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Proofs

Simple Deductive Proofs, LHS to RHS Proofs...

Paper

Paper 1

Difficulty

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Question 1

no calculator

easy

[Maximum mark: 4]

Prove that the sum of three consecutive positive integers is divisible by $3$.

easy

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Video

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Solutions

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Question 2

no calculator

easy

[Maximum mark: 4]

Consider two consecutive positive integers, $k$ and $k+1$.

Show that the difference of their squares is equal to the sum of the two integers.

easy

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Video

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Solutions

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Question 3

no calculator

easy

[Maximum mark: 4]

The product of three consecutive integers is increased by the middle integer.

Prove that the result is a perfect cube.

easy

Formula Booklet

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Video

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Solutions

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Question 4

no calculator

easy

[Maximum mark: 6]

1. Show that $(2n-1)^3 + (2n+1)^3 = 16n^3+12n$ for $n \in \mathbb{Z}$. [3]

2. Hence, or otherwise, prove that the sum of the cubes of any two consecutive odd integers is divisible by four. [3]

easy

Formula Booklet

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Video (a)

Video (b)

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Check with RV Newton

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Solutions

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Question 5

no calculator

easy

[Maximum mark: 5]

1. Prove that $\dfrac{5}{x^2} = \dfrac{5}{x(x-2)}-\dfrac{10}{x^2(x-2)}$. [3]

2. Determine the set of numbers $x$ for which the proof in part (a) is valid. [2]



easy

Formula Booklet

Mark Scheme

Video (a)

Video (b)

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Solutions

Revisit

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