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The mean and variance of seven observations are 8 and 16, respectively. If 5 of the observations are 2, 4, 10, 12, 14, then the product of the remaining two observations is :

JEE · Math · previous-year question

  1. A.40
  2. B.48correct
  3. C.49
  4. D.45

Answer

B. 48

Explanation

Given mean ($$\mu $$) = 8 variance ($${\sigma ^2}$$) = 16 No of observations (N) = 7 Let the two unknown observation = x and y We know, $${\sigma ^2} = {{\sum {x_i^2} } \over N} - {\mu ^2}$$ = 16 $$ \Rightarrow $$ $${{{2^2} + {4^2} + {{10}^2} + {{12}^2} + {{14}^2} + {x^2} + {y^2}} \over 7} - {\left( 8 \right)^2}$$ = 16 $$ \Rightarrow $$ x2 + y2 = 100 ........(1) We know, $$\mu $$ = $${{\sum {{x_i}} } \over N}$$ = 8 $$ \Rightarrow $$ $${{2 + 4 + 10 + 12 + 14 + x + y} \over 7}$$ = 8 $$ \Rightarrow $$ x + y = 14 ...........(2) As (x + y)2 = x2 + y2 + 2xy $$ \Rightarrow $$ (14)2 = 100 + 2xy $$ \Rightarrow $$ 196 = 100 + 2xy $$ \Rightarrow $$ xy = 48

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