Let's solve the questions step by step:

Question 1:

The equation given is $a_1 x^2 + b_1 x + c_1 = 0$, and the roots have the same ratio as the equation $a_2 x^2 + b_2 x + c_2 = 0$. This implies that the quadratic roots are proportional. The condition for proportionality of roots is:

$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}.$

Thus, the correct answer is (A) AP.

Question 2:

The roots of the first equation $x^2 + \alpha x + \beta = 0$ are given as 8 and 2. The roots of the second equation $x^2 + \alpha x + \beta = 0$ are given as 3 and 3. Using Vieta's formulas:

For the first equation:

• Sum of the roots: $8 + 2 = 10$
• Product of the roots: $8 \times 2 = 16$.

For the second equation:

• Sum of the roots: $3 + 3 = 6$
• Product of the roots: $3 \times 3 = 9$.

The resulting equation for the combined product of the roots is $x^2 + 2x + 1 = 0$. Therefore, the correct answer is (C).

Question 3:

For the given quadratic equation $a x^2 + b x + c = 0$, the relation for the value of $(ac)^{n+1} + (ac)^{n-1}$ is based on the general form of a geometric progression. Using the properties of roots and solving, the correct answer is (A).

Question 4:

Given $\alpha, \beta$ are the roots of the equation $x^2 - 2x + 3 = 0$, the equation whose roots are given by $\frac{\alpha - 1}{\alpha + 1}$ and $\frac{\beta - 1}{\beta + 1}$ can be found using transformations. The correct answer is (C).

Question 5:

The solution to the inequality $\frac{x+3}{x-2} \leq 2$ involves solving the quadratic inequality. Factoring and analyzing the sign of the resulting quadratic gives the solution as (A) (-∞, 2].

Question 6:

The equation $|x^2 + 1| - 6 = 0$ will have roots based on the fact that $|x^2 + 1| \geq 0$. Thus, real roots exist when $x^2 + 1 = 6$, which simplifies to $x^2 = 5$. Therefore, the roots are $\pm \sqrt{5}$, and the correct answer is (A).

Question 7:

The equation for $l x^2 + mx + n = 0$ with roots involving higher powers of the roots can be solved using substitution. After simplifying, the correct answer is (C).

Do you want further details or any clarifications? Here are some related questions for more practice:

1. How do you derive the quadratic formula from Vieta's relations?
2. What is the significance of the discriminant in determining the nature of roots?
3. Can you explain how transformations affect the roots of equations?
4. What is the general approach to solving quadratic inequalities?
5. How do modulus equations differ from standard quadratic equations?

Tip: Always analyze the behavior of the equation before solving inequalities by sketching graphs or considering the critical points.