Let `a,b,c in R. " If " f(x) =ax^(2)+bx+c` be such that `a+b+c=3 and f(x+y)=f(x)+f(y)+xy, AA x,y in R, " then " sum_(n=1)^(10)f(n)` is equal toA. 330B. 165C. 190D. 255

Let `a,b,c in R. " If " f(x) =ax^(2)+bx+c` be such that `a+b+c=3 and f

1.	Let `a,b,c in R. " If " f(x) =ax^(2)+bx+c` be such that `a+b+c=3 and f(x+y)=f(x)+f(y)+xy, AA x,y in R, " then " sum_(n=1)^(10)f(n)` is equal toA. 330B. 165C. 190D. 255
Answer» Correct Answer - A We have, `f(x)=ax^(2)+bx+c` Now, `f(x+y)=f(x) +f(y)+xy` Put `y=0 rArr f(x) = f(x)+f(0)+0` `rArr f(0)=0` `rArr c=0` Again, put `y= -x` ` therefore f(0)=f(x)+f(-x)-x^(2)` `rArr 0=ax^(2)+bx+ax^(2)-bx-x^(2)` `rArr 2ax^(2)-x^(2)=0` `rArr a=(1)/(2)` Also, `a+b+c=3` `rArr (1)/(2) +b+0=3 rArr b=(5)/(2)` ` therefore f(x)=(x^(2)+5x)/(2)` Now, `f(n)=(n^(2)+5n)/(2)=(1)/(2)n^(2)+(5)/(2)n` `therefore sum_(n=1)^(10)f(n)=(1)/(2)sum_(n=1)^(10)n^(2)+(5)/(2)sum_(n=1)^(10)n` `=(1)/(2)*(10xx11xx21)/(6)+(5)/(2)xx(10xx11)/(2)` `=(385)/(2)+(275)/(2)=(660)/(2)=330`