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KVPY SA Exam 2013 Question Paper
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© examsnet.com
Question : 1
Total: 80
PART I MATHEMATICS
Let x,y,z be three non-negative integers such that x + y + z = 10. The maximum possible value of xyz + xy + yz + zx is
52
64
69
73
Validate
Solution:
Taking three no's.
x + 1, y + 1, z + 1
AM ≥ GM
(
x
+
1
)
+
(
y
+
1
)
+
(
z
+
1
)
3
≥
{
(
x
+
1
)
(
y
+
1
)
(
z
+
1
)
}
1
∕
3
(
13
3
)
3
≥ xyz + xy + yz + zx + 11
(
13
3
)
3
−
11
≥ xyz + xy + yz + zx
equality hold when x = y = z but x + y + z = 0 and x, y, z are integers.
So maximum value when any two of x, y, z are equal to 3 and third is equal to 4.
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