Mathematics S1: free practice, theory and problems

1. Algebra and equations

What is it about?

In S1 you need solid algebra: fractions with letters, factorising, and first- and second-degree equations, also with fractions.

Concepts and formulas

• Fraction rules: $\dfrac{a}{b} + \dfrac{c}{d} = \dfrac{ad + bc}{bd}$, $\dfrac{a}{b} \cdot \dfrac{c}{d} = \dfrac{ac}{bd}$, $\dfrac{a}{b} : \dfrac{c}{d} = \dfrac{a}{b} \cdot \dfrac{d}{c}$.
• Factorise before cancelling: $\dfrac{x^2 - 4}{x - 2} = \dfrac{(x-2)(x+2)}{x-2} = x + 2$ (for $x \ne 2$).
• Equations with fractions: multiply every term by the common denominator, and check that the solution does not make a denominator zero.
• Inequalities: reverse the inequality sign when you multiply or divide by a negative number.

Practise algebra and equations in the app →

2. Functions and models

What is it about?

Many relationships in society and the economy can be described with functions: costs, income, population growth and prices.

Concepts and formulas

• Linear model: a fixed increase per unit, $f(x) = ax + b$.
• Exponential model: a fixed percentage increase, $f(x) = a \cdot b^x$. Growth factor $b = 1 + p/100$.
• Cost $K(x)$, income $I(x) = p \cdot x$ and profit $O(x) = I(x) - K(x)$.
• The zero of the profit function is where it breaks even.

Practise functions and models in the app →

3. Differentiation and optimisation

What is it about?

The derivative shows how fast something changes. In economics it is used to find which production level gives the largest profit.

Concepts and formulas

• $(ax^n)' = anx^{n-1}$.
• Marginal cost $K'(x)$: how much it costs to make one more unit. Marginal revenue $I'(x)$.
• Largest profit where $O'(x) = 0$, i.e. where marginal revenue = marginal cost.
• For $O(x) = -ax^2 + bx - c$ (with $a > 0$) the maximum is at $x = \dfrac{b}{2a}$.

Practise differentiation and optimisation in the app →

4. Probability

What is it about?

Probability describes how likely something is, from 0 (impossible) to 1 (certain).

Concepts and formulas

• Independent events: $P(A \cap B) = P(A) \cdot P(B)$.
• The complement rule: $P(\text{at least one}) = 1 - P(\text{none})$.
• Combinations: $\binom{n}{k}$ ways of choosing $k$ of $n$ without order.
• Hypergeometric: drawing without replacement from two groups.

Practise probability in the app →

5. Inequalities and sign charts

🔗 Shared curriculum with 1T. The questions are the same, but your progress counts separately for each subject.

What is it about?

An inequality asks for which $x$ something is greater or smaller than something else. The answer is a whole range of numbers, not a single number. The key tool is the sign chart: find where the expression is zero or undefined, and check the sign between these points.

Concepts and formulas

• A linear inequality is solved like an equation, but flip the inequality sign when you multiply or divide by a negative number: $-2x > 6 \Rightarrow x < -3$.
• Quadratic inequality: move everything to one side, factorise $ax^2 + bx + c = a(x - x_1)(x - x_2)$ and draw a sign chart for each factor.
• A product is positive when there is an even number of negative factors.
• Rational inequality $\dfrac{P(x)}{Q(x)} \ge 0$: the zeros of $Q$ are never part of the solution (the fraction is not defined there). Never multiply by an expression whose sign you do not know.
• Notation: $x < 2$ can also be written $(-\infty, 2)$, and $1 \le x \le 4$ as $[1, 4]$.

Example 1

Solve $x^2 - 5x + 4 < 0$. The zeros are $x = 1$ and $x = 4$, so $x^2 - 5x + 4 = (x - 1)(x - 4)$. Sign chart: both factors are negative for $x < 1$ (product positive), one is negative between 1 and 4 (product negative), none for $x > 4$. Solution: $1 < x < 4$.

Example 2

Solve $\dfrac{x + 3}{x - 1} \ge 0$. The numerator is zero at $x = -3$ (included), the denominator is zero at $x = 1$ (never included). Solution: $x \le -3$ or $x > 1$.

Practise inequalities and sign charts in the app →

6. Logarithms and exponential equations

🔗 Shared curriculum with R1. The questions are the same, but your progress counts separately for each subject.

What is it about?

The logarithm answers the question "which exponent do I need?". It is used to solve equations where the unknown is in the exponent, for example in growth and decay.

Concepts and formulas

• $\lg a = x \iff 10^x = a$, and $\ln a = x \iff e^x = a$.
• $\lg(ab) = \lg a + \lg b$, $\lg\dfrac{a}{b} = \lg a - \lg b$, $\lg a^n = n \lg a$ (similarly for $\ln$).
• $e^{kx} = a$ gives $x = \dfrac{\ln a}{k}$.
• Exponential growth: $f(x) = a \cdot b^x$. A growth factor $b > 1$ gives growth, $0 < b < 1$ gives decline.

Example

$3 \cdot 2^x = 48$ gives $2^x = 16$, so $x = \dfrac{\lg 16}{\lg 2} = 4$.

Practise logarithms and exponential equations in the app →

7. Exponential models

🔗 Shared curriculum with 1T. The questions are the same, but your progress counts separately for each subject.

What is it about?

When something increases or decreases by the same percentage each period, the growth is exponential: population, interest, the value of a car, medicine in the blood. The model is $f(x) = a \cdot b^x$, where $a$ is the initial value and $b$ is the growth factor.

Concepts and formulas

• Growth factor for an increase of $p$ %: $b = 1 + \dfrac{p}{100}$. For a decrease: $b = 1 - \dfrac{p}{100}$.
• After $x$ periods: $f(x) = a \cdot b^x$. Example: 20,000 NOK at 5 % interest for 3 years gives $20\,000 \cdot 1.05^3 = 23\,152.50$ NOK.
• Growth factor from two measurements: $b = \left(\dfrac{y_2}{y_1}\right)^{\frac{1}{x_2 - x_1}}$.
• When is a level reached? Solve $a \cdot b^x = c$ with logarithms: $x = \dfrac{\lg(c/a)}{\lg b}$.
• Linear model $ax + b$: the same number is added each period. Exponential: the same factor is multiplied.
• Percent and percentage points: if the rate goes from 2 % to 3 %, it increases by 1 percentage point, but by 50 %.

Example

A car costs 400,000 NOK and loses 15 % of its value each year. The value after $x$ years is $400\,000 \cdot 0.85^x$. After 4 years: $400\,000 \cdot 0.85^4 \approx 208\,800$ NOK.

Practise exponential models in the app →

8. Cost, revenue and profit

What is it about?

Businesses use functions to describe what it costs to produce $x$ units, what they earn, and when the profit is largest. The derivative is here called marginal cost and marginal revenue.

Concepts and formulas

• Cost function $K(x)$: fixed costs (the constant term) plus variable costs. Example: $K(x) = 0.02x^2 + 20x + 5000$.
• Revenue function at a fixed price $p$: $I(x) = p \cdot x$.
• Profit: $O(x) = I(x) - K(x)$.
• Marginal cost $K'(x)$: roughly what it costs to make one more unit. Marginal revenue $I'(x)$ likewise.
• Largest profit when $O'(x) = 0$, i.e. when marginal revenue = marginal cost: $I'(x) = K'(x)$.
• Unit cost $E(x) = \dfrac{K(x)}{x}$. It is smallest where $E(x) = K'(x)$.

Example

$K(x) = 0.02x^2 + 20x + 5000$ and the price is 100 NOK. $O'(x) = 100 - (0.04x + 20) = 0$ gives $x = 2000$ units. The profit is then $O(2000) = 200\,000 - (80\,000 + 40\,000 + 5000) = 75\,000$ NOK.

Practise cost, revenue and profit in the app →