BCA 4th Semester
Numerical Method Board Question Paper 2021

Compute the root of equation \( x^2 - 5x + 6 = 0 \) in the vicinity of \( x = 5 \) using Newton-Raphson method.

Estimate the value of \( \ln(2.5) \) using Newton-Gregory forward difference formula given the following data:

Write an algorithm to compute integration using Simpson's 1/3 rule. Evaluate integration of \[ \int_0^3 (3x^2 + 2x - 5) \, dx \] using Simpson's 1/3 rule.

What is meant by ill-conditioned system? Find the Cholesky decomposition of the following matrix: \[ \begin{bmatrix} 4 & 1 & 1 \\ 1 & 5 & 2 \\ 1 & 2 & 3 \end{bmatrix} \]

Use the classical RK method to estimate \( y(0.4) \) of the equation \[ \frac{dy}{dx} = x^2 + y^2 \] with \( y(0) = 0 \), assume \( h = 0.2 \).

Solve the Poisson equation \[ \nabla^2 f = 2x^2 y^2 \] over the square domain \[ 0 \leq x \leq 3, \quad 0 \leq y \leq 3 \] with \( f = 0 \) on the boundary and \( h = 1 \).

Write an algorithm and program to compute root of nonlinear equation using bisection method.

Given the data points:

a) Solve the following system of equations using Gauss-Seidel method:
2x - 7y - 10z = -17
5x + y + 3z = 14
x + 10y + 9z = 7
b) Write an expansion of Taylor’s Theorem. Solve the equation \[ y'(x) = x^2 + y^2, \quad y(0) = 1 \] for \( x = 0.5 \) using Taylor method.