COMPASSLearning

Talk to us Register Login

JEE Main · 2017 · 02 Apr 2017 · Offline · Relations And Functions XII

Let

a, b, c \in R

. If

f (x) = a x^{2} + b x + c

is such that

a + b + c = 3

and

f (x + y) = f (x) + f (y) + x y, \forall x, y \in R

, then

\sum_{n = 1}^{10} f (n)

is equal to :

(A)255
(B)330✓ Correct
(C)165
(D)190

Answer: (B)

Solution

Setting

x = y = 0

in

f (x + y) = f (x) + f (y) + x y

gives

f (0) = 0

, so

c = 0

. Comparing

f (x + 1) = f (x) + f (1) + x

with

f (x) = a x^{2} + b x

forces

2 a = 1

, so

a = \frac{1}{2}

; then

a + b + c = 3

gives

b = \frac{5}{2}

. Thus

f (n) = \frac{n ( n + 5 )}{2}

, and

n = 1 \sum 10 f (n) = \frac{1}{2} (\sum n^{2} + 5 \sum n) = \frac{1}{2} (385 + 275) = 330

.

JEE Main · 2017 · 02 Apr 2017 · Offline · Single correct option

Practise more: all JEE Main PYQs by year and chapter →

More JEE Main questions on Relations And Functions XII

Let $f : R - {0} \to (- \infty, 1)$ be a polynomial of degree 2 , satisfying $f (x) f (\frac{1}{x}) = f (x) + f (\frac{1}{x})$ . If $f (K) = - 2 K$ , then the sum of squares of all possible values of …
JEE Main 2025View solution →
Let $f (x) = \frac{2 ^{x + 2} + 16}{2 ^{2 x + 1} + 2 ^{x + 4} + 32}$ . Then the value of $8 (f (\frac{1}{15}) + f (\frac{2}{15}) + \dots + f (\frac{59}{15}))$ is equal to
JEE Main 2025View solution →
If $f (x) = \frac{( t a n 1 ^{o} ) x + l o g _{e} ( 123 )}{x l o g _{e} ( 1234 ) - ( t a n 1 ^{o} )}, x > 0$ , then the least value of $f (f (x)) + f (f (\frac{4}{x}))$ is
JEE Main 2023View solution →
Suppose $f : R \to (0, \infty)$ be a differentiable function such that $5 f (x + y) = f (x) \cdot f (y), \forall x, y \in R$ . If $f (3) = 320$ , then $\sum_{n = 0}^{5} f (n)$ is equal to:
JEE Main 2023View solution →
Let $f : R \to R$ be a function defined by $f (x) = (2 (1 - \frac{x ^{25}}{2}) (2 + x^{25}))^{\frac{1}{50}}$ . If the function $g (x) = f (f (f (x))) + f (f (x))$ , then the greatest integer less than or equal to …
JEE Main 2022View solution →