Binary operations

Correct answer: A. (-2)

Step 1: For identity (e), (a+e-3=a), so (e=3). Step 2: Put (8*b=3): (8+b-3=3), so (b=-2). Step 3: In shifted addition, the inverse is not simply (-a).

Correct answer: A. (8)

Step 1: First (2*5=2+5-3=4). Step 2: Then (4*7=4+7-3=8). Step 3: When brackets are given, follow that order even if the operation is associative.

Correct answer: A. (a+b+c-6)

Step 1: First (b*c=b+c-3). Step 2: (a*(b*c)=a+(b+c-3)-3=a+b+c-6). Step 3: Substitute the complete value of the inner operation into the next step.

Correct answer: A. (0)

Step 1: For identity (e), (a+e+\lambda ae=a). Step 2: This gives (e(1+\lambda a)=0). Step 3: The fixed element satisfying this for all (a) is (e=0).

Correct answer: A. (1+\lambda a\neq0)

Step 1: The identity is (0). Step 2: For inverse (b), (a+b+\lambda ab=0), so (b(1+\lambda a)=-a). Step 3: Such (b) exists only when (1+\lambda a\neq0).

Correct answer: A. (-\frac{4}{9})

Step 1: With (\lambda=2), (a*b=a+b+2ab). Step 2: From (4*b=0), (4+b+8b=0), so (4+9b=0). Step 3: (b=-\frac{4}{9}); substitute the parameter first to simplify the equation.

Correct answer: A. Identity is (1) and only (1) has an inverse

Step 1: (\operatorname{lcm}(a,1)=a), so (1) is the identity. Step 2: An inverse (b) of (a) must satisfy (\operatorname{lcm}(a,b)=1). Step 3: This is possible only for (a=1) and (b=1), so only (1) has an inverse.

Correct answer: A. It is a closed operation

Step 1: For closure, the result must again lie in (\mathbb{Z}). Step 2: In (a+b+ab), sums and products of integers remain integers. Step 3: In closure questions, focus only on the set of the result.

Correct answer: A. (-4)

Step 1: For identity (e), (a*e=a) must hold. Step 2: From (a+e+4=a), we get (e=-4). Step 3: In identity questions, (a) may be any value, so (e) must be fixed.

Correct answer: A. (-14)

Step 1: First find the identity: (a+e+4=a), so (e=-4). Step 2: If (x) is the inverse of (6), then (6+x+4=-4). Step 3: This gives (x=-14), so always find identity before inverse.

Correct answer: A. (0)

Step 1: For identity (e), write (a+e-ae=a). Step 2: This gives (e(1-a)=0), which is satisfied for all (a) by (e=0). Step 3: Subtracting (a) from both sides is a quick simplification.

Correct answer: A. (\frac{4}{3})

Step 1: The identity for this operation is (0). Step 2: From (4*x=0), (4+x-4x=0), so (4-3x=0). Step 3: Hence (x=\frac{4}{3}); be careful with negative signs.

Correct answer: A. (0)

Step 1: From (a*e=a), we get (a+e+ae=a). Step 2: Thus (e(1+a)=0), and (a\neq -1). Step 3: Hence (e=0); always read the set condition carefully.

Correct answer: A. (-\frac{2}{3})

Step 1: The identity is (0). Step 2: From (2*x=0), (2+x+2x=0), so (3x=-2). Step 3: Hence (x=-\frac{2}{3}), and it also belongs to (A).

Correct answer: A. (-1)

Step 1: The identity is (0). Step 2: From (a*x=0), (a+x+ax=0), so (x=\frac{-a}{1+a}). Step 3: At (a=-1), the denominator is zero, so inverse does not exist.

Correct answer: A. (5)

Step 1: For identity (e), (\frac{ae}{5}=a). Step 2: Since (a\neq0), divide by (a) to get (e=5). Step 3: In fraction-based operations, clear the denominator first.

Correct answer: A. (\frac{5}{2})

Step 1: The identity is (5). Step 2: From (10*x=5), (\frac{10x}{5}=5), so (2x=5). Step 3: Hence (x=\frac{5}{2}); find inverse by equating to the identity.

Correct answer: A. (2)

Step 1: For identity (e), (\max(a,e)=a) for all (a\in A). Step 2: This happens only when (e) is the smallest element. Step 3: The smallest element of this set is (2).

Correct answer: A. (8)

Step 1: For identity (e), (\min(a,e)=a) for all (a\in A). Step 2: This is possible only when (e) is the greatest element. Step 3: Hence the identity here is (8).

Correct answer: A. (6)

Step 1: For identity (e), (\gcd(a,e)=a) for every (a\in A). Step 2: This means every (a) must divide (e). Step 3: (6) is a multiple of all given elements, so (e=6).

Correct answer: A. (1)

Step 1: For identity (e), (\operatorname{lcm}(a,e)=a). Step 2: For this, (e) must divide every (a). Step 3: (1) divides all elements, so it is the identity.

Correct answer: A. It is closed but not commutative

Step 1: Difference of two integers is an integer, so closure holds. Step 2: (5-2\neq2-5), so it is not commutative. Step 3: Do not accept identity only from (a*0=a); also check (0*a=a).

Correct answer: A. Because the result is not always an integer

Step 1: For a binary operation, the result must remain in (\mathbb{Z}). Step 2: (1*2=\frac{3}{2}), which is not an integer. Step 3: One counterexample is enough to disprove closure.

Correct answer: A. It is commutative but not associative

Step 1: Since (a^2+b^2=b^2+a^2), it is commutative. Step 2: ((1*1)*2=8), while (1*(1*2)=26), so it is not associative. Step 3: Small values are very useful for testing associativity.

Correct answer: A. (1)

Step 1: For commutativity, (ka+b=kb+a) for all (a,b). Step 2: View it as ((k-1)a=(k-1)b). Step 3: This holds for all values only when (k=1).