JEE Main 2026 — Jan 23, Shift 2 — Mathematics solution

JEE Main2026Jan 23, Shift 2Mathematics

If the mean and the variance of the data

Class	4-8	8-12	12-16	16-20
Frequency	3	$λ$	4	7

are $μ$ and $19$ respectively, then the value of $λ + μ$ is

A
18
B
21
C
20
D
19

Solution

Class midpoints: $6, 10, 14, 18$ .

$μ = \frac{\sum f _{i} x _{i}}{\sum f _{i}} = \frac{6 ( 3 ) + 10 λ + 14 ( 4 ) + 18 ( 7 )}{14 + λ} = \frac{200 + 10 λ}{14 + λ} = 10 + \frac{60}{14 + λ}$ .

For $μ$ to be integer, $14 + λ$ must divide $60$ . Try $λ = 6$ : $μ = 10 + 3 = 13$ .

Check variance:

$σ^{2} = \frac{6 ^{2} ( 3 ) + 1 0 ^{2} ( 6 ) + 1 4 ^{2} ( 4 ) + 1 8 ^{2} ( 7 )}{20} - 1 3^{2}$

$= \frac{108 + 600 + 784 + 2268}{20} - 169 = \frac{3760}{20} - 169 = 188 - 169 = 19$ . ✓

$∴ λ + μ = 6 + 13 = 19$ .

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