If `lim_(xto0) (1-sqrt(cos2x).root(3)(cos3x).root(4)(cos4x)...root(n)(cosnx))/(x^(2))` has the value equal to 10, then the value of n equals____________.

Options

A. 4
B. 3
C. 2
D. `1//2`

Answer

`(6)`

Explanation

`L=-underset(xto0)lim((d)/(dx)prod_(r=2)^(n)(cosrx)^(1//r))/(2x)" "`(Using L'Hospital's rule)
Let`y=prod_(r=2)^(n)(cosrx)^(1//r)`
or `lny=sum_(r=2)^(n)((1)/(r)ln(cosrx))`
or `(1)/(y)(dy)/(dx)=-sum_(r=2)^(n)tan(rx)`
or `(d)/(dx)=-ysum_(r=2)^(n)tan(rx)`
or `L=underset(xto0)lim(y.sum_(r=2)^(n)tan(rx))/(2x)`
`=(1)/(2)[2+3+4+...+n]`
`=(1)/(2)[(n(n+1))/(2)-1]=(n^(2)+n-2)/(4)`
or `(n^(2)+n-2)/(4)=10`
or `n^(2)+n-42=0`
or `(n+7)(n-6)=0`
or `n=6`