KVPY SA Exam 2014 Question Paper

© examsiri.com

Question : 61 of 80

Marks: +1, -0

PART II

MATHEMATICS

Let a,b,c be non-zero real numbers such that

a + b + c = 0

; let

q = a^2+b^2+c^2

and

r = a^4+ b^4+ c^4

.
Then

$q^2 < 2r$ always
$q^2 = 2r$ always
$q^2 > 2 r$ always
$q^ 2 - 2r$ can take both positive and negative values

Solution:

a + b + c = 0

q = a^ 2 + b^2 + c^2 ; r = a^ 4 + b^4 + c^4

q^ 2 -2r = (a^2 + b^2 + c^2 )^ 2 - 2(a^4 + b^4 + c^4 )

= 2a^2 b^2 + 2b^2 c^ 2 + 2a^2 c^ 2 - a^ 4 - b^4- c ^4

= 2a^2 c^ 2 + 2b^2 c ^2 -(a^2 - b^2 )^ 2 -c ^4

= 2c^2 (a^2 + b^2 ) -c^2 (a -b)^2 -c^ 4

= c^ 2 [2a^2 + 2b^2 -(a -b)^2 -c ^2 ]

= c ^2 [2ab + a^2 + b^2 - c^ 2 ]

= c ^2 [(a + b)^2 -c^ 2 ]

= 0

© examsiri.com

Go to Question:

Prev Question

Next Question