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

---

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

---

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

---

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

---

5.	GCSE Higher

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

Worked Solution

---

6.	GCSE Higher

Use algebra to prove that \(0.3\dot1\dot8 \times 0.\dot8\) is equal to \( \frac{28}{99} \).

Worked Solution

---

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.

Worked Solution

---

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.

Worked Solution

---

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.

Worked Solution

---

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

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

Worked Solution

---

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.

Worked Solution

---

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

---

13.	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.

Worked Solution

---