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Question

Assertion :Let f(x) be a polynomial function satisfying f(x).f(1x)=f(x)+f(1x). If f(4)=65 and l1,l2,l3 are in G.P., then f(l1),f(l2),f(l3), are also in G.P. Reason: f(x)=±xn+1

A
Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
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B
Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
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C
Assertion is correct but Reason is incorrect
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D
Assertion is incorrect but Reason is correct
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Solution

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
f(x)+f(1x)=f(x)f(1x)(1)
We can write (1) as
f(x)=f(1x)f(1x)1(2)
We can also write (1) as
f(1x)=f(x)f(x)1(3)
Multiplying (1) and (2) we get
f(x)f(1x)=f(1x)f(x)[f(1x)1][f(x)1](f(x)1)(f(1x)1)=1(4)
Take g(x)=f(x)1g(1x)=f(1x)1
We can write (4) as
g(x)g(1x)=1
Now, if g(x) is a polynomial, g(x)=±xn
f(x)1=±xnf(x)=±xn+1
Hence, reason is correct
Now, f(4)=65f(x)=x3+1f(x)=3x2
Take l1=a,l2=ar,l3=ar2
f(l1)=3a2f(l2)=3a2r2f(l3)=3a2r4
f(l1),f(l2),f(l3) is also in GP
Hence, assertion and reason both are correct, and reason is the correct explanation for assertion.

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