# Leaving Cert Maths Questions: Algebra 3 Quiz

As the Leaving Cert exams edge closer, it is an important time to test your math’s skills. What better way to do this than with a self-assesment quiz that I have created – all you have to do it answer the leaving cert maths questions and get the correct answer and solution as soon as you click “submit”.

## What are H1Maths Quizzes?

Every H1Maths course comes with an end-of-topic exam. So for the Higher Level full bundle that’s 22 quick quizzes. It’s a great way to test your knowledge as well as helping you understand the areas you need to focus on to improve.

**The quizzes are based on real leaving cert maths questions** as well as a selection of the best questions from pre papers, maths books and revision handouts/ books, so you’re really getting a lot of variety and hopefully new questions than what you have seen before.

## What is Included in Algebra 3?

### Inequalities and Moduli

The inequality symbols \( >, \ge,<, \le \) are needed when solving problems in which a range of possible values satisfy the given conditions. Inequalities can be solved using algebra, by graphing the inequalities or sometimes using a number line). Quadratic inequalities appear often in exam papers and are generally found by substituting = for the inequality sign, and finding the roots of the equation – then sketching a graph of these roots and then using the graph to find the set of values of \(x\) that satisfy the inequality.

The modulus is represented like this, \(|x|\), and the modulus of a number is a measure of its size or magnitude. No matter if the \(x\) is – or + we are only interested in the size (so we take the positive value).

An example of a modulus inequality is shown below:

$$|x+3|<|3 x-7|$$

### Proofs

Proofs are a tricky topic on the exam. In algebra 3 we look at direct proofs, proofs by contradiction, proofs of abstract inequalities proofs by induction. That’s just paper 1, we of course have the 3 theorems in paper 2!

In mathematics a statement has to be proven to be *always* true and then becomes a theorem. **Direct proofs** use axioms, definitions and past theorems to construct a statement to prove something is always true. **Proof by contradiction** is when we set up a statement we know is false but claim it to be true, and by logically contradicting the statement we prove the statement must be false. **Proofs by induction **follows a specific procedure:

(i) The statement is proven true for some fixed value, usually \(n=1\) or \(n=2\).

(ii) The statement is then assumed true for \(n=k\).

(iii) Based on this assumption, we must show that the statement is true for \(n=k+1\).

(iv) In conclusion, a “rolling proof” is formed:

a. Since it was true for \(n=1\),

b. it is now true for \(n=1+1=2\).

c. Since it is true for \(n=2\), it is true for \(n=2+1=3\), etc.

d. It is therefore true for all values of \(n\).

### Indices, Exponentials and Logarithms

**Indices: **When a number is multiplied by itself a certain number of times, we use the index form to represent it: \(3^4 = 3\times3\times3\times3\). 3 is known as **the base** and 4 is **the index**. We use the rules of indices to solve a variety of index questions in the exams (pg 21 of formulae and tables)

**Exponentials:** Exponentials are equations or expressions where the index is a variable \(x\). When working with exponentials the most important thing is to set all values to the same base number.

**Logarithms:** The logarithm of a number is the power to which the base number must be raised to get that number (\(a^x=y\) is equivalent to \(\log_a{y}=x\). Just like with the indices there are **laws of logarithms **that are on page 21 of the formulae and tables book.

## Now Test your Skills!

If you require any assistance after the quiz or have any questions please reach out via the live-chat or email: [email protected]

You can always book a one-to-one grind here

Test your knowledge of indices, exponentials and logarithms with this short 5 question quiz. Best practice is to:

- Do the working-out on a sheet of paper
- Use the hint button if you require a little help to get you started
- Choose the right answer or fill in the blanks to answer the question
- Instantly be told if you are correct/ incorrect
- A pop-up will appear at the end of each question with a model solution to the question
- Do not get disheartened if you get a few incorrect answers, remember you can always review the on-demand videos or ask for help from the instructor!

Good luck! 😊

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- Algebra 3 0%

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- Question 1 of 5
##### 1. Question

### Find the Value of \(x\) in this equation:

### \(\log _{10}\left(x^{2}+6\right)-\log _{10}\left(x^{2}-1\right)=1\)

CorrectIncorrect##### Hint

Don’t forget all the rules of logarithms can be found in your formula book pg. 21

You will need: \(\log _{a}\left(\frac{x}{y}\right)=\log _{a} x-\log _{a} y\)

- Question 2 of 5
##### 2. Question

### Without using a calculator find the answer to:

### $$\log _{27} \frac{1}{3}$$

CorrectIncorrect##### Hint

Let \(\log _{27} \frac{1}{3} =x\)

Now how to get rid of the \(\log _{27}\)?

- Question 3 of 5
##### 3. Question

### Solve \(|x+2|=6\)

### There are two answers, the order does not matter when filling in the blanks!

x= or

CorrectIncorrect##### Hint

$$|x+2|=6 \Rightarrow x+2=6 or x+2=-6$$

### or

$$(|x+2|)^{2}=(6)^{2}$$

- Question 4 of 5
##### 4. Question

### Solve the equation \(2\left(4^{x}\right)+4^{-x}=3\).

### Choose the two answers for \(x\)

CorrectIncorrect##### Hint

First change the \(4^{-x}\) to \(\frac{1}{4^{x}}\) to get:

\(2\left(4^{x}\right)+\frac{1}{4^{x}}=3\)

- Question 5 of 5
##### 5. Question

### Find the value of the rational number \(p\) for which \(\frac{3^{\frac{1}{4}} \times 3 \times 3^{\frac{1}{6}}}{\sqrt{3}}=3^{p}\).

CorrectIncorrect##### Hint

Convert everything to indices and use the rules of indices to simplify:

The most important rules for this question are:

$$a^{p} a^{q}=a^{p+q}$$

$$\frac{a^{p}}{a^{q}}=a^{p-q}$$