Exercise 8.2 Class 12 Math Solutions - Unit 8 Numerical Solutions of Nonlinear Equations, Punjab Board
Step-by-step solutions to Exercise 8.2, Unit 8: Numerical Solutions of Nonlinear Equations, for Class 12 Punjab Board students. Covers iterative numerical algorithms to find roots of nonlinear equations, rules of accuracy and decimal tolerance, the Bisection Method, the Regula-Falsi (False Position) Method, and the Newton-Raphson Method, worked out question by question.
Exercise Overview
Exercise 8.2 of Unit 8: Numerical Solutions of Nonlinear Equations in Class 12 Mathematics under the Punjab Board PCTB / PECTAA curriculum introduces the core numerical iteration algorithms used to find real roots of nonlinear equations (f(x) = 0) to any required decimal accuracy. The exercise begins with the concepts of accuracy and error tolerance ((\epsilon)): an iterative algorithm terminates when the absolute error between successive approximations satisfies (|x_{n+1} - x_n| < \epsilon), or when the residual satisfies (|f(x_n)| < \epsilon), or when consecutive iterations agree to a specified number of decimal digits.
The exercise systematically compares bracketing and open root-finding techniques: 1. The Bisection Method continuously bisects the initial sign-change interval ([a, b]) (where (f(a) \cdot f(b) < 0)) by calculating the midpoint (c = \frac{a + b}{2}), offering guaranteed convergence with maximum error (\frac{b - a}{2^n}); 2. The Regula-Falsi (False Position) Method accelerates bracketing by connecting ((a, f(a))) and ((b, f(b))) with a straight secant line whose (x)-intercept is (c = \frac{a f(b) - b f(a)}{f(b) - f(a)}); 3. The Newton-Raphson Method constructs tangent lines at successive points ((x_n, f(x_n))) using the derivative (f'(x_n)), generating the quadratically converging open formula (x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}) (provided (f'(x_n) \neq 0)).
For Punjab Board HSSC Part 2 examinations across all nine educational divisions (Lahore, Rawalpindi, Gujranwala, Faisalabad, Multan, Bahawalpur, Sargodha, Sahiwal, and D.G. Khan), Exercise 8.2 is a major source of Section B Short Questions (performing 2-3 iterations of Bisection or Newton-Raphson; stating iteration formulas and convergence conditions) and Section C Long Questions (carrying out complete numerical convergence to 3 or 4 decimal places). These step-by-step solutions provide clean iteration tables, explicit derivative evaluations, and exact round-off checks for every problem.