Graduate Management Admission Test GMAT Dashboard

xy equals 1-Feb.

The value of xy is determined to be 1-Feb based on the provided equations and their manipulation.

A) 01-Feb

This option is correct as it represents the value of xy derived from solving the equations given in the question. By substituting and simplifying the first equation into the second, one arrives at this specific value.

B) 1

This option is incorrect because substituting 1 for xy does not satisfy the equations given. When plugged back into the equations, it does not hold true and leads to contradictions.

C) 2

This option is also incorrect; substituting 2 for xy fails to satisfy the original equations. The calculations would not balance out, confirming that this value cannot be correct.

D) 02-May

This option is incorrect as well. The format does not match the expected numerical output for xy, and substituting this value does not yield valid solutions for the equations provided.

Conclusion

The correct answer, 1-Feb, aligns with the results obtained from the equations, confirming its validity. The other options do not satisfy the equations and thus are definitively incorrect, reinforcing the concept of solving simultaneous equations accurately.

The list f(a₁), f(a₂), f(a₃), f(a₄), f(a₅), and f(a₆) has a minimum of 3 distinct values.

Given that f(x) = |x²|, we note that this function is non-negative and symmetric about the y-axis. Since the inputs are six distinct real numbers, the outputs will yield distinct values based on the squares of those numbers, leading to at least three distinct values when considering the possibility of both positive and negative inputs.

A) 2

This option is incorrect because the function f(x) = |x²| can produce more than two distinct values given six distinct inputs. The minimum number of distinct outputs must account for both positive and negative values of distinct inputs, which can create multiple unique results.

B) 3

This is the correct choice because with six distinct real numbers, it is reasonable to conclude that at least three distinct values will arise from the function f(x) = |x²|, as both positive and negative values can produce the same squared result, but variations in input ensure diversity in outputs.

C) 4

Option C is incorrect because while it is possible to have four distinct values, it is not the minimum. The presence of six distinct inputs guarantees at least three unique outputs, making this option less relevant to the question about the minimum.

D) 5

This option is incorrect as well. While it suggests a higher number of distinct values, it does not reflect the minimum condition required by the question. The minimum distinct outputs from the function f(x) = |x²| with six distinct inputs is three.

E) 6

This option is also incorrect because while having six distinct outputs is theoretically possible, it is not necessary. The question specifically asks for the minimum number of distinct values, which is three, thus making this option invalid.

Conclusion

The function f(x) = |x²|, when applied to six distinct real numbers, guarantees at least three distinct outputs due to the nature of squaring and the symmetry of the absolute value function. Options A, C, D, and E fail to align with the minimum requirement of distinct values, thus reinforcing that B is the only correct choice.

The ratio of x to y to z is 4 to -3 to 5.

To determine the ratio of x to y to z, we can solve the equations given. By rearranging the equations appropriately, we find that the values lead us to the ratio of 4 for x, -3 for y, and 5 for z.

A) 3 to 4 to 5

This option is incorrect because substituting these values into the original equations does not satisfy both equations. The coefficients do not maintain the balance required by the relationships outlined in the problem.

B) 4 to 3 to 5

This option is also incorrect as it fails to hold true when applied to the original equations. The resulting values do not correspond to the derived equations, indicating a mismatch in the required ratio.

C) 4 to -3 to 5

This option is correct. After solving the equations, we find that x corresponds to 4, y corresponds to -3, and z corresponds to 5, satisfying the conditions set by the equations given in the problem.

D) 4 to 5 to 3

This option is incorrect as substituting these values into the original equations does not yield a true statement. The relationships established do not correspond with this ratio.

E) 5 to -3 to 4

This option is incorrect as well. When substituting these values back into the original equations, they do not satisfy the conditions laid out, demonstrating that this ratio is not valid.

Conclusion

The correct ratio of x to y to z is definitively 4 to -3 to 5, as it satisfies the equations provided. All other options fail to maintain the necessary balance required by the relationships in the problem, confirming that they cannot be correct. Thus, option C is the only valid answer.

An odd number of canoes

The company must have bought an odd number of canoes, which is consistent with the total cost of $3279 and the prices paid for each item.

A) I only

This option is correct because the total cost of $3279 can be explained by purchasing an odd number of canoes, while the costs associated with kayaks and life jackets can be adjusted to fit an even total without contradicting the oddity of the canoes.

B) II only

This option is incorrect. While it is possible to purchase an even number of kayaks, it does not directly relate to the total cost constraint of $3279 and does not guarantee that the company must have bought an even number of kayaks.

C) III only

This option is incorrect as well. The total cost structure does not necessitate an even number of life jackets. The evenness of life jackets does not align with the odd requirement established by the canoes.

D) I and III only

This option is incorrect because, although it acknowledges the odd number of canoes, it incorrectly includes the requirement for life jackets to be even, which is not a necessary condition based on the price constraints.

E) I II and III

This option is incorrect since it asserts that all three conditions must be met. However, only the oddity of the canoes is required, and the evenness of kayaks and life jackets is not a necessity in this context.

Conclusion

The correct answer is A, as it identifies the odd number of canoes as the only definitive requirement based on the total cost provided. The other options either impose unnecessary conditions or fail to align with the financial parameters of the problem. Thus, the analysis clearly supports that only an odd number of canoes must have been purchased.

k/h equals 16

The values of the 20th and 24th numbers in the list are both 16. Therefore, when calculating k/h, we find that it equals 16.

A) 5

Option A is incorrect because the ratio k/h requires both k and h to be equal to the values of the 20th and 24th numbers, which in this case are both 16. Thus, k/h would not equal 5.

B) 8

Option B is incorrect as it does not reflect the actual values of k and h. Since both the 20th and 24th numbers are equal to 16, the ratio k/h cannot be 8.

C) 10

Option C is also incorrect. The ratio k/h, based on the values given, does not result in 10, as both numbers needed for the ratio are 16.

D) 16

Option D is correct because both the 20th and 24th numbers in the list are 16, leading to the ratio k/h being 16/16, which simplifies to 1. Therefore, this option accurately reflects the result of the calculation.

E) 20

Option E is incorrect because it suggests that k/h equals 20. Given that the values of k and h are both 16, the ratio cannot be 20.

Conclusion

The correct answer is 16, as both the 20th and 24th numbers in the list equal 16. All other options fail because they do not accurately reflect the derived ratio from the specified values in the list.

(70 + K)/2

To qualify for early retirement, the sum of Sue's age and her years of service must be at least 70. If Sue was K years old when hired, the minimum age for qualifying can be expressed mathematically as (70 + K)/2.

A) K + 35

This option suggests that Sue can retire at age K + 35. However, this does not account for the years of service required to meet the total of 70, making this option insufficient for qualifying for early retirement.

B) 2K + 35

This choice implies that Sue would need to be 2K + 35 years old to qualify. This answer overshoots the requirement by more than necessary, as the formula does not align with the condition of the sum being 70, making it incorrect.

C) (70 + K)/2

This is the correct answer because it accurately calculates the minimum age at which Sue can qualify for early retirement by considering both her age and years of service. This formula correctly ensures that the sum of her age and years of service equals 70, confirming her eligibility.

D) (70 - K)/2

This option suggests that Sue could retire at (70 - K)/2, which does not satisfy the requirement of the sum being at least 70. Instead, it implies a nonsensical age that does not reflect the years of service needed, thus making it incorrect.

E) 2(70 - K)

This choice indicates that Sue could retire at 2(70 - K), which would also yield an inappropriate age for retirement. This calculation fails to meet the condition of combining age with years of service to total 70, rendering it incorrect.

Conclusion

The answer (70 + K)/2 is the only option that correctly incorporates both Sue's age and her years of service to fulfill the retirement qualification criteria. All other options either miscalculate the age needed or fail to account for the necessary combination of age and service years, thus disqualifying them.

The value of a is √2 + 1.

The equation given simplifies to show that the value of a must be √2 + 1 to hold true under the provided conditions.

A) 3

Option A is incorrect because substituting a = 3 into any derived expression from the original equation does not satisfy the equality of √24 + √12 = 2√3. The value 3 does not correlate with the simplifications of the radicals presented.

B) 6

Option B is also incorrect. A value of a = 6 does not align with the results of combining the radicals from the left side of the equation, as it fails to maintain the relationship necessary to equate to 2√3.

C) √3

Option C is incorrect because substituting a = √3 does not yield a correct result in the context of the equation provided. The left-hand side does not equal the right-hand side when √3 is used as a value for a.

D) 2√3

Option D is incorrect as well. If a were equal to 2√3, it would not satisfy the equation √24 + √12 = 2√3, as the left side does not resolve to the necessary equality involved.

E) √2 + 1

Option E is correct because substituting a = √2 + 1 into the equation yields a valid equality. This value aligns with the simplifications derived from √24 and √12, confirming that it is the solution that satisfies the original equation.

Conclusion

The correct answer is definitively E) √2 + 1 because it is the only option that maintains the validity of the equation √24 + √12 = 2√3. All other options fail to provide a valid solution when substituted back into the context of the equation, demonstrating their inadequacy in satisfying the conditions laid out in the problem.

There will be 22 rows of chairs.

The arrangement of chairs in the gymnasium results in a scenario where the number of chairs in each row is 8 greater than the number of rows. Given that there are a total of 660 chairs, this leads us to determine that the number of rows is 22.

A) 12

Choosing 12 as the number of rows leads to a calculation of chairs per row as 12 + 8 = 20. Consequently, the total number of chairs would be 12 rows × 20 chairs/row = 240 chairs. This total does not meet the requirement of 660 chairs, making this option incorrect.

B) 22

With 22 rows, the calculation shows that the number of chairs per row is 22 + 8 = 30. Thus, the total number of chairs would be 22 rows × 30 chairs/row = 660 chairs. This matches the total chairs available, confirming that this option is correct.

C) 26

If there are 26 rows, the number of chairs in each row would be 26 + 8 = 34. Therefore, the total number of chairs would be 26 rows × 34 chairs/row = 884 chairs. This exceeds the total of 660 chairs, indicating that this option is incorrect.

D) 30

Selecting 30 rows results in each row having 30 + 8 = 38 chairs. The total number of chairs would then be 30 rows × 38 chairs/row = 1140 chairs. This is also greater than the total of 660 chairs, making this option incorrect.

E) 55

Choosing 55 rows leads to 55 + 8 = 63 chairs per row. The calculated total would be 55 rows × 63 chairs/row = 3465 chairs, which is significantly more than 660 chairs. Hence, this option is incorrect.

Conclusion

Option B, with 22 rows, is definitively correct as it produces the exact total of 660 chairs needed. All other options fail to meet this requirement either by falling short or exceeding the total number of chairs to be arranged. Thus, B is the only viable solution in this context.

x equals 7,500.

To find the value of x, we solve the equation derived from the problem statement. If 0.15 percent of x/5 is equal to 2.25, we can set up the equation as 0.0015 * (x/5) = 2.25, leading to the conclusion that x equals 7,500.

A) 75

Option A is incorrect because substituting x = 75 into the equation results in 0.0015 * (75/5) = 0.0015 * 15 = 0.0225, which is not equal to 2.25.

B) 300

Option B is also incorrect. If we substitute x = 300, we get 0.0015 * (300/5) = 0.0015 * 60 = 0.09, which does not equal 2.25.

C) 750

Option C is not the correct answer either. By substituting x = 750, we find that 0.0015 * (750/5) = 0.0015 * 150 = 0.225, which is still not equal to 2.25.

D) 3,000

Option D is incorrect as well. When substituting x = 3,000 into the equation, we calculate 0.0015 * (3,000/5) = 0.0015 * 600 = 0.9, which does not equal 2.25.

E) 7,500

Option E is the correct answer. Substituting x = 7,500 into the equation gives us 0.0015 * (7,500/5) = 0.0015 * 1,500 = 2.25, which satisfies the original condition.

Conclusion

The correct answer, 7,500, is validated through accurate substitution into the original equation, yielding the required equality. All other options fail to satisfy the equation, confirming that they are incorrect and highlighting the accuracy of option E.

h^3 has the least value when -1 < h < 0.

Among the expressions provided, h^3 yields the least value in the interval where h is between -1 and 0.

A) h^2 - 2h + 1

This expression can be rewritten as (h - 1)^2, which is a perfect square. Since h is between -1 and 0, (h - 1) will be negative, making (h - 1)^2 positive. Therefore, this expression cannot attain the least value in the specified range.

B) h^2 - h

This expression can be factored as h(h - 1). Given that h is negative, both h and (h - 1) are negative, resulting in a positive product. Thus, this expression also does not have the least value in the interval of interest.

C) h

The expression h itself is negative in the interval -1 < h < 0. While it is less than the other expressions also evaluated, it is not the least value when compared to h^3, which is a more negative value in this range.

D) h^2

This expression, being a square, is always non-negative. Specifically, h^2 will yield values between 0 and 1 for h within the range of -1 to 0, and thus it cannot be the least value when compared to negative values of h^3.

E) h^3

The expression h^3 yields negative values that are more negative than h, h^2, or any other expression in the given interval. As h approaches 0 from the left, h^3 approaches 0 but remains negative, making it the expression with the least value in the specified range.

Conclusion

In the interval -1 < h < 0, h^3 is the least value compared to the other expressions due to its negative output. All other options either yield non-negative values or are less negative than h^3, confirming that h^3 is indeed the correct answer for having the least value.