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JEE Main 2002
Exam
JEE Main
Year
2002
Subject
Mathematics
Chapter
Quadratic Equation

JEE Main 2002 · Mathematics · Quadratic Equation

If 2 a+3 b+6 c=0(a, b, c ∈ R) then the quadratic equation a x^2+b...

If 2a+3b+6c=0(a,b,c∈R)2 a+3 b+6 c=0(a, b, c \in R) then the quadratic equation ax2+bx+c=0a x^2+b x+c=0 has

Options

  1. Aat least one root in [0, 1]
  2. Bat least one root in [2, 3]
  3. Cat least one root in [4, 5]
  4. Dnone of these

Correct answer

A. at least one root in [0, 1]

Step-by-step solution

Let f(x)=ax33+bx22+cx⇒f(0)=0f(x)=\frac{a x^3}{3}+\frac{b x^2}{2}+c x \Rightarrow f(0)=0 and f(1)=a3+b2+c=2a+3b+6c6=0f(1)=\frac{a}{3}+\frac{b}{2}+c=\frac{2 a+3 b+6 c}{6}=0 Also f(x)f(x) is continuous and differentiable in [0,1]and[0,1][0,1] and \left[0,1]\right.. So by Rolle's theorem, f′(x)=0f^{\prime}(x)=0. i.e. ax2+bx+c=0a x^2+b x+c=0 has at least one root in [0,1][0,1]
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