Does the broadband speed depends on distance from the exchange?

Since BSNL uses ADSL broadband technology, the speed will definitely be sensitive to distance. The connection tends to be slower, the more you are away from the exchange.
This has a reason for it. The data is transmitted through the copper wires. These are the same wires that transmit your voice calls. As is well known, the quality of the signal gets degraded with the distance. This degradation is referred to as Noise in technical terms.
You might have experienced the noise you get while talking over the phone lines. It is the same phenomena that affects your data lines as well. Since the data is transmitted along the same lines, you get the same issues with your data speed. But, unlike voice signals, digital signals cannot handle missing data and thus any missing information is needed to be sent again. This is exactly what reduces your download/upload speed.
The noise is created because of the resistance, inductance and the capacitance of the copper wires. All these factors together contribute to the amount of noise and hence the speed.

In fact. BSNL uses ADSL ( Asynchronous digital subscriber link) technology to provide broadband connectivity. The copper wires used in this connection offer resistance to the data signals being sent through them. This resistance increases as the distance increases. The optimum distance for a better speed is woukld be between 3 to 6 km. Further distance would definitely slow down your broadband speed.

The noise can be avoided using micro filters. Check if your service center offers any such services like the micro filters. I do not think BSNL has any such plans.

There are multiple types of broadband connections and the type of their signal and the strength depends on the wiring and connectivity method used.

In case of BSNL which makes use ADSL/DSL for it's connection is likely to have some signal loss issue. It makes use of the connectivity points between exchange for the boosting of signal lost during transfer. So if any of the point reduces in strength, the boosting helps in signal but with some noise sometimes depending on wire condition. Think of it as water taps being setup from the filtering plant to distribution to the customer end.

The loss of signal in the DSL is also due to the copper wires used in the connectivity wire. The drop occurs in the each connectivity points and for that reason each of it requires boosting. With the T1 lines or the high speed lines the connectivity wires are made up of optical fiber so they have relatively low loss. And this makes signal strength higher, even the distance doesn't matter as long as the wire is in good condition.

It's not just BSNL that has this problem. Even Hathway, Airtel, Trikona, Tata Indicom etc have this drop issue in between the boosters. So often they change connectivity wires in between exchanges to rectify this.

There is no work around for the noise and the loss of signal in between the points of exchange due to copper wires used. The only work around is telecom providers adopting the optic fiber networks for the better signal strength. So far telecom providers advertise optical fiber network but their installation clearly shows the wires being normal telephone copper wires. As long as there are copper wired connections, the broadband connection drop and loss due to noise and wire ageing is likely to occur.