Exam-Style Questions.

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 \(c\).

Worked Solution

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2.	GCSE Higher

One is added to the product of two consecutive positive even numbers. Show that the result is a square number.

Worked Solution

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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.

Worked Solution

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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

Worked Solution

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5.	GCSE Higher

This expression can be used to generate a sequence of numbers.

$$n^2+n + 5$$

(a) Work out the first three terms of this sequence.

(b) What is the smallest value of \(n\) that produces a term of the sequence that is not a prime number?

(c) Is it true that odd square numbers have exactly three factors? Explain and justify your answer.

(d) Seymour is thinking of a number.

• It is a common factor of 144 and 192.
• It is a common multiple of 6 and 8.
• It is less than 100.

Find the two possible numbers that Seymour could be thinking of.

Worked Solution

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6.	GCSE Higher

Given that \(n\) can be any integer such that \(n \gt 1\), prove that \(n^2 + 3n\) is even.

Worked Solution

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7.	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.

Worked Solution

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8.	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

$$\frac{p-1}{q-1}-\frac pq\lt 0$$

Worked Solution

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9.	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.

Worked Solution

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10.	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.

Worked Solution

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11.	GCSE Higher

Prove that the expression below is always positive.

$$ x^2 - 5x + 9 $$

Worked Solution

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12.	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.

Worked Solution

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