Oxford University MAT 2016 Solved Papers

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Question : 6 of 10

Marks: +1, -0

Let

n

be a positive integer. Then

x^2+1

is a factor of

(3+x^4)^n-(x^2+3)^n(x^2-1)^n

for

all $n$ ,
even $n$ ,
odd $n$ ,
$n ≥ 3$ ,
no values of $n$ .

Solution:

Using the factor theorem, for

(x^2+1)

to be a factor,

(x^2+1)=0

, so

x^2=-1

. Then the equation given becomes

(4)^n-(2)^n(-2)^n=0

. This only holds when

(-2)^n

is positive, so the answer is (b).

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