Quadratic Equation Level - 2

In the Maths Olympiad of 2020 at Animal Planet, two representatives from the donkey’s side, while solving a quadratic equation, committed the following mistakes:
(i) One of them made a mistake in the constant term and got the roots as 5 and 9.
(ii) Another one committed an error in the coefficient of x and got the roots as 12 and 4.
But in the meantime, they realised that they are wrong and they managed to get it right jointly. Which of the following could be the correct equation?

If the roots of the equation a₁x² + a₂x + a₃ = 0 are in the ratio r₁ : r₂, then

For what value of a do the roots of the equation 2x² + 6x + a = 0, satisfy the conditions (α/β) + (β/α) < 2.

For what value of b and c would the equation x² + bx + c = 0 have roots equal to b and c.

The sum of a fraction and its reciprocal equals 85/18. Find the fraction.

A journey between Mumbai and Pune (192 km apart) takes two hours less by a car than by a truck. Determine the average speed of the car if the average speed of the truck is 16 km/h less than the car.

If both the roots of the quadratic equation ax² + bx + c = 0 lie in the interval (0, 3) then a lies in

 If the common factor of (ax² + bx + c) and (bx² + ax + c) is (x + 2) then

If P = 2^(2/3) + 2^(1/3) then which of the following is true?

If f(x) = x² + 2x – 5 and g(x) = 5x + 30, then the roots of the quadratic equation g[f(x)] will be