KVPY SA Exam 2014 Question Paper

Question : 5 of 80

Marks: +1, -0

Let

p (x) = x^ 2 - 5 x + a

and

q(x) = x^ 2 - 3 x + b

, where a and b are positive integers. Suppose

\hcf (p (x ), q(x)) = x -1

and

k(x) =1\cm (p (x ), q(x)).

If the coefficient of the highest degree term o f k(x) is 1, the sum of the roots of

(x - 1) + k(x)

Solution:

p(x) = x^2 - 5x + a,

q(x) = x^2- 3x + b

{ where, a, b ∈ N}

\hcf\ (p(x), q(x)) = x - 1

so x – 1 is root of both p(x) & q(x)

p(1) = 1 - 5 + a = 0 ⇒ a = 4

q(1) = 1 - 3 + b = 0 ⇒ b = 2

p(x) = x^2 - 5x + 4 = (x - 1) (x -4)

q(x) = x^2 -3x + 2 = (x- 1) (x - 2)

k(x) = \lcm (p(x), q(x))

= (x- 1) (x -2) (x -4)

(x - 1) + k(x) = (x - 1)

+ (x- 1) (x - 2) (x - 4)

= (x - 1) [1 + x^2 -6x + 8]

= (x - 1) (x^2-6x + 9)

= (x- 1) (x - 3)^2

roots 1, 3, 3
Sum of roots = 1 + 3 + 3 = 7

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