If t = – 2, then t 3 – 2 t 2 + 5 t – 3 = ? 1. – 29 2. – 27 3. – 13 4. – 9 5. 7

Answer Plug in for t: (– 2)3 – 2(– 2)2 + 5(– 2) – 3 – 8 – 10 – 3 – 29

What is the length, in feet, of the circumference of a circle whose diameter is 12 feet? 1. 2. 3. 4. 5. 6π 12π 6π 2 36π2

Answer Using the formula circumference = π × diameter, you get that the circumference = π × 12 = 12

If 3(x + 2) + 3 = 6 x, what is the value of x? 1. -3 2. -5/3 3. 1/3 4. 5/3 5. 3

Answer Distribute and then solve: 3 x + 6 + 3 = 6 x 9 = 3 x x = 3

If a drawer contains 7 navy socks, 4 white socks, and 9 black socks, what is the probability that the first sock randomly drawn out of the drawer will not be white? 1. 1/5 2. 1/4 3. 7/20 4. 4/5 5. 16

Answer There is a total of 20 socks, and 16 socks are navy or black, and, therefore, not white. The probability that the first sock drawn will not be white is 16 out of 20; 16/20 reduces to 4/5.

Which of the following shows the prime factorization for 720? 1. 24 × 32 × 5 2. 2 × 3 × 5 3. 24 × 3 × 15 4. 24 × 45 5. 23 × 90

Answer The answer must be expressed as the product of only prime numbers. (This makes choices #3, #4, and #5 wrong. ) 720 = 2 × 360 = 2 × 180 = 2 × 2 × 90 = 2 × 2 × 45 = 2 × 2 × 3 × 5 = 24 × 32 × 5, thus choice A is correct. Notice that choice B represents a value of 30.

The number of gallons of paint needed to cover the exterior of a house with one coat of paint is estimated by the formula [10 n(l + w) – 9 n] ÷ 350, where n is the number of stories, l is the length of the house in feet, and w is the width of the house in feet. Approximately how many gallon cans of paint should somebody buy in order to paint one coat on the exterior of a 30 × 50 -foot 2 -story house? 1. 3 2. 4 3. 5 4. 6 5. 7

Answer Plug in the variables: We know the number of stories, n, will be 2. We choose the larger number, 50, to represent the length, l. The width, or w, is 50. [10 n(l + w) – 9 n] ÷ 350 becomes: [10(2)(30 + 50) – 9(2)] ÷ 350 [20(80) – 18] ÷ 350 (1600 – 18) ÷ 350 1582 ÷ 350 4. 52 Since the paint must be purchased in gallon cans, 5 cans are necessary.

Which coordinate pair is a solution to the inequality 12 – 3 y > 6 x + 3? 1. (1, 1) 2. (2, 1) 3. (1, 2) 4. (-1, -2) 5. (2, -1)

Answer Plugging in (1, 1) gives 9 > 9, which is not a true statement. Plugging in (2, 1) gives 9 > 15, which is not a true statement. Plugging in (1, 2) gives 6 > 9, which is not a true statement. Plugging in (– 1, – 2) gives 18 > – 3, which IS a true statement. Plugging in (2, – 1) gives 15 > 15, which is not a true statement. Therefore (– 1, – 2) is the only correct solution.

Which linear equation has a slope of – 1/2 and a y-intercept of 6? 1. x = 6 y – 1/2 2. y = 6 x – 1/2 3. x = – 1/2 y + 6 4. y = – 1/2 x + 6 5. None of these

Answer Slope-intercept form for a linear equation is y = mx + b, where m is the slope and b is the y-intercept. The equation y = – 1/2 x + 6 has a slope of – 1/2 and a y-intercept of 6.

|– 5 · 2| – |(– 3)2| = ? 1. -19 2. -1 3. 1 4. 4 5. 19

Answer Simplify: |– 10| - |9| Absolute values result in a positive result: 10 – 9 1

The price of a $220 saw was reduced to $154. What was the percent discount? 1. 30% 2. 43% 3. 66% 4. 67% 5. 70% 1 2 3 4 5

Answer The percent discount is the difference in the prices divided by the original price: ($220 – $154) ÷ $220 $66 ÷ $220 0. 3 Multiply by 100 to get a percent: 30%

For all x, (3 x – 4)^2 = ? 1. 6 x – 8 2. 9 x 2 + 16 3. 9 x 2 - 16 4. 9 x 2 – 12 x + 16 5. 9 x 2 – 24 x + 16 1 2 3 4 5

Answer (3 x – 4)2 can be rewritten as (3 x – 4). Multiply using the distributive property or "FOIL" (first terms, outside terms, inside terms, last terms): 9 x 2 – 12 x + 16 9 x 2 – 24 x + 16

In a seminar, the ratio of men to women is 5: 9. If there are 56 people in the seminar, how many are women? 1. 4 2. 7 3. 20 4. 26 5. 36 1 2 3 4 5

Answer A ratio of 5 parts men to 9 parts women, means there are 14 parts in all. Thus, the number of women in the seminar are 9/14 of the total number of people in the seminar. Therefore, the number of women in the seminar is 9/14 of 56, or: 9/14 × 56 = 36.

(2· 3√ 9)3 = ? 1. 18 2. 24 3. 54 4. 72 5. 216 1 2 3 4 5

Answer (2· 3√ 9)3 is equivalent to 23 · (3√ 9)3. Because a cube root and a cube are inverses, (3√ 9)3 simplifies to 9. This results in 23 · 9 = 8 · 9 = 72.

If V = x^3 + y[r(3 s + 2) + 4 r] and x = 2, y = 3, r = 5, and s = 4, what is the value of V? 1. 269 2. 272 3. 275 4. 278 5. 281 1 2 3 4 5

Answer V= x^3 + y[r(3 s + 2) + 4 r] = (2)^3 + 3[5(3 × 4 + 2) + 4 × 5] = 8 + 3[5(14) + 20)] = 8 + 3[70 + 20] = 8 + 3[90] = 8 + 270 = 278 Remember to follow the order of operations: Parentheses, exponents, multiplication/division, addition/subtraction—otherwise known as PEMDAS.

What is the slope of the line determined by the equation 15 y – 10 x = – 9? 1. 3/2 2. 2/3 3. 3/5 4. -2/3 5. -3/2 1 2 3 4 5

Answer We need to make the equation look like y = mx + b, and our answer will be m, the slope. Solve the equation for y to write the equation in slopeintercept form: 15 y = 10 x – 9 y = ( 10/15)x – 9/15 y = 2/3 x + 3/5 The coefficient of the x term, 2/3 , is the slope of the line.

If –x > y and |x| > y, which values for x and y make both statements true? I. (0, 0) II. (– 2, 3) III. (– 5, 4) IV. (– 2, – 3) V. (5, – 5) 1. 2. 3. 4. 5. 1 2 All except II. All of them III. And IV. Only I. , II. , and IV. 3 4 5

Answer Plug in each of the possible answers to see which make true statements: I. – 0 > 0 is false. II. 2 > 3 is false. III. 5 > 4 is true and |– 5| > 4 is also true. IV. 2 > – 3 is true and |2| > – 3 is also true. V. – 5 > – 5 is false.

After a computer is discounted by 20%, its price is $960. What was the original price of the computer before the discount? 1. $768 2. $1066 3. $1120 4. $1200 5. $1220 1 2 3 4 5

Answer After the computer is discounted by 20%, its price is 80% of its original price, p. p × 0. 80 = $960 p = $960 ÷ 0. 80 p = $1200 Therefore, its original price was $1200.

In the standard (x, y) coordinate plane, point P is located at (16, 12) and point Q is located at (11, 7). What is the distance between points P and Q? 1. √ 5 2. 5 3. 5 √ 2 4. 10 5. 50 1 2 3 4 5

Answer Using the formula, Distance = =

If the circumference of a circle measures 10 feet, what is its area in square feet? 5 10 15 25 30 1. 2. 3. 4. 5. 1 2 3 4 5

Answer Using the formula, Circumference = × diameter, you get 10 = × diameter, or the diameter of the circle is 10 feet. Thus, its radius is 5 feet, and its Area = × Π radius 2 = × 52 = 25 square feet.

What is the x-intercept of the line given by the equation 5 x + 3 y = 12? 1. 12 2. 4 3. 12/5 4. -3/5 5. -5/3 1 2 3 4 5

Answer The x-intercept occurs when y = 0, so plug in 0 for y and solve for x: 5 x + 3(0) = 12 5 x = 12/5

For what value of b in the equation (2 z + b)(z – 3) = 0 will a solution for z be – 4? 1. -8 2. 1 3. 3 4. 8 5. 32/3 1 2 3 4 5

Answer To solve this quadratic equation for z, both factors would be set equal to 0: (2 z + b) = 0 and (z – 3) = 0 We will focus on the first factor that contains b. 2 z + b = 0 Substitute z = – 4 into the equation: – 8 + b = 0 Solve for b: b = 8

If the edge of a cube is doubled, the volume of the cube will be increased by what factor? 1. 2 2. 4 3. 6 4. 8 5. 16 1 2 3 4 5

Answer The volume of cube is given by the formula Volume = edge^3. Therefore, if the edge is doubled, you get Volume = (2 × edge)^3 = 2^3 × edge^3 = 8 × edge^3. Thus, the volume is increased by a factor of 8.

A train travels at a constant rate of 30 miles per hour. How long will it take the train to travel 100 miles? 1. 3 hours and 15 minutes 2. 3 hours and 20 minutes 3. 3 hours and 30 minutes 4. 3 hours and 33 1/3 minutes 5. 3 hours and 40 minutes 1 2 3 4 5

Answer Using the formula Distance = Rate × Time, you get 100 = 30 × Time, or Time = 100/30 = 3 1/3 hours = 3 hours and 20 minutes.

If 2(x – 1)^2 + 5 = 13, what are the two possible values of x? 1. 1 and – 3 2. 3 and – 1 3. 5 and – 3 4. 1 + √ 2 and 1 – √ 2 5. There is no solution. 1 2 3 4 5

Answer 2(x – 1)^2 + 5 =13 2(x – 1)^2=8 (x – 1)^2=4 x – 1= ± √ 4 x – 1 = ± 2 x – 1 = 2 or x – 1 = – 2 x = 3 or x = – 1

If |2 x – 3| >= 20 which shows the solutions for x? 1. -17/2 = x = 23/2 2. X -17/2 or x 23/2 3. X=23/2 4. x 17/2 5. There is no solution. 1 2 3 4 5

Answer The inequality |2 x - 3| 20 is the same as the two inequalities 2 x - 3 20 or 2 x + 3 20. Solving the first inequality, you get 2 x 23, or x 23/2. Solving the second inequality, you get 2 x 17, or x -17/2. These are both solutions.

At a recent sold-out concert of a 300 -seat theater, two types of tickets were sold: regular tickets for $8 each and discount tickets for $4 each. If $1, 600 in tickets were sold, how many were regular tickets? 1. 50 2. 100 3. 150 4. 200 5. 2, 500 1 2 3 4 5

Answer Let x = the number of regular tickets sold. Thus, 300 - x = the number of discount tickets sold. Since $1, 600 in tickets were sold, you get 8 x + 4(300 - x) = 1, 600. Solving for x, you get 8 x + 1, 200 - 4 x 1, 600, or 4 x = 400, and x = 100.

Which of the following is an equivalent expression for (5 x+4)/x – (x+1)/3 x ? 1. 4 x + 3 2. (4 x+3)/3 x 3. (14 x+13)/14 x 4. (14 x+11)/3 x 5. (4 x+3)/3 x^2 1 2 3 4 5

Answer multiplying by 1 doesn't change the value of the expression, we will multiply by 1, represented as 3/3. So, multiplying both the numerator and denominator of the first fraction by 3, you get

= ? -4 -1 -1/4 4 1. 2. 3. 4. 5. 1 2 3 4 5

Answer A number raised to a negative exponent can be represented by writing 1 over that number to the positive exponent. This means we can bring the 4^– 2 in the numerator down to the denominator as 4^2, and we can bring the 2^– 3 in the denominator up to the numerator as 2^3:

Kathy has test scores of 88, 90, 91, and 78. What must she score on her next test to have an average of 84 for all five tests? 1. 71 2. 73 3. 75 4. 77 5. 4 1 2 3 4 5

Answer The sum of the first four tests is 347. In order to have an average of 84 on all five tests, the sum of those tests must equal 5 × 84 = 420. Therefore, the fifth test score must be 420 - 347 = 73.

In the standard (x, y) coordinate plane, a rectangle has the coordinates (6, 5), (6, – 2), (– 3, 5), and (– 3, – 2). What is the length of a diagonal of the rectangle? 1. 4 2. √ 130 3. 16 4. 130 5. 256 1 2 3 4 5

Answer Explanation: Plot the points. Then, find the distance between two opposite points, (6, 5) and (– 3, – 2). (-3, 5) (6, 5)(3, -2) (6, -2) Use the Distance formula

If 3 y^2 + 24 y + 15 = 0, what are the possible values of y? 1. -8± 2√ 11 2. -4± 2√ 11 3. -4 ±√ 130 4. ± 6√ 11 5. ± √ 11 1 2 3 4 5

Answer First, reduce the equation by dividing all terms by 3 to give y^2 + 8 y + 5 = 0. Using the quadratic formula. where a and b are the coefficients of y^2 and y, respectively, and c is the constant.

What is the slope of a line perpendicular to the line 8 x + 4 y =12? -2 -1/2 2 12 1. 2. 3. 4. 5. 1 2 3 4 5

Answer Solving for y to write 8 x + 4 y =12 in slopeintercept form, you get: 4 y = -8 x + 12 y = -2 x + 3 Therefore, slope of this line is the coefficient of the x term, or -2. The slope of a line perpendicular to this line is the negative reciprocal of -2, or 1/2 .

What is the x-intercept of the circle (x – 3)^2 + (y + 4)^2 = 16? 1. (3, 0) 2. (4, 0) 3. (0, -3) 4. (0, -4) 5. The circle has no x-intercept 1 2 3 4 5

Answer The general equation for a circle is (x - a)^2 + (y - b)^2 = r^2, where (a, b) is the center of the circle, and r is the radius of the circle. Therefore, in the circle (x – 3)^2 + (y + 4)^2 = 16, the center of the circle is the point (3, -4), and the radius is 4. Thus, the since the center is 4 units below the xaxis, and the radius is 4, the circle will touch the x-axis 4 units directly above the center, at (3, 0).

If 4^(2 x + 1) = 1/64 , x = ? 1. -2 2. -3/2 3. 0 4. 3/2 5. 2 1 2 3 4 5

Answer Substitute 4^– 3 for 1/64 : 4(2 x + 1) = 4^– 3 Set the exponents equal to each other and solve for x: 2 x + 1 = – 3 2 x= – 4 x= – 2

What is the area, in square inches, of an isosceles trapezoid with side lengths 5, 5, 10, and 18 inches? 1. 21 2. 38 3. 42 4. 56 5. 70 1 2 3 4 5

Answer Because the trapezoid is isosceles, it can be broken into a rectangle and two right triangles, with the two side lengths of 5 applying to the legs of the trapezoid. Because the difference in the bases is 8, half that will be the length of the leg of each of the triangles. Therefore, the right triangles will have a hypotenuse of 5 and a leg of 4. The Pythagorean Theorem shows that the length of the second leg is 3. The area of the two right triangles is equivalent to a 3 by 4 rectangle, and the other rectangle that makes up the trapezoid is 3 by 10. This gives areas of 12 and 30, which adds up to 42.

The points (3, – 5), (1, – 4), and (– 5, k) lie on the same line. What is the value of k? 1. -4 2. -3 3. -2 4. -1 5. 0 1 2 3 4 5

Answer Because all the points are on the same line, the slope between any two points will be the same. Find the slope between the first two points: m = (– 5 –(– 4)) / (3 – 1) m = - 1/2 Set up an equation using the slope and a second pair of points and solve for k: – 1 / 2 = (– 4 – k) / (1 – (– 5)) – 1 / 2 = (– 4 – k) / 6 – 6 = 2(– 4 – k) – 6 = – 8 – 2 k 2 = – 2 k k = – 1

If the height of an equilateral triangle is 9, what is the perimeter of the triangle? 1. 18 √ 3 2. (81 x √ 2 )/4 3. 27 4. 27 x √ 2 5. 54 √ 3 1 2 3 4 5

Answer The height of an equilateral triangle creates two 30 -60 -90 triangles, with the height being the long leg of the special triangle. Using the special qualities of a 30 -60 -90 triangle, the short leg is which is rationalized to Because the short leg is half the base, double the short leg to find the base of the triangle, giving Multiply that length by 3 to get the perimeter,

For the circle represented by the equation x^2 + 6 x + y^2 - 2 y = -1 in the standard (x, y) coordinate plane, what are the coordinates of the center? 1. (-3, 1) 2. (-3, -1) 3. (3, -1) 4. (3, 1) 5. (-3, 1) 1 2 3 4 5

Answer The general equation for a circle is (x - a)^2 + (y - b)^2 = r^2, where (a, b) is the center of the circle, and r is the radius of the circle. In order to write the equation x^2 + 6 x y^2 - 2 y = -1 in this form, first add 9 to both sides of the equation to get x^2 + 6 x + 9 y^2 - 2 y = -1 + 9. This is the same as (x + 3)^2 + y^2 -2 y = 8. Then add 1 to both sides of the equation. This gives (x + 3)^2 + y^2 - 2 y + 1 = 8 + 1, which is the same as (x + 3)^2 + (y - 1)^2 = 9. Therefore, the coordinates of the center of the circle are (-3, 1).

A machine can produce N cans per minute. Another machine can produce 4 times as many cans per minute. How many minutes will it take the two machines working together to produce 1, 000 cans? 1. 500 N 2. 250/N 3. 250 N 4. 200/N 5. 200 N 1 2 3 4 5

Answer Since the first machine can produce N cans per minute, the other machine can produce 4 N cans per minute. Thus, working together, the two machines can produce 5 N cans per minute. We set up a proportion: Cross multiply to get: 5 Nx = 1000 Finally, isolate x: x= 1000/5 N x= 200/N

What is the smallest integer that is greater than ? 1. 1 2. 2 3. 3 4. 4 5. 5 1 2 3 4 5

Answer You can multiply to get: The total is 1. 5 + 1 + 0. 75 = 3. 25. The smallest integer larger than 3. 25 is 4.

If 3 x^2 + 7 x + 2 = 0, what is one possible value of x? 1. -2 2. -1 3. 0 4. 1/3 5. 3 1 2 3 4 5

Answer You can either factor this by trial and error, or you can actually solve it using the more general formulas for factoring a quadratic polynomial into (ax + b)×(cx + d). Here 3 x^2 + 7 x + 2 = (3 x + 1)×(x + 2) As this is zero only when one of the factors is zero, only x = -2 is a possible solution.

If the number 114 is written as a product of N numbers, where each of the N numbers is prime and where the number 1 is not included, what is the value of N? 1 2 3 4 5 1. 2. 3. 4. 5. 1 2 3 4 5

Answer Rewriting 114 as the product of prime numbers means breaking it into factors such that the factors themselves cannot be further subdivided. The factors 2, 3 and 19 combine to give 114, so the value of N must be three.

Let the function "deuce(x)" mean to add 2 to x. The process can be repeated, so that "deuce^2(x)" means to "deuce" the result of "deuce(x)". If deuce^n (2) is greater than 101, what is the smallest possible integer value of n? 102 101 100 51 50 1. 2. 3. 4. 5. 1 2 3 4 5

Answer The wording might be a little tricky, but all that's being described is a series 2, 4, 6, 8. . If 4 is the result of the first step, then 102 is reached after 50 steps.

0 < n^-1 + 3 m < 4 What is one possible pair of values for (n, m)? (-1, 0) (-1, 1) (1, -1) (1, 1) ( 1/2, 1) 1. 2. 3. 4. 5. 1 2 3 4 5

Answer Many of the answers are very close. But only the pair (-1, 1) works. (-1)^-1 + 3(1) = -1 + 3 = 2.

Which value is largest? 1. 150% of 32 2. 120% of 40 3. 80% of 60 4. 75% of 64 5. 50% of 98 1 2 3 4 5

Answer All the other choices were equal to 48. But 50% of 98 is 49, so this is the correct choice.

Two years ago, a rabbit breeder had 75 rabbits. Since then 500 rabbits have been born, 220 have died and 185 have been sold. How many rabbits does the breeder have now expressed as a percentage of the number she had two years ago? 21 % 44 % 133 % 200 % 227 % 1. 2. 3. 4. 5. 1 2 3 4 5

Answer She starts with 75 rabbits. 500 are born, 220 die and 185 are sold. The new number of rabbits is: 75 + 500 - 220 - 185 = 170 and 170 is indeed 227 % of 75.

What is the value of 1. 1, 000 2. 100, 000 3. 10, 000 4. 1, 000 5. 100 1 2 3 4 5

Answer The number inside the radical is 10^6 = 1, 000. If you know that 1000 is the square root of one million you're done. You might notice that taking the square root of a number is equivalent to raising it to the power 1/2. So (10^6)^1/2 = 10^3 = 1000.

How many positive integers less than 62 have 8 as a factor? 1. 9 2. 8 3. 7 4. 6 5. 2 1 2 3 4 5

Answer The first thing to notice is that 8*8 = 64. So the number has to be less that 8. But 8*7 = 56, which is less than 62. That means that there are 7 positive integers less than 62 with 8 as a factor

For which of the following is the sum of the digits in the tenths and hundredths places the greatest? 1. 510. 823 2. 620. 439 3. 126. 750 4. 982. 453 5. 563. 929 1 2 3 4 5

Answer For the number 126. 750, the sum of the digit in the tenths place (7) and the digit in the hundredths place (5) is 12. This is a greater sum than for any of the other numbers shown.