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Column-I Column-II
a. If the roots of x³ + x² + Kx + 8 = 0 are in G.P. then the value of K p. 6
b. The equation whose roots are K times the roots of
$3 x^{4} - \frac{5}{2} x^{3} + \frac{7}{6} x^{2} - x + \frac{7}{18} = 0$ is an equation with integral coefficients
then K = q. r
c. If the roots of the equation x³ –px²+ qx–r = 0 are in H.P.
Then the value of the mean root is r. 2
d. If the sum of two roots of x³ – px² + qx –r = 0 is zero then pq = s. $\frac{3 r}{q}$

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a-r,b-p,c-s,d-q

a-q,b-p,c-s,d-r

a-r,b-p,c-q,d-s

a-r,b-p,c-s,d-p

answer is A.

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Detailed Solution

A) $x^{3} + x^{2} + k x + 8 = 0, root β = - 2$
$- 8 + 4 - 2 k + 8 = 0 \Rightarrow k = 2$

B) $k = 6, f (\frac{x}{k}) = 0$
C) $α, β, γ$ are roots of $x^{3} - p x^{2} + q x - r = 0$
$\begin{array}{l} \frac{1}{α} + \frac{1}{β} + \frac{1}{γ} = \frac{q}{r}; \frac{2}{β} = \frac{1}{α} + \frac{1}{γ} \\ \frac{3}{β} = \frac{q}{r} \Rightarrow β = \frac{3 r}{q} \end{array}$
D) $α + β = 0, γ = p$
$\begin{array}{l} p^{3} - p^{3} + p q - r = 0 \\ p q = r \end{array}$

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