Correct Response: C. This claim may be verified with simple examples, such as π0 = 1pi to the 0 power equals 1 and (√ 2)2 = 2radical 2 squared equals 2.
Incorrect Response: A. The sum of an irrational number plus a rational number is irrational. Consider that a non-repeating, nonterminating decimal will remain as such when any terminating or repeating decimal is added to it.
Incorrect Response: B. The quotient of an irrational number divided by a rational number is irrational. Suppose that a ÷ bA divided by b formed a rational number, m/nm divided by n. Then it would follow that a = b m/nA equals b times the fraction m over n. That is, aA would also be a rational number. This result contradicts the statement that aA is an irrational number and so it cannot be true.
Incorrect Response: D. The sum of an irrational number plus a rational number is irrational. See the rationale for response A.
Correct Response: B. There are 3 sets of identical groups of x + 4x plus 4 shown, or (x + 4) + (x + 4) + (x + 4)the quantity x plus 4 plus the quantity x plus 4 plus the quantity x plus 4. This may be written as 3(x + 4)3 times the quantity x plus 4 by the distributive property of equality.
Incorrect Response: A. Since each of the three rods represents x, 3x is the correct term to represent the unknowns in the problem. However, this answer does not distribute the factor of 3 to the ones units shown—there are 12 ones units in the diagram, not 4.
Incorrect Response: C. The exponent signifies repeated multiplication of x + 4x plus 4. This expression would correspond to the total volume of a cube that has an edge length of x + 4x plus 4. This is not what is shown in the diagram.
Incorrect Response: D. This response may reflect a misconception that an exponent represents repeated addition (e.g.,the multiplication of 4 and 3). The expression 434 cubed denotes 4 × 4 × 4 = 644 times 4 times 4 equals 64 ones units.
Correct Response: D. Given that K is a multiple of x, then K = m • xK equals m times x for some whole number m. Similarly, given that K is a factor of y, then y/K = nY divided by K equals n for some whole number n. It follows from the substitution of K that y/m • x = n → y = m • x • nY divided by the product mx equals n implies y equals m times x times n. That is, y is a multiple of x.
Incorrect Response: A. Given that K is a factor of y, then y/K = nY divided by K equals n for some whole number n. Take the reciprocal of both sides of the equation to see that K/y = 1/nK divided by y equals 1 divided by n. As such, K/yK divided by y can only be a whole number if K = y.
Incorrect Response: B. Given that K is a multiple of x, then K = m • xK equals m times x for some whole number m. Dividing both sides of this equation by m • xm times x shows that x/Kx divided by K equals that unit fraction 1/m1 divided by m: K/m • K = m • x/m • K → 1/m = x/kK over the product m times K equals m times x divided by m times k implies 1 over m equals x over k. This means that x/kx divided by k is only a whole number when x = K.
Incorrect Response: C. For a proof by counterexample, consider the values K = 4, x = 2, and y = 8. These values are consistent with the information provided. In this case, y is not a factor of x, so the claim that "y is a factor of x" must not always be true.
Correct Response: D. The expanded form of the number represented as "1Y31 Y 3" is equal to 1 • 100 + Y • 10 + 3 • 11 times 100 plus Y times 10 plus 3 times 1. Multiplying the digit Y by 24 corresponds to a partial product of 24 • 10 • Y = 240 • Y24 times 10 times Y equals 240 times Y.
Incorrect Response: A. 20 • Y20 times Y does not reflect the contribution of the 4. The correct partial product can be written with the distributive property as (20 + 4) • 10 • YThe quantity 20 plus 4 times 10 times y.
Incorrect Response: B. The tens place value is not expressed with 24 • Y24 times Y. This response treats the digit Y as though it represented Y units of 1.
Incorrect Response: C. This response may represent 4 • 10 • Y4 times 10 times Y. This product does not account for the multiplication between Y and the digit 2 in 24.
Correct Response: D. Division by 2 will distribute over each of the terms in the parentheses: 2 × 66/2 + 10 × 64/2 + 1/2 × 62/2 = 66 + 5 × 64 + 1/4 × 622 times 6 to the sixth power divided by 2 plus 10 times 6 to the fourth power divided by 2 plus one half times 6 squared divided by 2 equals six to the sixth power plus 5 times six to the fourth power plus one fourth times 6 squared.
Incorrect Response: A. This answer is the result of a misapplication of the distributive property in which the base for each exponential expressing is divided by 2. (It does not hold, for instance, that 62 × 18/26 squared times 18 divided by 2 is equal to 32 × 93 squared times 9.)
Incorrect Response: B. This response demonstrates a misconception with the division of products in which the exponent for each exponential expression is halved. (It does not hold, for instance, that 62/26 squared divided by 2 is equal to 6.)
Incorrect Response: C. The response 63 + 5 × 62 + 1/4 × 66 cubed plus 5 times 6 squared plus one fourth times 6 could be obtained from multiple errors, such as dividing both the constants and the exponents for each expression by 2.
Correct Response: C. The value of a number does not change if a pair of inverse operations are applied to it. The pair of equations 4 × 8 = 324 times 8 equals 32 and 32 ÷ 8 = 432 divided by 8 equals 4 shows division by 8 reversing the effect of multiplication by 8.
Incorrect Response: A. In this response, multiplication occurs with one number and division with its integer opposite. This does not demonstrate the concept of inverse operations. The inverse operation of multiplication by 1 is division by 1, not –negative 1. However, even if this were addressed, the concept of inverse operations would not be well illustrated by multiplication and then division by 1.
Incorrect Response: B. This response demonstrates properties of zero, not inverse operations.
Incorrect Response: D. These equations multiply and divide by different numbers, making them inappropriate to use to illustrate the concept that multiplication and division are inverse operations.
Correct Response: B. Let a vertical position of 0 represent sea level and negative integers represent the depth of the submarine. In this reference system, the integer –negative 375 describes the initial position of the submarine. The expression –negative 375 + 183 − 228 represents new position of the submarine after it rises and then dives again. It evaluates to –negative 420 feet. If the submarine rises at 30 feet per second, then it reaches the surface in 420 ÷divided by 30 = 14 seconds.
Incorrect Response: A. This result can be obtained from the expression –negative 375 − 183 + 228 . In this case, the submarine starts at a depth of 375, sinks 183 more feet, and then rises 228 feet.
Incorrect Response: C. This result is approximately equal to (375 + 228) ÷divided by 30.
Incorrect Response: D. This result can be obtained from the expression 375 + 183 + 228. Because integer signs are not considered, this expression represents a submarine that dives by 183 feet and then by 228 feet.
Correct Response: D. Let x represent the original price of the television. The price after a 15% discount is 0.85x. The tax is paid on the price after discount, so it can be represented by (0.065)(0.85x)0 point 0 6 5 times 0 point 8 5 x. The total price paid including discount and tax is 0.85x + (0.065)(0.85x) = (0.85x)(1 + 0.065) = K0 point 8 5 x plus 0 point 0 6 5 times 0 point 8 5 x equals 0 point 8 5 x times the quantity 1 plus 0 point 0 6 5 equals k. Solving for x results in x = K/(0.85)(1 + 0.065)x equals k divided by the product of 0 point 8 5 and the quantity 1 plus 0 point 0 6 5.
Incorrect Response: A. This response provides the final cost of the television if K represents its initial amount. K, however, represents the final amount.
Incorrect Response: B. This expression attempts to reverse the effects of applying tax and providing a discount but does so incorrectly. Consider the claim that a 6.5% increase can be offset with a 6.5% decrease. The multiplication x(1 + 0.065)(1 – 0.065)x times the quantity 1 plus 0 point 0 6 5 times the quantity 1 minus 0 point 0 6 5 does not return x. Because the increase of 6.5% was applied with the multiplication of (1 + 0.065)the quantity 1 plus 0 point 0 6 5, it must be removed with the inverse operation of division by that same amount. The same argument applies to the factor of (1 + 0.15)the quantity 1 plus 0 point 1 5 that is used to compensate for the discount.
Incorrect Response: C. This equation is the result of solving the equation (0.85x)(1 – 0.065) = K0 point 8 5 x times the quantity 1 minus 0 point 0 6 5 equals K for x. The quantity (1 – 0.065)1 minus 0 point 0 6 5 treats the tax as if it discounted the original price, which is not correct.
Correct Response: C. If the GCF of x and y is 3, then x and y can be represented by 3k and 3m, respectively, where k and m are relatively prime (i.e.,they share no common factors, otherwise the GCF of x and y would reflect those factors and be greater than 3). Substituting this into 2/x2 over x = 5/y5 over y equals 2/3k2 over 3 k = 5/3m5 over 3 m. The least common denominator for these fractions is 3km. Given that 3k = x and m = y/3y over 3, the least common denominator can also be written as xy/3x times y over 3.
Incorrect Response: A. This response corresponds to a least common denominator that is the product of all the quantities shown: 2 • 5 • x • y2 times 5 times x times y.
Incorrect Response: B. This response incorrectly combines information about the GCF with the denominator values of x and y.
Incorrect Response: D. This response corresponds to the reciprocal of (2 • 5)/(x • y)2 times 5 over x times y.
Correct Response: B. Apply algebraic properties of inequality to test whether a + c/b + c > a/bA plus c over b plus c is greater than A over b is a true statement. The inequality can be rearranged as b(a + c) > a(b + c)b times the quantity A plus c is greater than A times the quantity b plus c through multiplication, expanded as ab + bc > ab + acA times b plus b times c is greater than A times b plus a times c by the distributive property, and then further reduced to the simple inequality b > ab is greater than A. This result may be interpreted to mean that a + c/b + c > a/bA plus c over b plus c is greater than A over b is a true statement when b > ab is greater than A. This must be true because the fraction a/bA over b is a proper fraction greater than zero.
Incorrect Response: A. The inequality a − c/b − c > a/bA minus c over b minus c is greater than A over b can be rearranged as b(a – c) > a(b – c)b times the quantity A minus c is greater than A times the quantity b minus c. This expands to ab – bc > ab – acA times b minus b times c is greater than A times b minus A times c and simplifies to –b > –anegative b is greater than negative A, which is also equivalent to b > ab is greater than A. However, this cannot be true because we are given that a/bA over b is a proper fraction greater than 0.
Incorrect Response: C. Simplify the left side of the inequality to obtain a/b > a/bA over b is greater than A over b, which is a false statement.
Incorrect Response: D. Observe that a ÷ c/b ÷ cA divided by c over b divided by c is equivalent to a/bA over b. The inequality shown is equivalent to the statement a/b > a/bA over b is greater than A over b, which is false.
Correct Response: C. The problem can be described by the linear equation 5 + 0.75x = 315 plus 0 point 7 5 times x equals 31, where x is the time in seconds that the cyclist has coasted down the hill and 5 is the initial speed for the cyclist at time x = 0. Solving the equation for x: 5 + 3/4 x = 31 → 3/4 x = 26 → x = 26 ÷ 3/4 → x = 26 × 4/3 → x = 104/3 = 34 2/35 plus 3 fourths x equals 31 implies 3 fourths x equals 26 implies x equals 26 divided by 3 fourths implies x equals 26 times 4 thirds implies x equals 104 over 3 equals 34 and 2 thirds.
Incorrect Response: A. This response is the result of the computation (31 – 5) × 3/4the quantity 31 minus 5 times 3 fourths, which represents a misinterpretation of the relationships in the problem.
Incorrect Response: B. This answer is the result of the computation 31 ÷ 3/431 divided by 3 fourths, which does not take the initial speed of the cyclist into account.
Incorrect Response: D. This answer is the result of the computation 36 ÷ 3/436 divided by 3 fourths, which may be the result of using the equation 3/4 x – 5 = 313 fourths times x minus 5 equals 31.
Correct Response: C. This response shows 3 • 6/8 = 18/8 = 2 1/43 times 6 eighths equals 18 over 8 equals 2 and 1 fourth. With these values, Abc = 2 • 1 • 4 = 8A b c equals 2 times 1 times 4 equals 8.
Incorrect Response: A. This response shows 3/5 + 5/5 + 3/5 = 11/5 = 2 1/53 fifths plus 5 fifths plus 3 fifths equals 11 fifths equals 2 and 1 fifth. With these values, Abc = 2 • 1 • 5 = 10A b c equals 2 times 1 times 5 equals 10.
Incorrect Response: B. This response shows 3 • 5/6 = 15/6 = 2 1/23 times 5 sixths equals 15 sixths equals 2 and one half. With these values, Abc = 2 • 1 • 2 = 5A b c equals 2 times 1 times 2 equals 5.
Incorrect Response: D. This response shows 1/4 + 3/4 + 1/4 = 5/4 = 1 1/41 fourth plus 3 fourths plus 1 fourth equals 5 fourths equals 1 and 1 fourth. With these values, Abc = 1 • 1 • 4 = 4A b c equals 1 times 1 times 4 equals 4.
Correct Response: B. The distance between 1/81 eighth and 3/83 eighths is 3/83 eighths −minus 1/81 eighth = 1/41 fourth unit. This distance is split into 6 segments such that each segment has a length of 1/41 fourth ÷divided by 6 = 1/241 twenty-fourth unit. Point D is located 3 segments away from 1/41 fourth + 3 •times 1/241 twenty-fourth = 1/81 eighth + 1/81 eighth = 1/41 fourth unit. Alternatively, consider that point D is located midway between 1/81 eighth and 3/83 eighths, which is 1/41 fourth.
Incorrect Response: A. In this response, Point D is located 3/73 sevenths of the way from point A to point H. That is, 2/24 + 3/7 2/3 − 2/24 = 1/32 twenty fourths plus 3 sevenths times the quantity 2 thirds minus 2 twenty fourths equals 1 third unit.
Incorrect Response: C. In this response, Point D is located 3/53 fifths of the way from point A to point F. That is, 1/12 + 3/5 1/2 − 1/12 = 1/31 twelfth plus 3 fifths tiems the quantity one half minus 1 twelfth equals 1 third unit.
Incorrect Response: D. In this response, Point D is located 3/43 fourths of the way from point A to point E. That is, 1/7 + 3/4 3/7 − 1/7 = 4/141 seventh plus 3 fourths times the quantity 3 sevenths minus 1 seventh equals 4 fourteenths unit.
Correct Response: C. The sequence 0, 4, 9.5, 16.5, 25, 35 increases in a quadratic pattern. This can be seen by the linear increase in the first differences: +plus4, +plus5.5, +plus7, +plus8.5, +plus10 (i.e.,the first differences increase by 1.5 with each term). Continuing this pattern leads to a diameter of 6 miles at day 35 + 11.5 = 46.5, a diameter of 7 miles at day 46.5 + 13 = 59.5, and a diameter of 8 miles at day 59.5 + 14.5 = 74.
Incorrect Response: A. This response may be based on the reasoning that the diameter of the spill increases at a constant rate of 4 miles every 25 days. This ratio cannot be doubled to 8 miles every 50 days because the diameter of the spill increases more slowly over time.
Incorrect Response: B. This response increases the number of days by 10 with each mile of diameter. This represents the difference between 35 days and 25 days. However, the sequence shows that this difference increases as the diameter grows. Repeated addition of 10 does not account for this dynamic.
Incorrect Response: D. This response may be based on the reasoning that the square of diameter is approximately equal to the number of days associated with the previous whole number diameter (i.e.,22 = 42 squared equals 4, 32 ≈ 9.53 squared is approximately 9 point 5, 42 ≈ 16.54 squared is approximately 16 point 5…). Continuing this pattern, 929 squared is approximately the number of days it takes for the spill to become 8 miles in diameter. This response does not describe the pattern as accurately as response C.
Correct Response: B. The value for the first term (when n = 1) is 2 and the sequence decreases by 8 with each new term. The expression that produces 2 for the first term and then decreases by 8 with each additional term is 10 − 8n.
Incorrect Response: A. This expression yields 2 − 8(1) = –62 minus 8 times 1 equals negative 6 for the first term. It would be correct if the first term corresponded to n = 0 (such as when x = 0 at the y-intercept of a linear equation), but it does not.
Incorrect Response: C. This expression yields 8(1) − 2 = 68 times 1 minus 2 equals 6 for the first term.
Incorrect Response: D. This expression yields 8(1) − 10 = –28 times 1 minus 10 equals negative 2 for the first term.
Correct Response: A. The amount of medication in the patient's bloodstream decreases by 10% each hour, which means that 90% of the medication present at a given moment will still remain in the bloodstream after 1 hour. After the first hour, 100(0.90) = 90 mg100 times 0 point 9 0 equals 90 milligrams remain. After the second hour, 90% of this value remains. This can be expressed as 90(0.90)90 times 0 point 9 0, but it is easier to connect back to the starting value to see the exponential nature of the relationship: 100(0.90)(0.90) = 100(0.90)2100 times 0 point 9 0 times 0 point 9 0 equals 100 times 0 point 9 0 squared. After t hours, there will be 100(0.90)t100 times 0 point 9 0 to the power of t milligrams of medication in the bloodstream.
Incorrect Response: B. A linear function changes by a constant rate, so the difference in the amount of medication present in the bloodstream between any two consecutive hours should be the same. This is not what happens. The data below show how a 100-mg100 milligram dose of medication will change when only 90% is retained each hour. The difference between any two consecutive hours varies.
A table with 2 columns and 5 rows. Column 1 head Number of hours since noon. Column 2 head amount of medication in bloodstream. Column 1 row 2 0 column 2 row 2 100. Column 1 row 3 1 column 2 row 3 90. Column 1 row 4 2 column 2 row 4 81. Column 1 row 5 3 column 2 row 5 72.9. Arrows to the right of the table show that data in column 2 changes by times 0.9 for each hour that passes.
Incorrect Response: C. A proportional function can be written as f(x) = kxf of x equals k times x. In this case, the amount of medication in the bloodstream divided by the number of hours since noon should always produce a constant value. They do not: 90 ÷divided by 1 = 90; 81 ÷divided by 2 = 40.5.
Incorrect Response: D. One characteristic of a quadratic function is the difference between two uniformly spaced values will change in a linear pattern. Consider the differences in the pattern below.
A table with 2 columns and 5 rows. Column 1 head Number of hours since noon. Column 2 head amount of medication in bloodstream. Column 1 row 2 0 column 2 row 2 100. Column 1 row 3 1 column 2 row 3 90. Column 1 row 4 2 column 2 row 4 81. Column 1 row 5 3 column 2 row 5 72.9. Arrows to the right of the table show that data in column 2 changes by minus 10 after the first hour, minus 9 after the second hour, and minus 8.1 after the third hour.
The first difference changes from –negative 10 to –negative 9 . If this were a linear pattern, the next value should be –negative 8, but it is –negative 8.1.
Correct Response: D. The width of the base of the monument can be represented by 12 −minus 4x, since 2x is needed for both the top and bottom sections of the stone path surrounding the base. The length can be represented by 20 −minus 2x, since x is needed for both the left and right sections of the stone path. The area of the base of the monument is (12 − 4x)(20 − 2x) = 240 − 24x − 80x + 8x2 = 8x2 − 104x + 240 = 8(x2 − 13x + 30)the quantity 12 minus 4 X times the quantity 20 minus 2 X equals 240 minus 24 X minus eighty X plus 8 X squared equals eight x squared minus 104 X plus 240 = 8 times the quantity X squared minus 13 X plus 30.
Incorrect Response: A. This expression represents an expansion of (20 − 2x)(12 − x)the quantity 20 minus 2 X times the quantity 12 minus X. This is the result of halving the width of the path rather than doubling it to account for the two pairs of sides.
Incorrect Response: B. This expression follows from (20 − x)(12 − 2x)the quantity 20 minus X times the quantity 12 minus 2 X. This does not account for the path having two pairs of sides.
Incorrect Response: C. This expression represents an expansion of (20 − 4x)(12 − 2x)the quantity 20 minus 4 X times the quantity 12 minus 2 X.
Correct Response: C. Three liters of a 3% acid solution contains 0.090 point 0 9 liters of acid. The ratio 0.09/30 point 0 9 divided by 3 compares the volume of acid to the total volume of this solution. When x liters of 15% acid solution are added to it, the ratio changes to 0.09 + 0.15x/3 + xthe quantity 0 point 0 9 plus 0 point 1 5 X all divided by 3 plus X. This ratio equals 0.120 point 1 2 when the correct amount of acid is added: 0.09 + 0.15x/3 + x = 0.12the quantity 0 point 0 9 plus 0 point 1 5 X all divided by 3 plus X equals 0 point 1 2. Apply the distributive property to obtain 0.15x + 0.09 = 0.12x + 0.360 point 1 5 x plus 0 point 0 9 equals 0 point 1 2 x plus 0 point 3 6.
Incorrect Response: A. The equation 0.09x + 0.15 = 0.12x0 point 0 9 x plus 0 point 1 5 equals 0 point 1 2 x has several errors. First, the quantity of 15% acid solution is fixed to 1 liter. To this, x liters of 9% acid is mixed in. The situation modeled by this equation answers a different question: "When is the total amount of acid in 1 liter of 15% acid solution and x liters of 9% acid solution the same as x liters of 12% acid solution?"
Incorrect Response: B. The equation 0.15x + 0.09 = 0.12x + 3 0 point 1 5 x plus 0 point 0 9 equals 0 point 1 2 x plus 3 could be the result of incorrectly applying the distributive property to 0.09 + 0.15x/3 + x = 0.12the quantity 0 point 0 9 plus 0 point 1 5 X all divided by 3 plus X equals 0 point 1 2.
Incorrect Response: D. As with incorrect response A, this response can be eliminated because the term 0.09x0 point 0 9 x would represent adding 9% acid solution to the mixture.
Correct Response: D. The solution set for the inequality can be found as follows:
Original problem: – 2/3 (3m − 2) < 1/2 (m + 1)negative 2 thirds times the quantity 3 m minus 2 is less than one half the quantity m plus 1
Apply the distributive property: –2m + 4/3 < 1/2 m + 1/2negative 2 m plus 4 thirds is less than one half m plus 1 half
Apply the addition property of inequalities: –2m − 1/2m < 1/2 − 4/3negative 2 m minus 1 half m is less than one half minus 4 thirds
Simplify: – 5/2m < – 5/6negative 5 halves m is less than negative 5 sixths
Apply the multiplicative property of inequalities: m > – 5/6 • – 2/5m is greater than negative 5 sixths times negative 2 fifths
Simplify: m > 1/3m is greater than 1 third
The graph should have an open circle at 1/31 third and shading to the right of 1/31 third. Graph D shows the solution set for the inequality.
Incorrect Response: A. One fast way to eliminate this response is by testing a point in the proposed solution set to see whether it satisfies the original inequality. Substituting the point m = –negative1 results in the inequality – 2/3 (3(–1) − 2) < 1/2 ((–1) + 1) → 10/3 < 0negative 2 thirds times the quantity 3 times negative 1 minus 2 is less than one half times the quantity negative 1 plus 1 implies 10 thirds is less than 0. This is a false statement, so this answer should be eliminated.
Incorrect Response: B. One fast way to eliminate this response is by testing a point in the proposed solution set to see whether it satisfies the original inequality. Substituting the point m = 0 results in the inequality – 2/3 (3(0) − 2) < 1/2 ((0) + 1) → 4/3 < 1/2negative 2 thirds times the quantity 3 times 0 minus 2 is less than one half times the quantity 0 plus 1 implies 4 thirds is less than one half. This is a false statement, so this answer should be eliminated.
Incorrect Response: C. This response could indicate a misconception with the multiplication property of inequalities. Following the steps from the correct response,
Original problem: – 2/3 (3m − 2) < 1/2 (m + 1)negative 2 thirds times the quantity 3 m minus 2 is less than one half the quantity m plus 1
Apply the distributive property: –2m + 4/3 < 1/2 m + 1/2negative 2 m plus 4 thirds is less than one half m plus 1 half
Apply the addition property of inequalities: –2m − 1/2m < 1/2 − 4/3negative 2 m minus 1 half m is less than one half minus 4 thirds
Simplify: – 5/2m < – 5/6negative 5 halves m is less than negative 5 sixths
The next step is to multiply both sides of the inequality by – 2/5negative 2 fifths, so the direction of the inequality must be reversed. The graph in the response could indicate that this change was not applied.
Correct Response: A. The middle graph shows that can C weighs less than can B, and the left graph shows that can B weighs less than can A. Therefore, can C weighs the least.
Incorrect Response: B. The left graph shows that cans A and B do not have the same height.
Incorrect Response: C. The middle graph shows that can C is more expensive than can B.
Incorrect Response: D. The right graph shows that can C is taller than can A.
Correct Response: D. The graph of f(x + 5)the function f of x plus 5 represents the horizontal translation of the graph of f(x)function f of x b units to the left. The function g(x)g of x = f(x + 5)function f of x plus 5 has a graph that is identical in shape to that of f(x)function f of x but is shifted to the left by 5 units.
Incorrect Response: A. This response may indicate a misconception in the direction of a shift. If f(x − 5)function f of x minus 5 were incorrectly interpreted to mean an upward shift of f(x)function f of x by 5 units, then (x + 5)2the quantity x plus 5 squared could be interpreted to mean that x2x squared was shifted down by 5 units.
Incorrect Response: B. This response may indicate a misconception in the direction of a shift. Interpreting f(x + 5)function f of x plus 5 to mean a shift of f(x)function f of x to the right by 5 units would lead to this result.
Incorrect Response: C. This response may indicate a misconception either in the direction of a shift or with the order of operations. Subtracting g(x) − 5function g of x minus 5 gives (x + 5)2 − 5 = x2 + 10x + 25 − 5 = x2 + 10x + 20the quantity x plus 5 squared minus 5 equals x squared plus 10 x plus 25 minus 5 equals x squared plus 10 x plus 20. This is not equal to f(x)function f of x.
Correct Response: A. The slope of each segment indicates the speed of the car in that segment. The car initially moves quickly and then slows down, speeds up, and slows down again. This is consistent with the description that the car began on the highway, slowed down in traffic, returned to highway speeds, and then slowed down on local roads. The constant slope back to the horizontal axis represents the steady rate home on local roads.
Incorrect Response: B. This response shows the distance from home always increases over time. However, the information states that the car returned home. A correct graph must show the car has 0 distance from home at the end of its journey.
Incorrect Response: C. This response shows the car returns home from its destination instantly. This is not what happened.
Incorrect Response: D. This response is similar to that of response B in that it shows the distance from home always increasing over time; however, the information states that the car returned home.
Correct Response: D. The ratio of # of students/# of teachers = x/y = 20/3 , so 3x = 20y, or y = 3/20 xnumber of students over number of teachers equals x over y equals 20 over 3, so 3 x equals 20 y or y equals 3 twentieths x. The graph of y = 3/20 xy equals 3 twentieths x is a line through the origin with slope equal to 3/203 twentieths. Graph D goes through (0, 0) and (100, 15) and has a slope of 15/100 = 3/2015 hundredths equals 3 twentieths.
Incorrect Response: A. This graph indicates that 12 teachers on a field trip would chaperone 0 students and that 80 students on a field trip would have 0 chaperones. This is not consistent with the situation.
Incorrect Response: B. This graph indicates that 60 teachers on a field trip would chaperone 0 students and that 9 students on a field trip would have 0 chaperones. This is not consistent with the situation.
Incorrect Response: C. This graph indicates that 200 teachers would chaperone 30 students, which is likely the result of interpreting the y-values to represent the number of students and the x-values to represent the number of teachers.
Correct Response: C. Line ℓL moves from (–2, k + 8)negative 2, k plus 8 to (2, k)2, k by moving down by 8 units and right by 4 units. Therefore, the slope of the line is –negative 2. The x-intercept of the line occurs where the value of the y-coordinate is 0. If the line starts from the point (–2, k + 8)negative 2, k plus 8, then it reaches its x-intercept by moving down by k + 8k plus 8 and to the right by half this distance. Its final position is –2 + 1/2 (k + 8)negative 2 plus 1 half k plus 8, and this expression is equivalent to k + 4/21 half the quantity k plus 4.
Incorrect Response: A. This expression represents the location of the y-intercept. The line moves 2 units to the right and 4 units down from the point (–2, k + 8)negative 2, k plus 8 to reach its y-intercept of (0, k + 4)0, k plus 4.
Incorrect Response: B. This expression may be the result of the reasoning described in the rationale for response A along with a sign error.
Incorrect Response: D. This expression may be the result of the reasoning described in the rationale for response C along with a sign error.
Correct Response: C. The slopes of parallel lines are equal. The slope of 3x + Ay = B3 x plus a y equals b can be found by solving the equation for y in terms of x: 3x + Ay = B → Ay = –3x + B → y = – 3/A x + B/A3 x plus a y equals b implies a y equals negative 3 x plus b implies y equals negative 3 over a times x plus b over a. Using point-slope form, y − y1 = m(x − x1)y minus y 1 equals m times the quantity x minus x 1, slope m = – 3/Anegative 3 over A and point (x1, y1) = (–4, 0)X 1, y 1 equals negative 4, 0 yields y = – 3/A (x + 4)y equals negative 3 over a times the quantity x plus 4 for the equation of line p.
Incorrect Response: A. The slope of a line written in the standard form ax + by = c is – a/bA x plus b y equals c is negative A over b. This response could be the result of calculating the slope of the first equation as a/bA divided by b.
Incorrect Response: B. This could be the result of combining the error described in response A and by misapplying the distributive property.
Incorrect Response: D. This could be the result of a distributive property error while applying the point-slope equation.
Correct Response: B. A family of lines is a set of lines that have something in common with each other. The set of lines shown in the graph all contain the point (2, –negative 3). The point-slope form of a linear equation is y − y1 = m(x − x1)y minus y 1 equals m times the quantity x minus x 1, where m is the slope of the line and (x1, y1)x 1, y 1 is a point on the line. In this case, the slope of the lines is the parameter k, and the point is (2, –negative 3). The equation for the family of lines is y − (–3) = k(x − 2)y minus negative 3 equals k times the quantity x minus 2, or y + 3 = k(x − 2)y plus 3 equals k times the quantity x minus 2.
Incorrect Response: A. The equation y + 3 = kx − 2y plus 3 equals k x minus 2 can be rearranged as y = kx − 5y equals k x minus 5. If this were true, then by the point-slope formula, the common point of intersection for the lines would be located at (0, –negative 5).
Incorrect Response: C. The equation y = kx + 2 − 3y equals k x plus 2 minus 3 simplifies as y + 1 = kxy plus 1 equals k x. If this were true, then the common point of intersection for the lines would be located at (0, –negative 1).
Incorrect Response: D. The equation y = k(x + 2) − 3y equals k times the quantity x plus 2 minus 3 can be rearranged as y + 3 = k(x + 2)y plus 3 equals k times the quantity x plus 2. If this were true, then the common point of intersection for the lines would be located at (–negative 2, –negative 3).
Correct Response: A. Let x represent the length of one side of the square pen and x2x squared the area of the pen. Increasing the length of one side by 4 meters and the other side by 2 meters creates a rectangle with length x + 4x plus 4 and width x + 2x plus 2. The new area, (x + 4)(x + 2)x plus 4 times x plus 2, is three times the old area, 3x23 x squared. Solving the equation (x + 4)(x + 2) = 3x2x plus 4 times x plus 2 equals 3 x squared will yield the value of x. First, distribute to multiply the binomials: x2 + 4x + 2x + 8 = 3x2x squared plus 4 x plus 2 x plus 8 equals 3 x squared. Then combine like terms and use the addition property of equality to set the equation equal to 0: 2x2 − 6x − 8 = 02 x squared minus 6 x minus 8 equals 0. Use the multiplication property of equality to simplify the equation by a factor of 2: x2 − 3x − 4 = 0x squared minus 3 x minus 4 equals 0.
Incorrect Response: B. This equation represents the expansion of (x + 4)(x + 2) = 0x plus 4 times x plus 2 equals 0. The area of the rectangular pen is not zero.
Incorrect Response: C. This equation may represent the sum of the new and expanded pens: x2 + (x + 4)(x + 2) = 0x squared plus the quantity x plus 4 times the quantity x plus 2 equals 0.
Incorrect Response: D. This equation may come from the equation (2x + 4) = 3x22 x plus 4 equals 3 x squared.
Correct Response: A. Substitute c, representing the y-intercept, into the equation to get y = x2 + bx − 36y equals x squared plus b x minus 36. The information given states that (3, 0) is also a point. Substitute this into the equation to determine the unknown value of b: 0 = 32 + b • 3 − 36 → 3b = 27 → b = 90 equals 3 squared plus b times 3 minus 36 implies 3 b equals 27 implies b equals 9. Obtain the other root by factoring the y = x2 + 9x − 36y equals x squared plus 9 x minus 36 as y = (x − 3)(x + 12)y equals the quantity x minus 3 times the quantity x plus 12. The other root is located at (–negative 12, 0).
Incorrect Response: B. A sign error, such as one developed from substituting c = 36 into the equation could lead to this response. For example, 0 = 32 + b • 3 + 36 → 3b = –27 → b = –90 equals 3 squared plus b times 3 plus 36 implies 3 b equals negative 27 implies b equals negative 9.
Incorrect Response: C. This response could represent the value obtained for bx when (3, 0) is substituted into the equation y = x2 + bx − 36y equals x squared plus b x minus 36: 0 = (3)2 + 3 • 3 − 360 equals 3 squared plus 3 times 3 minus 36.
Incorrect Response: D. A sign error, such as one developed from factoring y = x2 + 9x − 36y equals x squared plus 9 x minus 36 as y = (x + 3)(x − 12)y equals the quantity x plus 3 times the quantity x minus 12 could lead to this response.
Correct Response: C. The base of the cone is a circle. The net of the lateral surface of a cone is represented as a 60°60 degrees sector of a circle 10 cm10 centimeters in radius. It follows that the curved edge of the cone has a length of 1/6 × 2π(10) = 10π/31 sixth times 2 pi times 10 equals 10 pi divided by 3. This distance fully wraps around the circle at the base. To find the radius of that circle, set its circumference equal to the length of the curved edge of the lateral side and solve for r: 10π/3 = 2πr → r = 10/6 = 5/310 pi divided by 3 equals 2 pi r implies r equals 10 sixths which equals 5 thirds. The area of the circle is then π5/32 = 25/9 πpi times 5 thirds squared equals 25 ninths pi.
Incorrect Response: A. This answer is the result of not applying the square to both the numerator and the denominator in the final step of the solution shown for response C.
Incorrect Response: B. This response is the result of misinterpreting the radius of the circle to be 5/65 sixths, and then applying the power of 2 only to the numerator in the final step of the solution process shown for response C.
Incorrect Response: D. Making an error when solving for r in the solution shown for response C could lead to r = 5/65 sixths, which results in an answer of 25/36 π25 pi divided by 36.
Correct Response: A. The ratio for the smaller cube is Surface Area/Volume = 6x2/x3surface area divided by volume equals 6 x squared divided by x cubed. The ratio for the larger cube is Surface Area/Volume = 6(4x)2/(4x)3 = 16(6x2)/64x3 = 1/4 • 6x2/x3surface area divided by volume equals 6 times 4 x squared divided by 4x cubed equals 16 times 6 x squared divided by 64 x cubed equals 1 fourth times 6x squared divided by x cubed. In general, scaling the linear dimensions of a shape by k increases its surface area by a factor of k2k squared and its volume by a factor of k3k cubed. The ratio of these two values is k.
Incorrect Response: B. This response may indicate a misconception with the relationship between a linear dimension of a shape and its surface area and volume.
Incorrect Response: C. This response may indicate a misconception with the relationship between a linear dimension of a shape and its surface area and volume.
Incorrect Response: D. This response represents the reciprocal of the correct ratio.
Correct Response: D. When limited to a precision of 2 decimal places, the maximum overestimation possible in the edge of a cube is 10.49 inches. This leads to a maximum error in volume of 10.493 − 10310 point 49 cubed minus 10 cubed.
Incorrect Response: A. This represents the cube of the difference between the two side lengths. This does not represent the difference in the volumes of the two cubes.
Incorrect Response: B. At a precision of 0.01 inches, the greatest overestimate for the volume occurs when using 10 inches for a cube that has a true edge length of 9.5 inches and the greatest underestimate when the true edge length is 10.49 inches. In both cases, the error is relative to using the rounded value of 10 inches and not to each other. This misconception is further compounded with reasoning described in the rationale for response A.
Incorrect Response: C. A cube with an edge length of 9.5 inches is the smallest cube that would round up to an edge length of 10 inches. A cube of 9.4 inches would round to 9 inches, not 10, and so this calculation does not apply to this situation.
Correct Response: A. When solving geometric proofs, one can assume that points that appear to be collinear in a diagram are collinear. Thus, the assumption that can be made regarding the figure shown is that point A is on DCsegment D C.
Incorrect Response: B. This is true if triangle ABCA B C is equilateral, but there is no evidence in the diagram to support this claim.
Incorrect Response: C. There is no information provided in the diagram that supports that BDsegment B D is perpendicular to ACsegment A C.
Incorrect Response: D. There is no evidence to support this, such as congruency between alternate interior angles, in the image provided.
Correct Response: D. Consider the following diagram.
Right triangle A b c with line segment m n intersecting the vertical side and the opposite side to create a smaller triangle m b n. Side a c is 20, m n is 12, c n is 12, and n b is x.
Triangles ABCA B C and MBNM B N are similar by the triangle proportionality theorem (i.e.,the corresponding sides are parallel because MNM N is parallel to ACA C and intersects the sides of triangle ABCA B C). This establishes the proportion x/12 = x + 12/20x over 12 equals x plus 12 over 20 and the equation 12(x + 12) = 20x12 times the quantity x plus 12 equals 20 x. Determine x from the equation 12x + 144 = 20x → 8x = 144 → x = 1812 x plus 144 equals 20 x implies 8 x equals 144 implies x equals 18. The length of side CBsegment C B is 18 +plus 12 = 30.
Incorrect Response: A. This response may represent a misconception about similarity. If the triangles were considered to be related by subtraction, then 20 −minus 12 = 8 and 12 −minus 8 = 4; CNsegment C N = 12 + 4. However, similar triangles are related by multiplicative scale factors and not the subtraction shown here.
Incorrect Response: B. This represents the length of NBsegment N B. It must be added to the length of CNsegment C N to obtain the length of CBsegment C B.
Incorrect Response: C. This response may be based on an incorrect claim that ACsegment A C and CBsegment C B are congruent.
Correct Response: A. Using the diagram and the information given about the diagram, the following can be proven:
2 columns and 8 rows
| Statements | Reasons |
|---|---|
| 1.AD ≅ ABsegment A D is congruent to segment A B DC ≅ BCsegment D C is congruent to segment B C | 1.Given |
| 2.AC ≅ ACsegment A C is congruent to segment A C | 2.Reflexive Property |
| 3.△ADC ≅ △ABCtriangle A D C is congruent to triangle A B C | 3.SSSS S S |
| 4.∠DAX ≅ ∠BAXangle D A X is congruent to angle B A X | 4.CPCTCC P C T C |
| 5.AX ≅ AXsegment A X is congruent to segment A X | 5.Reflexive Property |
| 6.△DAX ≅ △BAXtriangle D A X is congruent to triangle B A X | 6.SASS A S |
| 7.DX ≅ BXsegment D X is congruent to segment B X | 7.CPCTCC P C T C |
The statement that is a true deduction based on the given information is DX ≅ BXsegment D X is congruent to segment B X.
Incorrect Response: B. Kite ABCDA B C D could satisfy the given information, which would mean that it is not necessarily true that both diagonals are perpendicular bisectors of each other. This means that AXsegment A X does not have to be congruent to CXsegment C X.
Incorrect Response: C. This statement is only true when ABCDA B C D is a square. Since a square is not the only shape to satisfy the given criteria, this statement is not necessarily true.
Incorrect Response: D. This statement is only true when ADsegment A D is parallel to BCsegment B C. If the figure is a kite, then these segments are not parallel.
Correct Response: C. The section marked D in the Venn diagram represents the quadrilaterals that have diagonals that are perpendicular bisectors of each other but are not congruent. A rhombus has perpendicular diagonals that bisect each other.
Incorrect Response: A. The diagonals of a kite are perpendicular, but only one diagonal must be bisected.
Incorrect Response: B. The diagonals of a rectangle are congruent, but not necessarily perpendicular.
Incorrect Response: D. The diagonals of a parallelogram are not necessarily perpendicular.
Correct Response: B. The volume of the cube is V = (2r)3 = 8r3 = 64 in3v equals 2 r cubed equals 8 r cubed equals 64 cubic inches. Divide 8r3 = 648 r cubed equals 64 by 8 to obtain r3 = 8r cubed equals 8. The value of r is the cube root of 8, which is 2. This is the radius of the sphere, which has a volume of V = 4/3 πr3 = 4/3 π(2)3 = 32π/3v equals 4 thirds pi r cubed equals 4 thirds pi times 2 cubed equals 32 pi over 3. The value of π is little more than the value of 3(π ÷ 3 ≈ 1.04)pi divided by 3 is approximately 1.04, so the approximation of the volume as 33 cubic inches is reasonable.
Incorrect Response: A. This response is the result of using the reciprocal of 4/34 thirds in the calculation for the volume of the sphere: V = 3/4 π(2)3 = 6π ≈ 19v equals 3 fourths times pi times 2 cubed equals 6 pi which is approximately 19.
Incorrect Response: C. This could be obtained by overestimating the value of π/3pi over 3 in the equation.
Incorrect Response: D. Using 4πr24 pi r squared for volume gives 4π • 22 = 16π4 pi times 2 squared equals 16 pi, which is approximately equal to this answer.
Correct Response: B. The top section of the rotated figure will be a hemisphere with radius z. The volume of the hemisphere is 1/2 • 4/3 • π • z3 = 2/3 πz31 half times 4 thirds times pi times z cubed equals 2 thirds pi times z cubed. The bottom section of the solid will be a cylinder with radius z and height 2z2 z. The volume of the cylinder is π • z2 • (2z) = 2πz3pi times z squared times 2 z equals 2 pi z cubed. The volume of the rotated figure is 2/3 πz3 + 2πz3 = 2/3 πz3 + 6/3 πz3 = 8πz3/32 thirds times pi times z cubed plus 2 pi z cubed equals 2 thirds pi z cubed plus 6 thirds pi z cubed equals 8 pi z cubed over 3.
Incorrect Response: A. This is the result of quartering the volume of the hemisphere instead of halving it. This leads to a volume for the hemisphere of 1/3 πz31 third times pi times z cubed, which results in a total volume calculation of 1/3 πz3 + 2πz3 = 7πz3/31 third pi z cubed plus 2 pi z cubed equals 7 pi z cubed divided by 3.
Incorrect Response: C. Forgetting to multiply the formula for the volume of a sphere by 1/21 half will result in the calculation of 4/3 πz34 thirds times pi times z cubed for the volume of the hemisphere, which leads to the result 4/3 πz3 + 2πz3 = 10πz3/34 thirds pi z cubed plus 2 pi z cubed equals 10 pi z cubed divided by 3.
Incorrect Response: D. If the 1/21 half is removed from the volume calculation for the hemisphere, and the radius of the hemisphere is interpreted to be 2z2 z, then the volume of the hemisphere is calculated as 8/3 πz38 thirds times pi times z cubed. This will result in the total volume calculation of 8/3 πz3 + 2πz3 = 14πz3/38 thirds pi z cubed plus 2 pi z cubed equals 14 pi z cubed divided by 3.
Correct Response: D. A circle that circ*mscribes a triangle has a center at the intersection of the perpendicular bisectors of the sides of the triangle. The slope of side ABsegment A B is 8 − 4/8 − 0 = 1/28 minus 4 over 8 minus 0 equals one half, making the slope of a perpendicular line to ABsegment A B = –negative 2. The midpoint of AB = 8 + 0/2, 8 + 4/2 = (4, 6)segment A B equals 8 plus 0 over 2, 8 plus 4 over 2 equals the point 4 6. The equation of the perpendicular bisector of ABsegment A B is y = –2x + 14y equals negative 2 x plus 14. The slope of side BCsegment B C is 0 − 4/2 − 0 = –20 minus 4 over 2 minus 0 equals negative 2, making the slope of a perpendicular line to BCsegment B C = 1/21 half. The midpoint of BC = 2 + 0/2, 0 + 4/2 = (1, 2)segment B C equals 2 plus 0 over 2, 0 plus 4 over 2 equals the point 1 2. The equation of the perpendicular bisector of BCsegment B C is y = 1/2 x + 1.5y equals 1 half x plus 1 point 5. The intersection of the perpendicular bisectors can be found by solving –2x + 14 = 1/2 x + 1.5 → 2.5x = 12.5x → x = 5negative 2 x plus 14 equals 1 half x plus 1 point 5 implies 2 point 5 x equals 12 point 5 x implies x equals 5 and y = –2(5) + 14 = 4y equals negative 2 times 5 plus 14 equals 4. The radius of the circle is the distance from (5, 4) to one of the vertices of the triangle. Using the vertex (0, 4),
= 5r equals the square root of the quantity 5 minus 0 squared plus 4 minus 4 squared equals 5 and the area of the circle is 52 • π = 25π5 squared times pi equals 25 pi. Incorrect Response: A. This may be the result of interpreting BCB C to be the diameter of the circle: 
= √ 20 = 2√ 5B C equals the square root of the quantity 4 minus 0 squared plus the quantity 0 minus 2 squared equals radical 20 equals 2 times radical 5. If this is the diameter of the circle, then the area is interpreted as πd/22 = π√ 52 = 5πpi times d over 2 squared equals pi times radical 5 squared equals 5 pi.
Incorrect Response: B. This answer may be the result of finding the circumference of the circle instead of the area.
Incorrect Response: C. This response may be the result of interpreting the segment ABA B to be the diameter of the circle: 
= √ 80 = 4√ 5A B equals the square root of the quantity 8 minus 0 squared plus the quantity 8 minus 4 squared equals radical 80 equals 4 times radical 5. If this is assumed to be the diameter of the circle, then the area is interpreted as πd/22 = π2 √ 52 = 20πpi times d over 2 squared equals pi times 2 radical 5 squared equals 20 pi.
Correct Response: C. Consider the median from point A to side BCB C: The midpoint of BCB C is 14 + 4/2, 0 + 12/2 = (9, 6)14 plus 4 over 2, 0 plus 12 over 2 equals the point 9 6. Since point A is the origin, the median from A to BCB C is included in the line 4 = 2/3 x → x = 64 equals 2 thirds x implies x equals 6. The center of mass (the intersection of the medians) must be a point on this line. Using substitution with y = 4y equals 4 (from the information given in the problem): 4 = 2/3 x → x = 64 equals 2 thirds x implies x equals 6.
Incorrect Response: A. This response may stem from writing the equation of the line as y = 9/6 xy equals 9 sixths x then substituting y = 4y equals 4 to get x = 8/3x equals 8 thirds.
Incorrect Response: B. This response may be the result of using a y-coordinate of 5 instead of 4 in the equation y = 3/2 xy equals 3 halves x.
Incorrect Response: D. This response may be the result of the misconception that the center of mass must be a point on the line x = 7x equals 7, since that line is the perpendicular bisector of side ABA B.
Correct Response: A. DE = 2D E equals 2 since E is the midpoint of DCsegment D C. GF = 2G F equals 2 and angle AGFA G F is a right angle since it is supplemental to angle DGFD G F, which is right for rectangles. AFsegment A F must be the hypotenuse of triangle AGFA G F since it is opposite the right angle, which makes AG = 2A G equals 2 since it is the other leg in isosceles triangle AGFA G F. A is given as (1, 1), therefore G is (1, 3) and DG = 5D G equals 5, so the area of the rectangle is 2 × 5 = 102 times 5 equals 10.
Incorrect Response: B. This response may be the result of interpreting point G to be located at (1, 2), which would lead to an area of 2 × 62 times 6 for the rectangle.
Incorrect Response: C. This answer may be the result of misinterpreting height of the rectangle to be 7 by comparing the y-values between points A and D.
Incorrect Response: D. This response may be the result of calculating the area of trapezoid ABCDA B C D: 1/2 (7 + 3) × 4 = 201 half times the quantity 7 plus 3 times 4 equals 20.
Correct Response: C. A bar graph is most appropriate for this situation because the data are comparing a single measure, the total sales, across six different brands. The total sales for each individual brand could be represented by the height of a bar, and the display would allow for easy visual comparisons of the total sales between each brand.
Incorrect Response: A. The most appropriate display for this set of data would be a circle graph or pie chart, since the goal is to show the percentages as parts of a whole.
Incorrect Response: B. The most appropriate display for this data set would be a scatter plot. This display would make it easiest to make observations about correlation between the variables.
Incorrect Response: D. The most appropriate display to show the change in a single variable over time is a line graph, so a bar graph is less appropriate for this situation than for the situation in response C.
Correct Response: A. The scores for the three seasons can be converted to a list:
7, 7, 9, 9, 9, 10, 10, 13, 14, 14, 18, 18, 19, 19, 20, 22, 25, 25, 28, 28, 29, 29, 32, 35, 35, 38, 40, 42, 42, 44
The median is the center value when the data set is ordered, which falls between 20 and 22 in this set, so the median is 21. The interquartile range is the difference between the first and third quartile. The first quartile, Q1Q 1, is the median for the lower half of the data, and the third quartile, Q3Q 3, is the median for the upper half of the data. Q1 = 13Q 1 equals 13, Q3 = 32Q 3 equals 32, so the interquartile range is the difference between those values, 32 − 13 = 1932 minus 13 equals 19.
Incorrect Response: B. This answer may be the result of miscalculating the median as 23, perhaps identifying the median between 22 and 25 and rounding or if copying the data into a list, making a copying error.
Incorrect Response: C. This answer may be the result of misinterpreting the interquartile range as the range. The interquartile range is the difference between the third quartile and the first quartile, but the range is the difference between the maximum value and the minimum value, in this case, 44 − 7 = 3744 minus 7 equals 37.
Incorrect Response: D. This answer may be the result of both errors from responses B and C. The median was determined incorrectly, and the interquartile range was misinterpreted for the range in this response.
Correct Response: C. The median for the data set is 4, and it remains 4 after 15 is added to the data set at the far right. (Generally, the median is less affected by outliers than the mean.)
Incorrect Response: A. The maximum changes from 8 to 15 once the value of 15 is included in the data set. The difference between those values is 7, which represents a more significant change than either the mean or the median.
Incorrect Response: B. The mean for the original data set is (0 + 0 + 1 + 1 + 1 + 2 + 4 + 4 + 4 + 4 + 5 + 6 + 6 + 7 + 8)/15 = 52/15 ≈ 3.5the sum of all values in the data set divided by 15 equals 52 over 15 which is approximately 3 point 5. After the inclusion of 15, the mean is (0 + 0 + 1 + 1 + 1 + 2 + 4 + 4 + 4 + 4 + 5 + 6 + 6 + 7 + 8 + 15)/16 = 67/16 ≈ 4.2the sum of all values in the data set plus 15 divided by 16 equals 67 over 16 which is approximately 4 point 2. (Generally, outliers influence the mean more than the median.)
Incorrect Response: D. The range for the original data set is 8 (8 − 08 minus 0), and it becomes 15 (15 − 015 minus 0) once the value of 15 is included in the data set. The change in the range is 7, which is greater than the change in the median and the mean.
Correct Response: A. For a normal distribution, the data are symmetrical about the mean with approximately 68% of the data existing within one standard deviation from the mean, 95% of the data existing within two standard deviations from the mean, and approximately 99% of the data existing within 3 standard deviations from the mean. The situation in the question is referring to the data that are greater than 1 standard deviation from the mean. Using the symmetry and percentage descriptions from a normal distribution, this implies that approximately 16% of the students have heights that fall in this range: 0.16 × 50,000 = 8,0000 point 1 6 times 50,000 equals 8,000.
Incorrect Response: B. This answer could have come from misinterpreting the meaning of 68% of data existing within one standard deviation of the mean and incidentally summing the percentages at both ends of the normal distribution. This number represents the number of students with heights that are at least 173 cm173 centimeters and the number of students with heights that are no greater than 163 cm163 centimeters: 0.16 × 2 × 50,000 = 0.32 × 50,0000 point 1 6 times 2 times 50,000 equals 0 point 3 2 times 50,000.
Incorrect Response: C. This value represents the number of students who are within 1 standard deviation from the mean. These students have heights between 163 cm163 centimeters and 173 cm173 centimeters: 0.68 × 50,000 = 34,0000 point 6 8 times 50,000 equals 34,000.
Incorrect Response: D. This response likely stems from a misinterpretation of the data that are specified in the question. This value represents the number of students who are no taller than 173 cm173 centimeters. It is the complement of the data set from what is specified in the question: 0.84 × 50,000 = 42,0000 point 8 4 times 50,000 equals 42,000.
Correct Response: C. Without knowing the demographics of the communications collected, this is a potential concern that could misrepresent the constituency. There is not enough information provided to know that the sample is random, so there is no way to guarantee that it is representative of the entire constituency. (The bias that can exist in a survey where the participants are volunteers is the same source of bias that may exist in this context.)
Incorrect Response: A. Generally, a larger sample is a better representation than a smaller sample, but only if the sample is random. There is no guarantee that the sample is random in this situation, so the results may be disproportionately impacted by a specific group, which could make the data an invalid representation of all constituents.
Incorrect Response: B. Even though access promotes broad participation, it does not ensure that an ideologically diverse group of people exercise the right to vote. The constituents who did vote may have been part of a specific, highly motivated group, which could have skewed the data. Access does not ensure that the sample is random.
Incorrect Response: D. The outcome of the bill should be the result of the opinions of the constituents, so this is not an accurate assessment of potential bias in the results.
Correct Response: C. The data can be represented with a Venn diagram, where C = the set of students who saw the comedy movie and A = the set of students who saw the animated movie. The intersection of both sets, which is the percentage of students who saw both movies, is 14%.
A venn diagram with circles labeled C and A that are partially overlapping. The overlapping region is labeled as 14 percent.
Since 55% of students saw the comedy movie, and this includes the 14% who saw both, then 55 − 1455 minus 14 or 41% of students saw only the comedy movie. Likewise, the 28% of students who saw the animated movie includes the 14% who saw both. So, 28 − 1428 minus 14 or 14% saw only the animated movie.
A venn diagram with circles labeled C and A that are partially overlapping. The overlapping region is labeled 14 percent, the region that represents only C is labeled 41 percent, and the region that represents only A is labeled as 14 percent.
Since the Venn Diagram includes 100% of the students, the percent of students who saw neither movie must be 100 − (the sum of the percentages within the sets)100 minus the sum of the percentages within the sets or (100 − (41 + 14 + 14)) = 100 − 69 = 31100 minus the quantity 41 plus 14 plus 14 equals 100 minus 69 equals 31.
A venn diagram with circles labeled C and A that are partially overlapping. The overlapping region is labeled 14 percent, the region that represents only C is labeled 41 percent, and the region that represents only A is labeled as 14 percent. The region outside of both circles is labeled as 31 percent.
Incorrect Response: A. This response may be the result of adding the 55% and 28% in the question without considering how the overlap affects those percentages. Then, 100 − 87100 minus 87 produces 13%.
Incorrect Response: B. This response may be the difference between the percentage of students who saw neither movie and the percentage of students who saw both movies, or 31 − 14 = 17%31 minus 14 equals 17 percent.
Incorrect Response: D. This response is the percentage of students who saw only the comedy movie.
Correct Response: B. Probabilities can be assigned to each of the first three branches of the tree diagram using the fact that there are 12 markers total in the box. Information regarding what fraction of each marker is dry can be used to assign probabilities to the second set of branches.
Red dry 1 half Green dry 1 fourth Blue dry 1 half To the right of each pair of branches at the final level of the diagram are the following probabilities: 1 twelfth, 1 twelfth, 1 fourth. 
The tree diagram is shown from the question with probabilities labeling the branches. On the first level, red, green, and blue are labeled 1 sixth, 1 third, and 1 half, respectively. On the second level. Which represents dry or not dry for each color, the probabilities are as follows:
The product of the probabilities along the branches in each path shows the probability that a given marker meets both criteria (e.g.,is green and dry = 1/121 twelfth). To find the probability that a randomly selected marker is dry, we must add the probabilities for each path that represents a dry marker. The sum of the possibilities for drawing a dry red, dry green, or dry blue marker is 1/12 + 1/12 + 1/4 = 5/121 twelfth plus 1 twelfth plus 1 fourth equals 5 twelfths.
Incorrect Response: A. This result could be due to a misinterpretation of the meaning of the first three branches in the tree diagram.
Incorrect Response: C. This result could be a misinterpretation of the probability specified in the question. This is the probability that a randomly selected marker will be blue.
Incorrect Response: D. This result could be a calculation of the complement for the probability specified in the question. This is the probability that a randomly selected marker is not dry.
Correct Response: A. The probability of selecting the first red number cube from the bag is 2/32 thirds, and the probability of selecting a second red number cube is 1/21 half, since the number cubes are drawn without replacement. The probability of selecting two red number cubes from the bag is then 2/3 × 1/2 = 1/32 thirds times 1 half equals 1 third.
Once the two red number cubes are selected, there are 2 possible ways to roll a resulting sum of 3: (1, 2) or (2, 1). There are 6 × 66 times 6 or 36 total possibilities for rolling two number cubes, so the probability of rolling a sum of 3 is 2/36 = 1/182 thirty-sixths equals 1 eighteenth. The probability of selecting two red dice and rolling a sum of three is 1/3 × 1/18 = 1/541 third times 1 eighteenth equals 1 fifty-fourth.
Incorrect Response: B. This answer may be the result of interpreting the probability of choosing two red number cubes as 1/21 half, and correctly calculating the probability of rolling a sum of three. This would lead to calculating the probability of selecting two red number cubes and rolling a sum of three as 1/2 × 2/36 = 1/361 half times 2 thirty-sixths equals 1 thirty-sixth.
Incorrect Response: C. This answer may be the result of errors related to the formation of the sample space, or it may be the result of incorrectly adding probabilities for different events in the problem.
Incorrect Response: D. This answer may be the result of calculating the probability of rolling a sum of 3 on two number cubes without considering the probability that the two chosen number cubes are red 2/36 = 1/182 thirty-sixths equals 1 eighteenth.
Correct Response: C. If we define the number of green grapes originally in the bag to be x, then the number of green grapes in the bag after 5 are removed is equal to x − 5x minus 5 and the total number of grapes in the bag is x − 5 + 25 = x + 20x minus 5 plus 25 equals x plus 20. We can then say that x − 5/x + 20 < 0.25 → x − 5 < 0.25(x + 20) → x − 5 < 0.25x + 5 → 0.75x < 10 → x < 13.3x minus 5 divided by x plus 20 is less than 0.25 implies x minus 5 is less than 0.25 times the quantity x plus 20 implies x minus 5 is less than 0.25 x plus 5 implies 0.75 x is less than 10 implies that x is less than 13 point 3 repeating.
Since x must be less than 13.313 point 3 repeating, and we are assuming there are only whole grapes in the bag, the greatest number of green grapes possible at the start is 13. This means that there could have been as many as 13 + 25 = 3813 plus 25 equals 38 grapes in the bag originally.
Incorrect Response: A. This response is the result of finding the correct answer, 38, but then adding the value of 5 back to it. The solution process outlined in the correct response finds the total number of grapes before 5 are removed, so adding 5 to that value is unnecessary.
Incorrect Response: B. The same solution process from the correct response is used, but the result is rounded up to the nearest whole value, which could lead to the answer of 14 + 2514 plus 25 or 39.
Incorrect Response: D. This may be the result of solving the original inequality, which leads to an answer of x < 6.6x is less than 6 point 6 repeating, which could lead to the answer 25 + 6 = 3125 plus 6 equals 31.
Correct Response: B. There is a one-to-one correspondence between the sample spaces for the range of numbers from 1–through 26 and the set of letters from A–to Z . This simulation samples from each set in the same manner, and each of the 100 sets can be easily compared to the criteria to check for at least one vowel.
Incorrect Response: A. The question does not specify that each of the random five letter samplings must be words, so this simulation does not accurately describe the scenario.
Incorrect Response: C. This simulation assumes that there is an equal probability of choosing a vowel or a consonant when randomly choosing a letter of the alphabet, which is not true.
Incorrect Response: D. The simulation described does not allow for the repetition of letters within a five-letter sample, which is not consistent with the scenario in the question. Additionally, the simulation involves drawing samples until "only one vowel" is drawn, rather than at least one vowel.
Shape and dimensions of the levee (Correct Response A) will affect how it functions in preventing flooding. On the other hand, cost (Incorrect Response B) and transportation (Incorrect Response C) of the materials have a smaller effect on the levees' ability to be breached, compared to its design and components. The devices' ability to support vehicles on the adjacent road (Incorrect Response D) is an important factor to consider; however, this question focuses more on problems that may arise from the effects of vehicles than how the levee will function to prevent flooding in the area.
Since the warning system has already been developed, the modeling and testing of prototypes (Correct Response C) must be the next step in the engineering design process. Defining the problem (Incorrect Response A), brainstorming a solution (Incorrect Response B), and researching criteria and constraints (Incorrect Response D) are steps that occur before the development of a prototype; therefore, they are not considered during the finalization part of this process.
The first action to take when evaluating a bridge design is calculating the necessary strength of the components based on its extreme load capacity (Correct Response B). The engineer's design must not only be able to withstand everyday traffic but also be prepared for the events of extreme stress that might arise while the bridge is functional, to protect people, property, and infrastructure within the area. Comparing tensile strength (Incorrect Response A) and selecting materials (Incorrect Response C) are parts of the engineering process that generally occur before the evaluation step and primarily occur during either the research or building a prototype stages. Testing to determine if the bridge has the ability to hold a variety of loads (Incorrect Response D) is important and must occur, but not necessarily before evaluation can begin.
To test the effectiveness of each mechanical arm, the students must determine its maximum capacity. Evaluating the limitations of the different prototypes is the best method for determining which design best achieves the goal of the project (Correct Response A). Calculating the durability of the materials (Incorrect Response B), moving the arm to perform various tasks (Incorrect Response C), and timing how long the arm can hold on to an object (Incorrect Response D) are important components to consider in the overall design. However, these tests focus on either prototype quality or other criteria in relation to functionality without specifically evaluating them against the goal of creating a device that can grasp and lift objects.
Steel has greater strength and sturdiness (Correct Response D) in comparison to the other materials that are commonly used to construct patio furniture. Even though steel can be resistant to corrosion (Incorrect Response A), other material, such as plastic, can also withstand outdoor conditions. Steel has the capability to be offered in a wide array of shapes and colors (Incorrect Response B), but the other materials can also be painted, tinted, and shaped. Lastly, comfort (Incorrect Response C) is a subjective measure that depends more on the shape of the chair rather than on its composition. Steel and aluminum can be an exception as they often become hot to the touch when left out in the sun.
Torque is a measure of rotational force. Of the options available, only the wrench (Correct Response B) is primarily used to cause an object to rotate on its axis. A mallet (Incorrect Response A) can be categorized as a third-class lever. In this type of simple machine, the force applied moves in the same direction as the force produced. A block and tackle device (Incorrect Response C) is also a simple machine, but the force applied is in the opposite direction as the force produced. A rivet gun (Incorrect Response D) does not fit easily into the different categories of simple machines, but it functions like the mallet in that the force is linear, not rotational.
Diamond-dust drill bits are used to drill holes in tile, porcelain, glass, or stone because diamonds are the hardest natural material. Hardness depends on the molecular structure of the material, and diamonds are particularly hard due to their lattice arrangement of carbon atoms double bonded to other carbon atoms (Correct Response B). Other properties, such as friction coefficient (Incorrect Response A), durability (Incorrect Response C), and melting point (Incorrect Response D), are more important to other physical or chemical changes that occur within structures of materials.
Hot-dip galvanization is an example of conditioning processes in manufacturing (Correct Response A) because the step consists of coating a product in order to improve its characteristics by changing its properties to become more resistant to oxidation and corrosion, compared to primary processes that focus on shaping and sizing a product from raw material. Finishing (Incorrect Response B) is the process of altering the surface of a product, such as cleaning, painting, or removal of defects. Unlike conditioning, the primary focus of this step is appearance or extension of product life during the final stages of completion of the product, rather than modifying its internal properties to a specified criteria of functionality of the final product. Assembling (Incorrect Response C) is the step in the manufacturing process that constructs the parts of the product into a whole unit through various fastening or jointing techniques. And forming (Incorrect Response D) is the process of shaping, bending, or stretching the product into a desired shape and does not add or remove material compared to the galvanization process that occurs in this scenario.
The fourth line in the table (Correct Response D) best matches the tool with the correct safety procedure because any time a person is handling any type of nail gun the tool needs to be pointing away to prevent accidental discharge. A handheld chisel (Incorrect Response A) is generally constructed out of one or two components. Since this equipment is quite solid, securing loose parts before use is not required. There are no guards on a handheld wrench (Incorrect Response B). Lastly, when using an electrical belt sander (Incorrect Response C) the tool does not need to be sharpened before use because there are no sharp edges.
The role of embedded electronic microcontrollers (Correct Response D) is to measure the exhaust gasses produced by combustion engines. This information allows the users or mechanics to adjust the engine in order to increase efficiency and reduce emissions. The laws of thermodynamics have not changed in recent years, compared to major advancements in technology (Incorrect Response A). The extraction of fuels (Incorrect Response B) has not changed much or as dramatically in the last 40 years and has no bearing on the efficiency of the engine, as octane fuels are still being used in the combustion process. Any advancement in aerodynamics (Incorrect Response C) primarily effects vehicle efficiency as a whole, rather than the efficiency of an engine that is protected within the vehicle.
Cell-phone technology relies on radio waves moving through the air to communicate; therefore, it does not require the extensive ground infrastructure (Correct Response B) that landline telephones, which transmit signals through fibers, rely on. Cell phones do have the capability to provide functionality of computers, radios, or television (Incorrect Response A), as well as the ability to transmit text, images, and videos (Incorrect Response C), but these features are not responsible for their spread into previously unconnected regions. The ability to interchange software components (Incorrect Response D) depends on the make and model of the phone and has little relevance to where the cell phone is being used.
In order for the airlines to determine which flights will be available at particular dates and times, the system needs to know the fleet's passenger capacity (Correct Response A). The system updates each time a seat is taken, allowing the airlines to use this information to chart which dates and times the planes fill the most often and compare that to the destinations that are the most popular. The number of flights to be scheduled in a day (Incorrect Response B) is important for air traffic controllers but does not inform the airlines which planes and routes are highest in demand. Net profits earned each day (Incorrect Response C) and price per seat on a flight (Incorrect Response D) depend upon what the airlines decide to charge. Airlines may consider supply and demand data in their calculations, but it is not a requirement when setting the costs of services and goods.
A processing device in a technical system is a chip within a computer component, used to understand and perform instructions based on incoming and outgoing data from operational systems or hardware. A microcontroller (Correct Response A) is the processor because it uses the incoming photo data to instruct the device to turn on or stay off within a set amount of time. The power supply (Incorrect Response B) functions to give energy to the system. Photoresistorsphoto re zis tors (Incorrect Response C) are a component used to affect photo resistance of any input luminosity information. And the lightbulb (Incorrect Response D) is the outgoing component that turns on or off based on the microcontroller; without the connection, the bulb would neither turn on nor stay off by itself.
The equation used to calculate the orbital velocity of a celestial body revolving around another body is Vorbit =V, for a circular orbit, is equal to
. G is equal to gravitational constant, M is the mass of the body at its center, and r is the radius of its orbit. Therefore, changes to gravitational force and change in radial distance between the two bodies are the best factors used to explain the difference in their orbital velocity (Correct Response B). Changes in density (Incorrect Response A), tilt of orbit (Incorrect Response C), and length of orbital path (Incorrect Response D) have effects on the planets, but they are not used to determine their velocity. A total lunar eclipse does not occur every month during the full moon phase because the Moon's orbital plane is tilted at an incline relative to Earth's plane of motion around the Sun (Correct Response A). A lunar eclipse occurs when all the bodies are aligned in the same plane and the Moon enters the shadow cast by the Earth. A lunar eclipse only appears to a viewer on the side of Earth that is currently experiencing night (Incorrect Response B). The distance between the Earth and the Sun has little effect on the probability of an eclipse (Incorrect Response C). Earth is larger than the Moon, so the shadow will completely cover the Moon when they are in the proper alignment (Incorrect Response D).
The oldest recorded fossils known today are prokaryote microorganisms, similar to bacteria or archaea. Eventually during the Proterozoic eon, eukaryotic cells evolved, leaving fossilized remains. All of these types of organisms lacked the hard body parts (Correct Response D), like bones and shells, that are resistant to decay and are more likely to fossilize. Having a large population size (Incorrect Response A) or body size (Incorrect Response C) can increase the chance of some members leaving behind fossil remnants, but the organisms are much more likely to fossilize if they have structures that more readily mineralize. Particularly during the Proterozoic eon, life on Earth thrived in shallow seas, where fossilization is more likely to occur than on land (Incorrect Response B).
Growing salt crystals best models the process of chemical sedimentary rock formation. The crystals form as the salt precipitates out of a supersaturated solution, which is a process that is very similar to how inorganic limestones (Correct Response A), such as oolitic limestones, stalactites, and stalagmites, form. Sandstone (Incorrect Response B) is formed by the cementation of sediments, grains, or fragments under the pressure of other layers of sediment and rock. Metamorphic granite (Incorrect Response C) is formed from its parent rock being subjected to high pressure and heat deep within the earth. Basalt (Incorrect Response D) is formed from the cooling of lava over a short period of time.
Chemical weathering occurs when there are changes made to the molecular structure of a rock or mineral. Water can act as a weak acid, breaking apart the calcium carbonate of the limestone (Correct Response C). Mechanical weathering leaves the molecular structure of the material intact but may cause the material to move (Incorrect Responses A and D) or break into smaller pieces (Incorrect Response B).
The Coriolis effect is driven by the differences in the amount of solar radiation combined with the rotation of the Earth on its axis. These two phenomena work together to cause masses of moving air to deflect right or left depending on their latitude. The prevailing direction of the wind causes the oceans to form circular gyres as the water continues to move and be deflected (Correct Response D). The transfer of heat from the tropics to the poles (Incorrect Response A) is a result of wind circulation and ocean gyres but is not the source for their deflection or formation. The movement of tides (Incorrect Response B) originates from forces related to the gravitational relationship between the Earth, Moon, and Sun. The evaporation of surface water (Incorrect Response C) occurs when solar radiation reaches the top of the water body and causes the liquid to turn into water vapor into the atmosphere. It can affect weather conditions but has nothing to do with deflection of winds and formation of ocean gyres.
The ozone layer is a component of the stratosphere, which is the layer that extends from 10 kmkilometers to 50 kmkilometers above the Earth's surface (Correct Response B). Air samples from altitudes below 10 kmkilometers (Incorrect Response A) or higher than 50 kmkilometers (Incorrect Responses C and D) would not give any indication of the health of the ozone layer.
Onshore breezes occur on warm summer afternoons because the land heats more rapidly than the water. The warm air over the land rises, and the cooler air over the water moves in to take its place (Correct Response D). Trade winds (Incorrect Response A) and jet streams (Incorrect Response C), on the other hand, are a result of the differences in heat energy at various latitudes. The variation in their strength is more affected by the tilt of the Earth than what is at the Earth's surface. Spring tornadoes (Incorrect Response B) result from the interactions of large, unstable air masses, rather than from the local, surface-level heating differences.
If there is a deep warm layer of air and only a thin, extremely cold layer of air, then the precipitation will not have time to freeze as it travels towards the surface. It will, however, freeze upon impact (Correct Response D). Snow pellets (Incorrect Response A) will fall if the layer of cold air is deeper, since the precipitation will have a chance to freeze while still in the atmosphere. Wet snow (Incorrect Response B) occurs when there is a thin layer of warm air near the surface, while sleet (Incorrect Response C) requires a shallow warm layer situated above a surface-level cold layer.
The difference between Doppler radar and traditional radar is that Doppler radar can use the Doppler effect to determine not only the presence of precipitation but its velocity as well (Correct Response B). The snow water equivalent (Incorrect Response A), total precipitation (Incorrect Response C), and precipitation intensity (Incorrect Response D) can all be directly measured or calculated using both traditional and Doppler radar.
The western part of the United States experiences greater tectonic activity compared to the eastern United States, due to the faulting and deformation that results from the interactions between the Pacific and North American plate (Correct Response B). Although some western states may have drier and hotter climates (Incorrect Response A), the surface temperatures only have a small effect on the temperatures of the water deep below the surface. The thickness of the crust (Incorrect Response C) and amount of vegetation (Incorrect Response D) vary across the continent and minimally affect the groundwater temperatures.
A positive climate feedback loop is where a change to one part of the system causes changes that amplify or increase the initial conditions. In this case, something causing an increase in global temperatures will trigger a series of events that causes global temperatures to continue to increase. This type of feedback loop is best exemplified through melting permafrost (Correct Response C). Melting of permafrost due to warming global temperatures will release more gases into the atmosphere, which contributes to the greenhouse effect, resulting in even warmer temperatures. Increased snowfall (Incorrect Response A), decomposing vegetation (Incorrect Response C), and slowing of thermohaline circulation (Incorrect Response D) represent negative climate feedback loops because these factors decrease the rate of change to the initial conditions.
Percolating water through active charcoal is most effective at removing toxins because charcoal has the ability to absorb and trap contaminates as the water passes through it (Correct Response D). Boiling water (Incorrect Response A) and exposing it to UV light (Incorrect Response C) are primarily used to disinfect and eliminate microbiological pathogens, but they do not physically remove other particles or chemicals that are harmful if consumed. While traces of ozone (O3O 3) present with oxygen are highly effective at killing biological contaminants and can react with certain heavy metals causing them to precipitate out, if an additional filtration step is not added those materials will still be present in the water (Incorrect Response B).
Meiosis is a cell division process that results in four haploid daughter cells, while mitosis produces two diploid cells (Correct Response C). The number of cells produced by each process is swapped in Incorrect Response A. The daughter cells produced through cell division (Incorrect Response B) may vary in size, but this is not a defining characteristic between these two processes. Both daughter cells of mitosis and meiosis contain a complete set of organelles and the structures needed in order to function properly (Incorrect Response D).
There are six defined characteristics of life: the ability to reproduce (Correct Response B), the ability to use energy, the ability to grow and change, having stable internal conditions, being organized, and being composed of at least one cell. It is possible that life does not need to possess carbon-containing molecules (Incorrect Response A). There are also many organisms that are at or colder than their surrounding temperatures (Incorrect Response C). All living things are organized but are not necessarily highly ordered (Incorrect Response D), as exemplified by organisms such as amoebas.
Structure X is the nucleus, which contains the genetic material of the cell (Correct Response C). The cellular membrane, which is the barrier of the cell, controls ion exchange (Incorrect Response A). The vacuoles of the cell, shown here as circles or ovals near the cell membrane, store nutrients of the cell (Incorrect Response B). Cellular waste products are broken down by lysosomes, which carry enzymes for that function (Incorrect Response D).
Fermentation is an anaerobic process, or one that is performed without oxygen, while cellular respiration is an aerobic process, which means that it requires oxygen to be performed (Correct Response C). All different sized cells can undergo either process (Incorrect Response A). Although aerobic processes produce more ATP, which can be used for the cell's energy needs, it takes time to switch between anaerobic and aerobic processes (Incorrect Response B). Light availability, or even lack of light, does not necessarily initiate fermentative processes (Incorrect Response D).
The data in the table indicates a slow growth of one or two birds a day and then jumps to double in size overnight, which is shown in the graph in Correct Response A. This is not a linear relationship, as seen in the graph in Incorrect Response B, and population is increasing over time rather than decreasing, which is depicted in the graphs in Incorrect Responses C and D.
Ecological succession is the gradual development of a complex ecosystem as a site moves through several stages of plant and animal communities (Correct Response D). Evolution (Incorrect Response A) is a change that happens within a species, not on an ecosystem scale. Biological magnification (Incorrect Response B) does occur at the ecosystem level, but it is referring to the increasing concentration of a contaminant, such as mercury, as it moves through the trophic pyramid. Exponential growth (Incorrect Response C) refers to a mathematic model. It could apply to an increase in the number of species or biomass in the field, but it does not explain the change in species presence over time.
The stigma is adapted for receiving pollen because during the pollination process, the grains transfer from the anther to the sticky stem of a stigma (Correct Response C). The petal (Incorrect Response A) functions to attract pollinators, the anther (Incorrect Response B) is the organ that produces the pollen, and the ovary (Incorrect Response D) of a flowering plant is where seed development and fertilization occur.
The phenotypic ratio can be found by building a Punnett square with one parent as Bbbig B little b and one parent as bblittle B little B, as shown.
| alleles | Bbig B | blittle B |
|---|---|---|
| blittle B | Bbbig B little B | bblittle B little B |
| blittle B | Bbbig B little B | bblittle B little B |
When DNA is replicating under normal conditions, A pairs with T, while C and G pair (Correct Response B). Incorrect Response A does not use this complementary rule and instead assumes that the complementary DNA strand is the same as the original sequence. Incorrect Response C represents a strand of complimentary RNA, while Incorrect Response D incorrectly pairs A and C, while T and G pair.
A variant gene can remain in a population, even if it poses no advantage, because these genes can be recessive and are not expressed (Correct Response C). Genetic drift (Incorrect Response A), mutations (Incorrect Response B), and natural selection (Incorrect Response D) may explain the existence of a variant gene, but these scenarios affect the communities through favoring an advantage or disadvantage of gene expression. Also, this does not explain why these phenotypes are not expressed in the majority of individuals.
Unique mitochondrial and chloroplast DNA indicate that these organelles must have, at one time, been individual prokaryotes. Over time, these prokaryotes would have formed a symbiosis that eventually became a dependency (Correct Response C). The formation of organic molecules was vital to the emergence of life, but this process does not directly explain why these organelles have their own DNA (Incorrect Response A). The emergence of oxygen in the atmosphere was the result of the waste product of photosynthetic organisms, which allowed for aerobic organisms to evolve (Incorrect Response B). However, if these organisms had chloroplasts, they would not necessarily need their own DNA. While there is much support for life arising in the oceans, this does not impact whether DNA is needed for an organelle to function (Incorrect Response D).
The theory of evolution states that: (1) the trait being selected for must have more than one allele, (2) that trait must be inherited, (3) there are more offspring under selection than can survive, and (4) different alleles confer different survival rates. With this in mind, the population of insects has a gene that is being selected for (the resistance allele) and it has selective pressure on it (those without the allele succumb to the pesticide) (Correct Response D). A hare's coat turning colors is the result of environmental changes, but there is no difference in trait shown across the population (Incorrect Response A). Becoming more skilled at hunting is an acquired, not inherited, trait (Incorrect Response B). The root system is not placed in context of how the rest of the population is surviving (Incorrect Response C).
"Pure substance" is a term used to describe chemical elements and compounds, substances with a fixed structure down to the atomic level that can be described with a single chemical formula, such as alcohol (Correct Response A). Milk (Incorrect Response B) contains both water- and fat-soluble compounds, as well as water itself. Gasoline (Incorrect Response C) is a mixture of hydrocarbons, each with its own chemical formula. Blood (Incorrect Response D) is a water-based solution of organic molecules that also contains a variety of living cells.
Hydrogen bonds are a type of intermolecular force that can form between the oxygen atom in one water molecule and a hydrogen atom in another water molecule. These forces are relatively strong compared to other molecules of similar molecular weights and are responsible for some properties of water, including surface tension (Correct Response C). The size of atoms (Incorrect Response A) can be a proxy for the strength of dispersion forces, another intermolecular force, but the size of the atoms in a water molecule is unremarkable. The solvent properties of water (Incorrect Response B) are produced by the molecule's ability to attack solute particles. The specific heat capacity of water (Incorrect Response D) is another effect of hydrogen bonding.
The reaction of hydrochloric acid with sodium hydroxide produces a solution of sodium chloride (Correct Response C). In Incorrect Responses A and B, a chemical reaction occurs so that a reactant will not be produced. The reactant oxygen atoms (Incorrect Response D) become part of the water produced in the products and so are unavailable.
A dependent variable in a scientific investigation is a measured effect of what is being tested or changed within the study. In this scenario, the amount of heat conserved is the independent variable because it is the effect that occurs when a different insolation material is applied to each iteration of the design (Correct Response D). The insulated material (Incorrect Response A) is the independent variable because it is the aspect of the experimental set up that the researcher is intentionally changing and is trying to test. The time (Incorrect Response B) and volume (Incorrect Response C) are the controls because they remain the same for each trial to create the same conditions across the experiment to produce comparable results.
Holding volume and temperature constant, there is a direct relationship between the amount of gas and its pressure. Decreasing the amount of gas in the chamber decreases the pressure (Correct Response D). Holding volume and the amount of gas constant, there is a direct relationship between the temperature of a gas and its pressure. Increasing the temperature of the gas in the chamber increases the pressure (Incorrect Response A). Increasing the amount of gas in the chamber increases its pressure (Incorrect Response B). Holding the amount of gas and its temperature constant, there is an inverse relationship between the volume of a gas and its pressure. Decreasing the volume of a gas would increase its pressure (Incorrect Response C).
The object with the greatest potential energy is object B (Correct Response B). Using the formula provided, the result of multiplying the three values together is equal to PE = (5)(9.8)(3)P E = 5 times 9.8 times 3 = 147 m/s2meters per second squared. Substituting the appropriate variables for objects A, C, and D within the same formula, the answers are 68.3 m/s2meters per second squared, 98 m/s2meters per second squared, 49 m/s2meters per second squared respectively.
The truck starts to move in reverse direction at 12 seconds because the displacement is a negative, based on its original position at zero (Correct Response D). The truck has not stopped moving at 24 seconds but has a zero displacement from its origin (Incorrect Response A). The truck is moving in a reverse, not a forward, direction at 36 seconds (Incorrect Response B). The slope of 0 at 6 seconds indicates that the truck is not moving and therefore is not accelerating (Incorrect Response C).
Work is the product of force applied to an object and its traveled distance. Since force is dependent on the mass and acceleration of an object, then the largest dog would be expected to demonstrate the most amount of work when climbing the same set of stairs, assuming a similar rate of acceleration (Correct Response A). The fastest dog (Incorrect Response B) and the slowest dog (Incorrect Response D) are traveling the same distance, but their mass is unknown. The smallest dog (Incorrect Response C) should apply the least amount of work because it has the lowest mass.
This classroom demonstration has two people pulling on opposite sides of a rope. This best represents net forces because the students can observe what happens if the people use the same force on each side or what happens if another person pulls with more force than the other (Correct Response B). Since this activity does not focus on the thermodynamic changes that may occur, it is not likely to be helpful for demonstrating entropy (Incorrect Response A). Energy transfers (Incorrect Response C) are occurring during the rope pull, but they are not immediately apparent. Additionally, there are no outward changes to the matter in the demonstration, so it is unlikely to help students visualize the conservation of matter (Incorrect Response D).
At constant velocity, the acceleration of the box must be zero. Therefore, all horizontal forces must balance each other and all vertical forces must balance each other. Forces II and III 2 and 3 must be equal in magnitude to produce balance vertically, and forces I and IV 1 and 4 (Correct Response B) must also be equal to produce balance horizontally. Forces I and III 1 and 3 (Incorrect Response A), forces II and IV 2 and 4 (Incorrect Response C), and forces III and IV 3 and 4 (Incorrect Response D) are not required to balance because each pair of forces is perpendicular to each other.
White light consists of many colors of light, and each color of light depends on its wavelength, λlambda. Refraction occurs when light passes from one material into another if the materials have different indices of refraction, n. The speed of light in the material is given by v = c/nc divided by n , where c = speed of light in a vacuum. Thus, a material with large refractive index will decrease the speed of light. Additionally, the speed, v = λf v = lambda f , also depends on the wavelength of light. As a result, different wavelengths, or colors, of light will have different speeds and therefore refract different amounts within a material. This dispersion of white light produces colors that refract different amounts and spread out into a full spectrum (Correct Response A). Amplitude (Incorrect Response B) is not affected by the transition from one material to another, nor does an amplitude change affect the color of light. Changes in reflection (Incorrect Response C) do not cause this phenomenon since prisms allow light to pass through them and refract. Light may become polarized entering a material (Incorrect Response D), but polarization does not cause color dispersion into a full spectrum of colors.
Parallel circuit branches connected to one voltage source will have the same voltage applied across the branch. The voltage of the source and the internal resistance of any light will not be affected by a light in a different branch burning out. Since the working lights will have the same internal resistance with the same voltage applied, by Ohm's Law—I = V/RI equals V over R—each branch will draw the same current (Correct Response B). Response A is misinterpreting the circuit to be a series circuit of light bulbs and incorrectly states that a burned bulb still completes a circuit and allows charge to flow. Response C is incorrect because it violates Ohm's Law for any parallel branch. Response D is describing the charge drawn from the battery to correctly decrease, but the charge drawn by each individual branch containing a working bulb will remain the same.
