 # 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

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

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

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.

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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