BioMedical Admissions Test (BMAT) Section 2: Scientific Knowledge and Applications — Mathematics Syllabus

How do you rationalise the denominator of $\dfrac{1}{\sqrt{3}}$?

Multiply top and bottom by $\sqrt{3}$: $\dfrac{1}{\sqrt{3}} \times \dfrac{\sqrt{3}}{\sqrt{3}} = \dfrac{\sqrt{3}}{3}$.

What is the formula for the $n$th term of an arithmetic sequence?

$a_{n} = a + (n-1)d$, where $a$ is the first term and $d$ is the common difference.

What is the formula for the $n$th term of a geometric sequence?

$a_{n} = a r^{\,n-1}$, where $a$ is the first term and $r$ is the common ratio.

Give the formula for the sum of the first $n$ terms of an arithmetic series.

$S_{n} = \dfrac{n}{2}\left(2a + (n-1)d\right) = \dfrac{n}{2}(a + l)$, where $l$ is the last term.

What is the formula for the $n$th triangular number?

$T_{n} = \dfrac{n(n+1)}{2}$. For example $T_{4} = \dfrac{4\times5}{2} = 10$.

State the area formulas for a triangle, a parallelogram, and a trapezium.

Triangle: $\tfrac{1}{2}bh$. Parallelogram: $bh$. Trapezium: $\tfrac{1}{2}(a+b)h$, where $a,b$ are the parallel sides.

Give the formulas for the circumference and area of a circle of radius $r$.

Circumference $= 2\pi r = \pi d$; Area $= \pi r^{2}$.

What are the arc length and sector area for an angle $\theta$ (in degrees) in a circle of radius $r$?

Arc length $= \dfrac{\theta}{360}\times 2\pi r$; Sector area $= \dfrac{\theta}{360}\times \pi r^{2}$.

Give the volume formulas for a cuboid, a cylinder, and a sphere.

Cuboid: $lwh$. Cylinder: $\pi r^{2} h$. Sphere: $\dfrac{4}{3}\pi r^{3}$.

Give the volume formulas for a cone and a pyramid.

Cone: $\dfrac{1}{3}\pi r^{2} h$. Pyramid: $\dfrac{1}{3} \times \text{base area} \times h$.

What is the curved surface area of a cylinder, and the total surface area of a sphere?

Cylinder curved surface area $= 2\pi r h$ (total $= 2\pi r h + 2\pi r^{2}$). Sphere surface area $= 4\pi r^{2}$.

What is the curved (lateral) surface area of a cone with slant height $l$?

Curved surface area $= \pi r l$; total surface area $= \pi r l + \pi r^{2}$.

State Pythagoras' theorem for a right-angled triangle.

$a^{2} + b^{2} = c^{2}$, where $c$ is the hypotenuse (the side opposite the right angle) and $a,b$ are the other two sides.

Define $\sin$, $\cos$, and $\tan$ for a right-angled triangle (SOHCAHTOA).

$\sin\theta = \dfrac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta = \dfrac{\text{adjacent}}{\text{hypotenuse}}$, $\tan\theta = \dfrac{\text{opposite}}{\text{adjacent}}$.

Give the exact values of $\sin$, $\cos$, and $\tan$ for $30^{\circ}$, $45^{\circ}$, and $60^{\circ}$.

$\sin30^{\circ}=\tfrac{1}{2},\,\cos30^{\circ}=\tfrac{\sqrt{3}}{2},\,\tan30^{\circ}=\tfrac{1}{\sqrt{3}}$; $\sin45^{\circ}=\cos45^{\circ}=\tfrac{1}{\sqrt{2}},\,\tan45^{\circ}=1$; $\sin60^{\circ}=\tfrac{\sqrt{3}}{2},\,\cos60^{\circ}=\tfrac{1}{2},\,\tan60^{\circ}=\sqrt{3}$.

State the sine rule and the cosine rule for a general triangle.

Sine rule: $\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}$. Cosine rule: $a^{2} = b^{2} + c^{2} - 2bc\cos A$.

What is the area of a triangle given two sides and the included angle?

$\text{Area} = \dfrac{1}{2}ab\sin C$, where $C$ is the angle between sides $a$ and $b$.

What do the interior angles of a triangle and of a quadrilateral sum to?

Triangle: $180^{\circ}$. Quadrilateral: $360^{\circ}$. In general an $n$-sided polygon has interior angle sum $(n-2)\times 180^{\circ}$.

What is the size of each interior and exterior angle of a regular $n$-sided polygon?

Each exterior angle $= \dfrac{360^{\circ}}{n}$; each interior angle $= 180^{\circ} - \dfrac{360^{\circ}}{n}$. Exterior angles always sum to $360^{\circ}$.

State the angle facts for parallel lines cut by a transversal.

Corresponding angles are equal; alternate angles are equal; co-interior (allied) angles sum to $180^{\circ}$.

Name two key circle theorems involving the angle at the centre and the angle in a semicircle.

The angle at the centre is twice the angle at the circumference on the same arc. The angle in a semicircle is $90^{\circ}$ (subtended by a diameter).

How do you calculate the mean, median, and mode of a data set?

Mean $= \dfrac{\text{sum of values}}{\text{number of values}}$. Median = middle value when data are ordered. Mode = most frequently occurring value.

What is the range, and what is the interquartile range (IQR)?

Range $= \text{highest} - \text{lowest}$. IQR $= Q_{3} - Q_{1}$ (upper quartile minus lower quartile); it measures spread of the middle $50\%$ and is resistant to outliers.

How do you estimate the mean from a grouped frequency table?

Use midpoints $x$ of each class: estimated mean $= \dfrac{\sum f x}{\sum f}$, where $f$ is the frequency of each class.

On a box plot, what five values are shown?

Minimum, lower quartile $Q_{1}$, median $Q_{2}$, upper quartile $Q_{3}$, and maximum. The box spans $Q_{1}$ to $Q_{3}$ with the median marked inside.

What does a cumulative frequency graph let you read off, and how do you find the median from it?

It gives the running total of frequencies. The median is read at the $\dfrac{n}{2}$ value, $Q_{1}$ at $\dfrac{n}{4}$, and $Q_{3}$ at $\dfrac{3n}{4}$, read across to the curve and down to the data axis.

In a histogram with unequal class widths, what represents frequency?

The area of each bar represents frequency; the vertical axis is frequency density, where $\text{frequency density} = \dfrac{\text{frequency}}{\text{class width}}$.

State the basic probability formula for equally likely outcomes and the range a probability can take.

$P(\text{event}) = \dfrac{\text{number of favourable outcomes}}{\text{total number of outcomes}}$, with $0 \leq P \leq 1$. $P(\text{not } A) = 1 - P(A)$.

State the AND and OR rules for combining probabilities.

For independent events: $P(A \text{ and } B) = P(A)\times P(B)$. For mutually exclusive events: $P(A \text{ or } B) = P(A) + P(B)$. Generally $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.

When multiplying or dividing without a calculator, how can powers of $10$ and estimation help?

Separate significant figures from powers of ten, e.g. $0.0006 \times 4000 = (6\times10^{-4})(4\times10^{3}) = 24\times10^{-1} = 2.4$. Round to $1$ significant figure first to sense-check the magnitude of an answer.