Question

The equation $10x^4-3x^2-1=0$ has

• A. at least one root in $(-1,0)$
• B. at least one root in $(0,1)$
• C. at least two roots in $(-1,1)$
• D. no root in $(-1,1)$

Answer 1

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

The correct options are
A at least one root in $(-1,0)$
B at least one root in $(0,1)$
C at least two roots in $(-1,1)$

There will be atleast one root in $(\alpha ,\beta )$ if $f\left(\alpha \right).f\left(\beta \right)<0$
$f(-1)=6,f\left(0\right)=-1$ and $f\left(1\right)=6$
In the interval $(-1,0)$,there will be atleast one root as
$f(-1).f\left(0\right)=-6<0$
In the interval $(0,1)$,there will be atleast one root as $f\left(0\right).f\left(1\right)=-6<0$

There will be two roots in $(-1,1)$ as
$-1<\frac{-b}{2a}<1$
$-1<\frac{3}{20}<1$
$-1<0.15<1$
Hence, options 'A', 'B' and 'C' are correct.