Question

What is the value of \( n \)?

• A. \( n^4 = 256 \)
• B. \( n^2>n^4 \)
• C.
• D.
• E.
• A. Copier A produced twice as many copies in its first 5 minutes of operation as copier B produced in its first 15 minutes.
• B. Copier B produces 10 copies per minute.
• C.
• D.
• J.
• A. \( c = 5a - 2b \)
• B. \( a = 2b \)
• C.
• D.
• O.
• A. If \( x \) were rounded to the nearest tenth, the result would be 0.6.
• B. If \( x \) were rounded to the nearest hundredth, the result would be 0.58.
• C.
• D.
• T.
• A. \( -(2 + y)>0 \)
• B. \( (2 + y)^2>0 \)
• C.
• D.
• Y.

Answer 1

The Correct Option is A

Step 1: Analyze statement (1).
From statement (1), we have the equation: \[ n^4 = 256. \] Taking the fourth root of both sides: \[ n = \pm 4. \] Thus, statement (1) gives us the value of \( n \), and is sufficient to answer the question.
Step 2: Analyze statement (2).
Statement (2) says \( n^2>n^4 \), but this provides no direct way to solve for \( n \). For example, \( n = 1 \) satisfies the inequality but does not solve for \( n \). Hence, statement (2) is not sufficient on its own.
Conclusion:
Statement (1) alone is sufficient to determine \( n \), so the correct answer is (A).

Answer 2

Step 1: Analyze statement (1).
Statement (1) tells us that copier A produced twice as many copies in its first 5 minutes as copier B produced in its first 15 minutes. While this gives a relation between the rates of the copiers, it does not give enough information to determine the total time it would take for the task if copier A had not broken down. Hence, statement (1) alone is insufficient.
Step 2: Analyze statement (2).
Statement (2) tells us that copier B produces 10 copies per minute. While this gives us the rate of copier B, we do not know how many copies were made by copier A, so statement (2) alone is insufficient.
Step 3: Combine both statements.
Together, we now know the rates of both copiers. From statement (1), we know copier A is twice as fast as copier B in the first 5 minutes, and from statement (2), we know copier B’s rate. Combining these facts allows us to calculate how long it would have taken copier A to complete the task alone.
Conclusion:
Thus, both statements together provide enough information to determine the total time, so the correct answer is (C).

Answer 3

Step 1: Analyze statement (1).
From statement (1), we know that \( c = 5a - 2b \), but this alone does not provide enough information about the values of \( a \) and \( b \) to determine if \( c \) is an integer.
Step 2: Analyze statement (2).
From statement (2), we know \( a = 2b \), but this alone does not help determine if \( c = 5a - 2b \) will be an integer, as we do not know the specific values of \( a \) and \( b \).
Step 3: Combine both statements.
Even when combining the two statements, we still do not have sufficient information to conclude whether \( c = 5a - 2b \) is an integer because we have no information about the actual values of \( a \) and \( b \).
Conclusion:
Thus, the statements together are not sufficient to answer the question, so the correct answer is (E).

Answer 4

Step 1: Analyze statement (1).
Statement (1) tells us that if \( x \) were rounded to the nearest tenth, the result would be 0.6. This implies that \( x \) is closer to 0.6 than 0.5, so \( x \) is between 0.55 and 0.65. This gives us some bounds on \( x \), but does not provide enough information to determine \( y \).
Step 2: Analyze statement (2).
Statement (2) tells us that if \( x \) were rounded to the nearest hundredth, the result would be 0.58. This implies that \( x \) is between 0.575 and 0.585.
Step 3: Combine both statements.
When both statements are combined, we know that \( x \) is between 0.575 and 0.585, and we also know that \( x \) is closer to 0.6 than to 0.5. This narrows down the possible value of \( y \), allowing us to conclude that \( y = 8 \).
Conclusion:
Thus, both statements together are sufficient to determine the value of \( y \), so the correct answer is (C).

Answer 5

Step 1: Analyze statement (1).
Statement (1) tells us that: \[ -(2 + y)>0 \implies 2 + y<0 \implies y<-2. \] Thus, from statement (1), we can conclude that \( y \) is a negative integer, so it is not greater than 0. Hence, statement (1) alone is sufficient to answer the question.
Step 2: Analyze statement (2).
Statement (2) gives us: \[ (2 + y)^2>0. \] Since the square of any real number is always non-negative, this inequality holds true for all values of \( y \neq -2 \). Therefore, statement (2) does not give us a specific answer about whether \( y \) is positive or negative. However, it does imply that \( y \neq -2 \), but it does not rule out \( y \) being positive. Hence, statement (2) alone is sufficient to answer the question.
Conclusion:
Both statements are sufficient to determine whether \( y>0 \), so the correct answer is (D).