Mathematical Proofs

Edexcel IAL WMA12/01/P2/June 2022/Q3 (Proof: Counter Example)

(i) Show that the following statement is false:​​ 

“n + 13-n3 is prime for all n ∈ N”

​​ (2)

(ii) Given that the points A(1, 0), B(3, −10) and C(7, −6) lie on a circle, prove that AB is a diameter of this circle.​​ 

(5)

SOLUTION​​ 

a-​​ 

n=1⇒1+13-13=23-1=7 (prime)

n=2  2+13-23=33-23=27-8=19 (prime)

n=3 ⇒3+13-33=43-33=64-27=37 (prime)

n=4 ⇒4+13-43=125-64=61 (prime)

n=5 ⇒5+13-53=216-125=91 (not prime)

The factors of​​ 91 are​​ 7 & 13; thus, 91 is not a prime number.​​ 

Hence, the statement that​​ n+13-n3 ​​​​ is prime for all​​ n∈N is false.

b-​​ 

We know that the angle in a semi-circle is a right angle.​​ 

So, If AB is the diameter the​​ AC^B=90o. Therefore, ​​ mAC x mBC= -1, because line AC & BC are perpendicular to each other.​​ 

mAC=0--61-7=6-6=-1

mBC=-10--63-7=-4-4=1

mAC x mBC=-1 x 1= -1

This proves that AC and BC are perpendicular to each other.​​ 

AC^B=90o 

And, the circle property is also fulfilled in this way.

Hence, AB is the diameter .​​