Assertion (A) : The value of k so that the factors of $\Big(x^2 - kx + \dfrac{121}{16}\Big)$ are same is $\dfrac{11}{2}$.

Reason (R) : (x + a) (x + b) = x² + (a + b)x + ab.

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

2. Both A and R are correct, and R is not the correct explanation for A.

3. A is true, but R is false.

4. A is false, but R is true.

Given; $\Big(x^2 - kx + \dfrac{121}{16}\Big)$

We know that this quadratic has equal factors if it's a perfect square trinomial, meaning :

$\Rightarrow \Big(x^2 - kx + \dfrac{121}{16}\Big) = \Big(x - \dfrac{11}{4}\Big)^2 \\[1em] \Rightarrow \Big(x^2 - kx + \dfrac{121}{16}\Big) = x^2 - 2 \times x \times \dfrac{11}{4} + \Big(\dfrac{11}{4}\Big)^2 \\[1em] \Rightarrow \Big(x^2 - kx + \dfrac{121}{16}\Big) = x^2 - \dfrac{11}{2}x + \dfrac{121}{16} \\[1em] \Rightarrow k = \dfrac{11}{2}.$

So, assertion (A) is true.

Solving,

(x + a)(x + b) = x(x + b) + a(x + b)

= x² + bx + ax + ab

= x² + x(a + b) + ab

So, reason (R) is true but, reason (R) is not the correct explanation of assertion (A).

Hence, option 2 is the correct option.