Algebra II Final Exam

See Question 14

Correct Answer: H Explanation: H. Solve the equation pq- 3r= 2 in terms of pand rby isolating the qon one side of the equal sign as follows:

See Question 15

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On a standard xy-graph, what is the distance between (-2, 4) and (1, -2)?

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See Question 7

Correct Answer: -5 ≤x< 5 Explanation: B. The number line includes all values that are both greater than or equal -5, but strictly less than 5, so the correct answer is Choice (B)

If the equation x^2+ mx+ n= 0 has two solutions, x = k and x = 2k,what is the value of mn in terms of k?

Correct Answer: -6k^3 Explanation: E. Given that the equation x2+ mx+ n= 0 has two solutions, x = kand x = 2k,you can work backward to build the original equation. Here's how: x= kx= 2k x- k= 0 x- 2k= 0 Now take the two equations and combine them: (x- k)(x- 2k) = 0 x2- 2kx- kx + 2k2= 0 x2- 3kx + 2k2= 0 As you can see, m= -3kand n = 2k2, so mn= -6k3.

See Question 13

Correct Answer: C Explanation: C. The secant of an angle equals the hypotenuse over the adjacent angle, so

What is the slope of a line on a standard xy-plane that passes through the point (1, 3) and (4, -3)?

Correct Answer: -2 Explanation: C. Plug the values for the coordinates into the formula for the slope of a line:

If pq+ 12 = 3p+ pr,and q- r= 7, what is the value of p?

Correct Answer: -3 Explanation: A. To begin, isolate the qand rterms in the first equation: pq+ 12 = 3p+ pr pq- pr + 12 = 3p pq- pr = 3p- 12 Now factor out pon the left side of the equation: p(q- r) = 3p- 12 Substitute 7 for q- rand solve for p: 7p= 3p- 12 4p= -12 p= -3

What is the x-intercept of a line that passes through the point (3, 4) and has a slope of 2?

Correct Answer: 1 Explanation: J. First, plot the point (3, 4) on a graph. A slope of 2 means "up 2, over 1," so plot this point on a graph, too.

Beth got a job painting dorm rooms at her college. At top speed, she could paint 5 identical rooms during one 6-hour shift. How long did she take to paint each room?

Correct Answer: 1 hour and 12 minutes Explanation: H. Six hours equals 360 minutes, so Beth paints 5 rooms in 360 minutes. She paints 1 room in 360 ÷ 5 = 72 minutes, which equals 1 hour and 12 minutes.

A rectangular box has two sides whose lengths are 3 centimeters and 9 centimeters and a volume of 135 ^. What is the area of its largest side?

Correct Answer: 45 cm^2 Explanation: D. The box has dimensions of 3 and 9 and a volume of 135, so plug these values into the formula for the volume of a box: So the remaining dimension of the box is 5. The two longest dimensions are 5 and 9, so the area of the largest side is 5 ×9 = 45.

Which of the following equals 12x^2y^3z^4- 30xy^2+ 24x^2y^5z?

Correct Answer: 6xy^2(2xyz^4- 5 + 4xy^3z) Explanation: B. The greatest common factor among the three terms in 12x^2y^3z^4- 30xy^2+ 24x^2y^5zis 6xy^2, so you can rule out Choices (C), (D), and (E). When you factor out 6xy^2, the first term reduces to 2xyz^4, so you also can rule out Choice (A).

William earned $3,200 per month as a teacher for the ten months from September to June. Then he took a job as a barista at a local café, where he earned $2,000 per month during July and August. What was his average monthly pay for the 12 months?

Correct Answer: $3,000 Explanation: E. William earned $3,200 ×10 = $32,000 as a teacher, and he earned $2,000 ×2 = $4,000 as a barista. So he earned $32,000 + $4,000 = $36,000 over the 12 months. Plug these values into the formula for the mean

Sebastian bought a meal at a restaurant and left a 15% tip. With the tip, he paid exactly $35.19. How much did the meal cost without the tip?

Correct Answer: $30.60 Explanation: D. The tip is a percent increase of 15%, which is 115%. Let x equal the price before the tip. Thus, 115% of this price equals $35.19: 1.15x= 35.19 Divide both sides by 1.15:

Anne and Katherine are both saving money from their summer jobs to buy bicycles. If Anne had $150 less, she would have exactly 1/3 as much as Katherine. And if Katherine had twice as much, she would have exactly 3 times as much as Anne. How much money have they saved together?

Correct Answer: $750 Explanation: E. If Anne had $150 less, Katherine would have three times more than Anne. Make this statement into an equation and simplify: 3(a- 150) = k 3a- 450 = k And if Katherine had twice as much, she would have three times more than Anne: 2k= 3a Substitute 3a - 450 for kinto the last equation and solve for a 2(3a- 450) = 3a 6a- 900 = 3a -900 = -3a 300 = a Now substitute 300 for ainto the same equation and solve for k: 2k= 3(300) 2k= 900 k= 450 Thus, together Anne and Katherine have 300 + 450 = 750, so the right answer is Choice (E).

On an xy-graph, three corners of a parallelogram are located at (3, 3), (4, - 4), and (-2, -1). Which of the following points could be the remaining corner?

Correct Answer: (-3, 6) or D

If you graph the lines 3x+ 2y= -2 and 5x- y= 14 on a standard xy-graph, at which point will the lines intersect?

Correct Answer: (2, -4) Explanation: G. Solve the two equations, 3x+ 2y= -2 and 5x- y= 14, as a system of equations. To do this, multiply the second equation by 2 and then add it to the first equation: Now solve for x: x= 2 Replace xwith 2 in either equation and solve for y: 3(2) + 2y= -2 6 + 2y= -2 2y= -8 y= -4 At the point of intersection, x= 2 and y= -4, so this point is (2, -4).

Which of the following is equal to (n- 3)2if n= 11?

Correct Answer: (n+ 5)(n- 7) Explanation: B. If n= 11, then: (n- 3)2= (11 - 3)2= 82= 64 Substitute 11 for n into each answer until you find one that equals 64: (n+ 6)(n- 6) = (17)(5) = 85 Wrong! (n+ 5)(n- 7) = (16)(4) = 64 You've found a match!

See Question 17

Correct Answer: (p+ t)° Explanation: B. ∠pand ∠r are opposite angles on the parallel lines, so they total 180°, which rules out Choice (A). The same is true of ∠qand ∠s,ruling out Choice (C). ∠tand ∠uare supplementary, so they add up to 180°, which rules out Choice (E). Finally, ∠r,∠s,and ∠tare all vertical angles with the three interior angles of a triangle, so they total 180°, ruling out Choice (D). By elimination, the correct answer is Choice (B).

Andrea wants to fill in two sections of her backyard with sod that must be purchased in 2-x-2-foot squares. If the two sections measure 30 x 40 feet and 60 x 80 feet, how many squares of sod does she need to buy?

Correct Answer: 1,500 Explanation: H. To begin, find the area of each of the two sections by multiplying the lengths of the sides: 30 feet ×40 feet = 1,200 square feet 60 feet ×80 feet = 4,800 square feet The total area that needs sod has an area of 1,200 + 4,800 = 6,000 square feet. Each individual square is 2 feet by 2 feet, so each has an area of 4 square feet. Because all the numbers are even, there will be no waste when the squares of sod are placed. Therefore, you just have to find the number of squares needed by dividing: 6,000 ÷ 4 = 1,500

When you multiply a number by 4 and then subtract 7, the result is the same as if you first subtracted 7 from the same number and then multiplied by 11. What is the number?

Correct Answer: 10 Explanation: F. Let x equal the number and then change the words into an equation and solve for x: 4x- 7 = 11(x- 7) 4x- 7 = 11x- 77 70 = 7x 10 = x

Doug, who runs track for his high school, was challenged to a race by his younger brother, Matt. Matt started running first, and Doug didn't start running until Matt had finished a quarter-mile lap on the school track. Doug passed Matt as they both finished their sixth lap. If both boys ran at a constant speed, with Doug running 2 miles an hour faster than Matt, what was Matt's speed?

Correct Answer: 10 miles per hour Explanation: G.Doug runs 2 miles an hour faster than Matt, so let Matt's speed equal x miles per hour. Then Doug's speed equals x + 2 miles per hour. Each lap is one-quarter of a mile, so Doug runs 1.5 miles in the time it takes Matt to run 1.25 miles.

Andrea and Zach are both waiting for an appointment with a guidance counselor. When they arrived, each received a card from the secretary, telling the hour and minute of his or her arrival. Two minutes ago, Andrea had been waiting exactly 1/2 as many minutes as Zach. Three minutes from now, Andrea will have been waiting exactly 2/3 as long as Zach. If the time is now 11:30, at what time did Andrea arrive?

Correct Answer: 11:23 Explanation: J. Let aequal the number of minutes that Andrea had been waiting two minutes ago. At that time, Zach had been waiting for twice as long as Andrea, so he had been waiting for 2aminutes. Three minutes from now, both Andrea and Zach will each have been waiting for 5 minutes longer:

The number 4 is the smallest positive integer that has exactly three factors: 1, 2, and 4. If kis the next-highest integer that also has exactly three factors, what is the sum of the three factors of k?

Correct Answer: 13 Explanation: F. The greatest of the five possible answers is 20, so kis less than 20. Use trial and error to find the value of k: Factors of 5:1 5 Factors of 6:1 2 3 6 Factors of 7:1 7 Factors of 8:1 2 4 8 Factors of 9:1 3 9 Thus, k= 9, and the sum of the factors of kis 1 + 3 + 9 = 13.

If you multiply two integers together and then add 4, the result is 40. Which of the following could NOT be the sum of the two numbers?

Correct Answer: 18 Explanation: J. Let the two integers equal x and y,and then create the following equation and simplify: xy+ 4 = 40 xy= 36 So x and y are a pair of integers that equal 36. Try adding all possible combinations of two integers that multiply out to 36: 1 ×36 = 36 1 + 36 = 37 2 ×18 = 36 2 + 18 = 20 3 ×12 = 36 3 + 12 = 15 4 ×9 = 36 4 + 9 = 13 6 ×6 = 36 6 + 6 = 12 This list of sums includes 12, 13, 15, and 20, but not 18. Thus, no pair of integers both satisfies the original equation and adds up to 18.

If a and b are the two values of t that satisfy the equation t2- 6t+ 8 = 0, with a> b,what is the value of a- b?

Correct Answer: 2 Explanation: F. Factor the left side of the equation: t2- 6t+ 8 = 0 (t- 2)(t- 4) = 0 Split this equation into two equations and solve: t- 2 = 0 t- 4 = 0 t= 2 t= 4 Thus, a= 4 and b= 2. So a- b= 4 - 2 = 2.

Given that i^2= -1, what is the value of (4 + 2i)(4 - 2i)?

Correct Answer: 20 Explanation: B. To begin, FOIL the values in the parentheses: (4 + 2i)(4 - 2i) = 16 - 8i+ 8i- 4i2= 16 - 4i2 Substitute -1 for i2: = 16 - 4(-1) = 16 + 4 = 20

A square field has an area of 22,500 square feet. To the nearest foot, what is the diagonal distance across the field?

Correct Answer: 212 feet Explanation: J. The area of the field is 22,500 square feet, so plug this value into the formula for the area of a square to find the length of the side: So the side of the field measures 150 feet. The diagonal of a square is the hypotenuse of a 45-45-90 triangle, which has three sides in a ratio of . Thus, the length of this diagonal is . (You can also use the Pythagorean theorem to find this result.)

If the least common multiple of 9, 10, 12, and v is 540, which of the following could be v?

Correct Answer: 27 Explanation: C. To begin, notice that 540 isn't a multiple of 24, so you can rule out Choice (B). Now find the least common multiple (LCM) of 9, 10, and 12. The LCM of 9 and 10 is 90, so the LCM of 9, 10, and 12 must be a multiple of 90. Here are the first six multiples of 90: 90, 180, 270, 360, 450, 540 The number 180 is a multiple of 12 as well, so the LCM of 9, 10, and 12 is 180. However, 180 also is a multiple of 18, 36, and 45. So if any of these numbers were v,the LCM of 9, 10, 12, and vwould be 180. As a result, you can rule out Choices (A), (D), and (E), leaving Choice (C) as your only answer.

If a cube has a volume of kcm^3 and a surface area of 10k cm^2, what is its height in centimeters?

Correct Answer: 3/5 or H

The equation y= axb+ c produces the following (x, y) coordinate pairs: (0, 2), (1, 7), and (2, 42).What is the value of abc?

Correct Answer: 30 Explanation: H. To begin, plug x= 0 and y= 2 into the equation y= axb+ c.Note that the first term drops out: 2 = a(0)b+ c 2 = c Now you can substitute 2 for cin the original equation, giving you y= axb+ 2. Next, plug in x= 1 and y= 7. Notice that everything drops out except for the coefficient of the first term: 7 = a(1)b+ 2 5 = a You can now substitute 5 for ain the equation, giving you y= 5xb+ 2. Now plug in x= 2 and y= 42: 42 = 5(2)b+ 2 40 = 5(2)b 8 = 2b 23= 2b 3 = b Finally, you can see that abc = (5)(3)(2) = 30.

Which of the following numbers completes the sequence 3, 8, 14, 21, 29, ___?

Correct Answer: 38 Explanation: D. The numbers increase at a somewhat steady rate, so you have to figure out how much you have to add to each number to produce the next in the sequence: 3 + 5= 8; 8 + 6= 14; 14 + 7= 21; 21 + 8= 29, and so on. The rule for the sequence is to add successively larger numbers to each number; therefore, the next number is 29 + 9= 38.

If 5x+ 3 = 10x- 17, what is the value of x?

Correct Answer: 4 Explanation: D. Solve for x:

When she chooses a password, Eloise always uses exactly ten different characters: five letters (A, B, C, D, and E) and five numbers (2, 3, 4, 5, and 6). Additionally, she always ensures that no pair of letters is consecutive and that no pair of numbers is consecutive. How many different passwords conform to these rules?

Correct Answer: between 10,000 and 100,000 Explanation: H. The first character of the password can be any letter or number, so Eloise has ten options. Her second choice must be from the set (letter or number) not yet used, so she has five options. Choice 3 is from the same set as Choice 1, and she has four options left. Choice 4 is the second item from the same set as Choice 2, so she has four options. Choices 5 and 6 are from different sets, each with three options; Choices 7 and 8 are from different sets, each with two options; Choices 9 and 10 are from different sets, each with only one option remaining. You can see this information in the following chart: 1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th 10 5 4 4 3 3 2 2 1 1 To find the total number of possible passwords, multiply these numbers together: 10 ×5 ×4 ×4 ×3 ×3 ×2 ×2 ×1 ×1 = 28,800

Anderson has a phone plan that charges a monthly rate of $50 for the first 1,000 minutes plus $0.25 for each additional minute. Which of the following functions models Anderson's plan for all m> 1,000, with mas the number of minutes per month and f(m) as the monthly charge?

Correct Answer: f(m) = 0.25m- 200 Explanation: C.For m > 1,000 — that is, beyond 1,000 minutes of usage — the plan charges a usage fee based on the number of minutes used plus a flat rate of $50. With this information, you can create the following function: f(m) = Usage fee + 50 The usage fee is $0.25 per minute; however, the first 1,000 minutes are already paid for. So the usage fee is 0.25(m- 1,000). Plug this value into the preceding function and simplify: f(m) = 0.25(m- 1,000) + 50 f(m) = 0.25m- 250 + 50 f(m) = 0.25m- 200

On his first day working out, Anthony did 30 push-ups. On each successive day, he did exactly 3 more push-ups than on the previous day. After completing his push-ups on the 30th day, how many push-ups had he completed on all 30 days?

Correct Answer: more than 2,000 Explanation: K. Anthony completed 30 push-ups the first day, 33 the second day, 36 the third day, and so on. Make a chart as follows: 1 2 3 4 5 . . . 10 . . . 15 . . . 20 . . . 25 . . . 30 30 33 36 39 42 57 72 87 102 117 To save time adding all these numbers, notice that the total of the first and 30th numbers is 30 + 117 = 147. This total is the same for the 2nd and 29th, the 3rd and 28th, and so on all the way to the 15th and 16th. Therefore, you have 15 pairings of days on which Anthony completed 147 pushups. You can simply multiply to find the total: 147 ×15 = 2,205.

See Question 8

Correct Answer: x+ 4 Explanation: J. The small triangle has a side with the length of 3, and the large triangle has a corresponding side with the length of 6, which is twice as long. The triangles are similar, so each side of the large triangle is twice the length of the corresponding side of the small triangle. One side of the small triangle has a length of 2, so the corresponding side of the large triangle is 4; thus, x= 4. One side of the small triangle has a length of 4, so the corresponding side of the large triangle is 8; thus, y= 8. Therefore, y= x+ 4.

If x^2- x- 2 > 0, which of the following is the solution set for x?

Correct Answer: x< -1 or x> 2 Explanation: J. Begin by treating the inequality x^2- x- 2 > 0 as if it were an equation. Factor to find the zeros: x2- x- 2 = 0 (x+ 1)(x- 2) = 0 x= -1 and x= 2 The graph of this function is a parabola that crosses the x-axis at -1 and 2. This graph is concave up because the coefficient of the x^2 term (that is, a) is positive. The parabola dips below the x-axis when -1 < x < 2, so these values do not satisfy the equation. Therefore, the correct answer is Choice (J). (If you doubt this answer, note that when x= 0, x2- x- 2 = -2, which is less than 0. So 0 is not in the solution set.)

What is the solution set for x for the inequality |2x+ 7| > 11?

Correct Answer: x< -9 or x> 2 Explanation: K. Split the inequality |2x+ 7| > 11 into two inequalities: 2x+ 7 > 11 and 2x+ 7 < -11. Solve both for x: 2x+ 7 > 11 2x+ 7 < -11 2x> 4 2x< -18 x> 2 x< -9 Therefore, x< -9 or x> 2.