First slide

Theory of equations

Question

$If the equation f (x) = (a + b - c) x^{2} + (b + c - a) x + (c + a - b) = 0 has only one root equal to one. Then the value of \frac{3 b + a}{c} is$

Difficult

A

0
B

-1
C

1
D

2

Solution

$\begin{array}{l} Clearly f (1) = a + b + c = 0 \\ Product of roots = \frac{c + a - b}{a + b - c} = \frac{b}{c} \\ I t h a s o n l y o n e r o o t = 1 \Rightarrow 2 n d r o o t a l s o = 1 \\ so \frac{b}{c} = 1 \Rightarrow b = c and ∴ a = - 2 c \\ ∴ \frac{3 b + a}{c} = \frac{3 c - 2 c}{c} = 1 \end{array}$

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