cube are each 3 inches long. What is the surface area, in square inches, of this cube? X 60° 30° D C B A E y x B O A E D C (2,5) 2 _ _ 3 x y 0 200 400 600 200 400 600 0 adult tickets sold student tickets sold x y 0 200 400 600 200 400 600 0 adult tickets sold student tickets sold x y 0 200 400 600 200 400 600 0 adult tickets sold student tickets sold x y 0 200 400 600 200 400 600 0 adult tickets sold student tickets sold x y 0 200 400 600 200 400 600 0 adult tickets sold student tickets sold GO ON TO THE NEXT PAGE. 2 2 ACT-61C-PRACTICE 31

Paper 75 · ACT January 2006 · Form 61C · Math Test

A number is increased by 25% and the resulting number is then decreased by 20%. The final number is what percent of the original number?

Paper 75 · ACT January 2006 · Form 61C · Math Test

Two numbers are reciprocals if their product is equal to 1. If x and y are reciprocals and x > 1, then y must be:

Paper 75 · ACT January 2006 · Form 61C · Math Test

The number line graph below is the graph of which of the following inequalities?

Paper 75 · ACT January 2006 · Form 61C · Math Test

All of the following graphs have equal scales on the axes. One of the graphs shows only points for which the y-coordinate is 1 less than the square of the x-coordinate. Which one?

Paper 75 · ACT January 2006 · Form 61C · Math Test

In teaching a lesson on the concept of thirds, Ms. Chu uses a divide-and-set-aside procedure. She starts with a certain number of colored disks, divides them into 3 equal groups, and sets 1 group aside to illustrate . She repeats the procedure by taking the disks she had NOT set aside, dividing them into 3 equal groups, and setting 1 of these groups aside. If Ms. Chu wants to be able to complete the divide-and-set-aside procedure at least 4 times (without breaking any of the disks into pieces), which of the following is the minimum number of colored disks she can start with?

Paper 75 · ACT January 2006 · Form 61C · Math Test

Which of the following is true for all consecutive inte- gers m and n such that m < n ? – m2 is odd + n2 is even

Paper 75 · ACT January 2006 · Form 61C · Math Test

A function P is defined as follows: for x > 0, P(x) = x5 + x4 – 36x – 36 for x < 0, P(x) = –x5 + x4 + 36x – 36 What is the value of P(–1) ?

Paper 75 · ACT January 2006 · Form 61C · Math Test

For a project in Home Economics class, Kirk is making a tablecloth for a circular table 3 feet in diameter. The finished tablecloth needs to hang down 5 inches over the edge of the table all the way around. To finish the edge of the tablecloth, Kirk will fold under and sew down 1 inch of the material all around the edge. Kirk is going to use a single piece of rectan- gular fabric that is 60 inches wide. What is the shortest length of fabric, in inches, Kirk could use to make the tablecloth without putting any separate pieces of fabric together? 1 _ _ 3 O x y H. O x y G. O x y O x y F. O x y J. x –1 3 GO ON TO THE NEXT PAGE. 2 2 ACT-61C-PRACTICE 32

Paper 75 · ACT January 2006 · Form 61C · Math Test

The equations of the 2 graphs shown below are y1(t) = a1 sin(b1t) and y2(t) = a2 cos(b2t), where the con- stants b1 and b2 are both positive real numbers. Which of the following statements is true of the con- stants a1 and a2 ?

Paper 75 · ACT January 2006 · Form 61C · Math Test

For x such that 0 < x < , the expression + is equivalent to:

Paper 75 · ACT January 2006 · Form 61C · Math Test

Consider the functions f(x) = 公僓 x and g(x) = 7x + b. In the standard (x,y) coordinate plane, y = f 冸g(x)冹 passes through (4,6). What is the value of b ?

Paper 75 · ACT January 2006 · Form 61C · Math Test

The triangle, 䉭XYZ, that is shown below has side lengths of x, y, and z inches and is not a right triangle. Let X′ be the image of X when the triangle is reflected across YZ ___ . Which of the following is an expression for the perimeter, in inches, of quadrilateral X′YXZ ?

Paper 75 · ACT January 2006 · Form 61C · Math Test

A function f is an odd function if and only if f(–x) = –f(x) for every value of x in the domain of f. One of the functions graphed in the standard (x,y) coordinate plane below is an odd function. Which one?

Paper 75 · ACT January 2006 · Form 61C · Math Test

What is the real value of x in the equation log224 – log23 = log5 x ? O y x x O y y O x y O x O x X Y Z x y z 公僓僓僓僓 僓僓 僓僓 1 – sin2 x ________ ____ cos x 公僓僓僓僓 僓僓 僓僓 1 – cos2 x ________ ____ sin x π _ _ 2 O y1(t) t y y2(t) 2 2 ACT-61C-PRACTICE END OF TEST 2 STOP! DO NOT TURN THE PAGE UNTIL TOLD TO DO SO. DO NOT RETURN TO THE PREVIOUS TEST. 33