Question

The mean and standard deviation of marks of $10$ students in a class test of $10$ marks is $7$ and $1$ respectively. The marks of $A$ is not known and the standard deviation of the remaining $9$ students is $0$. If $A$ does not score full marks, then the marks of $A$ is

• A. $2$
• B. $4$
• C. $6$
• D. $8$

Answer 1

The correct option is B $4$

$\overline{x}=7$
Total marks of $10$ students,
$\sum x_i=10\overline{x}=70$

$\sigma =1$
$\Rightarrow \frac{\sum x_i^2}{10}-7^2=1\Rightarrow \sum x_i^2=500$

Let $A$ scored $x$ marks.
Now, the mean of the remaining students is, $\frac{70-x}{9}$
and the standard deviation is,
$\frac{500-x^2}{9}-{\left(\frac{70-x}{9}\right)}^2=0\Rightarrow 9(500-x^2)=(70-x{)}^2\Rightarrow 4500-9x^2=4900+x^2-140x\Rightarrow 10x^2-140x+400=0\Rightarrow x^2-14x+40=0\Rightarrow (x-4\left)\right(x-10)=0\Rightarrow x=4,10$
As $A$ does not score full marks,
hence, the marks scored by $A$ is $4$.