Mathematics 1T: free practice, theory and problems

1. Algebra and equations

What is it about?

Algebra is calculating with letters. You must be able to simplify expressions, factorise, and solve first- and second-degree equations and systems of equations.

Concepts and formulas

• Power rules: $a^m \cdot a^n = a^{m+n}$, $\dfrac{a^m}{a^n} = a^{m-n}$, $(a^m)^n = a^{mn}$, $a^0 = 1$, $a^{-n} = \dfrac{1}{a^n}$.
• Standard form: $a \cdot 10^n$ with $1 \le a < 10$.
• The square identities: $(a+b)^2 = a^2 + 2ab + b^2$, $(a-b)^2 = a^2 - 2ab + b^2$, $(a+b)(a-b) = a^2 - b^2$.
• Quadratic equation $ax^2 + bx + c = 0$: $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
• The discriminant $b^2 - 4ac$: positive gives two solutions, zero gives one, negative gives none.
• Systems with two unknowns are solved by substitution or elimination.

Example

Solve $x^2 - 5x + 6 = 0$. Here $b^2 - 4ac = 25 - 24 = 1$, so $x = \dfrac{5 \pm 1}{2}$, i.e. $x = 2$ or $x = 3$.

Practise algebra and equations in the app →

2. Functions

What is it about?

A function gives one $y$-value for each $x$-value. In 1T you work with linear functions, quadratic functions and polynomials, and read off zeros, maxima and minima.

Concepts and formulas

• Linear function: $f(x) = ax + b$. The slope $a = \dfrac{y_2 - y_1}{x_2 - x_1}$, and the constant term $b$ is where the graph crosses the $y$-axis.
• Quadratic function: $f(x) = ax^2 + bx + c$. The graph is a parabola, opening upwards when $a > 0$ and downwards when $a < 0$.
• The axis of symmetry (and the $x$-value of the maximum or minimum): $x = -\dfrac{b}{2a}$.
• Zeros are where $f(x) = 0$, i.e. where the graph crosses the $x$-axis.

Example

$f(x) = x^2 - 4x + 1$ has its minimum at $x = -(-4)/2 = 2$, and $f(2) = 4 - 8 + 1 = -3$. The minimum point is $(2, -3)$.

Practise functions in the app →

3. Differentiation and rate of change

What is it about?

The derivative tells how fast a function is changing at a particular point: the slope of the tangent. The average rate of change is the slope between two points.

Concepts and formulas

• Average rate of change from $x_1$ to $x_2$: $\dfrac{f(x_2) - f(x_1)}{x_2 - x_1}$.
• Instantaneous rate of change at $x$: the derivative $f'(x)$.
• Differentiation rules for polynomials: $(x^n)' = n x^{n-1}$, $(k)' = 0$, $(k \cdot f)' = k f'$, $(f + g)' = f' + g'$.
• Where $f'(x) = 0$ the function may have a maximum or minimum. The sign of $f'$ tells whether the function is increasing ($f' > 0$) or decreasing ($f' < 0$).

Example

$f(x) = x^3 - 3x$ gives $f'(x) = 3x^2 - 3$. $f'(x) = 0$ when $x = \pm 1$.

Practise differentiation and rate of change in the app →

4. Trigonometry

What is it about?

Trigonometry connects angles and sides in triangles. In right-angled triangles you use sine, cosine and tangent. In other triangles you use the sine rule, the cosine rule and the area formula.

Concepts and formulas

• Right-angled triangle: $\sin v = \dfrac{\text{opposite}}{\text{hypotenuse}}$, $\cos v = \dfrac{\text{adjacent}}{\text{hypotenuse}}$, $\tan v = \dfrac{\text{opposite}}{\text{adjacent}}$.
• Pythagoras: $a^2 + b^2 = c^2$.
• Area formula: $A = \tfrac12 ab \sin C$.
• Sine rule: $\dfrac{\sin A}{a} = \dfrac{\sin B}{b} = \dfrac{\sin C}{c}$.
• Cosine rule: $c^2 = a^2 + b^2 - 2ab\cos C$.

Practise trigonometry in the app →

5. Inequalities and sign charts

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. Factorisation and polynomial division

What is it about?

To factorise is to write an expression as a product. This makes it easy to find zeros, simplify fractions and draw sign charts. Polynomial division is "long division" with polynomials.

Concepts and formulas

• Common factor: $6x^2 + 9x = 3x(2x + 3)$.
• Square identities backwards: $x^2 - 9 = (x + 3)(x - 3)$, $x^2 + 6x + 9 = (x + 3)^2$.
• With the zeros $x_1, x_2$: $ax^2 + bx + c = a(x - x_1)(x - x_2)$.
• Rational expressions are simplified by factorising numerator and denominator and cancelling factors (never terms): $\dfrac{x^2 - 4}{x - 2} = \dfrac{(x + 2)(x - 2)}{x - 2} = x + 2$ for $x \ne 2$.
• The remainder theorem: the remainder when $P(x)$ is divided by $(x - a)$ is $P(a)$. If $P(a) = 0$, the division is exact and $(x - a)$ is a factor.
• Polynomial division: divide the leading term, multiply back, subtract, and repeat.

Example

$P(x) = x^3 - 2x^2 - x + 2$. Try $x = 1$: $P(1) = 1 - 2 - 1 + 2 = 0$, so $(x - 1)$ is a factor. Polynomial division gives $P(x) : (x - 1) = x^2 - x - 2 = (x - 2)(x + 1)$. So $P(x) = (x - 1)(x - 2)(x + 1)$, with zeros $1$, $2$ and $-1$.

Practise factorisation and polynomial division in the app →

7. Powers, roots and logarithms

What is it about?

Powers are repeated multiplication, roots are powers with a fraction as exponent, and the logarithm answers the question "which exponent do I need?". With logarithms you can solve equations where $x$ is in the exponent.

Concepts and formulas

• $a^{\frac{1}{n}} = \sqrt[n]{a}$ and $a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m$. Example: $8^{\frac{2}{3}} = 2^2 = 4$.
• Root rules: $\sqrt{ab} = \sqrt{a}\sqrt{b}$, so $\sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}$.
• The common logarithm: $\lg x = y \iff 10^y = x$. So $\lg 1000 = 3$ and $\lg 0.01 = -2$.
• The logarithm rules: $\lg(ab) = \lg a + \lg b$, $\lg\dfrac{a}{b} = \lg a - \lg b$, $\lg a^n = n\lg a$.
• The equation $a \cdot b^x = c$ is solved like this: $b^x = \dfrac{c}{a}$, so $x = \dfrac{\lg(c/a)}{\lg b}$.

Example

Solve $500 \cdot 1.04^x = 800$. Divide by 500: $1.04^x = 1.6$. Take lg of both sides: $x \lg 1.04 = \lg 1.6$, so $x = \dfrac{\lg 1.6}{\lg 1.04} \approx 12.0$.

Practise powers, roots and logarithms in the app →

8. Exponential models

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 →