Proof – Year 12 A level maths worksheet with answers to download as PDF

Let \(n\) be any integer. Even integers can be written as \(2n\) and odd integers as \(2n+1\).

1. Prove that the sum of any two odd numbers is even. [3 marks]
2. Prove that the product of an even number and any integer is even. [2 marks]
3. Prove that the square of an odd number is odd. [3 marks]

Three consecutive integers can be written \(n\), \(n+1\) and \(n+2\).

1. Prove that the sum of three consecutive integers is a multiple of 3. [2 marks]
2. Prove that the sum of four consecutive integers is never a multiple of 4. [3 marks]
3. Prove that the product of two consecutive integers is always even. [3 marks]

Find a counter-example to show that each statement is false.

1. "\(n^2+n+11\) is prime for every positive integer \(n\)." [2 marks]
2. "If \(x^2>9\) then \(x>3\)." [2 marks]
3. "The sum of two prime numbers is always even." [2 marks]
4. "\(\sqrt{a^2+b^2}=a+b\) for all positive \(a\) and \(b\)." [2 marks]

Use exhaustion to prove each statement.

1. The square of any integer ends in 0, 1, 4, 5, 6 or 9. [3 marks]
2. Every cube of an integer is either a multiple of 9, or one more than a multiple of 9, or one less than a multiple of 9. [4 marks]

Prove each identity.

1. \((n+3)^2-(n-3)^2\equiv12n\) [3 marks]
2. \((2n+1)^2-(2n-1)^2\equiv8n\) [3 marks]
3. \(n^3-n\equiv n(n-1)(n+1)\) [3 marks]

To prove \(\sqrt2\) is irrational, assume the opposite and look for a contradiction.

1. Assume \(\sqrt2=\dfrac ab\) where \(\dfrac ab\) is a fraction in its simplest form. Show that \(a^2=2b^2\). [1 mark]
2. Explain why \(a\) must be even. [2 marks]
3. Write \(a=2k\), show that \(b\) is also even, and state the contradiction. [3 marks]

A kitchen calendar shows each month in rows of 7 days. Amira draws a square around four numbers in a \(2\times2\) block, for example 9, 10, 16 and 17. She notices that \(9+17=26\) and \(10+16=26\). She wonders whether this always happens.

1. Let the top-left number be \(n\). Write the other three numbers of the block in terms of \(n\). [2 marks]
2. Prove that the two diagonal sums are always equal. [2 marks]
3. Prove that the sum of the four numbers is always a multiple of 4. [2 marks]
4. Amira says the total of the block could be 70. Is she right? [2 marks]

A square lawn has side \(x\) metres, where \(x\) is a whole number. A path of width 1 m is laid around all four sides, so the lawn plus path is a larger square. The path is covered with paving slabs.

1. Show that the area of the path is \(4x+4\) square metres. [3 marks]
2. Prove that the area of the path is always a multiple of 4. [2 marks]
3. The area of the path is 84 m\(^2\). Find \(x\). [2 marks]
4. Explain why the path area can never be an odd number. [1 mark]

At a school reunion, \(n\) people each shake hands once with every other person.

1. Show that the total number of handshakes is \(\dfrac{n(n-1)}{2}\). [2 marks]
2. Prove that \(n(n-1)\) is always even, so the formula always gives a whole number. [3 marks]
3. There were 66 handshakes. How many people attended? [3 marks]
4. Can the number of handshakes ever be 50? Justify your answer. [2 marks]

Ben claims: "Take any two-digit number, reverse its digits, and add the two numbers. The answer is always a multiple of 11." A two-digit number with tens digit \(a\) and units digit \(b\) equals \(10a+b\).

1. Test Ben\'s claim with 37 and 82. [2 marks]
2. Prove the claim. [3 marks]
3. Prove that the difference between a two-digit number and its reverse is always a multiple of 9. [3 marks]
4. Is the difference always a multiple of 11? Give a counter-example if not. [2 marks]

A tiler arranges square tiles in a square pattern of \(n\) by \(n\) tiles. She then adds a border of tiles to make an \((n+1)\) by \((n+1)\) square. She claims the number of extra tiles is always odd.

1. Show that the number of extra tiles is \(2n+1\). [2 marks]
2. Prove her claim. [2 marks]
3. The extra tiles cost £0.85 each and she has a £30 budget. What is the largest square she can extend? [3 marks]
4. Prove that the sum of the first \(n\) odd numbers is \(n^2\) using the tile pattern or by algebra. [3 marks]

A taxi company charges £3 plus £2 per kilometre (whole kilometres only). A bus company charges £5 plus £4 per kilometre.

1. Show that the taxi fare for \(n\) km is always an odd number of pounds. [2 marks]
2. Show that the bus fare is always odd. [2 marks]
3. Prove that the difference between the two fares is always even. [2 marks]
4. For which distance are the two fares equal? Explain. [2 marks]

A magician asks a volunteer: "Think of a whole number. Double it, add 6, halve it, then subtract your original number. The answer is 3."

1. Show algebraically that the answer is always 3. [3 marks]
2. The volunteer instead starts with \(n\), squares it, adds \(n\), and divides by \(n\) (with \(n\ne0\)), then subtracts \(n\). Show the answer is always 1. [3 marks]
3. The magician asks: "Pick a three-digit number, repeat it to make a six-digit number (such as 247247). It is always divisible by 7, 11 and 13." Prove it. [4 marks]

Jaya puts £1 in a jar on day 1, £2 on day 2, £3 on day 3 and so on, adding one pound more each day.

1. Show that after \(n\) days she has saved £\(\tfrac12n(n+1)\). [2 marks]
2. Prove that after \(n\) days the amount is a whole number of pounds. [2 marks]
3. On which day does her total first exceed £500? [3 marks]
4. Prove that the total after \(n\) days plus the total after \(n+1\) days is a square number. [3 marks]

A Pythagorean triple is a set of three positive integers \((a,b,c)\) with \(a^2+b^2=c^2\). A carpenter uses the 3-4-5 triangle to check that corners are square. Jess finds a rule: for any positive integer \(n\), the numbers \(2n+1\), \(2n^2+2n\) and \(2n^2+2n+1\) form a triple.

1. Check the rule for \(n=4\). [2 marks]
2. Prove the rule works for every positive integer \(n\). [4 marks]
3. Use the rule to find a triple with the longest side 85. [3 marks]
4. Show that the carpenter\'s triple can be scaled to give a triangle with perimeter 60 m. [2 marks]

Euclid proved that there are infinitely many prime numbers. His proof is a famous proof by contradiction. Suppose there is a finite list of all primes \(p_1,p_2,\dots,p_k\). Let \(N=p_1p_2\cdots p_k+1\).

1. Explain why \(N\) leaves remainder 1 when divided by each prime \(p_i\). [2 marks]
2. Explain why \(N\) is either prime or has a prime factor not on the list. [3 marks]
3. State the contradiction and conclude. [2 marks]
4. Calculate \(N\) for the list 2, 3, 5, 7, 11, 13 and show that \(N\) is not prime. Does this break the argument? Explain. [3 marks]

Four friends compare their results on a number puzzle. Each claims a statement about integers \(n\). For each statement decide whether it is true or false. If true, prove it. If false, give a counter-example.

1. "\(n^2-n\) is always even." [3 marks]
2. "\(n^2+n+1\) is always odd." [3 marks]
3. "\(n^3+n\) is always divisible by 3." [3 marks]
4. "\(2^n+1\) is prime for all \(n\ge1\)." [3 marks]

A teacher claims some facts about means. Let \(n\) be a positive integer.

1. Prove that the mean of three consecutive integers is the middle integer. [2 marks]
2. Prove that the sum of the squares of three consecutive integers is always 2 more than a multiple of 3. [3 marks]
3. Prove that the sum of the squares of two consecutive integers is always odd. [3 marks]
4. Find two consecutive integers whose squares add to 421. [3 marks]

Prove each inequality for all real values stated.

1. \(x^2-6x+10>0\) for all real \(x\). [3 marks]
2. \(a^2+b^2\ge2ab\) for all real \(a\) and \(b\). [3 marks]
3. \(x+\dfrac1x\ge2\) for all \(x>0\). [4 marks]
4. Hence show that the sum of a positive number and its reciprocal is never less than 2, and state when it equals 2. [2 marks]

Prove that no square number can be written in the form \(4k+2\) or \(4k+3\), where \(k\) is a non-negative integer. A student starts to fill in a table.

1. Explain why every integer has one of the two forms in the table. [1 mark]
2. Complete the proof. [3 marks]
3. Use your result to show that 1 000 003 is not a perfect square. [2 marks]
4. Show that \(x^2-4y=3\) has no integer solutions. [3 marks]

Use proof by contradiction to answer this question.

1. Prove that if \(x\) is rational and \(y\) is irrational then \(x+y\) is irrational. [4 marks]
2. Prove that \(\sqrt2+\sqrt3\) is irrational, given that \(\sqrt6\) is irrational. [5 marks]
3. Prove that \(\log_2 3\) is irrational. [4 marks]