Verified Solution • 2025–26 Session moderate

QJEEM23Mac588: JEE Main 2023 Mathematics - Class 11 JEEM23Mac588 Solution: Three rotten apples are mixed accidently with seven good apples …

Three rotten apples are mixed accidently with seven good apples and four apples are drawn one by one without replacement. Let the random variable X denote the number of rotten apples. If μ\mu and σ2\sigma^2 represent mean and variance of X, respectively, then 10(μ2+σ2)10(\mu^2+\sigma^2) is equal to :

Multiple Choice Options

  • A 20
  • B 30
  • C 250
  • D 25

Step-by-Step Verified Solution

3 rotten apples are mixed with 7 good apples.

∴\therefore Total apples = 10

Among those 10 apples 4 are chosen randomly.

xi{x_i} pi{p_i}
pixi{p_i}{x_i} pi(xi)2{p_i}{({x_i})^2}
0 7C410C4=35210{{{}^7{C_4}} \over {{}^{10}{C_4}}} = {{35} \over {210}} 0 0
1 3C1×7C310C4=105210{{{}^3{C_1} \times {}^7{C_3}} \over {{}^{10}{C_4}}} = {{105} \over {210}} 105210{{105} \over {210}} 105210{{105} \over {210}}
2 3C2×7C210C4=63210{{{}^3{C_2} \times {}^7{C_2}} \over {{}^{10}{C_4}}} = {{63} \over {210}} 126210{{126} \over {210}} 252210{{252} \over {210}}
3 3C3×7C110C4=7210{{{}^3{C_3} \times {}^7{C_1}} \over {{}^{10}{C_4}}} = {7 \over {210}} 21210{{21} \over {210}} 63210{{63} \over {210}}

xi{x_i} = Number of rotten apples drawn.

pi{p_i} = Probability of rotten apple.

We know,

Mean (μ)=∑pixi(\mu ) = \sum {{p_i}{x_i}}

=0+105210+126210+21210= 0 + {{105} \over {210}} + {{126} \over {210}} + {{21} \over {210}}

=252210=65= {{252} \over {210}} = {6 \over 5}

Also,

Variance (σ2)=(∑pi(xi)2)−μ2({\sigma ^2}) = \left( {\sum {{p_i}{{({x_i})}^2}} } \right) - {\mu ^2}

=105210+252210+63210−3625= {{105} \over {210}} + {{252} \over {210}} + {{63} \over {210}} - {{36} \over {25}}

=12+1210+310−3625=1425= {1 \over 2} + {{12} \over {10}} + {3 \over {10}} - {{36} \over {25}} = {{14} \over {25}}

∴\therefore 10(μ2+σ2)10(\mu^2 + {\sigma ^2})

=10((65)2+1425)= 10\left( {({6 \over 5})^2 + {{14} \over {25}}} \right)

=10(36+1425)= 10\left( {{{36 + 14} \over {25}}} \right)

=10×5025=20= 10 \times {{50} \over {25}} = 20

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