Lesson 51: Additional Maths with Mr. Barrow

Overview

• The lesson covers questions 1 to 10 of an OCR sample paper.
• Students should attempt the questions before reviewing solutions.

Question 1: Recurrence Relation

• Concept: Sequence defined by ( u_{n+1} = 2u_n - 1 ).
• Task: Find the 6th term given the 3rd term is 12.
• Process:
  • 4th term: ( 2 \times 12 - 1 = 23 )
  • 5th term: ( 2 \times 23 - 1 = 45 )
  • 6th term: ( 2 \times 45 - 1 = 89 )
• Marks: Method mark for finding intermediate terms, accuracy for the 6th term.
• Practice: Recurrence relations - Lesson 8._

Question 2: Binomial Expansion

• Task: Find the coefficient of ( x^3 ) in ((2 + 3x)^5).
• Concept: Binomial expansion involves choosing components to form a specific term.
• Process:
  • Use "5 choose 3" to find combinations.
  • Calculation: ( 3x^3 \times 2^2 \times 10 = 1080x^3 ).
• Final Answer: Coefficient is 1080.
• Practice: Binomial expansion - Lessons 23 & 24.

Question 3: Differentiation

• Task: Differentiate ( y = x^3 + 2x - 7 ) and show there are no turning points.
• Solution:
  • Differentiation: ( \frac{dy}{dx} = 3x^2 + 2 )
  • Turning Points: Solve ( 3x^2 + 2 = 0 ), no real solutions.
• Marks: Method for differentiation, showing no turning points.
• Practice: Differentiation and stationary points - Lesson 38.

Question 4: Integration

• Task: Integrate ( x^2 + 3 ) from 1 to 2.
• Process:
  • Integrate: ( \int (x^2 + 3) dx = \frac{x^3}{3} + 3x )
  • Evaluate from 1 to 2: ( (\frac{8}{3} + 6) - (\frac{1}{3} + 3) = \frac{16}{3} ).
• Marks: Integration and substitution method.
• Practice: Definite integration - Lesson 41.

Question 5: Trigonometry

• Task: Sketch ( h ) vs ( \theta ) graph, find expression for ( h ), solve for ( h = 100 ).
• Concept: Circular function and cosine relationship.
• Process:
  • Graph: Sketch sine-like curve from 0 to 360°.
  • Expression: ( h = 67.5 - 67.5 \cos \theta ).
  • Solve: ( \theta = 118.8° ) or ( 241.2° ) when ( h = 100 ).
• Practice: Trigonometric equations - Lesson 14.

Question 6: Algebraic Fractions

• Part A Task: Simplify ( \frac{x}{x+2} - \frac{6}{x-1} ).
• Solution:
  • Common denominator: ( (x+2)(x-1) )
  • Result: ( \frac{x^2 - 7x - 12}{x^2 + x - 2} )
• Part B Task: Solve ( \frac{x^2 - 7x - 12}{x^2 + x - 2} = 4 ).
• Solution: Use quadratic formula to solve resulting equation.
• Practice: Algebraic fractions - Lesson 1.

Question 7: Completing the Square

• Task: Convert ( 2x^2 + 8x - 12 ) to completed square form, find minimum value.
• Solution:
  • Completed form: ( 2(x+2)^2 - 20 ).
  • Minimum value is (-20).
• Practice: Completing the square - Lesson 5.

Question 8: Cosine Rule

• Task: Show one angle in triangle is 60°.
• Solution:
  • Use cosine rule for angle opposite 7 cm side.
  • ( \cos \theta = \frac{1}{2} ), hence ( \theta = 60° ).

Question 9: Factor Theorem

• Part A: Show ( x-3 ) is a factor of ( x^3 - 5x^2 + x + 15 ).
  • Substitute 3 into the polynomial; it equals 0.
• Part B: Solve cubic equation using factorization and quadratic formula.
• Practice: Factor theorem - Lesson 4.

Question 10: Permutations

• Task: Calculate number of passcodes with specific conditions.
• Part A: No restrictions: ( 7^4 = 2,401 ).
• Part B: With exactly 2 letters and 2 digits:
  • Calculate possible choices: 18 options for character selection.
  • Arrange in ( 4! ) ways: Total = 432.
• Practice: Permutations and combinations - Lessons 20-22.

Next Steps

• Next lesson will cover questions 11-16 of the OCR sample paper. Ensure to attempt these questions before reviewing solutions.