If the equations $4 x^{2} - 11 x + 2 k = 0$ and $x^{2} - 3 x - k = 0$ have a common root and obtain the common root for this value of $' k'$ is

see full answer

21% of IItians & 23% of AIIMS delhi doctors are from Sri Chaitanya institute !!

An Intiative by Sri Chaitanya

$\frac{- 7}{36}$

$\frac{36}{7}$

$\frac{- 17}{36}$

$\frac{17}{36}$

answer is C.

(Unlock A.I Detailed Solution for FREE)

Ready to Test Your Skills?

Check your Performance Today with our Free Mock Test used by Toppers!

Take Free Test

Detailed Solution

Let $(x - α)$ is common root for $4 x^{2} - 11 x + 2 k = 0 \to (1)$

$x^{2} - 3 x - k = 0 \to (2)$

From (1) and (2),

$4 α^{2} - 11 α + 2 k = 0$ and $α^{2} - 3 α - k = 0$

$4 α^{2} = 11 α - 2 k$

$α^{2} = \frac{11 α - 2 k}{4} \to (3)$

$α^{2} = 3 α + k \to (4)$

$\frac{11 α - 2 k}{4} = \frac{3 α + k}{1}$

$\begin{array}{l} 11 α - 2 k = 4 (3 α + k) \\ 11 α - 2 k = 12 α + 4 k \end{array}$

$- 2 k - 4 k = 12 α - 11 α$

$- 6 k = α$ sub equation (4)

${(- 6 k)}^{2} = 3 (- 6 k) + k$

$\begin{array}{l} 36 k^{2} = - 18 k + k \\ 36 k^{2} = - 17 k \end{array}$

$k = \frac{- 17}{36}$ sub in above equation $α = 0, \frac{17}{6}$ .

Watch 3-min video & get full concept clarity