The value(s) of k for which equations $3 x^{2} + 4 k x + 2 = 0 a n d 2 x^{2} + 3 x - 2 = 0$ will have a common root can be:

\frac{- 11}{8}

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\frac{11}{8}

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\frac{7}{4}

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\frac{- 7}{4}

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Solution

The correct option is C $\frac{7}{4}$
Let $α$ be the common root of both the given equations.
$∴ 3 α^{2} + 4 k α + 2 = 0 a n d 2 α^{2} + 3 α - 2 = 0$
On solving by cross multiplication method we get:
$\frac{α^{2}}{- 8 k - 6} = \frac{- α}{- 6 - 4} = \frac{1}{9 - 8 k} \Rightarrow α = \frac{10}{9 - 8 k} . . . (i) a n d α^{2} = \frac{8 k + 6}{8 k - 9} . . . (i i) O n s i m p l i f y i n g (i) a n d (i i) w e g e t : 32 k^{2} - 12 k - 77 = 0 \Rightarrow 32 k^{2} - 56 k + 44 k - 77 = 0 \Rightarrow 8 k (4 k - 7) + 11 (4 k - 7) = 0 \Rightarrow (8 k + 11) (4 k - 7) = 0 \Rightarrow k = \frac{- 11}{8}, \frac{7}{4}$

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