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Exam-Style Questions on ProofProblems on Proof adapted from questions set in previous Mathematics exams. |
1. | GCSE Higher |
State whether each of the following statements is true or false. Give reasons for your answers.
(a) When \(a^2 = 25\) the only value that \(a\) can have is 5.
(b) When \(b\) is a positive integer, the value of \(3b\) is always a factor of the value of \(12b\).
(c) When \(c\) is positive, the value of \(c^2\) is always greater than the value of \(c\).
2. | GCSE Higher |
One is added to the product of two consecutive positive even numbers. Show that the result is a square number.
3. | GCSE Higher |
(a) Give a reason why 0 is an even number.
(b) The lengths of the sides of a right-angled triangle are all integers. Prove that if the lengths of the two shortest sides are even, then the length of the third side must also be even.
4. | GCSE Higher |
Betsy thinks that \((3x)^2\) is always greater than or equal to \(3x\).
Is she is correct?
Show your working to justify your decision
5. | GCSE Higher |
Given that \(n\) can be any integer such that \(n \gt 1\), prove that \(n^2 + 3n\) is even.
6. | GCSE Higher |
Use algebra to prove that \(0.3\dot1\dot8 \times 0.\dot8\) is equal to \( \frac{28}{99} \).
7. | GCSE Higher |
The diagram shows a quadrilateral ABCD in which angle DAB equals angle CDA and AB = CD.
Prove that the diagonals of this quadrilateral are of equal length.
8. | GCSE Higher |
m and n are positive whole numbers with m > n
m2 – n2 = (m + n)(m – n)
If m2 – n2 is a prime number prove that m and n are consecutive numbers.
9. | GCSE Higher |
(a) Prove that the recurring decimal \(0.\dot2 \dot1\) has the value \(\frac{7}{33}\)
(b) The value of \(x\) is given as:
$$x=\frac{1}{5^{120}\times2^{123}}$$Show that, when \(x\) is written as a terminating decimal, there are 120 zeros after the decimal point.
(c) The reciprocal of any prime number \(p\) (where \(p\) is neither 2 nor 5) when written as a decimal, is always a recurring decimal.
A theorem in mathematics states:
The period of a recurring decimal is the least value of \(n\) for which \(p\) is a factor of \(10^n – 1\)
Marilou tests this theorem for the reciprocal of eleven.
She uses her calculator to show that 11 is a factor of \(10^2 – 1\) then makes this statement:
"The period of the recurring decimal is 2 because 11 is a factor of \(10^2-1\). This shows the theorem to be true in this case."
Explain why Marilou's statement is incomplete.
10. | GCSE Higher |
Express as a single fraction and simplify your answer.
$$\frac{p-1}{q-1}-\frac pq$$Using your answer to part (a), prove that if \(p\) and \(q\) are positive integers and \(p \lt q\), then
$$\frac{p-1}{q-1}-\frac pq\lt 0$$11. | GCSE Higher |
(a) Prove that the product of two consecutive whole numbers is always even.
(b) Prove, by giving a counter example, that the sum of four consecutive integers is not always divisible by 4.
12. | GCSE Higher |
The number \(T\) can be expressed as \(T = 4k + 3\) where \(k\) is a positive integer.
(a) Show that \(T^2\) is always an odd number.
\(T\) and \(U\) are consecutive odd numbers where \(U > T\).
(b) Write down an expression for \(U\), in terms of \(k\).
(c) Show that \(U^2 - T^2\) is always a multiple of 16.
13. | GCSE Higher |
Prove that the expression below is always positive.
$$ x^2 - 5x + 9 $$14. | IB Analysis and Approaches |
Consider two consecutive positive even numbers, \(2n\) and \(2n + 2\).
Show that the difference of their squares is equal to twice their sum.
15. | IB Analysis and Approaches |
Use mathematical induction to show that:
$$\sum_{r=1}^n \dfrac{1}{r(r+1)} = \dfrac{n}{n+1} $$for all \(n \in \mathbb{Z}^+\).
16. | IB Analysis and Approaches |
Consider the sum of the squares of any two consecutive odd integers.
(a) Show that \((2n + 1)^2 + (2n + 3)^2 = 8n^2 +16n + 10\) , where \(n \in \mathbb{Z} \)
(b) Hence, or otherwise, prove that the sum of the squares of any two consecutive odd integers is even.
17. | IB Analysis and Approaches |
Using mathematical induction and the definition \( ^nC_r = \frac{n!}{r!(n-r)!} \), prove that
$$ \sum_{r=2}^{n} \; ^rC_2 = \; ^{n+1}C_3 $$for all \( n \in \mathbb{Z}^+ \).
18. | IB Analysis and Approaches |
Prove that the integers \(a\) and \(b\) cannot both be odd if \(a^2+b^2\) is exactly divisible by 4.
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