Lecture Notes

Key Concepts

1. Logarithmic Equation

   • Given equation: (\log_y x^2 + \log_x y^2 = 1)
   • Simplifying gives (\log_x y + \log_y x = 2)
   • (\log_x y) and (\log_y x) are reciprocals.
   • Solution for (k + 1/k = 2) implies (k = 1).
   • Conclusion: (x = y).

2. Quadratic Equation

   • Solving equation (x^2 - x - 30 = 0) leads to roots: (x = 6, -5).
   • Since (x) cannot be negative, (x = 6).
   • Thus, (y = 6), and (x^2 + y^2 = 72).

Speed and Distance Problem

1. Speed Increase Scenario
   • Geeta’s speed increase from 12 km/hr to 20 km/hr leads to 1 hour saved.
   • Speed ratio 3:5 leads to time ratio 5:3.
   • Time difference = 2x = 1 hour.
   • Distance = 30 km.

Combinatorial Geometry

1. Triangles from Points
   • Points on two lines: 7 points on one, 5 on another.
   • Calculate triangles using combinations: (\binom{7}{2} \cdot \binom{5}{1} + \binom{7}{1} \cdot \binom{5}{2} = 175).

Profit and Loss Problem

1. Profit and Loss Balancing
   • Apples at 10% profit, mangoes at 20% loss.
   • No overall profit or loss means profit = loss.
   • Apples to mango ratio: 5:2.
   • Aruna buys 50 apples and 20 mangoes.

Sequence and Series

1. Binomial Expansion

   • Expansion of ((5x - 9)^4).
   • Coefficients' sum found using (P(1)) for polynomials: (256).

2. Sequence without Perfect Squares

   • New sequence by removing squares from (1, 2, 3, ...).
   • Calculate 2022nd term: Use close square 2025, result is 2067.

Arithmetic and Geometric Progression

1. Connection between AP and GP

   • AP terms form consecutive GP terms.
   • Solve for the next term in GP as nth in AP: (n = 344).

2. AP with Logarithms

   • Terms of (2^x + 3y), identify (x = 3).

Probability and Arrangement

1. Circular Arrangement with Conditions
   • Count possibilities considering adjacent seating restrictions.
   • Apply subtraction principle: Final count = 960.

Maximum and Inequalities

1. Maximizing Sum of Integers

   • Highest integer is 100, average difference constraint.
   • Optimal configuration yields a sum of 3150.

2. Inequality Solving with Logarithms

   • Solve (\log_3 x ) constraints: Solution range ([2, 81)).

Concurrent Lines and Geometry

1. Finding Concurrent Lines

   • Solve for common point, find (k = 0).

2. Triangle Side Lengths

   • Longest side is 40, viable shortest side (x > 19).

Modulus and Geometry

1. Distance and Modulus
   • Insight into modulus and line equations.
   • Calculate minimum value using symmetry: Result = 26.

Exponential Properties

1. Exponential Function Constraints
   • For (81^x + 81^{f(x)} = 3), bound (f(x) < 0.25).

Algebra and Number Theory

1. **Variable Substitution and Solving
   • Use substitution and factorization, results in solving quadratic equations and validating solutions.