IB DP Math MAA HL : IB Style Mock Exams – Set 5 Paper 1

Question

Using the method of proof by contradiction, prove that $\sqrt{3}$ is irrational.

Answer/Explanation

To prove that $\sqrt{3}$ is irrational using proof by contradiction, we assume the opposite, namely that $\sqrt{3}$ is rational. That is, there exist integers $a$ and $b$ with no common factors other than 1 such that $\sqrt{3} = \frac{a}{b}$.

We can assume that $a$ and $b$ are positive integers (since if $\sqrt{3} = \frac{a}{b}$, we can always multiply both sides by $-1$ to obtain $-\sqrt{3} = -\frac{a}{b}$, which is equivalent to $\sqrt{3} = \frac{-a}{b}$).

Squaring both sides of $\sqrt{3} = \frac{a}{b}$, we get:

$$3 = \frac{a^2}{b^2}$$

Multiplying both sides by $b^2$, we get:

$$3b^2 = a^2$$

This means that $a^2$ is divisible by 3, and therefore $a$ must also be divisible by 3 (since the square of an integer is only divisible by 3 if the integer itself is divisible by 3).

Let $a = 3c$, where $c$ is also a positive integer. Substituting into the equation above, we get:

$$3b^2 = (3c)^2 = 9c^2$$

Dividing both sides by 3, we get:

$$b^2 = 3c^2$$

This means that $b^2$ is divisible by 3, and therefore $b$ must also be divisible by 3.

However, we have now shown that both $a$ and $b$ are divisible by 3, which contradicts our assumption that $a$ and $b$ have no common factors other than 1. Therefore, our initial assumption that $\sqrt{3}$ is rational must be false, and $\sqrt{3}$ is indeed irrational.

Question

The graph of \(y = \frac{{a + x}}{{b + cx}}\) is drawn below.

(a)     Find the value of a, the value of b and the value of c.

(b)     Using the values of a, b and c found in part (a), sketch the graph of \(y = \left| {\frac{{b + cx}}{{a + x}}} \right|\) on the axes below, showing clearly all intercepts and asymptotes.

▶️Answer/Explanation

Markscheme

(a)     an attempt to use either asymptotes or intercepts     (M1)

\(a = – 2,{\text{ }}b = 1,{\text{ }}c = \frac{1}{2}\)     A1A1A1

(b)       A4

Note: Award A1 for both asymptotes,

A1 for both intercepts,

A1, A1 for the shape of each branch, ignoring shape at \((x = – 2)\).

[8 marks]

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