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Median with unknown middle terms in an ordered list

This type of question, asked 3 times: CDS 2 2022, CDS 1 2026, CDS 2 2026. All 3 questions below, oldest first.

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  1. Question 1

    CDS 2 2022

    65
    The data of different natural numbers 4, 7, 10, 14, 2x+3, 2x+5, 22, 23, 30, 504,\ 7,\ 10,\ 14,\ 2x + 3,\ 2x + 5,\ 22,\ 23,\ 30,\ 50 are in ascending order. How many possible values are there for the median of the data for various values of xx ?

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  2. Question 2

    CDS 1 2026

    40
    The numbers 59, 53, 51, 43, 36, (8x+1)(8x + 1), (x2+1)(x^2 + 1), 13, 12, 9, 7, 3 are written in descending order and their median is 25. What is/are the values of x ?

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  3. Question 3

    CDS 2 2026

    93
    Consider the observations x, x + 4, 30, 33, 74, 78, 49, 52, 98, 85. If the median of the data is 67, then what is the value of x ?

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