Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app

JEE Advanced 2010Paper 2

Exam: JEE Advanced
Year: 2010
Paper: Paper 2
Subject: Mathematics
Chapter: Application of Derivatives

JEE Advanced 2010 · Paper 2 · Mathematics · Application of Derivatives

Paragraph: Consider the polynomial f(x)=1+2 x+3 x^2+4 x^3 . Let s...

Paragraph: Consider the polynomial

f(x)=1+2 x+3 x^2+4 x^3

. Let

s

be the sum of all distinct real roots of

f(x)

and let

t=|s|

.Question: The real number

s

lies in the interval

Options

A $\left(-\frac{1}{4}, 0\right)$
B $\left(-11,-\frac{3}{4}\right)$
C $\left(-\frac{3}{4},-\frac{1}{2}\right)$
D $\left(0, \frac{1}{4}\right)$

Correct answer

C. $\left(-\frac{3}{4},-\frac{1}{2}\right)$

Step-by-step solution

Given,

f(x)=4 x^3+3 x^2+2 x+1

\begin{aligned} & f^{\prime}(x)=2\left(6 x^2+3 x+1\right) \\ & D=9-24 < 0 \end{aligned}

Hence,

f(x)=0

has only one real root.

\begin{aligned} f\left(-\frac{1}{2}\right) & =1-1+\frac{3}{4}-\frac{4}{8}>0 \\ f\left(-\frac{3}{4}\right) & =1-\frac{6}{4}+\frac{27}{16}-\frac{108}{64} \\ & =\frac{64-96+108-108}{64} < 0 \end{aligned}

f(x)

changes its sign in \left(-\frac{3}{4}, \frac{-1}{2}\right)

, hence

f(x)=0

has a root in \left(\frac{-3}{4}, \frac{-1}{2}\right)

.

Practice Application of Derivatives on Quantrex Academy →

More from Application of Derivatives

Consider the function f: (0, ∞) arrow (-∞, ∞) given by f(x) = x \, _e(x) - x + 1 . Then… JEE Advanced 2026 Let P be the point on the parabola y = x^2 such that the slope of the tangent to the… JEE Advanced 2026 Let f: ℝ arrow ℝ be a differentiable function such that f((x+y)/(3)) = (f(x) + f(y))/(3)… JEE Main 2026 _ 0 ≤ x ≤ π (16 ((x)/(2)) ^3((x)/(2))) is equal to: JEE Main 2026 Let f(x) be a polynomial of degree 5 , and have extrema at x = 1 and x = -1 . If _ x → 0… JEE Main 2026 The number of critical points of the function f(x) = cases |( x)/(x)|, & x ≠ 0 1, & x = 0… JEE Main 2026 Let f be a differentiable function satisfying f(x)=1-2 x+∫_ 0 ^ x e^ (x-t) f(t) dt, x ∈ R… JEE Main 2026 Consider the following three statements for the function f:(0, ∞) arrow ℝ defined by… JEE Main 2026 Let (2 α, α) be the largest interval in which the function f(t)= |t+1| t^ 2 , t 2 , is… JEE Main 2026 Let f(x)=x^ 2025 -x^ 2000 , x ∈[0,1] and the minimum value of the function f(x) in the… JEE Main 2026 Let f: ℝ → ℝ be a twice differentiable function such that f''(x) > 0 for all x ∈ ℝ and… JEE Main 2026 Let f: R arrow R be a twice differentiable function such that the quadratic equation f(x)… JEE Main 2026 Let ℝ denote the set of all real numbers. Define the function f: R arrow R by f(x)=\… JEE Advanced 2025 Let ℝ denote the set of all real numbers. For a real number x , let [x] denote the… JEE Advanced 2025 Let ℝ denote the set of all real numbers. Let f: ℝ arrow ℝ be defined by f(x)= cases (6… JEE Advanced 2025 Let the function f(x)=(x)/(3)+(3)/(x)+3, x ≠ 0 be strictly increasing in (-∞, α_1) U(α_2… JEE Main 2025 Full Application of Derivatives list Application of Derivatives JEE Maths PYQs JEE Mathematics PYQs All JEE Advanced PYQs