KVPY SA Exam 2016 Question Paper

Question : 3 of 80

Marks: +1, -0

Let t be real number such that

t^2=at+b

for some positive integers a and b. Then for any choice of positive integers a and b,

t ^3

is never equal to

Solution:

t^2 = at + b ; a, b ∈ I^{+}

t^3 = at^2 + bt

= a(at + b) + bt

= a^2t + bt + ab

⇒ t^3 = (a^2 + b)t + ab

, check possibility for a, b ∈

I^{+}

from options.
(A)

a^2 + b = 4

ab = 3 possible
(B)

a^2 + b = 8

ab = 5 not possible
(C)

a^2 + b = 10

ab = 3 possible
(D)

a^2 + b = 6

ab = 5 possible

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