XAT 2011 — QA & DI Question 32
Answer the following question based on the information given below.
From a group of 545 contenders, a party has to select a leader. Even after holding a series of meetings, the politicians and the general body failed to reach a consensus. It was then proposed that all 545 contenders be given a number from 1 to 545. Then they will be asked to stand on a podium in a circular arrangement, and counting would start from the contender numbered 1. The counting would be done in a clockwise fashion. The rule is that every alternate contender would be asked to step down as the counting continued, with the circle getting smaller and smaller, till only one person remains standing. Therefore the first person to be eliminated would be the contender numbered 2.
In a list of 7 integers, one integer, denoted as x is unknown. The other six integers are 20, 4, 10, 4, 8, and 4. If the mean, median, and mode of these seven integers are arranged in increasing order, they form an arithmetic progression. The sum of all possible values of x is
Answer & solution
- A
26
- B
32
- C
34
- D
38
40
It can be observed that, irrespective of the value of x, mode of these numbers will be 4.
Now, the median of these numbers will depend on the value of x,
If x < 4 then the median of these seven numbers will be 4.
Now, as the mode is 4, the median cannot be 4.
(the question states that mean, median and mode are arranged in ascending order.)
Hence x cannot be less than 4.
Now,
If 4 < x ≤ 8
the median will be x & the mean will be,
Now > 7 and x ≤ 8
∴ 4, x and will form an AP only if > x
∴ - x = x - 4
∴ + 4 = 2x
∴ x = 6
Hence, x = 6 is a possible answer.
Now, if x > 8 then median will be 8 & mean will be
Now, if x > 8 then is greater than 8.
∴ Increasing order of mean, median and mode will be,
4, 8,
Now, they are in A.P.
∴ 8 - 4 = - 8
∴ 12 =
∴ x = 12 × 7 – 50
∴ x = 34
Hence, sum of all possible values of x = 6 + 34 = 40
Hence, option (e).