Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main 202528 JanuaryMorning Shift
Exam
JEE Main
Year
2025
Date
28 January
Shift
Morning Shift
Subject
Mathematics
Chapter
Functions

JEE Main 2025 · 28 January · Morning Shift · Mathematics · Functions

Let f: ℝ arrow ℝ be a function defined by f(x)=(2+3 a)...

Let f:R→Rbeafunctiondefinedbyf: \mathbb{R} \rightarrow \mathbb{R} be a function defined byf(x)=(2+3 a) x^2+\left(\frac{a+2}{a-1}\right)x+b,a≠1.x+b, a \neq 1 .\text{If } f(x+\mathrm{y})=f(x)+f()=f(x)+f(\mathrm{y})+1−)+1-\frac{2}{7}xx\mathrm{y}, then the valueof28∑i=15∣f(i)∣value of 28 \sum_{i=1}^5|f(i)| is

Options

  1. A545
  2. B715
  3. C735
  4. D675

Correct answer

D. 675

Step-by-step solution

 Put y=0f(x)=f(0)+f(x)+1−0f(0)=−1f(0)=0+0+b⇒b=−1f(−1+1)=f(−1)+f(1)+1+27f(0)=f(−1)+f(1)+97\begin{aligned} & \text{ Put } y=0 \\ & f(x)=f(0)+f(x)+1-0 \\ & f(0)=-1 \\ & f(0)=0+0+b \\ & \Rightarrow b=-1 \\ & f(-1+1)=f(-1)+f(1)+1+\frac{2}{7} \\ & f(0)=f(-1)+f(1)+\frac{9}{7}\end{aligned}−1=(2+3a)+(a+2a−1)(−1)+b+(2+3a)+a+2a−1+b+97\begin{aligned}-1=(2+3 a)+\left(\frac{a+2}{a-1}\right)(-1)+b+(2+3 a) & \\ & +\frac{a+2}{a-1}+b+\frac{9}{7}\end{aligned} −1=4+6a−2+97−1=2+97+6a6a=−1−2−97a=−57f(x)=−x27+97−127x−1f(x)=−x27−34x−1∑i=15f(i)=−17(5×6×116)−34(5×62)−5=−557−454−5=67528⇒28∣∑i−15f(i)∣=675\begin{aligned} & -1=4+6 a-2+\frac{9}{7} \\ & -1=2+\frac{9}{7}+6 a \\ & 6 a=-1-2-\frac{9}{7} \\ & a=\frac{-5}{7} \\ & f(x)=\frac{-x^2}{7}+\frac{\frac{9}{7}}{\frac{-12}{7}} x-1 \\ & f(x)=\frac{-x^2}{7}-\frac{3}{4} x-1 \\ & \sum_{i=1}^5 f(i)=-\frac{1}{7}\left(\frac{5 \times 6 \times 11}{6}\right)-\frac{3}{4}\left(\frac{5 \times 6}{2}\right)-5 \\ & =\frac{-55}{7}-\frac{45}{4}-5 \\ & =\frac{675}{28} \\ & \Rightarrow 28\left|\sum_{i-1}^5 f(i)\right|=675\end{aligned}$
Practice Functions on Quantrex Academy →

More from Functions

Let ℕ denote the set of all positive integers. Consider the sets A = \ 1, 2, 3, 4, 5\ and… JEE Advanced 2026Let f:(1,∞)→ℝ be a function defined as f(x) = (x-1)/(x+1) . Let f^ i+1 (x) = f(f^i(x))… JEE Main 2026Let f be a polynomial function such that _2(f(x)) = ( _2(2+(2)/(3)+(2)/(9)+…∞))·… JEE Main 2026Let f: ℝ arrow ℝ be defined as f(x) = (2x^2 - 3x + 2)/(3x^2 + x + 3) . Then f is : JEE Main 2026Let e be the base of natural logarithm and let f: \ 1, 2, 3, 4\ arrow \ 1, e, e^2, e^3\… JEE Main 2026Let f: R arrow R be a function such that f(x) + 3f((π)/(2) - x) = x , x ∈ R . Let the… JEE Main 2026The sum of all the integral values of p such that the equation 3 ^2 x + 12 x - 3 = p , x… JEE Main 2026Let A = \ 1, 2, 3, 4, 5, 6\ . The number of one-one functions f: A arrow A such that f(1)… JEE Main 2026For the function f:[1,∞) arrow [1,∞) defined by f(x)=(x-1)^4+1 , among the two… JEE Main 2026Let for some α ∈ ℝ , f:ℝarrowℝ be a function satisfying f(x+y)=f(x)+2y^2+y+α xy for all… JEE Main 2026Let [·] denote the greatest integer function. If the domain of the function f(x) = ^ -1… JEE Main 2026The number of functions f: \ 1, 2, 3, 4\ arrow \ a, b, c\ , which are not onto, is: JEE Main 2026If the domain of the function f(x) = _ (0.6) (|(2x-5)/(x^2-4)|) is (-∞, a] \ b\ [c, d)… JEE Main 2026Given below are two statements: Statement I: The function f: R arrow R defined by… JEE Main 2026The sum of all the elements in the range of f(x)=Sgn( x)+Sgn( x)+Sgn( x)+Sgn( x) , x ≠ n… JEE Main 2026If g(x)=3 x^ 2 +2 x-3, f(0)=-3 and 4 g(f(x))=3 x^ 2 -32 x+72 , then f(g(2)) is equal to: JEE Main 2026 Full Functions list Functions JEE Maths PYQsJEE Mathematics PYQs All JEE Main PYQs