WMA11 Pure Mathematics 1

Lessons in WMA11

1. The discriminant
   b² - 4ac > 0: two real roots; = 0: one repeated root; < 0: no real roots.
2. Differentiating powers of x
   y = axⁿ gives dy/dx = anxⁿ-1. Rewrite roots and fractions as powers first.
3. Laws of indices
   a^m x aⁿ = a^m+n, a^m / aⁿ = a^m-n, (a^m)ⁿ = a^mn; a^-n = 1/aⁿ; a^m/n = (nth root of a)^m.
4. Surds and rationalising the denominator
   Simplify with the largest square factor; rationalise sqrt(a) by multiplying by sqrt(a), and a + sqrt(b) by multiplying by a - sqrt(b).
5. Solving quadratics and their graphs
   Rearrange to = 0, then factorise or use x = (-b +/- sqrt(b² - 4ac))/(2a). Roots are the x-intercepts; c is the y-intercept.
6. Completing the square
   x² + bx + c = (x + b/2)² - (b/2)² + c. The turning point of a(x + p)² + q is (-p, q).
7. Simultaneous equations
   Substitute the linear equation into the other, solve, then use the linear equation for the partner values. The discriminant counts intersections.
8. Linear inequalities and regions
   Reverse the sign when multiplying or dividing by a negative. Dashed boundary for < and >, solid for ≤ and ≥.
9. Quadratic inequalities
   Find critical values, sketch, then read off: between the roots for < 0, outside the roots for > 0 (when x² is positive).
10. Polynomials and cubic graphs
    Factorise, mark the roots and the y-intercept, and use the sign of the x³ term for the shape. A squared factor touches the axis.
11. Reciprocal graphs and intersections
    y = k/x and y = k/x² have asymptotes x = 0 and y = 0. The number of intersections is the number of solutions.
12. Transformations of graphs
    f(x) + a moves up; f(x + a) moves left; af(x) stretches y by a; f(ax) stretches x by 1/a.
13. Equations of straight lines
    m = (y_2 - y_1)/(x_2 - x_1); y - y_1 = m(x - x_1); rearrange to ax + by + c = 0 with integers if asked.
14. Parallel and perpendicular lines
    Parallel: equal gradients. Perpendicular: m_1 m_2 = -1. Length by Pythagoras; midpoint by averaging.
15. The sine rule and the ambiguous case
    a/sin A = b/sin B = c/sin C. When finding an angle from two sides and a non-included angle, check whether 180° minus your answer also works.
16. The cosine rule and the area of a triangle
    a² = b² + c² - 2bc cos A; cos A = (b² + c² - a²)/(2bc); area = (1/2) ab sin C.
17. Radians, arc length and sector area
    π rad = 180°; s = rθ; A = (1/2) r² θ; segment = (1/2) r² (θ - sin θ); θ must be in radians.
18. Graphs of sine, cosine and tangent
    sin and cos: period 360°, range -1 to 1. tan: period 180°, asymptotes at 90° + 180°n. Learn the exact values for 30°, 45°, 60°.
19. Differentiation from first principles
    f'(x) = limit as h → 0 of (f(x + h) - f(x))/h. The derivative is both a gradient and a rate of change.
20. Simplifying before differentiating, and second derivatives
    Expand and split into powers of x before differentiating. d^2y/dx² means differentiate twice.
21. Tangents and normals
    Tangent gradient = dy/dx at the point; normal gradient = -1/(tangent gradient); then use y - y_1 = m(x - x_1).
22. Integration: reversing differentiation
    ∫ xⁿ dx = xⁿ+1/(n + 1) + c (n not -1). Use a known point to find c.

Next exams for WMA11

• January 2027: Fri 8 Jan 2027, morning, 1h 30m
• Summer 2027: Thu 6 May 2027, afternoon, 1h 30m

From Pearson's final timetables. All exam dates. Always confirm your entries with your exams officer.