KVPY SA Exam 19-Nov-2017 Question Paper

Question : 62 of 80

Marks: +1, -0

Let S be the set of all points

(a/b,c/d)

on the circle with radius 1 centred at (0,0) where a and b are relatively prime integers, c and d are relatively prime integers (that is HCF (a, b) = HCF (c,d) = 1), and the integers b and d are even. Then the set S

is empty
has four elements
has eight elements
is infinite

Validate

Solution:

circle is

x_2 + y_2 = 1

y±√{1-a^2/b^2}

\;\;\;∵x=a/b

y=±1/b√{b^2-a^2}

As y is retional so

\table b^2, -,a^2, =, p^2;↓ ,,↓,,↓ ;\even ,,\odd ,,\odd

b^2 = a^2 + p^2

= (2k + 1)^2 + (2λ + 1)^2

= 4k^2 + 4k + 1 + 4λ^2 + 4λ + 1

b^2 = 4 (k^2 +λ^2 + k + λ) + 2

impossilbe
as L.H.S. is multiple of 4 but R.H.S is not multiple of 4

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