Pure Mathematics 1

Worked examples (5)

• 1Rationalise the denominator (4 marks, P1)
• 2Fractional and negative indices (3 marks, P1)
• 3Expand three brackets (3 marks, P1)
• 4Factorise a cubic using the factor theorem (5 marks, P1)
• 5Simplify an algebraic fraction (4 marks, P1)

Worked examples (5)

• 1Completing the square (4 marks, P1)
• 2Discriminant — find the range of values (5 marks, P1)
• 3Solve using the quadratic formula — exact form (3 marks, P1)
• 4Hidden quadratic — substitution (5 marks, P1)
• 5Completing the square to sketch a parabola (6 marks, P1)

Worked examples (5)

• 1Find the equation of a line through two points (4 marks, P1)
• 2Equation of perpendicular line (5 marks, P1)
• 3Distance and midpoint (4 marks, P1)
• 4Show two lines are parallel and find distance (5 marks, P1)
• 5Modelling with a linear equation (6 marks, P1)

Worked examples (5)

• 1Cosine rule — find a side (4 marks, P1)
• 2Sine rule — find an angle (4 marks, P1)
• 3Area of triangle = $\\tfrac{1}{2}ab \\sin C$ (3 marks, P1)
• 4Ambiguous case of the sine rule (5 marks, P1)
• 5Sine graph (4 marks, P1)

Worked examples (5)

• 1Solve a pair of linear simultaneous equations (3 marks, P1)
• 2Linear-quadratic simultaneous equations (5 marks, P1)
• 3Linear inequality with set notation (3 marks, P1)
• 4Quadratic inequality (4 marks, P1)
• 5Combined linear and quadratic (6 marks, P1)

Worked examples (5)

• 1Sketch a factorised cubic (4 marks, P1)
• 2Sketch $y = k/x$ (3 marks, P1)
• 3Translation transformation (3 marks, P1)
• 4Vertical and horizontal stretches (4 marks, P1)
• 5Intersection of cubic and line (6 marks, P1)

Worked examples (6)

• 1Differentiate $x^{2}$ from first principles (5 marks, P1)
• 2Differentiate using the power rule (4 marks, P1)
• 3Equation of a tangent (5 marks, P1)
• 4Stationary points and nature (7 marks, P1)
• 5Increasing and decreasing intervals (5 marks, P1)
• 6Equation of a normal (5 marks, P1)

Worked examples (5)

• 1Convert between degrees and radians (3 marks, P1)
• 2Arc length (3 marks, P1)
• 3Sector area (3 marks, P1)
• 4Segment area (6 marks, P1)
• 5Small-angle approximations (5 marks, P1)

Worked examples (6)

• 1Indefinite integral of a polynomial (4 marks, P1)
• 2Integrate mixed powers (5 marks, P1)
• 3Definite integral (4 marks, P1)
• 4Area under a curve (6 marks, P1)
• 5Find a curve given its gradient function (5 marks, P1)
• 6Trapezium rule (6 marks, P1)