Mathematics S2: free practice, theory and problems

1. Sequences and series

🔗 Shared curriculum with R2. The questions are the same, but your progress counts separately for each subject.

What is it about?

A sequence is a list of numbers. A series is the sum of the terms of a sequence.

Concepts and formulas

• Arithmetic sequence (constant difference $d$): $a_n = a_1 + (n-1)d$, sum $S_n = \dfrac{n(a_1 + a_n)}{2}$.
• Geometric sequence (constant ratio $k$): $a_n = a_1 k^{n-1}$, sum $S_n = a_1\dfrac{k^n - 1}{k - 1}$.
• An infinite geometric series converges when $|k| < 1$, with sum $S = \dfrac{a_1}{1 - k}$.

Practise sequences and series in the app →

2. Savings, loans and present value

What is it about?

Savings, loans and investments are geometric series in disguise. Each payment grows (or is discounted) by the same factor, so the sum becomes a geometric series.

Concepts and formulas

• Growth factor per period: $k = 1 + r$, where $r$ is the interest rate per period as a decimal.
• Savings (future value): $a$ NOK is deposited at the end of each period for $n$ periods. Right after the last deposit the balance is $S = a \cdot \dfrac{k^n - 1}{k - 1}$.
• Present value: an amount $B$ in $n$ periods is worth $\dfrac{B}{k^n}$ today. The present value of $n$ equal amounts $a$ (the first in one period) is $N = a \cdot \dfrac{1 - k^{-n}}{r}$.
• Annuity loan: equal instalments. The loan $L$ is the present value of all instalments, so $a = \dfrac{L \cdot r}{1 - (1 + r)^{-n}}$.
• Serial loan: equal repayments $L/n$. Interest is charged on the remaining loan, so the instalment decreases over time.

Example

A loan of 1,000,000 NOK, 5 % interest, 20 annual instalments (annuity): $a = \dfrac{1\,000\,000 \cdot 0.05}{1 - 1.05^{-20}} \approx 80\,243$ NOK per year. As a serial loan, the first instalment is $50\,000 + 50\,000 = 100\,000$ NOK, and it decreases by 2500 NOK each year.

Practise savings, loans and present value in the app →

3. Differentiation rules in practice

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

What is it about?

Here you practise the rules you need to differentiate everything in R1: powers, $e^x$, $\ln x$, and combinations with the product, quotient and chain rules.

Concepts and formulas

• $(x^n)' = nx^{n-1}$, $(e^x)' = e^x$, $(\ln x)' = \dfrac{1}{x}$.
• The chain rule: $\big(e^{kx}\big)' = ke^{kx}$, $\big(\ln u\big)' = \dfrac{u'}{u}$, $\big((ax + b)^n\big)' = na(ax + b)^{n-1}$.
• The product rule: $(uv)' = u'v + uv'$. Example: $(x e^x)' = e^x + xe^x = (1 + x)e^x$.
• The quotient rule: $\left(\dfrac{u}{v}\right)' = \dfrac{u'v - uv'}{v^2}$. Example: $\left(\dfrac{x}{x + 1}\right)' = \dfrac{1}{(x + 1)^2}$.

Method

1. Look at the structure: is it a product, a quotient or a function of a function?
2. Choose the rule and write down $u$, $v$ (or the inner function $u$) and their derivatives.
3. Put it together and simplify. Insert numbers at the very end.

Practise differentiation rules in practice in the app →

4. Cost, revenue and profit

🔗 Shared curriculum with S1. The questions are the same, but your progress counts separately for each subject.

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 →

5. Probability distributions

What is it about?

A probability distribution tells you how likely each possible outcome is. With the expected value and the standard deviation you describe the "centre" and the "spread". The two most important distributions are the binomial distribution and the normal distribution.

Concepts and formulas

• Expected value: $E(X) = \sum x \cdot P(X = x)$. For a die: $E(X) = 3.5$.
• Variance $\text{Var}(X) = \sum (x - \mu)^2 P(X = x)$, standard deviation $\sigma = \sqrt{\text{Var}(X)}$.
• Binomial distribution ($n$ independent trials, probability $p$ of success): $P(X = k) = \binom{n}{k}p^k(1 - p)^{n - k}$, $E(X) = np$, $\sigma = \sqrt{np(1 - p)}$.
• The normal distribution $N(\mu, \sigma)$: bell-shaped. About 68 % lies within $\mu \pm \sigma$, 95 % within $\mu \pm 2\sigma$ and 99.7 % within $\mu \pm 3\sigma$.
• Standardisation: $Z = \dfrac{X - \mu}{\sigma}$, and $P(X < a) = \Phi\!\left(\dfrac{a - \mu}{\sigma}\right)$. Key values: $\Phi(1.645) = 0.95$, $\Phi(1.96) = 0.975$.
• The central limit theorem: sums and averages of many independent variables become approximately normally distributed.

Example

The height of 18-year-olds is $N(175, 7)$ cm. The share below 182 cm is $\Phi\!\left(\dfrac{182 - 175}{7}\right) = \Phi(1) \approx 0.841$.

Practise probability distributions in the app →

6. Hypothesis testing

What is it about?

A hypothesis test decides whether data give enough reason to reject a claim. Is a coin biased? Does a new medicine work? Has a new teaching method raised the average grade?

Concepts and formulas

• The null hypothesis $H_0$: "no change", for example $p = 0.5$.
• The alternative hypothesis $H_1$: what we want to show, for example $p > 0.5$ (one-sided) or $p \ne 0.5$ (two-sided).
• Significance level $\alpha$: usually 5 %. It is the probability of rejecting $H_0$ even though it is true (type I error).
• P-value: the probability of a result at least as extreme as the one observed, **given that $H_0$ is true**.
• Decision: if the p-value is less than $\alpha$, we reject $H_0$. Otherwise we do not have enough evidence (that does not mean $H_0$ is proved).
• Test statistic for a mean: $Z = \dfrac{\bar x - \mu_0}{\sigma/\sqrt{n}}$. For a proportion: $Z = \dfrac{X - np_0}{\sqrt{np_0(1 - p_0)}}$.
• One-sided test at the 5 % level: reject when $Z > 1.645$. Two-sided: when $|Z| > 1.96$.

Example

A coin gives 60 heads in 100 tosses. $H_0: p = 0.5$, $H_1: p > 0.5$. $Z = \dfrac{60 - 50}{\sqrt{100 \cdot 0.5 \cdot 0.5}} = \dfrac{10}{5} = 2$. The p-value is $1 - \Phi(2) \approx 0.023 < 0.05$, so we reject $H_0$: the coin appears to be biased.

Practise hypothesis testing in the app →