Mathematical Proofs

Edexcel IAL WMA12/01/P2/June 2019/Q3 (Proof)

(i) Use algebra to prove that for all real values of x​​ 

x – 42≥2x – 9 

(3)

(ii) Show that the following statement is untrue.​​ 

2n + 1 is a prime number for all values of n, n∈N 

(1)

SOLUTION​​ 

i-​​ 

First, we are going to use jotting method to find a good starting point, but remember, jotting is not the part of the proof.​​ 

Jotting

x2-8x+16≥2x-9

x2-10x+25≥0

x-52≥0

Proof​​ 

We always start a proof from a known or settled fact like in this case, we know that the expression squared is always​​ ≥​​ 0.​​ 

Consider,​​ For add real x values 

x-52≥0 

x2-2x5+52≥0

x2-10x+25≥0

The statement we have to proof has​​ x – 42​​ on the left side of the inequality therefore, we will be using eth completing square method to solve this inequality. ​​ 

x2-8x-2x+16+9≥0

x2-8x+16≥2x-9

x-42≥2x-9

Hence, proved.

ii-​​ 
For​​ n=2,

2n + 1=22+1=5 (prime number)

For​​ n=3,

2n + 1=23+1=9 (not a prime number)

Hence, the statement is untrue.​​