What is the gradient relationship for parallel and perpendicular lines?

Parallel lines have equal gradients ($m_1=m_2$). Perpendicular lines have gradients whose product is $-1$, i.e. $m_1 m_2=-1$, so $m_2=-\dfrac{1}{m_1}$.

Give the equation of a straight line in gradient–point form and in general form.

Gradient–point: $y-y_1=m(x-x_1)$. Slope–intercept: $y=mx+c$. General form: $ax+by+c=0$. The $y$-intercept is $c$ in $y=mx+c$.

State the equation of a circle with centre $(a,b)$ and radius $r$.

$(x-a)^{2}+(y-b)^{2}=r^{2}$. Expanded general form is $x^{2}+y^{2}+2gx+2fy+c=0$ with centre $(-g,-f)$ and radius $\sqrt{g^{2}+f^{2}-c}$.

State three key circle theorems relating angles.

The angle at the centre is twice the angle at the circumference on the same arc. The angle in a semicircle is $90^{\circ}$. Angles in the same segment are equal. Opposite angles of a cyclic quadrilateral sum to $180^{\circ}$.

What is the relationship between a tangent to a circle and the radius at the point of contact?

The tangent is perpendicular to the radius at the point of contact (angle $=90^{\circ}$). Also, two tangents drawn from an external point are equal in length.

State the laws of logarithms for $\log_a(xy)$, $\log_a\!\left(\dfrac{x}{y}\right)$ and $\log_a(x^{k})$.

$\log_a(xy)=\log_a x+\log_a y$; $\log_a\!\left(\dfrac{x}{y}\right)=\log_a x-\log_a y$; $\log_a(x^{k})=k\log_a x$. Also $\log_a a=1$ and $\log_a 1=0$.

What is the relationship between exponentials and logarithms, and the change of base formula?

$y=a^{x}\iff x=\log_a y$ (logarithm is the inverse of exponentiation). Change of base: $\log_a x=\dfrac{\log_b x}{\log_b a}$.

What are the key features of the exponential graph $y=a^{x}$ for $a>1$?

It passes through $(0,1)$, is always positive, increasing, with the $x$-axis ($y=0$) as a horizontal asymptote. For $0<a<1$ the curve is decreasing instead.

State the angle facts: angles on a straight line, around a point, and vertically opposite.

Angles on a straight line sum to $180^{\circ}$. Angles around a point sum to $360^{\circ}$. Vertically opposite angles are equal. The angles in a triangle sum to $180^{\circ}$.

What is the sum of interior angles of a polygon with $n$ sides, and the sum of exterior angles?

Sum of interior angles $=(n-2)\times 180^{\circ}$. Sum of exterior angles $=360^{\circ}$ for any convex polygon. Each interior angle of a regular polygon $=\dfrac{(n-2)\times 180^{\circ}}{n}$.

🇬🇧 Engineering and Science Admissions Test (ESAT) · flashcards

Engineering and Science Admissions Test (ESAT) Mathematics 1 (Core, Compulsory Module) Flashcards

Q: How many Mathematics 1 (Core, Compulsory Module) flashcards are in this Engineering and Science Admissions Test (ESAT) deck?

50 cards. This page previews 24 of them, sampled evenly across the deck so you can judge the difficulty before installing anything.

Q: Are these Engineering and Science Admissions Test (ESAT) flashcards free?

Yes. The preview here is free to read with no signup, and the full 50-card deck is free inside the Examius app.

Q: What do the Mathematics 1 (Core, Compulsory Module) cards cover?

They follow the Engineering and Science Admissions Test (ESAT) Mathematics 1 (Core, Compulsory Module) syllabus — 5 chapters and 20 topics — so the questions track what is actually examinable.

Q: How should I use these flashcards?

Read the syllabus first so you know the shape of the subject, then drill the deck. Examius schedules each card with spaced repetition, so cards you keep missing come back sooner and ones you know drift further apart.

50 question-and-answer cards covering Mathematics 1 (Core, Compulsory Module) as it is examined in Engineering and Science Admissions Test (ESAT). 24 of them are printed below, taken from across the deck — no signup, no paywall on the preview.

50Cards in deck

24Free preview

20Syllabus topics

~174Chars per answer

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24 sample cards from the Mathematics 1 (Core, Compulsory Module) deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

What is the gradient relationship for parallel and perpendicular lines?
Parallel lines have equal gradients ($m_1=m_2$). Perpendicular lines have gradients whose product is $-1$, i.e. $m_1 m_2=-1$, so $m_2=-\dfrac{1}{m_1}$.
Give the equation of a straight line in gradient–point form and in general form.
Gradient–point: $y-y_1=m(x-x_1)$. Slope–intercept: $y=mx+c$. General form: $ax+by+c=0$. The $y$-intercept is $c$ in $y=mx+c$.
State the equation of a circle with centre $(a,b)$ and radius $r$.
$(x-a)^{2}+(y-b)^{2}=r^{2}$. Expanded general form is $x^{2}+y^{2}+2gx+2fy+c=0$ with centre $(-g,-f)$ and radius $\sqrt{g^{2}+f^{2}-c}$.
State three key circle theorems relating angles.
The angle at the centre is twice the angle at the circumference on the same arc. The angle in a semicircle is $90^{\circ}$. Angles in the same segment are equal. Opposite angles of a cyclic quadrilateral sum to $180^{\circ}$.
What is the relationship between a tangent to a circle and the radius at the point of contact?
The tangent is perpendicular to the radius at the point of contact (angle $=90^{\circ}$). Also, two tangents drawn from an external point are equal in length.
State the laws of logarithms for $\log_a(xy)$, $\log_a\!\left(\dfrac{x}{y}\right)$ and $\log_a(x^{k})$.
$\log_a(xy)=\log_a x+\log_a y$; $\log_a\!\left(\dfrac{x}{y}\right)=\log_a x-\log_a y$; $\log_a(x^{k})=k\log_a x$. Also $\log_a a=1$ and $\log_a 1=0$.
What is the relationship between exponentials and logarithms, and the change of base formula?
$y=a^{x}\iff x=\log_a y$ (logarithm is the inverse of exponentiation). Change of base: $\log_a x=\dfrac{\log_b x}{\log_b a}$.
What are the key features of the exponential graph $y=a^{x}$ for $a>1$?
It passes through $(0,1)$, is always positive, increasing, with the $x$-axis ($y=0$) as a horizontal asymptote. For $0<a<1$ the curve is decreasing instead.
State the angle facts: angles on a straight line, around a point, and vertically opposite.
Angles on a straight line sum to $180^{\circ}$. Angles around a point sum to $360^{\circ}$. Vertically opposite angles are equal. The angles in a triangle sum to $180^{\circ}$.
What is the sum of interior angles of a polygon with $n$ sides, and the sum of exterior angles?
Sum of interior angles $=(n-2)\times 180^{\circ}$. Sum of exterior angles $=360^{\circ}$ for any convex polygon. Each interior angle of a regular polygon $=\dfrac{(n-2)\times 180^{\circ}}{n}$.
Give the area and circumference formulas for a circle of radius $r$.
Area $=\pi r^{2}$. Circumference $=2\pi r=\pi d$. Arc length (angle $\theta$ in radians) $=r\theta$; sector area $=\dfrac{1}{2}r^{2}\theta$.
State the volume and surface area formulas for a sphere, cylinder and cone.
Sphere: $V=\dfrac{4}{3}\pi r^{3}$, $A=4\pi r^{2}$. Cylinder: $V=\pi r^{2}h$, curved area $=2\pi r h$. Cone: $V=\dfrac{1}{3}\pi r^{2}h$, curved area $=\pi r l$ ($l$ slant height).
State Pythagoras' theorem and the three basic trigonometric ratios.
Pythagoras: $a^{2}+b^{2}=c^{2}$ (hypotenuse $c$). In a right triangle: $\sin\theta=\dfrac{\text{opp}}{\text{hyp}}$, $\cos\theta=\dfrac{\text{adj}}{\text{hyp}}$, $\tan\theta=\dfrac{\text{opp}}{\text{adj}}$.
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: $a^{2}=b^{2}+c^{2}-2bc\cos A$. Triangle area $=\dfrac{1}{2}ab\sin C$.
State the fundamental trig identities $\sin^{2}\theta+\cos^{2}\theta$ and $\tan\theta$ in terms of sine and cosine.
$\sin^{2}\theta+\cos^{2}\theta=1$ and $\tan\theta=\dfrac{\sin\theta}{\cos\theta}$. Dividing the first identity by $\cos^{2}\theta$ gives $1+\tan^{2}\theta=\sec^{2}\theta$.
Give the exact trig values for $30^{\circ}$, $45^{\circ}$ and $60^{\circ}$ (sine and cosine).
$\sin 30^{\circ}=\dfrac{1}{2}$, $\cos 30^{\circ}=\dfrac{\sqrt{3}}{2}$; $\sin 45^{\circ}=\cos 45^{\circ}=\dfrac{\sqrt{2}}{2}$; $\sin 60^{\circ}=\dfrac{\sqrt{3}}{2}$, $\cos 60^{\circ}=\dfrac{1}{2}$.
How do you convert between degrees and radians?
$\pi$ radians $=180^{\circ}$. To convert degrees to radians multiply by $\dfrac{\pi}{180}$; to convert radians to degrees multiply by $\dfrac{180}{\pi}$. For example $90^{\circ}=\dfrac{\pi}{2}$ rad.
What is the period and range of $\sin x$, $\cos x$ and $\tan x$?
$\sin x$ and $\cos x$ have period $2\pi$ and range $[-1,1]$. $\tan x$ has period $\pi$, range all real numbers, with asymptotes at $x=\dfrac{\pi}{2}+n\pi$.
How do you calculate the mean, median and mode of a data set?
Mean $=\dfrac{\sum x}{n}$ (sum divided by count). Median is the middle value when ordered (average of the two middle values if $n$ is even). Mode is the most frequently occurring value.
Define range, interquartile range (IQR), and how to find the quartiles.
Range $=\text{max}-\text{min}$. IQR $=Q_3-Q_1$ (upper quartile minus lower quartile), measuring the spread of the middle $50\%$ and resistant to outliers. Quartiles split the ordered data into four equal parts.
Give the formula for variance and standard deviation of a data set.
Variance $\sigma^{2}=\dfrac{\sum (x-\bar{x})^{2}}{n}=\dfrac{\sum x^{2}}{n}-\bar{x}^{2}$. Standard deviation $\sigma=\sqrt{\sigma^{2}}$, measuring average spread about the mean.
What does a box plot display, and how is an outlier commonly identified?
A box plot shows the minimum, $Q_1$, median, $Q_3$ and maximum (five-number summary). An outlier is typically a value below $Q_1-1.5\times\text{IQR}$ or above $Q_3+1.5\times\text{IQR}$.
State the probability rules for the complement, mutually exclusive events, and independent events.
Complement: $P(\text{not }A)=1-P(A)$. Mutually exclusive: $P(A\cup B)=P(A)+P(B)$. Independent: $P(A\cap B)=P(A)\times P(B)$. General addition: $P(A\cup B)=P(A)+P(B)-P(A\cap B)$.
State the conditional probability formula and what tree diagrams represent.
$P(A\mid B)=\dfrac{P(A\cap B)}{P(B)}$. A tree diagram shows successive outcomes with probabilities on each branch; multiply along branches for combined outcomes and add across separate branches for an 'or' result. Branches from a node sum to $1$.

What this deck covers

The Mathematics 1 (Core, Compulsory Module) deck follows the Engineering and Science Admissions Test (ESAT) Mathematics 1 (Core, Compulsory Module) syllabus — 5 chapters and 20 topics — so questions land on material that is genuinely examinable rather than trivia around it. That works out to roughly 10.0 cards per chapter.

Answers are written to be recallable, not just readable — averaging about 174 characters, which is long enough to carry the reasoning and short enough to say out loud.

A deck like this earns its keep on the second and third pass. Read the syllabus first so you know the shape of the subject, then use the cards to find the specific facts that have not stuck.

Mathematics 1 (Core, Compulsory Module) flashcards FAQ

How many Mathematics 1 (Core, Compulsory Module) flashcards are in this Engineering and Science Admissions Test (ESAT) deck?

50 cards. This page previews 24 of them, sampled evenly across the deck so you can judge the difficulty before installing anything.

Are these Engineering and Science Admissions Test (ESAT) flashcards free?

Yes. The preview here is free to read with no signup, and the full 50-card deck is free inside the Examius app.

What do the Mathematics 1 (Core, Compulsory Module) cards cover?

They follow the Engineering and Science Admissions Test (ESAT) Mathematics 1 (Core, Compulsory Module) syllabus — 5 chapters and 20 topics — so the questions track what is actually examinable.

How should I use these flashcards?

Read the syllabus first so you know the shape of the subject, then drill the deck. Examius schedules each card with spaced repetition, so cards you keep missing come back sooner and ones you know drift further apart.