Assertion (A) : If radius of a right circular cylinder is double and the height is reduced to $\dfrac{1}{2}$ of the original, the ratio of volume of new cylinder thus formed to the volume of the original cylinder is 1 : 1.

Reason (R) : Volume of a cylinder = πr²h where r is the radius of the circular base and h is height.

1. Both A and R are correct, and R is the correct explanation for A.

2. Both A and R are incorrect.

3. A is true, but R is false.

4. A is false, but R is true.

Let originally r be the radius of the circular base and h be height of the circular cylinder, then

Volume of a cylinder (v) = πr²h.

So, reason (R) is true.

Given,

Radius of new cylinder (R) = 2r

Height of new cylinder (H) = $\dfrac{1}{2}h$

Ratio of volume of new cylinder thus formed to the volume of the original cylinder,

$\Rightarrow \dfrac{V}{v} = \dfrac{πR^2H}{πr^2h} \\[1em] = \dfrac{π \times (2r)^2 \times \dfrac{1}{2}h}{πr^2h} \\[1em] = \dfrac{π \times 4r^2 \times \dfrac{1}{2}h}{πr^2h} \\[1em] = \dfrac{2πr^2h}{πr^2h} \\[1em] = \dfrac{2}{1}.$

So, assertion (A) is false.

∴ A is false, but R is true.

Hence, option 4 is the correct option.