Browse Proof

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Proof by deduction

30 questions

LessonNot started
SECTION A: PURE MATHEMATICS
(a) Disprove the following statement:

"2n+12^n + 1 is prime for all positive integers nn." (2 marks)
(b) Prove that the sum of the squares of any two consecutive odd integers is an even number that is not divisible by 44. (3 marks)
●●●●●Level 25 marksStart
Prove, using algebra, that

(2n+5)2−(2n−1)2(2n + 5)^2 - (2n - 1)^2

is a multiple of 24 for all n∈Nn \in \mathbb{N}

(3 marks)
●●●●●Level 33 marksStart
(a) Find a counterexample to disprove the statement that, for all non-zero real numbers aa and bb, if a<ba < b then 1a>1b\dfrac{1}{a} > \dfrac{1}{b}.

(1 mark)
(b) In each of the following cases, state which one of the symbols ⇒\Rightarrow, ⇐\Leftarrow or ⇔\Leftrightarrow should replace □\square to make the statement correct.

(i) (x−3)(x+5)=0□x=−5(x - 3)(x + 5) = 0 \quad \square \quad x = -5

(ii) x3<−27□x<−3x^3 < -27 \quad \square \quad x < -3

(iii) cos⁡x°=−12□x=120\cos x° = -\tfrac{1}{2} \quad \square \quad x = 120

(3 marks)
●●●●●Level 34 marksStart
(a) A student states

"if n2n^2 is a multiple of 8 then nn is a multiple of 8"

Show, by counter example, that this statement is not always true.

(1 mark)
(b) Prove, using algebra, that for all odd integers nn, the value of the expression

n2+4n+7n^2 + 4n + 7

is always a multiple of 4 but never a multiple of 8

(4 marks)
●●●●●Level 35 marksStart
Prove that the sum of the squares of any two consecutive odd integers is even but not divisible by 44. (4 marks)
●●●●●Level 34 marksStart
(a) Prove that the sum of any three consecutive integers is divisible by 33. (2 marks)
(b) Show, by means of a counter-example, that the sum of any four consecutive integers need not be divisible by 44. (2 marks)
●●●●●Level 34 marksStart
Prove that n3−nn^3 - n is divisible by 66 for all positive integers nn. (4 marks)
●●●●●Level 34 marksStart
(a) Show, by means of a counter-example, that the following statement is false.

"For all real numbers xx and yy, if x>yx > y then x2>y2x^2 > y^2." (2 marks)
(b) Prove that, for all real numbers xx and yy, if x>y>0x > y > 0 then x2>y2x^2 > y^2. (2 marks)
●●●●●Level 34 marksStart
Prove that the difference between the squares of any two consecutive odd numbers is a multiple of 88. (4 marks)
●●●●●Level 34 marksStart
(a) Given that nn is a positive integer, express

2n+32n−3+2n−32n+3\frac{2\sqrt{n} + 3}{2\sqrt{n} - 3} + \frac{2\sqrt{n} - 3}{2\sqrt{n} + 3}

as a single fraction not involving surds.

(3 marks)
(b) Hence, deduce that

2n+32n−3+2n−32n+3\frac{2\sqrt{n} + 3}{2\sqrt{n} - 3} + \frac{2\sqrt{n} - 3}{2\sqrt{n} + 3}

is a rational number for all positive integer values of nn

(1 mark)
●●●●●Level 34 marksStart
Prove, using algebra, that

n4−n2n^4 - n^2

is a multiple of 12 for all n∈Nn \in \mathbb{N}

(4 marks)
●●●●●Level 44 marksStart
A student attempts to answer the following question.

“Use proof by contradiction to prove that there is no greatest prime number.”

The student starts the proof with

Assume that there is a greatest prime number, NN.

⇒\Rightarrow the complete list of prime numbers is p1, p2, p3, …, Np_1,\ p_2,\ p_3,\ \ldots,\ N

Let Q=p1×p2×p3×…×N+1Q = p_1 \times p_2 \times p_3 \times \ldots \times N + 1

Complete the proof.

(4 marks)
●●●●●Level 44 marksStart
A student is asked to prove the following statement.

“For all real values of aa and bb, a2+b2+2≥2(a+b)a^2 + b^2 + 2 \geq 2(a + b)”

The student's proof is shown below.

a2+b2+2≥2a+2ba^2 + b^2 + 2 \geq 2a + 2b

⇒a2−2a+1+b2−2b+1≥0\Rightarrow a^2 - 2a + 1 + b^2 - 2b + 1 \geq 0

⇒(a−1)2+(b−1)2≥0\Rightarrow (a - 1)^2 + (b - 1)^2 \geq 0

A square is never negative, so the statement is true.
(a) Explain the error in the student's proof.

(1 mark)
(b) Prove the statement.

(3 marks)
(c) Deduce the least possible value of a2+b2−2a−2ba^2 + b^2 - 2a - 2b

(1 mark)
●●●●●Level 45 marksStart
Prove, by exhaustion, that

n3+2n^3 + 2

is not a multiple of 9 for any integer nn

(4 marks)
●●●●●Level 44 marksStart
Two arithmetic progressions, PP and QQ, each have 6060 terms, denoted by pip_i and qiq_i respectively, where i=1,2,3,…,60i = 1, 2, 3, \ldots, 60.

The common difference of PP is dd, where dd is a positive integer. The progressions are related as follows.
  • p1=q60p_1 = q_{60}
  • q1=p60q_1 = p_{60}
(a) You are given that there is at least one value of ii for which qi=pi+24q_i = p_i + 24.

Show that, in this case, i=612−12di = \dfrac{61}{2} - \dfrac{12}{d}.

(6 marks)
(b) Hence show that it is impossible for the equation qi=pi+24q_i = p_i + 24 to hold unless dd takes certain values, which should be stated.

(2 marks)
●●●●●Level 48 marksStart
Aled makes the following claim.

“nn is a positive integer that is a multiple of 33 ⇒\Rightarrow 2n+12^n + 1 is not prime”
(a) Prove that Aled's claim is true.

(4 marks)

Bethan makes a different claim.

“nn is a positive integer ⇒\Rightarrow 2n+12^n + 1 is not prime”
(b) Show that Bethan's claim is false.

(1 mark)
●●●●●Level 45 marksStart
The number KK is defined by K=n4+4K = n^4 + 4, where nn is an integer greater than 11.
(a) Given that n4+4≡(n2+an+2)(n2−an+2)n^4 + 4 \equiv (n^2 + an + 2)(n^2 - an + 2), where aa is a positive constant, find aa.

(1 mark)
(b) Prove that KK has at least two distinct factors other than 11 and KK.

(5 marks)
●●●●●Level 46 marksStart
(a) Find a counterexample to disprove the statement that n2+n+11n^2 + n + 11 is prime for all positive integers nn.

(1 mark)
(b) In each of the following cases, state which one of the symbols ⇒\Rightarrow, ⇐\Leftarrow or ⇔\Leftrightarrow should replace □\square to make the statement correct.

(i) nn is an integer multiple of 6□n6 \quad \square \quad n is an integer multiple of 33

(ii) ∣x∣<2□x2<4|x| < 2 \quad \square \quad x^2 < 4

(iii) ln⁡x<0□x<1\ln x < 0 \quad \square \quad x < 1

(3 marks)
(c) Prove that the difference between the squares of any two odd numbers is always a multiple of 88.

(4 marks)
●●●●●Level 48 marksStart
(a) Find a counterexample to disprove the statement that x2=x\sqrt{x^2} = x for all real numbers xx.

(1 mark)
(b) A student writes the statement cos⁡x°=0⇔x=90\cos x° = 0 \Leftrightarrow x = 90.

(i) Explain why this statement is incorrect.

(ii) Write a corrected version of this statement.

(2 marks)
(c) Prove that the sum of any three consecutive multiples of 55 is always a multiple of 1515.

(3 marks)
●●●●●Level 46 marksStart
A student wishes to prove that, for all positive integers aa and bb, a2−6ab≠10a^2 - 6ab \ne 10.
(a) Prove that a2−6ab=10⇒aa^2 - 6ab = 10 \Rightarrow a is even.

(2 marks)
(b) Hence or otherwise prove that, for all positive integers aa and bb, a2−6ab≠10a^2 - 6ab \ne 10.

(3 marks)
●●●●●Level 45 marksStart
In this question you must show detailed reasoning.
(a) Given that xx and yy are positive numbers such that

x2(x+3y)<y2(y+3x)x^2(x + 3y) < y^2(y + 3x)

prove that y>xy > x

(4 marks)
(b) Using a counter example, show that the result in part (a) is not true for all real numbers.

(2 marks)
●●●●●Level 46 marksStart
(a) Show, by means of a counter-example, that the following statement is false:

'For all positive integers nn, the number 2n+12^n + 1 is prime.' (1 mark)
(b) Prove that the square of any odd number is one more than a multiple of 88. (4 marks)
●●●●●Level 45 marksStart
Prove by contradiction that there are no positive integers aa and bb for which

a2−b2=10a^2 - b^2 = 10. (4 marks)
●●●●●Level 44 marksStart
The polynomial P(x)\text{P}(x) is given by

P(x)=3x3+11x2+15x+6\text{P}(x) = 3x^3 + 11x^2 + 15x + 6
(a) Use the factor theorem to show that (3x+2)(3x + 2) is a factor of P(x)\text{P}(x)

(2 marks)
(b) Express P(x)\text{P}(x) in the form

P(x)=(3x+2)(ax2+bx+c)\text{P}(x) = (3x + 2)(ax^2 + bx + c)

where aa, bb and cc are constants to be found.

(2 marks)
(c) Given that nn is a positive integer, use your answer to part (b) to explain why

3n3+11n2+15n+63n^3 + 11n^2 + 15n + 6

is never prime.

(2 marks)
●●●●●Level 46 marksStart
The polynomial P(x)\text{P}(x) is given by

P(x)=4x3−8x2+9x−3\text{P}(x) = 4x^3 - 8x^2 + 9x - 3
(a) Use the factor theorem to show that (2x−1)(2x - 1) is a factor of P(x)\text{P}(x)

(2 marks)
(b) Express P(x)\text{P}(x) in the form

P(x)=(2x−1)(ax2+bx+c)\text{P}(x) = (2x - 1)(ax^2 + bx + c)

where aa, bb and cc are integers to be found.

(2 marks)
(c) Given that nn is an integer and n≥2n \ge 2, use your answer to part (b) to explain why

4n3−8n2+9n−34n^3 - 8n^2 + 9n - 3

is never prime.

(2 marks)
●●●●●Level 46 marksStart
Given that xx is a real number such that

x3+x=12x^3 + x = 12
(a) use proof by contradiction to prove that x<3x < 3

(4 marks)
(b) Show that 2k2−12k+212k^2 - 12k + 21 is positive for all real values of kk

(2 marks)
●●●●●Level 56 marksStart
(a) Prove, using algebra, that 5n+1−5n5^{n+1} - 5^n is a multiple of 20 for all positive integers nn

(3 marks)
(b) A student was asked to use proof by contradiction to prove the following statement.

“For all integers nn, if n2n^2 is a multiple of 4 then nn is even”

The start of the student's proof is shown below.

Assume that n2n^2 is not a multiple of 4 and that nn is odd.

⇒n=2k+1\Rightarrow n = 2k + 1 for some integer kk

⇒n2=4k2+4k+1=4(k2+k)+1\Rightarrow n^2 = 4k^2 + 4k + 1 = 4(k^2 + k) + 1

Explain why the student's proof cannot be completed.

(1 mark)
(c) Use proof by contradiction to prove the statement.

(3 marks)
●●●●●Level 57 marksStart
Prove that the difference between the squares of any two consecutive odd numbers is a multiple of 88.
●●●●●Level 54 marksStart
Prove that if nn is odd then n2−1n^2 - 1 is a multiple of 88.
●●●●●Level 53 marksStart
(a) In an arithmetic series, the first term is aa and the common difference is dd.

Prove that the sum of the first nn terms of the series is given by

Sn=n2[2a+(n−1)d]S_n = \frac{n}{2}\left[2a + (n - 1)d\right]

(3 marks)

Dev borrows £30003000 from a relative. He repays the loan in monthly instalments.

Dev repays £5555 in month 11, £6565 in month 22, £7575 in month 33 and so on, so that the monthly repayments form an arithmetic sequence.
(b) Show that Dev repays £145145 in month 1010.

(1 mark)

Given that Dev takes exactly nn months to repay the loan completely,
(c) show that

n2+10n−600=0n^2 + 10n - 600 = 0

(2 marks)
(d) Solve the equation in part (c).

(2 marks)
(e) Hence state the number of months Dev takes to repay the loan, giving a brief reason for your answer.

(1 mark)
●●●●●Level 59 marksStart

Proof by exhaustion

8 questions

LessonNot started
Given that aa, bb and cc are integers greater than 0 such that
  • c=2a−1c = 2a - 1
  • a+b+c=14a + b + c = 14
prove, by exhaustion, that the product abcabc is always a multiple of 6

You may use the table below to illustrate your answer.

You may not need to use all rows of this table.
aabbccabcabc
(3 marks)
●●●●●Level 23 marksStart
(a) Use a counter example to show that the following statement is false

"n2−n+17n^2 - n + 17 is prime for all n∈Nn \in \mathbb{N}"

(2 marks)
(b) Use algebra to prove by exhaustion that for all n∈Nn \in \mathbb{N}

"n2+2n+3n^2 + 2n + 3 is not a multiple of 4"

(4 marks)
●●●●●Level 36 marksStart
(a) A student states

"if n2n^2 is a multiple of 8 then nn is a multiple of 8"

Show, by counter example, that this statement is not always true.

(1 mark)
(b) Prove, using algebra, that for all odd integers nn, the value of the expression

n2+4n+7n^2 + 4n + 7

is always a multiple of 4 but never a multiple of 8

(4 marks)
●●●●●Level 35 marksStart
A student was asked to prove by exhaustion that if nn is an integer then

n3+5n+7n^3 + 5n + 7

is not divisible by 3

The start of the student's proof is shown in the box below.
Student's proof
Consider the case when n=3kn = 3k
n3+5n+7=27k3+15k+7=3(9k3+5k+2)+1n^3 + 5n + 7 = 27k^3 + 15k + 7 = 3(9k^3 + 5k + 2) + 1 which is not divisible by 3
Complete this proof.

(4 marks)
●●●●●Level 34 marksStart
In this question you must show detailed reasoning.

Use algebra to prove by exhaustion that, for all integers mm that are not multiples of 3, the value of

2m2+12m^2 + 1

is always a multiple of 3

(4 marks)
●●●●●Level 34 marksStart
Prove that n5−nn^5 - n is divisible by 55 for every integer nn, by exhausting the possible remainders of nn on division by 55.
●●●●●Level 34 marksStart
Prove, by exhaustion, that

n3+2n^3 + 2

is not a multiple of 9 for any integer nn

(4 marks)
●●●●●Level 44 marksStart
It is given that any integer can be expressed in the form 5m+r5m + r, where mm is an integer and rr is an integer with 0≤r≤40 \le r \le 4.

Use this fact to answer the following.
(a) By considering the different values of rr, prove that the square of any integer cannot be expressed in the form 5n+25n + 2 or 5n+35n + 3, where nn is an integer.

(4 marks)
(b) Two integers are chosen at random from the integers 11 to 500500 inclusive. The two integers are not necessarily different.

By considering the different values of rr, determine the probability that the sum of the squares of these two integers is divisible by 55.

(4 marks)
●●●●●Level 48 marksStart

Disproof by counter-example

14 questions

LessonNot started
A student writes the following two statements.
(a) “For all positive integers nn, 2n>n22^n > n^2”

Prove by counter example that this statement is not true.

(2 marks)
(b) “For all real numbers xx, if x3>xx^3 > x then x>1x > 1”

Prove by counter example that this statement is not true.

(2 marks)
●●●●●Level 24 marksStart
SECTION A: PURE MATHEMATICS
(a) Disprove the following statement:

"2n+12^n + 1 is prime for all positive integers nn." (2 marks)
(b) Prove that the sum of the squares of any two consecutive odd integers is an even number that is not divisible by 44. (3 marks)
●●●●●Level 25 marksStart
(a) Find a counterexample to disprove the statement that, for all non-zero real numbers aa and bb, if a<ba < b then 1a>1b\dfrac{1}{a} > \dfrac{1}{b}.

(1 mark)
(b) In each of the following cases, state which one of the symbols ⇒\Rightarrow, ⇐\Leftarrow or ⇔\Leftrightarrow should replace □\square to make the statement correct.

(i) (x−3)(x+5)=0□x=−5(x - 3)(x + 5) = 0 \quad \square \quad x = -5

(ii) x3<−27□x<−3x^3 < -27 \quad \square \quad x < -3

(iii) cos⁡x°=−12□x=120\cos x° = -\tfrac{1}{2} \quad \square \quad x = 120

(3 marks)
●●●●●Level 34 marksStart
(a) Use a counter example to show that the following statement is false

"n2−n+17n^2 - n + 17 is prime for all n∈Nn \in \mathbb{N}"

(2 marks)
(b) Use algebra to prove by exhaustion that for all n∈Nn \in \mathbb{N}

"n2+2n+3n^2 + 2n + 3 is not a multiple of 4"

(4 marks)
●●●●●Level 36 marksStart
(a) A student states

"if n2n^2 is a multiple of 8 then nn is a multiple of 8"

Show, by counter example, that this statement is not always true.

(1 mark)
(b) Prove, using algebra, that for all odd integers nn, the value of the expression

n2+4n+7n^2 + 4n + 7

is always a multiple of 4 but never a multiple of 8

(4 marks)
●●●●●Level 35 marksStart
Disprove, by counter-example: "for all real numbers aa and bb, if a2=b2a^2 = b^2 then a=ba = b".
●●●●●Level 32 marksStart
Disprove, by counter-example: "the product of two irrational numbers is always irrational".
●●●●●Level 32 marksStart
(a) Prove that the sum of any three consecutive integers is divisible by 33. (2 marks)
(b) Show, by means of a counter-example, that the sum of any four consecutive integers need not be divisible by 44. (2 marks)
●●●●●Level 34 marksStart
(a) Show, by means of a counter-example, that the following statement is false.

"For all real numbers xx and yy, if x>yx > y then x2>y2x^2 > y^2." (2 marks)
(b) Prove that, for all real numbers xx and yy, if x>y>0x > y > 0 then x2>y2x^2 > y^2. (2 marks)
●●●●●Level 34 marksStart
Aled makes the following claim.

“nn is a positive integer that is a multiple of 33 ⇒\Rightarrow 2n+12^n + 1 is not prime”
(a) Prove that Aled's claim is true.

(4 marks)

Bethan makes a different claim.

“nn is a positive integer ⇒\Rightarrow 2n+12^n + 1 is not prime”
(b) Show that Bethan's claim is false.

(1 mark)
●●●●●Level 45 marksStart
(a) Find a counterexample to disprove the statement that n2+n+11n^2 + n + 11 is prime for all positive integers nn.

(1 mark)
(b) In each of the following cases, state which one of the symbols ⇒\Rightarrow, ⇐\Leftarrow or ⇔\Leftrightarrow should replace □\square to make the statement correct.

(i) nn is an integer multiple of 6□n6 \quad \square \quad n is an integer multiple of 33

(ii) ∣x∣<2□x2<4|x| < 2 \quad \square \quad x^2 < 4

(iii) ln⁡x<0□x<1\ln x < 0 \quad \square \quad x < 1

(3 marks)
(c) Prove that the difference between the squares of any two odd numbers is always a multiple of 88.

(4 marks)
●●●●●Level 48 marksStart
(a) Find a counterexample to disprove the statement that x2=x\sqrt{x^2} = x for all real numbers xx.

(1 mark)
(b) A student writes the statement cos⁡x°=0⇔x=90\cos x° = 0 \Leftrightarrow x = 90.

(i) Explain why this statement is incorrect.

(ii) Write a corrected version of this statement.

(2 marks)
(c) Prove that the sum of any three consecutive multiples of 55 is always a multiple of 1515.

(3 marks)
●●●●●Level 46 marksStart
In this question you must show detailed reasoning.
(a) Given that xx and yy are positive numbers such that

x2(x+3y)<y2(y+3x)x^2(x + 3y) < y^2(y + 3x)

prove that y>xy > x

(4 marks)
(b) Using a counter example, show that the result in part (a) is not true for all real numbers.

(2 marks)
●●●●●Level 46 marksStart
(a) Show, by means of a counter-example, that the following statement is false:

'For all positive integers nn, the number 2n+12^n + 1 is prime.' (1 mark)
(b) Prove that the square of any odd number is one more than a multiple of 88. (4 marks)
●●●●●Level 45 marksStart

Proof by contradiction

31 questions

LessonNot started
In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

The curve C1C_1 has equation

y=2x4+5x2−1x∈Ry = 2x^4 + 5x^2 - 1 \qquad x \in \mathbb{R}

The curve C2C_2 has equation

y=−4x2−11x∈Ry = -4x^2 - 11 \qquad x \in \mathbb{R}

Use algebra to prove by contradiction that C1C_1 and C2C_2 do not intersect.

(4 marks)
●●●●●Level 34 marksStart
Use proof by contradiction to prove that the curve with equation

y=x5+4x−3sin⁡xy = x^5 + 4x - 3\sin x

has no stationary points.

(4 marks)
●●●●●Level 34 marksStart
In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

Use proof by contradiction to show that for all angles xx, where 180∘<x<270∘180^\circ < x < 270^\circ

tan⁡x+cot⁡x⩾2\tan x + \cot x \geqslant 2

(4 marks)
●●●●●Level 34 marksStart
Given that n∈Zn \in \mathbb{Z}, use algebra to prove by contradiction that

“if n3−6n+7n^3 - 6n + 7 is odd then nn is even”

(4 marks)
●●●●●Level 34 marksStart
A student was asked to prove by contradiction that

“there are no positive integers xx and yy such that 2x2+5xy+2y2=262x^2 + 5xy + 2y^2 = 26”

The start of the student's proof is shown below.

Assume that positive integers xx and yy exist such that 2x2+5xy+2y2=262x^2 + 5xy + 2y^2 = 26

⇒(2x+y)(x+2y)=26\Rightarrow (2x + y)(x + 2y) = 26

If 2x+y=12x + y = 1 and x+2y=26x + 2y = 26

⇒3x=−24⇒x=−8, y=17\Rightarrow 3x = -24 \Rightarrow x = -8, \ y = 17 \quad xx is not positive

Show the calculations and statements that are needed to complete the proof.

(4 marks)
●●●●●Level 34 marksStart
Given that m3+5m^3 + 5 is odd, use proof by contradiction to show, using algebra, that mm is even.
●●●●●Level 33 marksStart
Use proof by contradiction to show that if n2n^2 is even, then nn is even.
●●●●●Level 33 marksStart
Use proof by contradiction to show that if n2n^2 is odd, then nn is odd.
●●●●●Level 33 marksStart
Use proof by contradiction to show that there are no integers mm and nn satisfying

6m+9n=16m + 9n = 1 (3 marks)
●●●●●Level 33 marksStart
Use proof by contradiction to show that there is no largest positive integer whose square ends in the digit 66. (3 marks)
●●●●●Level 33 marksStart
Prove by contradiction that 3\sqrt{3} is irrational. You may use the fact that if a2a^2 is a multiple of 33, then aa is a multiple of 33. (6 marks)
●●●●●Level 36 marksStart
Prove by contradiction that log⁡23\log_2 3 is irrational. (4 marks)
●●●●●Level 34 marksStart
A student attempts to answer the following question.

“Use proof by contradiction to prove that there is no greatest prime number.”

The student starts the proof with

Assume that there is a greatest prime number, NN.

⇒\Rightarrow the complete list of prime numbers is p1, p2, p3, …, Np_1,\ p_2,\ p_3,\ \ldots,\ N

Let Q=p1×p2×p3×…×N+1Q = p_1 \times p_2 \times p_3 \times \ldots \times N + 1

Complete the proof.

(4 marks)
●●●●●Level 44 marksStart
A student wishes to prove that, for all positive integers aa and bb, a2−6ab≠10a^2 - 6ab \ne 10.
(a) Prove that a2−6ab=10⇒aa^2 - 6ab = 10 \Rightarrow a is even.

(2 marks)
(b) Hence or otherwise prove that, for all positive integers aa and bb, a2−6ab≠10a^2 - 6ab \ne 10.

(3 marks)
●●●●●Level 45 marksStart
A student was asked to prove, for p∈Zp \in \mathbb{Z}, that

“if p3p^3 is a multiple of 55, then pp must be a multiple of 55”

The start of the student's proof by contradiction is shown below.

Assumption: there exists a number pp, p∈Zp \in \mathbb{Z}, such that p3p^3 is a multiple of 55, and pp is NOT a multiple of 55

Let p=5k+1p = 5k + 1, k∈Zk \in \mathbb{Z}

Consider p3=(5k+1)3=125k3+75k2+15k+1=5(25k3+15k2+3k)+1p^3 = (5k + 1)^3 = 125k^3 + 75k^2 + 15k + 1 = 5(25k^3 + 15k^2 + 3k) + 1, which is not a multiple of 55
(a) Show the calculations and statements that are required to complete the proof.

(3 marks)
(b) Hence prove, by contradiction, that 253\sqrt[3]{25} is an irrational number.

(5 marks)
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Use proof by contradiction to show that there is no greatest even integer.
●●●●●Level 43 marksStart
Use proof by contradiction to prove that 2\sqrt{2} is irrational.
●●●●●Level 45 marksStart
Use proof by contradiction to show that the sum of a rational number and an irrational number is irrational.
●●●●●Level 43 marksStart
Use proof by contradiction to show that if n2n^{2} is divisible by 33, then nn is divisible by 33. (4 marks)
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Use proof by contradiction to show that the product of a non-zero rational number and an irrational number is irrational. (4 marks)
●●●●●Level 44 marksStart
Use proof by contradiction to prove that 5\sqrt{5} is irrational. You may use the fact that if n2n^{2} is divisible by 55 then nn is divisible by 55. (5 marks)
●●●●●Level 45 marksStart
Use proof by contradiction to show that there are no POSITIVE integers xx and yy satisfying

x2−y2=1x^{2} - y^{2} = 1 (4 marks)
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Prove by contradiction that there are no integers aa and bb such that

4a2−b2=64a^2 - b^2 = 6

(4 marks)
●●●●●Level 44 marksStart
Prove by contradiction that log⁡23\log_2 3 is irrational. (5 marks)
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Prove by contradiction that there are no positive integers aa and bb for which

a2−b2=10a^2 - b^2 = 10. (4 marks)
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SECTION A: PURE MATHEMATICS

You may assume that 6\sqrt6 is irrational.

Prove by contradiction that 2+3\sqrt2 + \sqrt3 is irrational. (5 marks)
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Given that xx is a real number such that

x3+x=12x^3 + x = 12
(a) use proof by contradiction to prove that x<3x < 3

(4 marks)
(b) Show that 2k2−12k+212k^2 - 12k + 21 is positive for all real values of kk

(2 marks)
●●●●●Level 56 marksStart
(a) Prove, using algebra, that 5n+1−5n5^{n+1} - 5^n is a multiple of 20 for all positive integers nn

(3 marks)
(b) A student was asked to use proof by contradiction to prove the following statement.

“For all integers nn, if n2n^2 is a multiple of 4 then nn is even”

The start of the student's proof is shown below.

Assume that n2n^2 is not a multiple of 4 and that nn is odd.

⇒n=2k+1\Rightarrow n = 2k + 1 for some integer kk

⇒n2=4k2+4k+1=4(k2+k)+1\Rightarrow n^2 = 4k^2 + 4k + 1 = 4(k^2 + k) + 1

Explain why the student's proof cannot be completed.

(1 mark)
(c) Use proof by contradiction to prove the statement.

(3 marks)
●●●●●Level 57 marksStart
Use proof by contradiction to prove that 3\sqrt{3} is irrational. (You may use the result that if a2a^2 is a multiple of 33 then aa is a multiple of 33.)
●●●●●Level 55 marksStart
Use proof by contradiction to show that there is no smallest positive rational number.
●●●●●Level 53 marksStart
Use proof by contradiction to show that log⁡23\log_{2} 3 is irrational. (5 marks)
●●●●●Level 55 marksStart