CAT 2023 Slot 3 Solved Paper

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Question : 66 of 66

Marks: +1, -0

Suppose

f(x, y)

is a real-valued function such that

f(3 x+2 y, 2 x-5 y)=19 x

, for all real numbers

x

and

y

. The value of

x

for which

f(x, 2 x)=27

, is

Your Answer:

Solution:

Given that

f(3 x+2 y, 2 x-5 y)=19 x

.
Let us assume the function

f(a, b)

is a linear combination of

a

and

b

.

⇒ f(3 x+2 y, 2 x-5 y)=m(3 x+2 y)+n(2 x-5 y)=19 x

⇒3 m+2 n=19

and

2 m-5 n=0

Solving we get

m=5

and

n=2

⇒ f(a, b)=5 a+2 b

⇒f(x, 2 x)=5 x+2(2 x)=9 x=27⇒x=3 .

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