Algebra

Step 1: Concept
Use the relationship between roots and coefficients: and .
Step 2: Analysis
Divide the second equation by to get .
Step 3: Evaluation
The term must be equal to the roots of the original equation, and . Thus, and .
Step 4: Conclusion
Substituting , the roots are and .
Final Answer: (a)

Step 1: Concept
Use algebraic identities for sums of powers.
Step 2: Analysis
. Substituting values: .
Step 3: Evaluation
Using , we get .
Step 4: Conclusion
. After further calculation involving and , is found to be 18.
Final Answer: (b)

Step 1: Understanding the Concept:
If divides the cubic, then the cubic must be for some .

Step 2: Detailed Explanation:
. Comparing with : , , . Thus and . Since , is fixed, and can be any of the 10 numbers from 1 to 10. So 10 polynomials.

Step 3: Final Answer:
10, which corresponds to option (D).

Step 1: Understanding the Concept:
Express in terms of and form a quadratic.

Step 2: Detailed Explanation:
.
Let .
Minimum occurs when , i.e., , .
Minimum value = .

Step 3: Final Answer:
Option (A) .

Step 1: Understanding the Concept:
means for all .
Step 2: Detailed Explanation:
Take any . Then . Multiply on left by : . Multiply on right by : . So , hence G is abelian.
Step 3: Final Answer:
G is abelian.

Step 1: Understanding the Concept:
First find the identity element such that .
Step 2: Detailed Explanation:
Let be the identity. Then .
Now, inverse of means . So .
Step 3: Final Answer:
The inverse of is .

Step 1: (bigger).}
Step 2: . Not matching. Option (B) is .}

Step 1: Understanding the Concept:
(the identity element).
Step 2: Detailed Explanation:
(the identity under ).
The inverse of the identity element is the identity itself: .
Step 3: Final Answer:
The inverse of is .

Step 1: Understanding the Concept:
A fundamental result in group theory states that every group whose order is a prime number must be cyclic.
Step 2: Detailed Explanation:
The number 7 is prime. By Lagrange's theorem and the corollary for prime-order groups, every group of prime order is cyclic (generated by any non-identity element) and therefore also abelian.
Step 3: Final Answer:
Every group of order 7 is cyclic (and hence abelian).