The rational roots of the cubic equation $x^{3} + 14 k x^{2} + 56 k x - 64 k^{3} = 0$ are in the ratio $1 : 2 : 4$ . The possible value of $k$ are

0

only

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1

only

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2, 0

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- 2, - 1

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Solution

The correct option is A $0$ only
$x^{3} = 4 k x^{2} + 56 k x - 64 x^{3} = 0$
Let the root be $P, 2 P, 4 P$
$P + 2 P + 4 P = - 14 k$
$7 P = - 14 k$
$P = - 2 k$
also, $(P) (2 P) (48) = 64 k^{3}$
$8 P^{3} = 64 k^{3}$
$- 64 k^{3} = 64 k^{3}$
$- k^{3} = k^{3}$
$k = 0$
$∵= P = - 2 k$
$P = 0$
But since root of the roots are $1 : 2 : 4$ and we got $P = 0$ , so none of the option satisfy (Bonus).

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Question 6
Value(s) of k for which the quadratic equation $2 x^{2}$ - kx + k = 0 has equal roots is/are
(A) 0 only
(B) 4
(C) 8 only
(D) 0, 8

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