KVPY SA Exam 2015 Question Paper

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Question : 1 of 80

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PART I

MATHEMATICS

Two distinct polynomials

f ( x )

and

g (x )

are defined as follows:

f ( x ) = x ^2 + ax + 2

;

g(x) = x ^2 + 2 x +a.

If the equations

f ( x) = 0

and

g(x) = 0

have a common root then the sum of the roots of the equation

f ( x ) + g(x) = 0

is

$-1/2$
0
$1/2$
1

Solution:

Let 'α' is the common root
So,

α^2 + aα + 2 = 0

....(i)

α^2+ 2α+ a = 0

...(ii)
⇒

(a -2) α + 2 – a = 0

⇒ α = 1 is common root.
∴

1^2 + a + 2 = 0⇒ a = - 3.

f(x) + g(x) = 0
⇒

2x^2+ (a + 2) x + (a + 2) = 0

⇒

2x^2- x-1 = 0

⇒ Sum of roots =

1/2

.

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