BGCSE Mathematics 0563 cheat sheet

Place value: a digit’s value is digit × 10ⁿ, where n is its position relative to the decimal point (positive left, negative right).

Order of operations: brackets, indices, multiplication and division from left to right, then addition and subtraction from left to right.

Standard form: N = a × 10ⁿ with 1 ≤ a < 10 and n an integer; to add or subtract, convert to the same power of 10.

Divisibility: 2 if last digit even; 3 if digit sum divisible by 3; 5 if last digit 0 or 5; 9 if digit sum divisible by 9; 10 if last digit 0.

Every integer greater than 1 has a unique prime factorisation.

HCF is the product of common prime factors using the lowest power of each; LCM uses the highest power of every prime factor present.

Rounding: identify the required place; if the next digit is 5 or more, increase the required digit by 1, otherwise leave it.

For directed numbers: same signs give a positive product or quotient; different signs give a negative product or quotient.

To convert a recurring decimal to a fraction, subtract equations for the decimal and its shifted form to eliminate the repeating part.

A ratio a:b is equivalent to ka:kb; simplify by dividing both parts by their highest common factor.

To share Q in ratio a:b:c, total parts = a+b+c; shares are Q×a/(a+b+c), Q×b/(a+b+c), Q×c/(a+b+c).

Direct proportion: y = kx; inverse proportion: y = k/x, where k is a constant.

Percentage change = (new value − original value) ÷ original value × 100%.

Percentage of a quantity = (percentage ÷ 100) × quantity.

Simple interest I = PRT ÷ 100, where P is principal, R is annual rate percent, T is time in years.

Compound amount A = P(1 + r/100)ⁿ, where n is the number of compounding periods; compound interest = A − P.

Profit = selling price − cost price; percentage profit = (profit ÷ cost price) × 100%.

Discount = marked price − sale price; VAT or sales tax is calculated on the price before tax and added to it.

Expand a(b+c)=ab+ac and (a+b)(c+d)=ac+ad+bc+bd.

Factorise by taking out the highest common factor; a²−b²=(a+b)(a−b).

Quadratic formula: if ax²+bx+c=0 with a≠0, x=[−b±√(b²−4ac)]/(2a).

To solve a linear equation, perform the same operation on both sides and collect like terms.

Simultaneous linear equations: eliminate a variable by matching coefficients, or substitute one equation into the other.

Index laws: aᵐ×aⁿ=aᵐ⁺ⁿ; aᵐ÷aⁿ=aᵐ⁻ⁿ; (aᵐ)ⁿ=aᵐⁿ; a⁰=1 (a≠0).

Change of subject: undo operations in reverse order, keeping the equation balanced.

Multiplying or dividing an inequality by a negative number reverses the inequality sign.

Algebraic fractions: factorise numerators and denominators, then cancel common factors or use a common denominator.

Function notation: f(a) means substitute x=a into the rule for f(x).

Straight-line graph: y=mx+c has gradient m and y-intercept c.

Gradient between (x₁,y₁) and (x₂,y₂): m=(y₂−y₁)/(x₂−x₁).

Line through (x₁,y₁) with gradient m: y−y₁=m(x−x₁).

Parallel lines have equal gradients; perpendicular lines have gradients whose product is −1.

On a distance-time graph, gradient = speed; on a speed-time graph, gradient = acceleration and area = distance.

Quadratic y=ax²+bx+c is a parabola; its axis of symmetry is x=−b/(2a).

Domain is the set of input x-values; range is the set of output y-values.

The inverse function f⁻¹ reverses f; its graph is the reflection of y=f(x) in the line y=x.

Angles on a straight line sum to 180°; angles at a point sum to 360°; vertically opposite angles are equal.

Interior angles of a triangle sum to 180°; an exterior angle equals the sum of the two opposite interior angles.

Interior angle sum of an n-sided polygon is (n−2)×180°; exterior angles sum to 360°.

Circle: circumference = 2πr; area = πr².

Areas: triangle = ½bh; parallelogram = bh; trapezium = ½(a+b)h.

Volume of a prism = cross-sectional area × length; volume of a cylinder = πr²h.

Pythagoras: in a right triangle with hypotenuse c, a²+b²=c².

Similar figures: corresponding angles are equal; lengths scale by k, areas by k², volumes by k³.

Arc length = (θ/360)×2πr; sector area = (θ/360)×πr².

SOHCAHTOA: sinθ=opposite/hypotenuse; cosθ=adjacent/hypotenuse; tanθ=opposite/adjacent.

Sine rule: a/sinA = b/sinB = c/sinC.

Cosine rule: a²=b²+c²−2bc cosA.

Area of triangle = ½ab sinC.

Bearings are measured clockwise from north and written as three digits, e.g. 045°.

Back bearing = forward bearing +180° if less than 180°, or −180° if 180° or more.

Angle of elevation is above the horizontal; angle of depression is below the horizontal.

Exact values: sin30°=½, cos60°=½, tan45°=1, sin45°=cos45°=√2/2.

Identity: sin²θ+cos²θ=1.

Column vector (x,y) represents x units horizontally and y units vertically; magnitude = √(x²+y²).

Add or subtract vectors by adding or subtracting corresponding components; multiply by scalar k by multiplying each component by k.

Position vector AB = b − a, where a and b are position vectors of A and B from the origin.

Translation moves every point by the same column vector and does not change shape or size.

Reflection uses a mirror line; rotation uses a centre, angle and direction; enlargement uses a centre and scale factor.

Enlargement with scale factor k: lengths ×k, area ×k², volume ×k³.

Rotation 90° clockwise about origin: (x,y)→(y,−x); 90° anticlockwise: (x,y)→(−y,x); 180°: (x,y)→(−x,−y).

Reflections: x-axis (x,y)→(x,−y); y-axis (x,y)→(−x,y); y=x→(y,x); y=−x→(−y,−x).

A transformation is an isometry if it preserves lengths and angles; translations, rotations and reflections are isometries.

Mean = sum of values ÷ number of values.

Median = middle value of ordered data; for an even number of values, average the two middle values.

Mode = most frequent value; a data set may have no mode or more than one mode.

Range = largest value − smallest value.

Frequency-table mean = Σfx ÷ Σf, where x is each value or midpoint and f is its frequency.

Pie-chart sector angle = (frequency ÷ total frequency) × 360°.

Histogram with unequal class widths: frequency density = frequency ÷ class width; bar area represents frequency.

Cumulative frequency curve: median at 50%, lower quartile at 25%, upper quartile at 75%; interquartile range = Q3−Q1.

Scatter graph: positive correlation slopes upward, negative slopes downward, zero correlation shows no linear trend.

Probability P(E) = number of favourable outcomes ÷ total number of equally likely outcomes.

0 ≤ P(E) ≤ 1; P(not E)=1−P(E).

Mutually exclusive events: P(A or B)=P(A)+P(B).

Independent events: P(A and B)=P(A)×P(B).

Tree diagrams: multiply probabilities along a branch and add probabilities of mutually exclusive branches.

Experimental probability = relative frequency = number of successes ÷ number of trials.

Expected frequency = probability of event × number of trials.

Without replacement, the denominator and favourable counts change after each selection.

The sum of probabilities of all mutually exclusive outcomes in a sample space is 1.