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JEE Main 202624 JanuaryMorning Shift
Exam
JEE Main
Year
2026
Date
24 January
Shift
Morning Shift
Subject
Mathematics
Chapter
Statistics

JEE Main 2026 · 24 January · Morning Shift · Mathematics · Statistics

The mean and variance of a data of 10 observations are 10 and 2,...

The mean and variance of a data of 10 observations are 10 and 2, respectively. If an observations α\alpha in this data is replaced by β\beta, then the mean and variance become 10.1 and 1.99, respectively. Then α+β\alpha+\beta equals

Options

  1. A15
  2. B5
  3. C10
  4. D20

Correct answer

D. 20

Step-by-step solution

n=10n=10, mean =10=10, variance =2=2.
So ∑xi=100\sum x_i = 100 and ∑xi2=10(2+100)=1020\sum x_i^2 = 10(2+100)=1020.
After replacing α\alpha by β\beta:
new mean =10.1⇒∑xi−α+β=101⇒β−α=1=10.1 \Rightarrow \sum x_i - \alpha + \beta = 101 \Rightarrow \beta - \alpha = 1
New variance =1.99⇒∑xi2−α2+β210−(10.1)2=1.99=1.99 \Rightarrow \frac{\sum x_i^2 - \alpha^2 + \beta^2}{10} - (10.1)^2 = 1.99
∑xi2−α2+β2=10(1.99+102.01)=1040\sum x_i^2 - \alpha^2 + \beta^2 = 10(1.99+102.01) = 1040
β2−α2=1040−1020=20\beta^2 - \alpha^2 = 1040 - 1020 = 20
(β−α)(β+α)=20⇒1⋅(α+β)=20(\beta-\alpha)(\beta+\alpha) = 20 \Rightarrow 1 \cdot (\alpha+\beta) = 20
α+β=20\alpha + \beta = 20
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