Question

Directions: A statement is followed by three conclusions. Choose the answer from the options below.
(AA) Using the given statement, only conclusion I can be derived.
(BB) Using the given statement, only conclusion II can be derived.
(CC) Using the given statement, only conclusion III can be derived.
(DD) Using the given statement, all conclusions can be derived.
(EE) Using the given statement, none of the three conclusions I, II and III can be derived.

An operation # is defined by \(a \# b = 1 - \dfrac{b}{a}\).
Conclusion I: \((2 \# 1) \# (4 \# 3) = -1\)
Conclusion II: \((3 \# 1) \# (4 \# 2) = -2\)
Conclusion III: \((2 \# 3) \# (1 \# 3) = 0\)

• A. Using the given statement, only conclusion I can be derived.
• B. Using the given statement, only conclusion II can be derived.
• C. Using the given statement, only conclusion III can be derived.
• D. Using the given statement, none of the three conclusions I, II and III can be derived.

Hint:

Work out the value of each conclusion's expression using the given rule for #, then compare it with the number the conclusion claims.

Answer 1

The Correct Option is D

Step 1: Understand the rule.
The operation \(a \# b\) is defined as \(1 - \dfrac{b}{a}\), so whenever \(a \# b\) appears, plug the two numbers into this formula in that exact order, first number as \(a\), second as \(b\).
Each conclusion must be checked by actually computing the value on the left side and comparing it with the number the conclusion claims.

Step 2: Test Conclusion I.
First, \(2 \# 1 = 1 - \dfrac{1}{2} = \dfrac{1}{2}\).
Next, \(4 \# 3 = 1 - \dfrac{3}{4} = \dfrac{1}{4}\).
Combine these two results with the same rule, treating \(\dfrac{1}{2}\) as the new \(a\) and \(\dfrac{1}{4}\) as the new \(b\):
\[ \left(\frac{1}{2}\right) \# \left(\frac{1}{4}\right) = 1 - \frac{1/4}{1/2} = 1 - \frac{1}{2} = \frac{1}{2} \]
The conclusion claims this equals \(-1\), but the value works out to \(\dfrac{1}{2}\), so Conclusion I cannot be derived.

Step 3: Test Conclusion II.
First, \(3 \# 1 = 1 - \dfrac{1}{3} = \dfrac{2}{3}\).
Next, \(4 \# 2 = 1 - \dfrac{2}{4} = \dfrac{1}{2}\).
Combine:
\[ \left(\frac{2}{3}\right) \# \left(\frac{1}{2}\right) = 1 - \frac{1/2}{2/3} = 1 - \frac{3}{4} = \frac{1}{4} \]
The conclusion claims this equals \(-2\), but the value works out to \(\dfrac{1}{4}\), so Conclusion II cannot be derived either.

Step 4: Test Conclusion III.
First, \(2 \# 3 = 1 - \dfrac{3}{2} = -\dfrac{1}{2}\).
Next, \(1 \# 3 = 1 - \dfrac{3}{1} = -2\).
Combine, being careful with the negative numbers:
\[ \left(-\frac{1}{2}\right) \# (-2) = 1 - \frac{-2}{-1/2} = 1 - 4 = -3 \]
The conclusion claims this equals \(0\), but the value works out to \(-3\), so Conclusion III cannot be derived either.

Step 5: Decide between the answer options.
Since none of Conclusion I, II or III actually comes out true, the correct choice is the option saying none of the three conclusions can be derived.
The options claiming only I, only II, only III or all conclusions hold are all wrong, since the calculations show every one of the three is false.

Final Answer:
None of the three conclusions I, II and III can be derived from the given rule.
\[ \boxed{\text{None of the three conclusions can be derived}} \]