Verified Solution • 2025–26 Session moderate

QJEEM21Maa4b6: JEE Main 2021 Mathematics - Class 11 JEEM21Maa4b6 Solution: Consider a set of 3n numbers having variance 4. In this set,…

Consider a set of 3n numbers having variance 4. In this set, the mean of first 2n numbers is 6 and the mean of the remaining n numbers is 3. A new set is constructed by adding 1 into each of first 2n numbers, and subtracting 1 from each of the remaining n numbers. If the variance of the new set is k, then 9k is equal to __________.

Step-by-Step Verified Solution

Let first 2n observations are x1, x2 ...................., x2n

and last n observations are y1, y2 ....................., yn

Now, ∑xi2n=6{{\sum {{x_i}} } \over {2n}} = 6 , ∑yin=3{{\sum {{y_i}} } \over n} = 3

⇒∑xi=12n,∑yi=3n\Rightarrow \sum {{x_i}} = 12n,\sum {{y_i}} = 3n

∴\therefore ∑xi+∑yi3n=15n3n=5{{\sum {{x_i}} + \sum {{y_i}} } \over {3n}} = {{15n} \over {3n}} = 5

Now, ∑xi2+∑yi23n−52=4{{\sum {x_i^2} + \sum {y_i^2} } \over {3n}} - {5^2} = 4

⇒∑xi2+∑yi2=29×3n=87n\Rightarrow \sum {x_i^2} + \sum {y_i^2} = 29 \times 3n = 87n

Now, mean is ∑(xi+1)+∑(yi−1)3n=15n+2n−n3n=163{{\sum {({x_i} + 1) + \sum {({y_i} - 1)} } } \over {3n}} = {{15n + 2n - n} \over {3n}} = {{16} \over 3}

Now, variance is ∑(xi+1)2+∑(yi−1)23n−(163)2{{{{\sum {{{({x_i} + 1)}^2} + \sum {({y_i} - 1)} } }^2}} \over {3n}} - {\left( {{{16} \over 3}} \right)^2}

=∑xi2+∑yi2+2(∑xi−∑yi)+3n3n−(163)2= {{\sum {x_i^2 + \sum {y_i^2} + 2\left( {\sum {{x_i}} - \sum {{y_i}} } \right) + 3n} } \over {3n}} - {\left( {{{16} \over 3}} \right)^2}

=87n+2(9n)+3n3n−(163)2= {{87n + 2(9n) + 3n} \over {3n}} - {\left( {{{16} \over 3}} \right)^2}

= 29+6+1−(163)229 + 6 + 1 - {\left( {{{16} \over 3}} \right)^2}

=324−2569=689=k= {{324 - 256} \over 9} = {{68} \over 9} = k

⇒\Rightarrow 9k = 68

Therefore, the correct answer is 68.

Related Learning Resources & Exam Preparation

JEE Main PYQ Question Bank
Browse year-wise authentic previous years' questions with solutions
Authentic CBT Mock Tests
Simulate official exam environment with live timer and percentile scoring