JEE MAIN - Mathematics (2018 - 15th April Evening Slot - No. 7)
If f(x) is a quadratic expression such that f (1) + f (2) = 0, and $$-$$ 1 is a root of f (x) = 0, then the other root of f(x) = 0 is :
$$-$$ $${5 \over 8}$$
$$-$$ $${8 \over 5}$$
$${5 \over 8}$$
$${8 \over 5}$$
Explanation
Let $$\alpha $$ and $$\beta $$ = - 1 are the roots of the polynomial, then we get
f(x) = x2 + (1 - $$\alpha $$)x - $$\alpha $$
$$ \therefore $$ f(1) = 2 - 2$$\alpha $$
and f(2) = 6 - 3$$\alpha $$
Also given,
f (1) + f (2) = 0
$$ \therefore $$ 2 - 2$$\alpha $$ + 6 - 3$$\alpha $$ = 0
$$ \Rightarrow $$ $$\alpha $$ = $${8 \over 5}$$
f(x) = x2 + (1 - $$\alpha $$)x - $$\alpha $$
$$ \therefore $$ f(1) = 2 - 2$$\alpha $$
and f(2) = 6 - 3$$\alpha $$
Also given,
f (1) + f (2) = 0
$$ \therefore $$ 2 - 2$$\alpha $$ + 6 - 3$$\alpha $$ = 0
$$ \Rightarrow $$ $$\alpha $$ = $${8 \over 5}$$
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