Statement 1 is the definition of a graded-index fibre and statement 3 is its purpose, so the answer is 1 & 3, option 1.

The index profile. A graded-index fibre has

\(n(r)=n_{1}\sqrt{1-2\Delta\left(\dfrac{r}{a}\right)^{\alpha}}\)

with \(\alpha\approx2\) giving a near-parabolic profile — highest at the axis, falling smoothly to the cladding value at the core edge. Statement 2's "sudden change" is the defining feature of the other kind of fibre, in which the index is uniform across the core and drops abruptly at the boundary.

Why grading improves the bandwidth — an elegant piece of physics. In a step-index multimode fibre, a ray bouncing at a steep angle travels a much longer path than one running near the axis, so a launched pulse arrives smeared over time. Grading fixes this without eliminating the modes. The off-axis rays spend most of their journey in the outer, lower-index region, where the phase velocity \(v=c/n\) is higher. Travelling further but faster, they arrive at almost the same instant as the axial ray, which travels the shortest path but through the slowest region. The rays also follow smooth sinusoidal curves rather than zig-zags, since the continuously falling index refracts them gradually back towards the axis. Intermodal dispersion falls by roughly two orders of magnitude — from about 50 ns/km to well under 1 ns/km.

Why statement 4 is nevertheless false. Grading reduces intermodal dispersion; only single-mode fibre eliminates it, by permitting one path alone. The bandwidth-distance product tells the story plainly:

Where graded-index fibre is used is the middle ground its numbers suggest: its 50 µm core couples easily to inexpensive LED or VCSEL sources and tolerates loose connectors, so it dominates in-building and data-centre links, where the distances are short and single mode's precision splicing would not repay its cost.

Hence, statements 1 and 3 are true.