Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Advanced 2026Paper 2
Exam
JEE Advanced
Year
2026
Paper
Paper 2
Subject
Mathematics
Chapter
Statistics

JEE Advanced 2026 · Paper 2 · Mathematics · Statistics

Consider a data consisting of 10 observations x_1, x_2, , x_ 10 ,...

Consider a data consisting of 1010 observations x1,x2,…,x10x_1, x_2, \ldots, x_{10}, whose mean is 55 and variance is 77. If the mean and the variance of the first 88 observations x1,x2,…,x8x_1, x_2, \ldots, x_8 are 44 and 3.53.5, respectively, and x9<x10x_9 < x_{10}, then the value of 3x9+2x103x_9 + 2x_{10} is ___________.

Correct answer

0

Step-by-step solution

Given ∑i=110xi10=5⇒∑i=110xi=50\dfrac{\sum_{i=1}^{10} x_i}{10} = 5 \Rightarrow \sum_{i=1}^{10} x_i = 50

∑i=110xi210−52=7⇒∑i=110xi2=320\dfrac{\sum_{i=1}^{10} x_i^2}{10} - 5^2 = 7 \Rightarrow \sum_{i=1}^{10} x_i^2 = 320

For the first 88 observations:

∑i=18xi8=4⇒∑i=18xi=32\dfrac{\sum_{i=1}^{8} x_i}{8} = 4 \Rightarrow \sum_{i=1}^{8} x_i = 32

∑i=18xi28−42=3.5⇒∑i=18xi2=8(3.5+16)=156\dfrac{\sum_{i=1}^{8} x_i^2}{8} - 4^2 = 3.5 \Rightarrow \sum_{i=1}^{8} x_i^2 = 8(3.5 + 16) = 156

Subtracting the sums, we get:

x9+x10=50−32=18x_9 + x_{10} = 50 - 32 = 18

x92+x102=320−156=164x_9^2 + x_{10}^2 = 320 - 156 = 164

Using (x9+x10)2=x92+x102+2x9x10(x_9 + x_{10})^2 = x_9^2 + x_{10}^2 + 2x_9 x_{10}:

182=164+2x9x10⇒2x9x10=324−164=160⇒x9x10=8018^2 = 164 + 2x_9 x_{10} \Rightarrow 2x_9 x_{10} = 324 - 164 = 160 \Rightarrow x_9 x_{10} = 80

The numbers x9x_9 and x10x_{10} are roots of the equation t2−18t+80=0t^2 - 18t + 80 = 0.

(t−8)(t−10)=0⇒t=8,10(t - 8)(t - 10) = 0 \Rightarrow t = 8, 10

Since x9<x10x_9 < x_{10}, we have x9=8x_9 = 8 and x10=10x_{10} = 10.

Therefore, 3x9+2x10=3(8)+2(10)=24+20=443x_9 + 2x_{10} = 3(8) + 2(10) = 24 + 20 = 44.

Answer: 4444
Practice Statistics on Quantrex Academy →

More from Statistics

A set of four observations has mean 1 and variance 13 . Another set of six observations… JEE Main 2026Let the mean and the variance of seven observations 2, 4, α, 8, β, 12, 14 , α < β , be 8… JEE Main 2026A data consists of 20 observations x_1, x_2, …, x_ 20 . If ∑_ i=1 ^ 20 (x_i + 5)^2 = 2500… JEE Main 2026A variable X takes values 0, 0, 2, 6, 12, 20, …, n(n-1) with frequencies ^nC_0, ^nC_1… JEE Main 2026The mean deviation about the mean for the data x_i 5 7 9 10 12 15 f_i 8 6 2 2 2 6 is… JEE Main 2026For 10 observations x_1, x_2, …, x_ 10 , if ∑_ i=1 ^ 10 (x_i+2)^2=180 and ∑_ i=1 ^ 10… JEE Main 2026Suppose that the mean and median of the non-negative numbers 21, 8, 17, a, 51, 103, b… JEE Main 2026The mean and variance of n observations are 8 and 16 , respectively. If the sum of the… JEE Main 2026If the mean of the data Class 5-10 10-15 15-20 20-25 25-30 30-35 Frequency 2 k 28 54 k+1… JEE Main 2026The mean and variance of 10 observations are 9 and 34.2, respectively. If 8 of these… JEE Main 2026Let X=\ x ∈ ~N: 1 ≤ x ≤ 19\ and for some a, b ∈ ℝ, Y=\ a x+b: x ∈ X\ . If the mean and… JEE Main 2026The mean and variance of a data of 10 observations are 10 and 2, respectively. If an… JEE Main 2026If the mean and the variance of the data Class 4-8 8-12 12-16 16-20 Frequency 3 λ 4 7 are… JEE Main 2026Let the mean and variance of 8 numbers -10,-7,-1, x, y, 9,2,16 be (7)/(2) and (293)/(4)… JEE Main 2026If the mean deviation about the median of the numbers k, 2 k, 3 k, …. ., 1000 k is 500… JEE Main 2026Let the mean and variance of 7 observations 2,4,10, x, 12,14, y, x>y , be 8 and 16… JEE Main 2026 Full Statistics list Statistics JEE Maths PYQsJEE Mathematics PYQs All JEE Advanced PYQs