# IPMAT Question Paper 2019 | IPM Indore Quants

###### IPMAT Sample Paper | IPMAT Question Paper | Question 3

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Question 3: A real-valued function f satisfies the relation f(x)f(y) = f(2xy + 3) + 3f(x + y) - 3f(y) + 6y, for all real numbers x and y, then the value of f(8) is

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f (x) f (y) = f (2xy + 3) + 3f (x + y) - 3f (y) + 6y

Putting x = y = 0,
f (0) f (0) = f (2.0.0 + 3) + 3f (0 + 0) - 3f (0) + 6.0
f (0)2= f (3) + 3f (0) - 3f (0) + 0

f (3) = f (0)2

Putting y = 0,
f (x) f (0) = f (2.x.0 + 3) + 3f (x + 0) - 3f (0) + 6.0
f (x) f (0) = f (3) + 3f (x) - 3f (0)

f (x) f (0) = f (0)2+ 3f (x) - 3f (0)
f (x) f (0) - f (0)2= 3f (x) - 3f (0)
f (0) * (f (x)- f (0)) = 3 * (f (x)- f (0))

Either f(x) = f(0) or f(0) = 3

Since all functions can’t have the same value, f(0)=3.

Putting x=0, y=3,
f (0) f (3) = f (2.0.3 + 3) + 3f (0 + 3) - 3f (3) + 6.3
f(0) f(3) = f(3) + 3f(3) -3f(3) +18 = f(3) +18.
3 f(3) = f(3) + 18. Therefore, f(3) =9.

f(3) = 9.

Putting x=0, y=8.

f (0) f (8) = f (2.0.8 + 3) + 3f (0+ 8) - 3f (8) + 6.8
3.f(8) = f(3) + f(8) – f(8) + 48
3. f(8) = 9 + 48.
f(8) = 19

The question is"A real-valued function f satisfies the relation f(x)f(y) = f(2xy + 3) + 3f(x + y) - 3f(y) + 6y, for all real numbers x and y, then the value of f(8) is"

##### Hence, the answer is 19

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