Lecture Notes on Additional MTH Paper (May/June 2024 Paper 2-3)

Question 1: Coordinate Geometry

• Points A(1, 4) and B(5, 6) given.
• Find the perpendicular bisector of line segment AB.
• Midpoint Calculation:
  • Midpoint M = ((x1 + x2)/2, (y1 + y2)/2)
  • M = ((1 + 5)/2, (4 + 6)/2) = (3, 5)
• Gradient Calculation:
  • Gradient (m) of AB = (y2 - y1) / (x2 - x1) = (6 - 4) / (5 - 1) = 1/2
  • Gradient of Perpendicular Bisector (m_perp) = -2
• Equation of Perpendicular Bisector:
  • y = -2x + c
  • Use midpoint to find c:
  • 5 = -2(3) + c => c = 11
  • Equation: y = -2x + 11
• Intercepts:
  • Y-intercept D(0, 11)
  • X-intercept C(5.5, 0)
• Area of Triangle DBC:
  • Area = 1/2 * base * height
  • Base OC = 5.5, Height OD = 11
  • Area = 1/2 * 5.5 * 11 = 30.25

Question 2: Discriminant

• Given equation has no roots if b² - 4ac < 0.
• Identify coefficients:
  • a = k, b = 2k - 1, c = k + 1
• Discriminant:
  • (2k - 1)² - 4(k)(k + 1) < 0
  • Solve to find valid range for k:
  • k > 1/8

Question 3: Modulus Function

• Draw graphs for y = |2x - 5| and y = |4 - x|.
• X-intercept and Y-intercepts calculated.
• Identify regions where |2x - 5| < |4 - x| by observing graph intersections.
  • Solution: x < 1 or x > 3.

Question 4: Binomial Expansion

• Find the term independent of x in the expansion of (3 - 1/2x)⁶.
• Use combinatorial calculations to identify the relevant powers.
• Result: (105/8)

Question 5: Trigonometric Ratios

• Solve for x in terms of tan.
  • tan x = ±1 => x = π/4, -π/4.
• Find the gradient of the curve at these points.

Question 6: Quadratic Equations

• Given roots and factorization method to solve polynomials.

Question 7: Circular Measure

• Use sector area formula to find area of shaded region between two circular arcs.
  • Area = Area of larger sector - Area of smaller sector.
  • Find angle theta = π/7.

Question 8: Integration

• Find y when given the first and second derivative conditions.
• Use integration to find the curve equation.

Question 9: Area Application

• Find total area under the curve and subtract area of triangle formed to find area of shaded region.

Question 10: Vectors

• Find expressions for vectors and show how to derive ratios between segments.
  • Use similarity and properties of lines in vector forms to establish relationships.

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Summary

• This lecture covered key areas in coordinate geometry, quadratic equations, trigonometry, integration, circular measure, and vectors. Each question emphasized problem-solving techniques and practical applications of mathematical theory.
• Practice past papers to solidify understanding.