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4-2.Quadratic Equations and Inequations
normal
The number of integers $k$ for which the equation $x^3-27 x+k=0$ has at least two distinct integer roots is
A
$1$
B
$2$
C
$3$
D
$4$
(KVPY-2016)
Solution
(b)
We have,
$x^3-27 x+k=0$
$f(x) =x^3-27 x$
$f^{\prime}(x) =3 x^2-27=3\left(x^2-9\right)$
Now, sum of roots of $x^3-27 x+k=0$ is Zero.
$\therefore$ Third root is also integer.
Now, put $x=6 t$
$216 t^3-27(6 t)+k=0$
$54\left(4 t^3-3 t\right) =-k$
$54 \cos 3 \theta =-k \quad[\because t=\cos \theta]$
$\cos 3 \theta =-\frac{k}{54}$
Now, we get integral solution on $\theta=0, \pi$
$\therefore$ T'wo values of $k$ are possible.
Standard 11
Mathematics
