1. Identify compound $Y$ from the bromine water test

   Bromine water oxidizes an aldehyde group of an aldose to the corresponding aldonic acid.

   Since $Y$ gives gluconic acid on treatment with bromine water, $Y$ must be glucose.

   $\text{Glucose} \xrightarrow{Br_2/H_2O} \text{Gluconic acid}$

2. Determine what polysaccharide $X$ gives on hydrolysis

   The question says that polysaccharide $X$ on boiling with dilute $H_2SO_4$ gives $Y$.

   Since $Y = \text{glucose}$, $X$ must be a polymer of glucose.

3. Use the nature of glycosidic linkage

   The question further states that $X$ contains only $\beta$-glycosidic linkages.

   Now examine the options:

   • Starch: polymer of $\alpha$-D-glucose; contains $\alpha$-linkages
   • Amylose: component of starch; has $\alpha(1\to4)$ linkages
   • Amylopectin: has $\alpha(1\to4)$ and $\alpha(1\to6)$ linkages
   • Cellulose: polymer of $\beta$-D-glucose; has only $\beta(1\to4)$ glycosidic linkages

4. Match all conditions

   Cellulose satisfies both conditions:

   • On hydrolysis, it gives glucose
   • It contains only $\beta$-glycosidic linkages

5. Evaluate options

   • A: starch $\to$ incorrect
   • B: cellulose $\to$ correct
   • C: amylose $\to$ incorrect
   • D: amylopectin $\to$ incorrect

Therefore, the compound $X$ is:

$\boxed{\text{Cellulose}}$