JEE MAIN - Mathematics (2017 (Offline) - No. 1)
Let $$a$$, b, c $$ \in R$$. If $$f$$(x) = ax2 + bx + c is such that
$$a$$ + b + c = 3 and $$f$$(x + y) = $$f$$(x) + $$f$$(y) + xy, $$\forall x,y \in R,$$
then $$\sum\limits_{n = 1}^{10} {f(n)} $$ is equal to
$$a$$ + b + c = 3 and $$f$$(x + y) = $$f$$(x) + $$f$$(y) + xy, $$\forall x,y \in R,$$
then $$\sum\limits_{n = 1}^{10} {f(n)} $$ is equal to
165
190
255
330
Explanation
f(x) = ax2 + bx + c
f(1) = a + b + c = 3 $$ \Rightarrow $$ f (1) = 3
Now f(x + y) = f(x) + f(y) + xy ...(1)
Put x = y = 1 in eqn (1)
f(2) = f(1) + f(1) + 1
= 2f(1) + 1
$$ \Rightarrow $$ f(2) = 7
Similarly f(3) = 12
f(4) = 18
$$\sum\limits_{n = 1}^{10} {f(n)} $$ = 3 + 7 + 12 + 18 + 25 + 33 + 42 + 52 + 63 + 75 = 330
f(1) = a + b + c = 3 $$ \Rightarrow $$ f (1) = 3
Now f(x + y) = f(x) + f(y) + xy ...(1)
Put x = y = 1 in eqn (1)
f(2) = f(1) + f(1) + 1
= 2f(1) + 1
$$ \Rightarrow $$ f(2) = 7
Similarly f(3) = 12
f(4) = 18
$$\sum\limits_{n = 1}^{10} {f(n)} $$ = 3 + 7 + 12 + 18 + 25 + 33 + 42 + 52 + 63 + 75 = 330
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