Statement II: The quadratic equations and have one root in common. The other roots of the first and second

Question

Statement I: If $a_{1} x^{2} + b_{1} x + c_{1} = 0$ and $a_{2} x^{2} + b_{2} x + c_{2} = 0$ $(a_{1} \neq 0, a_{2} \neq 0)$ have a common root.

Then ${(c_{1} a_{2} - c_{2} a_{1})}^{2} = (b_{1} c_{2} - b_{2} c_{1}) (a_{1} b_{2} - a_{2} b_{1})$

Statement II: The quadratic equations $x^{2} - 6 x + a = 0$ and $x^{2} - c x + 6 = 0$ have one root in common. The other roots of the first and second equations are integers in the ratio 4:3, if the common root is $α$ then $α^{3}$ is $8$

Smart Achievers · Accepted Answer

(1)

Let root of $x^{2} - 6 x + a = 0$ are $α, 4 β = 4 α β = a$

$x^{2} - c x + 6 = 0$ are $α, 3 β = 3 α, β = 6 \Rightarrow a = 8$

$\Rightarrow$ $x^{2} - 6 x + 8 = 0 \Rightarrow x^{2} - c x + 6 = 0$

Has a root common $\Rightarrow c = 5$ $o r$ $c = \frac{11}{2}$

Integral roots are 2, 4 and 2, 3. Common root is 2

Solution

1 Statement I is true and Statement II is true

2 Statement I is false and Statement II is true

3 Statement I is true and Statement II is false

4 Statement I is false and Statement II is false

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