Lecture Notes on Exam Paper 31, May/June 2013

Question 1: Solving Exponential Equations

• Problem: Solve an equation involving ( e^{2x} ), answer to 3 decimal places.
• Method:
  • Let ( y = e^{2x} ).
  • Quadratic form: ( 3y - \frac{4}{y} = 5 ) becomes ( 3y^2 - 5y - 4 = 0 ).
  • Use quadratic formula: ( y = 3.257 ) or ( y = -0.5 ).
  • Solve for ( x ):
    • Use ( y = e^{2x} = 3.257 ) to find ( x ).
    • Discard ( y = -0.5 ) (no solution for negative exponential).
  • Solution: ( x = 0.407 ).

Question 2: Sketching Graphs

• Task: Sketch ( y = |2x + 3| ).
• Steps:
  • Sketch ( y = 2x + 3 ) first.
  • Find intercepts: ( y = 3 ) when ( x = 0 ), ( x = -1.5 ) when ( y = 0 ).
  • Reflect negative part of line above x-axis.
  • Label axes and intercepts.

Question 3: Binomial Expansion

• Objective: Find coefficient of ( x^3 ) in ((3 + x)(1 + 4x)^{1/2} ).
• Approach:
  • Use binomial expansion on ( (1 + 4x)^{1/2} ).
  • Calculate terms involving ( x^3 ) from expansion.
  • Final coefficient: 10.

Question 4: Trigonometric Equations

• Proof: Show given trigonometric identity.
  • Use double angle formulas for sine and cosine.
  • Rearrange to match target equation.
• Solve: Angle solutions for ( cos^2(\theta) + 2sin\theta cos\theta - 3sin^2(\theta) = 0 ).
  • Factorize and solve quadratic form.
  • Solutions: ( \theta = 45^\circ ) and ( \theta = 161.6^\circ ).

Question 5: Implicit Differentiation

• Task: Differentiate implicitly to find ( \frac{dy}{dx} ).
• Process:
  • Apply product rule and differentiate implicitly.
  • Rearrange to express ( \frac{dy}{dx} ).
• Tangent Line: Find coordinates where tangent is parallel to y-axis.
  • Condition: ( \frac{dy}{dx} = \infty ) leads to set denominator to zero.
  • Solve system of equations for coordinates.

Question 6: Vectors in a Parallelogram

• Part A: Find position vector of D.
  • Use midpoint property of parallelogram diagonals.
• Part B: Angle between vectors BA and BC.
  • Use dot product: ( cos\theta = \frac{\text{BA} \cdot \text{BC}}{||\text{BA}|| ||\text{BC}||} ).
• Part C: Area of parallelogram.
  • Use cross product relation or known side lengths and angle.
  • Express in form ( P\sqrt{Q} ) with integers.

Question 7: Differential Equations

• Given: ( \int y \sin^2(3y) , dy = 4 \tan(2x) , dx ).
• Steps:
  • Separate variables and integrate both sides.
  • Use initial condition to find constant.
  • Find ( x ) when ( y = \frac{\pi}{6} ).

Question 8: Partial Fractions and Integration

• Objective: Express rational function in partial fractions.
  • Linear and repeated linear factors.
• Integration: Use partial fractions to integrate.
  • Express result in form ( A + B \ln(C) ).

Question 9: Iterative Solutions

• Part A: Show expression for ( a ) using integration by parts.
• Part B: Verify ( a ) lies between 0.5 and 1 using sign change method.
• Part C: Use iteration to find ( a ) to two decimal places.
  • Show each step to four decimal places.

Question 10: Polynomial Roots

• Part A: Show ( x + 3 ) is a factor using factor theorem.
• Part B: Show complex root ( Z = -1 + 2\sqrt{6}i ).
  • Substitute and verify zero remainder.
• Part C: Solve ( p(Z^2) = 0 ) for complex roots.
  • Find expressions for roots using square roots and conjugates.