Question

Which of the following could be the possible integral value(s) of 'a' for which the equation $x^3-3x+a=0$ has three distinct roots?

• A. 1
• B. -1
• C. \N
• D. 3
• E. -3

Answer 1

Answer: (A), (B), (C)

\N

$\mathrm{Let}f\left(x\right)=x^3-3x+a\mathrm{Let}f\text{'}\left(x\right)=0\Rightarrow 3x^2-3=0\Rightarrow x=\pm 1\mathrm{For}\mathrm{three}\mathrm{distinct}\mathrm{real}\mathrm{roots},f\left(1\right)f(-1)<0\Rightarrow (1-3+a)(-1+3+a)<0\Rightarrow \left(a+2\right)\left(a-2\right)<0\Rightarrow -2<a<2\mathrm{Possible}\mathrm{values}\mathrm{of}a\mathrm{are}-1,0,1.$