Comprehension
Directions for questions 94 to 96: Answer the questions on the basis of the tables given below.
Two binary operations \( \oplus \) and \( * \) are defined over the set \( \{a, e, f, g, h\} \) as per the following tables:
| \( \oplus \) | a | e | f | g | h |
|---|---|---|---|---|---|
| a | a | e | f | g | h |
| e | e | f | g | h | a |
| f | f | g | h | a | e |
| g | g | h | a | e | f |
| h | h | a | e | f | g |
| \( * \) | a | e | f | g | h |
|---|---|---|---|---|---|
| a | a | a | a | a | a |
| e | a | e | f | g | h |
| f | a | f | h | e | g |
| g | a | g | e | h | f |
| h | a | h | g | f | e |
Thus, according to the first table \( f \oplus g = a \), while according to the second table \( g * h = f \), and so on. Also, let \( f^2 = f * f \), \( g^3 = g * g * g \), and so on.
Question: 1
What is the smallest positive integer \(n\) such that \(g^n = e\)?
What is the smallest positive integer \(n\) such that \(g^n = e\)?
Show Hint
Remember that \(g^2 = g \ast g\), \(g^3 = g \ast g \ast g\), and so on. Compute the powers of g one at a time from the second table.
Updated On: Nov 6, 2026
- 3
- 5
- 2
- 4
Show Solution
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The Correct Option is D
Solution and Explanation
Step 1: Powers of g (use the * table):
\(g^1 = g\). \(g^2 = g \ast g\): row g, column g of the * table gives h. So \(g^2 = h\).
Step 2: Next powers:
\(g^3 = g^2 \ast g = h \ast g\): row h, column g gives f. So \(g^3 = f\). \(g^4 = g^3 \ast g = f \ast g\): row f, column g gives e. So \(g^4 = e\).
Step 3: Smallest n:
\(g^1 = g,\ g^2 = h,\ g^3 = f\) are all different from e, and \(g^4 = e\). So the smallest n is 4, option (4).
Final Answer:
The correct option is (4): 4
\(g^1 = g\). \(g^2 = g \ast g\): row g, column g of the * table gives h. So \(g^2 = h\).
Step 2: Next powers:
\(g^3 = g^2 \ast g = h \ast g\): row h, column g gives f. So \(g^3 = f\). \(g^4 = g^3 \ast g = f \ast g\): row f, column g gives e. So \(g^4 = e\).
Step 3: Smallest n:
\(g^1 = g,\ g^2 = h,\ g^3 = f\) are all different from e, and \(g^4 = e\). So the smallest n is 4, option (4).
Final Answer:
The correct option is (4): 4
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Question: 2
Upon simplification, \(f \oplus [f \ast \{f \oplus (f \ast f)\}]\) equals
Upon simplification, \(f \oplus [f \ast \{f \oplus (f \ast f)\}]\) equals
Show Hint
Work from the innermost bracket outwards, using the * table for \(\ast\) and the \(\oplus\) table for \(\oplus\).
Updated On: Nov 6, 2026
- e
- h
- g
- f
Show Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
Step 1: Innermost:
\(f \ast f\): row f, column f of the * table is h.
Step 2: Next bracket:
\(f \oplus (f \ast f) = f \oplus h\): row f, column h of the \(\oplus\) table is e.
Step 3: Curly bracket:
\(f \ast \{\cdot\} = f \ast e\): row f, column e of the * table is f.
Step 4: Outermost:
\(f \oplus f\): row f, column f of the \(\oplus\) table is h. So the value is h, option (2).
Final Answer:
The correct option is (2): h
\(f \ast f\): row f, column f of the * table is h.
Step 2: Next bracket:
\(f \oplus (f \ast f) = f \oplus h\): row f, column h of the \(\oplus\) table is e.
Step 3: Curly bracket:
\(f \ast \{\cdot\} = f \ast e\): row f, column e of the * table is f.
Step 4: Outermost:
\(f \oplus f\): row f, column f of the \(\oplus\) table is h. So the value is h, option (2).
Final Answer:
The correct option is (2): h
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Question: 3
Upon simplification, \(\{a^{10} \ast (f^{10} \oplus g^{9})\} \oplus e^{8}\) equals
Upon simplification, \(\{a^{10} \ast (f^{10} \oplus g^{9})\} \oplus e^{8}\) equals
Show Hint
Powers are taken with the * operation. Note that a * anything = a, and e behaves like an identity for *. Simplify each power first.
Updated On: Nov 6, 2026
- e
- f
- g
- h
Show Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
Step 1: Powers with *:
\(a^{10} = a\) because \(a \ast a = a\). \(e^8 = e\) because \(e \ast e = e\).
Step 2: Power of f:
\(f^2 = f \ast f = h\), \(f^3 = h \ast f = g\), \(f^4 = g \ast f = e\), \(f^5 = e \ast f = f\). The powers repeat with period 4, so \(f^{10} = f^2 = h\).
Step 3: Power of g:
From the previous question, \(g, h, f, e\) repeat with period 4, so \(g^9 = g^1 = g\) (since 9 = 4 x 2 + 1).
Step 4: Inner bracket:
\(f^{10} \oplus g^{9} = h \oplus g\): row h, column g of the \(\oplus\) table gives g.
Step 5: Apply *:
\(a^{10} \ast g = a \ast g = a\) (the row of a in the * table is all a).
Step 6: Final:
\(a \oplus e^8 = a \oplus e\): row a, column e of the \(\oplus\) table is e. So the value is e, option (1).
Final Answer:
The correct option is (1): e
\(a^{10} = a\) because \(a \ast a = a\). \(e^8 = e\) because \(e \ast e = e\).
Step 2: Power of f:
\(f^2 = f \ast f = h\), \(f^3 = h \ast f = g\), \(f^4 = g \ast f = e\), \(f^5 = e \ast f = f\). The powers repeat with period 4, so \(f^{10} = f^2 = h\).
Step 3: Power of g:
From the previous question, \(g, h, f, e\) repeat with period 4, so \(g^9 = g^1 = g\) (since 9 = 4 x 2 + 1).
Step 4: Inner bracket:
\(f^{10} \oplus g^{9} = h \oplus g\): row h, column g of the \(\oplus\) table gives g.
Step 5: Apply *:
\(a^{10} \ast g = a \ast g = a\) (the row of a in the * table is all a).
Step 6: Final:
\(a \oplus e^8 = a \oplus e\): row a, column e of the \(\oplus\) table is e. So the value is e, option (1).
Final Answer:
The correct option is (1): e
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