1.

Let `f(x)={{:((alphacotx)/(x)+(beta)/(x^(2))",",0ltxle1),((1)/(3)",",x=0):}` If f(x) is continuous at x =0, then the value of `alpha^(2)+beta^(2)` isA. 1B. 2C. 5D. 9

Answer» Correct Answer - B
`underset(xrarr0)(lim)f(x)=(1)/(3)`
`rArr" "underset(xrarr0)(lim)(x.alphacotx+beta)/(x^(2))=(1)/(3)`
`rArr" "underset(xrarr0)(lim)(xalpha+betatanx)/(x^(2).tanx)=(1)/(3)`
`rArr" "underset(xrarr0)(lim)(alphax+beta(x+(x^(3))/(3)+...oo))/(x^(3)((tanx)/(x)))=(1)/(3)`
`rArr" "underset(xrarr0)(lim)((alpha+beta)x+((beta)/(3))x^(3)+...oo)/(x^(3))=(1)/(3)`
So, `" "alpha+beta=0`
Also,`" "beta=1rArr alpha=-1`


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