Mathematics Question Review (2025-12-14))

Questions:

1. Prove that ln(x) < x - 1 for x=/=1

2. Prove that e^x> x + 1 for x=/= 1

3. Prove that for m being a real perfect square, the expressions m + 1 and m - 1 must not be perfect squares.

Solution:

1. By the method of contrapositive,

assume ln(x) > x - 1 is true,

then put any real value of x, say e,

then we have 1 > e - 1

Since e > 2, e - 1 > 1 which is contradiction.

2. By the method of contrapositive again,

assume e^x < x+1 is true,

then put any real value of x, say -1,

then we have 1/e < 0

since e > 0, contradiction again.

3. Asume there exists a pefect square n^2 such thay n^2= (m + 1) (m + 1)

Since m + 1 and m - 1 are not equal,

and n^2 = m^2 - 1,

then we have n^2 + 1 = m^2,

which is impossible for any m>0, n>0.

QED

such thay n^2= (m + 1) (m - 1)

You may iterate 16 = 2 x 8

Them m-1=2 and m+1 = 8,

which is no solution.