Most likely it's either the tolerance of the RAM (cheap Value RAM tends to be sloppy and aggressive HyperX RAM too fast).  What  is the exact model of the RAM you're trying?

Preinstalled 4GB Samsung RAM model: M471B5173BH0-YK0

New 8GB Kingston RAM model: KVR16LS11/8

That's Value RAM -- which is the likely issue.  

The memory Kingston sells for Dell systems start with KCP in the part number.  Return the memory and order that.

Dell has a long tradition of building diagnostic indicators into our systems. These indicators can be really useful in helping to diagnose and resolve any issues your PC might experience whilst starting up. I've given a guide to the various codes for each family of notebook below.

If the display cannot show any errors your computer might emit a series of beeps during start-up.

1. 3 means 3-3-3-3...That is 3 beeps and 3 beeps and ...: Endless looping until user presses the power button to shut down the system.

2. The delay between each beep is 300ms. The delay between each set of beeps is 3 sec, and the beep sound lasts 300ms.

3. After each beep and each set of beeps, the BIOS should detect if the user presses the power button, if so, BIOS will jump out from looping and execute the normal shutdown process and power system.

Please refer the below article for understand and troubleshooting dell beep sound.

https://www.dell.com/support/article/en-us/sln309623/understanding-beep-codes-on-a-dell-notebook-pc?lang=en

No. I can't return the product now as I purchased it from Amazon. Under the reviews there had been many customer reviews who are using the same kingston RAM for the same laptop in various configurations viz. only Kingston 8 GB, Kingston 8GB + 8 GB and Kingston 8 GB+ pre-installed Samsung 4 GB.

I tried inserting both RAM's again and booting the laptop. It's working perfectly fine.

I suspect the problem is with pre-installed Samsung 4 GB RAM because earlier a few times it had shown "Check Media Fail" upon boot.

Anyways thanks for the efforts.