Exam-Style Questions on Number

1.	GCSE Higher

Find the highest common factor (HCF) of 60 and 96.

Worked Solution

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2.	GCSE Higher

Without using a calculator:

(a) Express 68 as the product of its prime factors.

(b) Divide 68 by 5 giving your answer in decimal format.

(c) Multiply 13.06 by 2.1

Worked Solution

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3.	GCSE Higher

The number, \(N\), can be written as the product of prime factors in index form as:

$$N = 3 × 5^3 × x^4$$

Work out \(5N^2\) as a product of prime factors in index form giving your answer in terms of \(x\).

Worked Solution

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4.	GCSE Higher

The difference between the areas of the two squares is 51 cm².

When measured in cm the lengths of the sides of the squares are integers and both less than 20cm.

Find the lengths of the sides of the two squares.

 

Worked Solution

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5.	IB Studies

This Venn diagram shows the relationship between the sets of numbers

$$\mathbb N, \mathbb Z, \mathbb Q \text{ and } \mathbb R$$

Write down the following numbers in the appropriate place in the Venn diagram.

(a) \(5\)

(b) \(\frac23\)

(c) \(\pi\)

(d) \(0.91\)

(e) \(\sqrt 7\)

(f) \(-0.75\)

Worked Solution

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6.	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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7.	IB Analysis and Approaches

Consider \( w = a - bi \) where \( |w| = 1 \)

Show that \( \text{Re} \left( \frac{w + i}{w - i} \right) \) is always equal to 0.

Worked Solution

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8.	IB Analysis and Approaches

A complex number, \(c\), has a real part \(1\) and an imaginary part \( -\sqrt{3} \).

(a) Show that \(c = 2e^{i\frac{5 \pi}{3}}\).

(b) Find the smallest positive integer \(n\) such that \( c^n\) is a real number.

(c) Find the value of \(c^n\) when \(n\) takes the value found in part (b).

Consider the equation \(z^3-9z^2+18z-28=0\), where \(z \in \mathbb{C} \).

(d) Given that \(c\) is a root of \(z^3-9z^2+18z-28=0\), find the other roots.

(e) By using a suitable transformation from \(z\) to \(w\), or otherwise, find the roots of the equation \(1-9w+18w^2-28w^3=0\), where \(w \in \mathbb{C} \).

Consider the equation \(z^2 = 2z^*\) , where \(z \in \mathbb{C}, z \neq 0 \).

(f) By expressing \(z\) in the form \(a + bi\), find the roots of the equation.

Worked Solution

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9.	IB Analysis and Approaches

Consider \( P(z) = z^3 +2nz - \frac{40z^2}{n} - 10n\), where \( z \in \mathbb{C} \) and \( n \in \mathbb{R}^+ \).

Given that \( z - 4i \) is a factor of \( P(z) \), find the roots of \( P(z) = 0 \).

Worked Solution

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10.	IB Analysis and Approaches

Consider the complex numbers \(z_1 = c + 3i\) and \(z_2 = c^2-2-ci\)

(a) Find an expression for \(z_1z_2 \) in terms of \(c\).

(b) Hence, given that arg(\(z_1z_2 \)) = \( \tan^{-1} 2 \) show that:

$$c^3 - c^2 + c + 3 = 0 $$

Worked Solution

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