Pearson Edexcel IAL Mathematics P2.1 Proof Question Bank
Practise algebraic proof, exhaustion and counterexamples using integer forms, tables and clear conclusions from assumptions.
- Syllabus
- First assessment 2019
- Course
- Mathematics YMA01
- Level
- AS
Exam points
- represent odd or non-divisible integers as algebraic cases before proving divisibility
- complete an exhaustion table for all possible integer triples, then state the result
- give a counterexample that calculates to a rational or composite value and rejects the claim
Covered syllabus points
P2.1 - Proof question 1
[Maximum number: 6]
In this question you must show detailed reasoning.
Question (a)
(a)
Given that x and y are positive numbers such that
(x−y)3>x3−y3
prove that
y>x
[ 4 ]
(x−y)3=x3−3x2y+3xy2−y3(x−y)3>x3−y3⇒−3x2y+3xy2>03xy(y-x)>0
As x and y are positive, (y-x)>0, so y>x.
M1
A1
dM1
A1*
M1: Attempts expansion of (x−y)3.
A1: Correct simplified inequality with cubed terms cancelled.
dM1: Takes out or divides by a common factor of kxy.
A1*: Correct working and reasoning using positivity of x and y.
Question (b)
(b)
Using a counter example, show that the result in part (a) is not true for all real numbers.
[ 2 ]
M1: Chooses a suitable counterexample, for example x=3, y=-1. Any positive x and negative y will work; substitution is not needed for this mark.
A1: Shows that the result is not true for their values. For example, (3−(−1))3=64 and 33−(−1)3=28, so 64>28 but -1<3.
P2.1 - Proof question 2
[Maximum number: 5]
Question (a)
(a)
Prove by counter example that the statement
"if p is a prime number then 2 p+1 is also a prime number" is not true.
[ 1 ]
E.g. p=7⇒2p+1=15
Which is not a prime number (so the statement is not true)
Identifies a counter example and
makes a conclusion/shows it is not prime.
(1)
Question (b)
(b)
Use proof by exhaustion to prove that if n is an integer then
5n2+n+12
is always even.
[ 4 ]
For n odd, let n=2k+1:
5n2+n+12=5(2k+1)2+2k+1+12
=20k2+22k+18=2(10k2+11k+9)
For n even, let n=2k:
5n2+n+12=5(2k)2+2k+12
=20k2+2k+12=2(10k2+k+6)
Both expressions are even, so 5n2+n+12 is even for all integers n.
M1: starts proof by considering odd or even n.
M1: considers both odd and even cases.
A1: both cases attempted with at least one correct even expression.
A1: complete proof with suitable conclusion.
P2.1 - Proof question 3
[Maximum number: 5]
Question (a)
(a)
A student states
"If x and y are irrational numbers, x=y, then x y is also irrational."
Show, by counter example, that this statement is not always true.
[ 1 ]
Provides a counterexample, for example
2×8=4
B1: provides a counterexample and shows that the statement is not always true. The result must be written in rational form.
[1]
Question (b)
(b)
Prove, using algebra, that for all odd integers n, the value of the expression
n3+3n+2
is always even but never a multiple of 4
[ 4 ]
A proof attempted by substituting odd numbers n=1,3,5 etc into n^3+3 n+2 will not score any marks
B1: Knows that odd numbers are of the form 2 k +/- 1 using any variable apart from n
Other variations may be used, for example 2 k +/- 3,2 k +/- 5, etc but will be much less common. Note that use of 4 k+1 is flawed as it doesn't pick up all the odd numbers so on its own scores B0
M1: Attempts to expand (2 k+1)^3+3(2 k+1)+2 or (2 k-1)^3+3(2 k-1)+2 using any variable including n.
Generally, look for a minimum of (2 k+1)^3+3(2 k+1)+2 a cubic expression in k
As with the B mark, other variations may be used, for example 2 k +/- 3 but will be much less common. The use of n=4 k+1 in n^3+3 n+2 and expanded to a cubic form can be awarded this M mark only
A1: A correct expansion for (2 k+1)^3+3(2 k+1)+2 Allow the use of any variable including n Look for 8 k^3+12 k^2+12 k+6 but allow any correct expression that allows the problem to be solved e.g. 4(2 k+1)(k^2+k+1)+2 FYI the correct expansion for (2 k-1)^3+3(2 k-1)+2 is 8 k^3-12 k^2+12 k-2
A1*: Requires the candidate to have fully correct algebra, scored all 3 previous marks and
1) prove/show that the result is even
2) prove/show that the result is not a multiple of 4
3 ) give a minimal conclusion
Examples of expressions, for n=2 k+1, that show the result is even;
- 8 k^3+12 k^2+12 k+6 with a statement saying that it is a sum of even numbers
- (8 k^3+12 k^2+12 k+6) / 2=4 k^3+6 k^2+6 k+3
- 8 k^3+12 k^2+12 k+6=2(4 k^3+6 k^2+6 k+3)
Examples of expressions, for n=2 k+1, that show the result is not a multiple of 4
- 8 k^3+12 k^2+12 k+6=2(4 k^3+6 k^2+6 k+3) with some statement alluding to the fact that another factor of 2 cannot be taken out of the (4 k^3+6 k^2+6 k+3)
- (8 k^3+12 k^2+12 k+6) / 4=2 k^3+3 k^2+3 k+3/2
- 8 k^3+12 k^2+12 k+6=4(2 k^3+3 k^2+3 k+1.5)
Examples of expressions, for n=2 k+1, that show the result is both even but not a multiple of 4
- 8 k^3+12 k^2+12 k+6=4(2 k^3+3 k^2+3 k+1)+2
- 8 k^3+12 k^2+12 k+6=4 x (2 k^3+3 k^2+3 k)+6
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