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GCSE Higher Paper 1 (Non-Calculator)
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Trigonometry
Exact trigonometric values
\[ \sin 60° = \frac{\sqrt{3}}{2} \]
Recall exact trig values for standard angles.
Number – Indices
Index law (multiplication)
\[ 5^4 \times 5^3 = 5^7 \]
Add indices when multiplying same base.
Multiple index laws
\[ \frac{2^6 \times 2^{-2}}{2^3} \]
Apply multiple index laws to simplify.
Negative indices
\[ 3^{-2} = \frac{1}{9} \]
Write negative indices as fractions.
Algebraic indices
\[ \frac{12x^5y^2}{4x^2y^4} \]
Simplify algebraic fractions using index laws.
Number – Fractions
Subtracting mixed numbers
\[ 4\frac{5}{6} – 2\frac{1}{4} \]
Subtract mixed numbers with different denominators.
Dividing mixed numbers
\[ 2\frac{1}{4} \div 9 \]
Divide a mixed number by an integer.
Multiplying fractions
\[ 1\frac{1}{2} \times \frac{2}{5} \]
Multiply a mixed number by a proper fraction.
Number – Decimals
Decimal division
\[ 7.2 \div 0.8 \]
Divide decimals without a calculator.
Decimal multiplication
\[ 0.03 \times 0.4 \]
Multiply decimals using place value.
Number – Primes
Prime factorisation
\[ 120 = 2^3 \times 3 \times 5 \]
Express as product of prime powers.
Algebra – Sequences
nth term of arithmetic sequence
\[ 5, 11, 17, 23, … \rightarrow 6n – 1 \]
Find the nth term formula.
Algebra – Expanding
Expanding double brackets
\[ (3x – 2)(x + 4) \]
Expand and simplify using FOIL.
Algebra – Factorising
Factorising ax² + bx + c
\[ 2x^2 + 5x – 3 \]
Factorise quadratics with a ≠ 1.
Algebra – Equations
Simultaneous equations
\[ 2x + 3y = 12, \; 5x – 3y = 9 \]
Solve by elimination.
Algebraic fractions equation
\[ \frac{4}{x} + \frac{3}{x+1} = 2 \]
Solve equations with algebraic fractions.
Algebra – Inequalities
Quadratic inequalities
\[ x^2 – 4x – 5 < 0 \]
Solve and express as compound inequality.
Algebra – Surds
Simplifying surds
\[ \sqrt{48} = 4\sqrt{3} \]
Express surds in simplest form.
Rationalising simple denominators
\[ \frac{8}{\sqrt{2}} = 4\sqrt{2} \]
Rationalise single-term surd denominators.
Rationalising with conjugates
\[ \frac{6}{2 + \sqrt{3}} \]
Rationalise using the conjugate.
Algebra – Proof
Algebraic proof
\[ (n+3)^2 – (n+1)^2 \]
Prove divisibility statements.
Algebra – Quadratics
Completing the square
\[ x^2 + 8x + 10 \rightarrow (x+4)^2 – 6 \]
Write in completed square form.
Ratio
Combining ratios
\[ a:b = 3:4, \; b:c = 2:5 \]
Combine two ratios into a:b:c.
Reverse percentages
\[ \text{After 20\% increase: 84} \]
Find the original value.
Proportion
\[ y \propto x \text{ or } y \propto \frac{1}{x} \]
Apply direct or inverse proportion.
Combined ratio and percentage
\[ \text{Boys:Girls = 2:3, 60\% of…} \]
Multi-step ratio and percentage problems.
Geometry – Angles
Missing angle in polygon
\[ \text{Pentagon: sum = 540°} \]
Use angle sum formula to find missing angle.
Geometry – Circles
Circle theorem application
\[ \text{Centre} = 2 \times \text{circumference} \]
Apply circle theorems to find angles.
Sector area
\[ A = \frac{\theta}{360} \times \pi r^2 \]
Calculate sector area in terms of π.
Geometry – 3D
Prism volume
\[ V = \text{Area} \times \text{length} \]
Calculate volume using cross-section area.
Cylinder surface area
\[ 2\pi r^2 + 2\pi rh \]
Calculate total surface area in terms of π.
Geometry – Trigonometry
Cosine rule for sides
\[ c^2 = a^2 + b^2 – 2ab\cos C \]
Find a side using exact trig values.
Sine rule for sides
\[ \frac{a}{\sin A} = \frac{b}{\sin B} \]
Find a side using the sine rule.
Solving trig equations
\[ \cos x = 0.5, \; 0° \leq x \leq 360° \]
Find all solutions in a given range.
Geometry – Transformations
Describing rotations
\[ \text{Rotation, 90° CW, centre (a,b)} \]
Fully describe a rotation transformation.
Probability
Venn diagram probability
\[ P(A \text{ but not } B) \]
Calculate probabilities from Venn diagrams.
Tree diagram probability
\[ P(\text{exactly one}) \]
Calculate combined event probabilities.
Relative frequency
\[ P \approx \frac{\text{freq}}{\text{trials}} \]
Estimate probability from experimental data.
Statistics
Reading histograms
\[ \text{Freq} = \text{density} \times \text{width} \]
Calculate frequency from frequency density.
Graphs
Equation of a line
\[ y = mx + c \]
Find the equation given two points.
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