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EXPLAIN THESE QUESTIONS... PLEASE
1. If n ∈ N, solve the equation cosn x − sinn x = 1, for x ∈ R.
2. Let a, b, c ∈ R, a2+b2 6= 0 and L = {(x, y) ∈ Q×Q | ax+by = c}.
a) Can you find an example of (a, b, c) such that L = ∅ ?
b) Can you find an example of (a, b, c) such that L has a unique
element?
c) Prove that if L contains two elements, then L has infinitely
many.
3. Show that the set
A = {±1 ± 2 ± 3 ± ... ± 2006}
contains an even number of elements.
4. Find a polynomial P(X) of degree n , with real coefficients, satis-
fying
P(X)|P(X2 − 2007X + 2007).
5. A n× n matrix contains elements only +1 or −1, such that on any
row, on any column, and on the two diagonals of the matrix the
product of elements is −1. What are the possible values of n (n ∈ N)?
1. If n ∈ N, solve the equation cosn x − sinn x = 1, for x ∈ R.
2. Let a, b, c ∈ R, a2+b2 6= 0 and L = {(x, y) ∈ Q×Q | ax+by = c}.
a) Can you find an example of (a, b, c) such that L = ∅ ?
b) Can you find an example of (a, b, c) such that L has a unique
element?
c) Prove that if L contains two elements, then L has infinitely
many.
3. Show that the set
A = {±1 ± 2 ± 3 ± ... ± 2006}
contains an even number of elements.
4. Find a polynomial P(X) of degree n , with real coefficients, satis-
fying
P(X)|P(X2 − 2007X + 2007).
5. A n× n matrix contains elements only +1 or −1, such that on any
row, on any column, and on the two diagonals of the matrix the
product of elements is −1. What are the possible values of n (n ∈ N)?
