How many solutions does \cos ^n(x)+\cos ^{2 n}(x)=0 have in the range 0 \leq x \leq 2 \pi for an integer n \geq 1 ?

Correct Answer is : 2 for even n and 3 for odd n,

Oxford University MAT 2016 Solved Papers

Question : 4 of 10

Marks: +1, -0

How many solutions does

\cos ^n(x)+\cos ^{2 n}(x)=0

have in the range

0 ≤ x ≤ 2 π

for an integer

n ≥ 1

Solution:

\cos ^n(x)+\cos ^{2 n}(x)=\cos ^n(x)(1+\cos ^n(x))=0

. For this to be true, if

n

is even,

\cos (x)=0

has two roots, but when

n

is odd either

\cos (x)=0

\cos (x)=-1

, which is three roots. Hence the answer is (d).

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