How many number of questions you want to show to one respondent. If you want show only 2 question and Q1 will always appear then you can do one thing.

Put all 12 questions in different block.

Step 1- Q1 block should be outside the randomizer in survey flow.

Step 2: Create randomizer in survey flow. Set 1 at box and check evenly present checkbox.

Step 3: put Q2 to Q12(all 11 blocks inside the randomizer)

Step4: Duplicate Q2 block 7 more times so Q2 block should be appeared under randomizer 8 times. Q3 and Q8 Block duplicate 3 more times so Q3 block should be appeared under randomizer 4 times.

Test it properly. I'm sure it will work.

@ArunDubey - Hey, thanks. I know this way, but this is too long to apply. More so, when I have to get done the same thing in different proportion. I am looking for far efficient way involving less of manual work.

As you mentioned initially that you were not able to identify the uneven randomization so that was the simplest way to do achieve this requirement in Qualtrics. If you already know this so I think, you can try other way two ways. As follows:

1) Using embedded field instead of blocks under randomization and then apply branch logic respective to each embedded field on each question.

2) Use web services and apply external database. Then you can code with the help of sorting algorithm. But this is more complex way to achieve the same task and for my view if something can be achieved using same survey programming tool so always go with it.

Hey @ArunDubey, thanks for the response. I understand that copying (multiple times) is the simplest way, but since I need to use this at multiple places and with different ratios such as 7:31:11:13, it might become very laborious task. Hope you understand the concern.

1) I tried the first method by generating a random number as embedded data (in the survey flow). And then using the display logic to direct to the questions. However, I am not sure about the equi-probable nature of random numbers! As far as I know, there is no guarantee of getting an even distribution using random numbers.

Please help if I am doing it wrong or the method can be bettered!

2) I am sorry I am not at all aware about these things.

Thanks again.

Rand function will not give you equal distribution. You will need to use sorting algorithm and DB to store and to pick values so that you will be able to apply least fill randomization. It's completely time wasting and too much effort. Better to go with first option. 

@ArunDubey Exactly, that’s what was my worry. Thanks for confirming.

By first you mean this
“Using embedded field instead of blocks under randomization and then apply branch logic respective to each embedded field on each question.”

Right? I will try this and get back to you.

Meanwhile, I also got to learn about quotas and was able to restrict the number of responses for each question using them. What do you think about that idea? (Obviously, I need to start with the total number of responses to begin with, which is something I know as I plan to run the experiment through Qualtrics.)

Regarding your on first point. Yes, use branch logic respective to embedded field.

About the quota condition, if you can create quota and can play with quotafull branch logic then you can use that one too. 

@ArunDubey : would be grateful if you can given an example of the first way you have suggested.

And yes, quotas are working.

Thanks.