Edexcel iGCSE Higher Tier Mathematics Revision

Comprehensive practice flashcards covering key rules, formulas, procedures, and theorems across all topics of the Edexcel iGCSE Higher Tier Mathematics syllabus.

How do you split a number such as $240$ into its prime factors using a factor tree?

Repeatedly divide the number by prime factors (circling each prime found) until only primes remain. For $240$, this produces $2^4 \times 3 \times 5$.

How are the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) calculated from a Venn diagram of prime factors?

The HCF is calculated by multiplying all the numbers in the intersection (middle) of the Venn diagram. The LCM is calculated by multiplying all the numbers in the entire Venn diagram.

What rule should be used when dividing one fraction by another?

Use "keep, flip, change": keep the first fraction the same, flip the second fraction upside down (take its reciprocal), and change the division sign to multiplication.

In probability, what arithmetic operation is performed when encountering the word "or" versus the word "and"?

When you see the word "or", add the probabilities; when you see the word "and", multiply the probabilities.

How do you calculate an estimate or expected frequency of an event occurring over multiple trials?

Multiply the total number of times the event takes place by the probability of that particular outcome.

How can you determine if a given number, such as $-30$, is a term in an arithmetic sequence with $n\text{th}$ term $-4n + 39$ or $-4n + 35$?

Set the $n\text{th}$ term expression equal to the number and solve for $n$. If $n$ is a whole number (integer), the value is in the sequence; if $n$ is not a whole number, it is not.

What three details must always be stated to fully describe an enlargement?

1. State that the transformation is an enlargement; 2. State the center of enlargement as coordinates; 3. State the scale factor of enlargement.

In a column translation vector, what do the top and bottom values represent?

The top value indicates horizontal movement (right if positive, left if negative), and the bottom value indicates vertical movement (up if positive, down if negative).

How is the interquartile range (IQR) calculated from an ordered dataset?

Subtract the lower quartile ($Q_1$) from the upper quartile ($Q_3$): $\text{IQR} = Q_3 - Q_1$.

When finding the estimated mean from a grouped frequency table, what common error must be avoided regarding the denominator?

You must divide the total of $\text{midpoint} \times \text{frequency}$ by the total frequency, never by the number of rows/classes in the table.

What condition must the coefficient $A$ satisfy for a number to be in standard form $A \times 10^n$?

$A$ must be greater than or equal to $1$ and strictly less than $10$ ($1 \le A < 10$), and $n$ must be an integer.

What formula is used to calculate percentage profit?

$$\text{Percentage Profit} = \frac{\text{Change}}{\text{Original}} \times 100$$

How do you calculate the original price in a reverse percentage problem where an item has been reduced by $20\%$ to a sale price of £$4.80$?

Calculate the multiplier ($\frac{100 - 20}{100} = 0.8$) and divide the sale price by this multiplier: $4.80 \div 0.8 = £6.00$.

What is the general formula for compound interest or repeated percentage change?

$$\text{Final Amount} = \text{Initial Amount} \times (\text{Multiplier})^{\text{Number of Years}}$$

How do you factorize an algebraic expression that is a difference of two squares, such as $x^2 - 16$?

Take the square root of each term and write the product of their sum and difference: $(x - 4)(x + 4)$, using the identity $a^2 - b^2 = (a - b)(a + b)$.

What is the key recommendation for solving linear equations where the variable appears on both sides?

Always eliminate or remove the smallest value of the variable first from both sides.

On a number line, what is the graphical difference between an inequality involving strictly less than ($<$) versus less than or equal to ($\le$)?

A hollow (unshaded) circle is used when the endpoint is not included ($<$ or $>$), while a solid (filled-in) circle is used when the endpoint is included ($\le$ or $\ge$).

When solving simultaneous equations by elimination, how do you decide whether to add or subtract the equations?

If the signs of the matching coefficients are different, add the equations; if the signs of the matching coefficients are the same, subtract the equations.

How is pressure calculated from force and area using formula triangles?

$\text{Pressure} = \frac{\text{Force}}{\text{Area}}$, measured in units such as $\text{N}\,\text{m}^{-2}$.

What are the four parallel line angle rules and their characteristic shapes?

1. Vertically opposite angles are equal (X-shape); 2. Alternate angles are equal (Z-shape); 3. Corresponding angles are equal (F-shape); 4. Co-interior angles add up to $180^\circ$ (C-shape).

What are the formulas connecting the number of sides $n$, exterior angles, and interior angles of a regular polygon?

$\text{Exterior Angle} = \frac{360^\circ}{n}$ and $\text{Interior Angle} + \text{Exterior Angle} = 180^\circ$.

What are the three essential conventions that must always be followed when giving bearings?

1. Always measured from North; 2. Always measured in a clockwise direction; 3. Always written using three figures (e.g., $045^\circ$ or $150^\circ$).

Which trigonometric ratio from SOHCAHTOA is used to find the opposite side when the adjacent side and an angle are known?

The tangent ratio (TOA): $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$, so $\text{Opposite} = \tan(\theta) \times \text{Adjacent}$.

What is the formula for the area of a trapezium?

$\text{Area} = \frac{1}{2}(a + b)h$, where $a$ and $b$ are the lengths of the parallel sides and $h$ is the perpendicular height between them.

How do you calculate the volume of any uniform prism?

Multiply the cross-sectional area by the depth, length, or height that the shape is extended into: $\text{Volume} = \text{Cross-Sectional Area} \times \text{Length}$.

Which points are plotted on the horizontal ($x$) axis when constructing a cumulative frequency diagram?

Always plot the upper endpoints (upper boundaries) of each class interval against the running cumulative frequency totals.

What is the formula for the gradient $m$ of a straight line connecting coordinates $(x_1, y_1)$ and $(x_2, y_2)$?

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

What is the quadratic formula used to solve equations of the form $ax^2 + bx + c = 0$?

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

How do you calculate the upper bound of a quantity $c$ defined by subtraction, $c = b - a$?

Take the upper bound of the starting value and subtract the lower bound of the subtracted value: $c_{\text{upper}} = b_{\text{upper}} - a_{\text{lower}}$.

What circle theorem relates the angle subtended at the center of a circle to the angle at the circumference from the same arc?

The angle subtended at the center of a circle is twice the angle subtended at the circumference ($\text{angle at center} = 2 \times \text{angle at circumference}$).

What property is true regarding opposite angles in a cyclic quadrilateral?

Opposite angles in a cyclic quadrilateral always sum to $180^\circ$.

What is the intersecting chords theorem for two line segments intersecting inside a circle?

The product of the two parts of one chord is equal to the product of the two parts of the other chord: $a \times b = c \times d$.

What is the recommended strategy when filling in values in a three-set Venn diagram?

Always start in the very center with the intersection of all three sets, and subtract already-counted elements as you move outwards to double intersections and single sets.

What four systematic steps are used to rearrange a formula to make $x$ the subject when $x$ appears twice?

1. Multiply both sides by any denominator to remove fractions; 2. Expand all brackets; 3. Move all terms containing $x$ to one side and all other terms to the opposite side; 4. Factorize out $x$ and divide through by the remaining bracketed factor.

In algebraic proof, how are even, odd, and consecutive integers defined algebraically?

An even number is defined as $2n$, an odd number as $2n + 1$, and consecutive integers as $n, n + 1, n + 2, \dots$ (where $n$ is an integer).

What are the standard algebraic equations for direct proportion and inverse proportion between variables $y$ and $x$?

Direct proportion is expressed as $y = kx$, and inverse proportion is expressed as $y = \frac{k}{x}$, where $k$ is a constant of proportionality.

In a histogram, what do the area of each bar and the vertical axis represent?

The area of each bar represents the frequency ($\text{Frequency} = \text{Area}$), and the vertical axis represents the frequency density ($\text{Frequency Density} = \frac{\text{Frequency}}{\text{Class Width}}$).

If two mathematically similar solids have a linear length scale factor of $k$, what are their area and volume scale factors?

Their area scale factor is $k^2$, and their volume scale factor is $k^3$.

How does the transformation $y = f(x - a)$ affect the graph of $y = f(x)$?

It translates the graph horizontally to the right by $a$ units (affecting the $x$-coordinates by adding $a$).

How does the transformation $y = f(-x)$ affect the graph of $y = f(x)$?

It reflects the graph across the vertical $y$-axis (negating all $x$-coordinates while keeping $y$-coordinates the same).

How do you rationalize the denominator of a fraction containing a binomial surd, such as $\frac{5 + 2\sqrt{3}}{2 + \sqrt{3}}$?

Multiply both numerator and denominator by the conjugate of the denominator with its middle sign reversed ($2 - \sqrt{3}$), causing the surds in the denominator to cancel out.

What are the formulas for the arc length and area of a sector of a circle with angle $\theta$ and radius $r$?

$\text{Arc Length} = \frac{\theta}{360} \times 2\pi r$ and $\text{Sector Area} = \frac{\theta}{360} \times \pi r^2$.

When should the sine rule be used instead of the cosine rule in a non-right-angled triangle?

Use the sine rule when you know a side and its directly opposite angle; use the cosine rule when you do not have an angle and its opposite side.

What is the trigonometric formula for the area of a non-right-angled triangle given two sides and the included angle?

$\text{Area} = \frac{1}{2}ab\sin(C)$, where $a$ and $b$ are two adjacent sides and $C$ is the angle between them.

How do you complete the square for a quadratic expression of the form $x^2 + bx + c$?

Halve the $b$ term to form $\left(x + \frac{b}{2}\right)^2$, subtract the square of that halved term, and add $c$: $\left(x + \frac{b}{2}\right)^2 - \left(\frac{b}{2}\right)^2 + c$.

What is the relationship between the gradient of a straight line $m_1$ and the gradient of a line perpendicular to it $m_2$?

The perpendicular gradient is the negative reciprocal of the original gradient: $m_2 = -\frac{1}{m_1}$.

What procedure is used to find the inverse function $f^{-1}(x)$ from a function $f(x)$?

1. Replace $f(x)$ with $y$; 2. Swap the positions of $x$ and $y$; 3. Rearrange the equation to make $y$ the subject; 4. Replace $y$ with $f^{-1}(x)$.

What rule is used to find the derivative $\frac{dy}{dx}$ of a term $a x^n$ with respect to $x$?

Multiply the coefficient by the current power, and reduce the power by one: $\frac{d}{dx}(a x^n) = an x^{n-1}$. Any constant term differentiates to zero.

What mathematical condition identifies turning points (stationary points) on a curve $y = f(x)$?

Stationary points occur where the gradient function is zero: $\frac{dy}{dx} = 0$.

In kinematics using differentiation, how are velocity and acceleration obtained from displacement $s$ with respect to time $t$?

Velocity is the first derivative of displacement ($v = \frac{ds}{dt}$), and acceleration is the first derivative of velocity or second derivative of displacement ($a = \frac{dv}{dt} = \frac{d^2s}{dt^2}$).

What are the standard formulas for the $n\text{th}$ term and the sum of the first $n$ terms ($S_n$) of an arithmetic series?

The $n\text{th}$ term is $a_n = a + (n - 1)d$, and the sum is $S_n = \frac{n}{2}[2a + (n - 1)d]$, where $a$ is the first term and $d$ is the common difference.