Let a and b be two positive real numbers such that a√+b√=32 and
a√+b√=31. What is the value of (+)
Answers
Answer:
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if I could
Step-by-step explanation:
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Answer:
9
Step-by-step explanation:
Given :
a√a + b√b = 32,
a√b + b√a = 31.
Solution :
a√a + b√b = 32 ...(1)
a√b + b√a = 31 ...(2)
let a = x² & b = y².
Then,
(1) can be re-written as,
⇒ (x)³ + (y)³ = 32 ....(3)
(2) can be re-written as,
⇒ x²y + xy² = 31 ...(4)
by adding 3×(4) + (3),
⇒ x³ + y³ + 3x²y + 3xy² = 32 + 31×3 = 125
⇒ (x + y)³ = 125
⇒ x + y = 5 ( ∵ -5 is not possible, as (-5)³ = -125 )
from (4)
⇒ x²y + xy² = 31,
⇒ x²y + xy² = xy(x + y) = 31
⇒ 5xy = 31
⇒ xy=\frac{31}{5}xy=
5
31
⇒ a + b = x² + y² = (x + y)² - 2xy
= 5^2-2(\frac{31}{5})=25-\frac{62}{5}=\frac{125-62}{5}=\frac{63}{5}5
2
−2(
5
31
)=25−
5
62
=
5
125−62
=
5
63
so,
\frac{5(a+b)}{7} = \frac{5(\frac{63}{5})}{7}= \frac{63}{7} = 9
7
5(a+b)
=
7
5(
5
63
)
=
7
63
=9
Is this question from PRMO 2017 (I remember like solving this Q.)