 # Exam-Style Questions.

## Problems adapted from questions set for 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?

### 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.

### 6.

GCSE Higher

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

### 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.

### 8.

GCSE Higher

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

Using your answer to part (a), prove that if $$p$$ and $$q$$ are positive integers and $$p \lt q$$, then

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

### 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.

### 10.

GCSE Higher

Prove that the expression below is always positive.

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

### 11.

IB Analysis and Approaches

(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.

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