Algebra

Step 1: General form of the nth term of an AP.
The nth term of an arithmetic progression (AP) is given by: where is the first term and is the common difference.
Step 2: Use the given terms to find and .
We are given that the 3rd term and the 9th term . For the 3rd term: For the 9th term: Now solve these two equations simultaneously. Subtract Equation 1 from Equation 2: Substitute into Equation 1:
Step 3: Find the term that is zero.
We want to find the term where : Thus, the 5th term of the AP is zero.

Step 1: Recall the standard form of a quadratic polynomial.
A quadratic polynomial is given by the equation: The sum and product of the zeros and of the polynomial are related to the coefficients as: - Sum of zeros: - Product of zeros:
Step 2: Use the given sum and product of zeros.
We are given that the sum of zeros is 6 and the product is -2. Therefore: - Sum: - Product:
Step 3: Write the quadratic polynomial.
Using the relationships above, the quadratic polynomial can be written as:
Step 4: Conclusion.
Thus, the required quadratic polynomial is:

Step 1: Substitute into the polynomial.
We are given . To find the value when , substitute into the equation:
Step 2: Simplify the expression.

Step 3: Conclusion.
Thus, the value of when is:

Step 1: Recognize the condition for no solution.
For the system of linear equations to have no solution, the lines must be parallel. The condition for two lines to be parallel is that the ratio of the coefficients of and in both equations must be equal.
Step 2: Apply the condition.
The first equation is , and the second equation is . The condition for parallel lines is:
Step 3: Solve for .
Solving the above equation gives:
Step 4: Conclusion.
Thus, the value of for which the system has no solution is .

The given arithmetic progression is . The common difference in an arithmetic progression is the difference between two consecutive terms. The common difference is: Final Answer: -1.

The given equation is . This is a quadratic equation of the form , which can be factored as . The roots of the equation are and . Final Answer: -2, 2.

The given equation is . Substituting into the equation: Final Answer: 7.