Given f(x) = 16x³ - 8x² + 4x + 7

Assertion (A): When we subtract 1 from f(x), the resulting polynomial is divisible by (2x + 1).

Reason (R): $f\Big(-\dfrac{1}{2}\Big)$ = 1.

1. Assertion (A) is true, but Reason (R) is false.

2. Assertion (A) is false, but Reason (R) is true.

3. Both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are correct, and Reason (R) is incorrect reason for Assertion (A).

Given f(x) = 16x³ - 8x² + 4x + 7

Subtract 1 from f(x), we get :

g(x) = 16x³ - 8x² + 4x + 6

By Factor Theorem,

(x - r) is a factor of f(x) if and only if f(r) = 0.

(2x + 1) is a factor of g(x) if and only if $g\Big(-\dfrac{1}{2}\Big)$ = 0.

$g\Big(-\dfrac{1}{2}\Big) = 16 \times \Big(-\dfrac{1}{2}\Big)^3 - 8 \times \Big(-\dfrac{1}{2}\Big)^2 + 4 \times \Big(-\dfrac{1}{2}\Big) + 6\\[1em] = 16 \times \Big(-\dfrac{1}{8}\Big) - 8 \times \Big(\dfrac{1}{4}\Big) + 4 \times \Big(-\dfrac{1}{2}\Big) + 6\\[1em] = -2 - 2 - 2 + 6\\[1em] = 0.$

So, assertion (A) is true.

$f\Big(-\dfrac{1}{2}\Big) = 16 \times \Big(-\dfrac{1}{2}\Big)^3 - 8 \times \Big(-\dfrac{1}{2}\Big)^2 + 4 \times \Big(-\dfrac{1}{2}\Big) + 7\\[1em] = 16 \times \Big(-\dfrac{1}{8}\Big) - 8 \times \Big(\dfrac{1}{4}\Big) + 4 \times \Big(-\dfrac{1}{2}\Big) + 7\\[1em] = -2 - 2 - 2 + 7\\[1em] = 1$

So, reason (R) is true.

Thus, both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct reason for Assertion (A).

Hence, option 3 is the correct option.