Algebra

Step 1: Use the system of equations.
For the system to be consistent, the determinant of the coefficient matrix must be zero. Solving the system will give the value of .
Step 2: Conclusion.
Thus, .
Final Answer:

Step 1: Understand the definition of determinant.
A determinant is a scalar value that can be calculated from a square matrix.
Step 2: Conclusion.
Thus, the correct answer is that the determinant is a number associated with a square matrix.
Final Answer:

Step 1: Understand the properties of groups.
In a group, the equation has a unique solution, which is given by .
Step 2: Conclusion.
Thus, the unique solution is .
Final Answer:

Step 1: Applying the AM-GM inequality.
By the Arithmetic Mean-Geometric Mean (AM-GM) inequality, we know that for positive real numbers , the arithmetic mean is always greater than or equal to the geometric mean. This gives the relation .

Step 2: Conclusion.
The correct statement is option (2).

Since the identity element does not exist for the group, it is classified as a semi-group.

The given values satisfy the equation , so the correct answer is .
Step 2: Conclusion.
The correct answer is (D), 4.

Step 1: Simplify the expression.
Given , we can substitute the value of and raise it to the power of 4.
Step 2: Conclusion.
Thus, .
Final Answer:

Step 1: Solve the given system of equations.
By solving the system of equations using basic algebraic methods, the value of is found to be .
Step 2: Conclusion.
Thus, the value of is .
Final Answer:

Step 1: Understanding the Given Condition
The number of subsets of a set with elements is given by:
Similarly, the number of subsets of a set with elements is:
It is given that:
Step 2: Expressing the Equation in Factorized Form
Factorizing the given equation:
Expressing 56 as a product of powers of 2:
Step 3: Comparing Both Sides
By comparing, we get:
Thus:
So:
Step 4: Finding the Sum
The sum of and is:
Step 5: Matching with Options
The correct answer is . Final Answer: (A) 9.

Given: The equation is: Step 1: Rearranging the equation To solve for , we first move all the -terms to one side and constant terms to the other side. Subtract from both sides: Now, add 8 to both sides: Step 2: Solve for Now, divide both sides by 5: Answer: The correct answer is option (2): .

Step 1: Write the given equation. The given equation is: Step 2: Find a common denominator. The common denominator of and is . So, rewrite the equation: Step 3: Simplify the equation. Now, combine the fractions on the left-hand side: Simplify the numerator: Step 4: Cross-multiply to eliminate the denominator. Now, cross-multiply: Step 5: Expand and solve for . Expand both sides: Rearrange the terms: Now solve the quadratic equation . The solutions are: Simplifying gives: Therefore, the value of is approximately . Answer: Therefore, .

Step 1: Write the given quadratic equation. The quadratic equation is: Step 2: Factorize the quadratic equation. We need to factorize . We look for two numbers that multiply to give and add up to . These numbers are and . Thus, the factorized form is: Step 3: Solve for . Set each factor equal to zero: Answer: Therefore, the solutions are and .

Step 1: Rearranging the equation The given equation is: To solve for , first add 2 to both sides: Step 2: Solve for Now, divide both sides by 4: Answer: The correct answer is option (2): .