Maths Q10 – JEE-MAIN 2026

Suppose that the mean and median of the non-negative numbers

21, 8, 17, a, 51, 103, b, 13, 67, (a > b)

are 40 and 21, respectively. If the mean deviation about the median is 26, then

2 a

is equal to:

Clue 1 of 3: Understand the definitions

The mean is the sum of all values divided by the count. The median is the middle value when the data is sorted. Mean deviation about the median is the average of the absolute differences between each data point and the median.

🧠 Full Solution Path

Step 1: Determine

a + b

using the mean and establish conditions for

a, b

using the median.✦ Active

The given numbers are $21, 8, 17, a, 51, 103, b, 13, 67$ . There are 9 numbers. The sum of these numbers is $21 + 8 + 17 + a + 51 + 103 + b + 13 + 67 = 280 + a + b$ . Given mean is 40, so:

\frac{280 + a + b}{9} = 40 ⟹ 280 + a + b = 360 ⟹ a + b = 80

The sorted known numbers are $8, 13, 17, 21, 51, 67, 103$ . Since there are 9 numbers, the median is the $5^{t h}$ term. Given median is 21. For 21 to be the $5^{t h}$ term, there must be 4 numbers less than or equal to 21. The known numbers less than 21 are $8, 13, 17$ . This implies $b \leq 21$ . Since $a > b$ and $a + b = 80$ , we have $a = 80 - b$ . If $b \leq 21$ , then $a \geq 80 - 21 = 59$ . Thus, $b \leq 21$ and $a \geq 59$ . This also satisfies $a > b$ .

Step 2: Form an equation for

a - b

using the mean deviation about the median.○ Expand

The median $M = 21$ . The mean deviation about the median is $M D = \frac{1}{N} \sum_{i = 1}^{N} | x_{i} - M |$ . Given $M D = 26$ and $N = 9$ , so:

\sum_{i = 1}^{9} | x_{i} - 21 | = 9 \times 26 = 234

Calculate deviations for known numbers: $| 8 - 21 | = 13$ , $| 13 - 21 | = 8$ , $| 17 - 21 | = 4$ , $| 21 - 21 | = 0$ , $| 51 - 21 | = 30$ , $| 67 - 21 | = 46$ , $| 103 - 21 | = 82$ . Sum of these deviations $= 13 + 8 + 4 + 0 + 30 + 46 + 82 = 183$ . For $a$ and $b$ : Since $b \leq 21$ , $| b - 21 | = 21 - b$ . Since $a \geq 59$ , $| a - 21 | = a - 21$ . So, the total sum of deviations is:

183 + (a - 21) + (21 - b) = 234 ⟹ 183 + a - b = 234 ⟹ a - b = 234 - 183 = 51

Step 3: Solve the system of equations for

a

and find

2 a

.○ Expand

We have two equations:

1) a + b = 80

2) a - b = 51

Adding (1) and (2): $(a + b) + (a - b) = 80 + 51 ⟹ 2 a = 131$ . Subtracting (2) from (1): $(a + b) - (a - b) = 80 - 51 ⟹ 2 b = 29 ⟹ b = 14.5$ . The value of $a = 131 / 2 = 65.5$ . These values satisfy the conditions $a > b$ ( $65.5 > 14.5$ ), $b \leq 21$ ( $14.5 \leq 21$ ), and $a \geq 59$ ( $65.5 \geq 59$ ). The question asks for $2 a$ .

2 a = 131

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Maths Question 10 – JEE-MAIN 2026

Clue 1 of 3: Understand the definitions

🧠 Full Solution Path