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ENSH 251Bachelor in Geomatics Engineering · Semester 44 Papers Available

Numerical Methods

Past examination question papers and complete curriculum syllabus for Numerical Methods (ENSH 251), Bachelor in Geomatics Engineering Semester 4 under Institute of Engineering (IOE), Tribhuvan University.

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Most Frequently Asked Questions

Top recurring IOE board exam questions for Numerical Methods with verified mark schemes, formula notation, and recurrence frequency.

Showing 30 of 30 top repeated questions

Solution of Non-Linear Equations

4 Questions
#1Repeated 4 Times[8 Marks]Solution of Non-Linear Equations
Derive the Newton-Raphson iterative formula using Taylor series expansion. Discuss the limitations and failure conditions of the Newton-Raphson method (e.g., $f'(x) \approx 0$, oscillating/cycling, inflection points). Find a real root of $x \log_{10} x = 1.2$ correct to 4 decimal places.
Appeared in:2080 Chaitra2078 Chaitra2075 Baisakh2073 Bhadra
#2Repeated 4 Times[8 Marks]Solution of Non-Linear Equations
Write an algorithm and pseudo-code for the Secant method. Find a real root of the equation $e^x - 3x = 0$ or $x^3 - 2x - 5 = 0$ correct to 4 decimal places using Secant method.
Appeared in:2082 Kartik2079 Chaitra2075 Baisakh2071 Bhadra
#3Repeated 2 Times[6 Marks]Solution of Non-Linear Equations
Prove that the order of convergence of the Newton-Raphson method is quadratic ($p = 2$). Under what conditions does the Newton-Raphson method fail to converge?
Appeared in:2082 Kartik2080 Chaitra
#4Repeated 2 Times[6 Marks]Solution of Non-Linear Equations
Explain the Bisection method and False Position (Regula Falsi) method. Find the root of $x e^x - \cos x = 0$ correct to 3 decimal places using Regula Falsi method.
Appeared in:2081 Chaitra2079 Chaitra

Solution of System of Linear Algebraic Equations

5 Questions
#1Repeated 5 Times[8 Marks]Solution of System of Linear Algebraic Equations
Solve the system of linear equations using LU Factorization (LU Decomposition) method: $$\begin{aligned} 2x + 3y + z &= 9 \\ x + 2y + 3z &= 6 \\ 3x + y + 2z &= 8 \end{aligned}$$
Appeared in:2082 Kartik2080 Chaitra2078 Chaitra2075 Baisakh2072 Ashwin
#2Repeated 5 Times[8 Marks]Solution of System of Linear Algebraic Equations
Write an algorithm for the Power Method. Determine the largest (dominant) eigenvalue and the corresponding eigenvector of the matrix $\begin{bmatrix} 2 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 2 \end{bmatrix}$ using the Power method.
Appeared in:2082 Kartik2080 Chaitra2078 Chaitra2075 Baisakh2073 Bhadra
#3Repeated 4 Times[8 Marks]Solution of System of Linear Algebraic Equations
Apply the Gauss-Seidel iterative method to solve the following system of equations correct to 3 decimal places: $$\begin{aligned} 10x_1 + x_2 + 2x_3 &= 44 \\ 2x_1 + 10x_2 + x_3 &= 51 \\ x_1 + 2x_2 + 10x_3 &= 61 \end{aligned}$$
Appeared in:2080 Chaitra2078 Chaitra2074 Chaitra2072 Ashwin
#4Repeated 2 Times[6 Marks]Solution of System of Linear Algebraic Equations
Explain the Power Method for finding the dominant eigenvalue and its corresponding eigenvector of a square matrix. Find the largest eigenvalue of $A = \begin{pmatrix} 4 & 1 & 0 \\ 1 & 2 & 1 \\ 0 & 1 & 1 \end{pmatrix}$ using 4 iterations.
Appeared in:2082 Kartik2081 Chaitra
#5Repeated 2 Times[8 Marks]Solution of System of Linear Algebraic Equations
Explain Cholesky factorization method for symmetric positive-definite matrices. Solve the linear system using Cholesky decomposition: $4x_1 + 2x_2 + 14x_3 = 14$, $2x_1 + 17x_2 - 5x_3 = -101$, $14x_1 - 5x_2 + 83x_3 = 155$.
Appeared in:2081 Chaitra2079 Chaitra

Interpolation

7 Questions
#1Repeated 5 Times[8 Marks]Interpolation
Fit a curve of the form $y = a e^{bx}$ (or $y = a b^x$) to the given set of data points using the principle of least squares regression. State the linearized normal equations.
Appeared in:2082 Kartik2080 Chaitra2076 Bhadra2073 Bhadra2071 Bhadra
#2Repeated 4 Times[8 Marks]Interpolation
Develop the Lagrange interpolation formula. From the given data points $(1, 3), (2, 12), (3, 31), (5, 127)$, estimate the value of $y$ at $x = 4$ using Lagrange's interpolation polynomial.
Appeared in:2082 Kartik2079 Chaitra2076 Bhadra2072 Ashwin
#3Repeated 4 Times[8 Marks]Interpolation
Estimate $y(2.5)$ using the Natural Cubic Spline interpolation technique from the following table of data: $(1, 1), (2, 5), (3, 11), (4, 8)$. Explain boundary conditions for natural cubic splines.
Appeared in:2080 Chaitra2078 Chaitra2075 Baisakh2073 Bhadra
#4Repeated 2 Times[6 Marks]Interpolation
Derive Lagrange's Interpolating Polynomial formula for unequally spaced data points. Write an algorithm to find the value of $y$ at a given $x$ using Lagrange interpolation.
Appeared in:2082 Kartik2081 Chaitra
#5Repeated 2 Times[6 Marks]Interpolation
Derive Newton's divided difference interpolation formula. Calculate $f(2.5)$ using Newton's divided difference table for the given set of points: $(1, 0), (2, 2), (3, 12), (4, 40), (5, 90)$.
Appeared in:2081 Chaitra2078 Bhadra
#6Repeated 2 Times[8 Marks]Interpolation
What is a cubic spline? Explain natural cubic spline interpolation and derive the tridiagonal system of equations for natural cubic splines.
Appeared in:2082 Kartik2080 Chaitra
#7Repeated 2 Times[6 Marks]Interpolation
Fit an exponential curve of the form $y = a e^{b x}$ to the given experimental data using the principle of least squares: $(1, 1.5), (2, 4.5), (3, 13.8), (4, 40.2), (5, 125.0)$.
Appeared in:2082 Kartik2079 Chaitra

Numerical Differentiation and Integration

6 Questions
#1Repeated 4 Times[8 Marks]Numerical Differentiation and Integration
Derive Simpson's $1/3$ rule for numerical integration from Newton-Cotes quadrature formula. Write an algorithm to evaluate $\int_a^b f(x) dx$ using Composite Simpson's $1/3$ rule.
Appeared in:2080 Chaitra2078 Chaitra2074 Chaitra2071 Bhadra
#2Repeated 4 Times[8 Marks]Numerical Differentiation and Integration
Evaluate the integral $I = \int_0^2 e^{-x^2} dx$ (or $\int_1^3 \frac{\sin^2 x}{x} dx$) using 3-point Gauss-Legendre Quadrature formula and compare with standard trapezoidal rule.
Appeared in:2082 Kartik2080 Chaitra2076 Bhadra2073 Bhadra
#3Repeated 4 Times[8 Marks]Numerical Differentiation and Integration
Evaluate the integral $\int_0^1 \frac{1}{1+x^2} dx$ using Romberg's integration method with $h = 0.5, 0.25, 0.125$ to obtain an accuracy of four decimal places.
Appeared in:2078 Chaitra2075 Baisakh2073 Bhadra2072 Ashwin
#4Repeated 2 Times[8 Marks]Numerical Differentiation and Integration
Derive the three-point Gaussian quadrature formula $\int_{-1}^1 f(x)\,dx = \frac{5}{9}f\left(-\sqrt{\frac{3}{5}}\right) + \frac{8}{9}f(0) + \frac{5}{9}f\left(\sqrt{\frac{3}{5}}\right)$. Use it to evaluate $\int_1^3 \frac{1}{x}\,dx$.
Appeared in:2082 Kartik2081 Chaitra
#5Repeated 2 Times[8 Marks]Numerical Differentiation and Integration
Derive Simpson's 1/3 rule and Simpson's 3/8 rule from Newton-Cotes quadrature formula. What are their respective local and global truncation error terms?
Appeared in:2081 Chaitra2080 Chaitra
#6Repeated 2 Times[6 Marks]Numerical Differentiation and Integration
Evaluate the double integral $\int_0^1 \int_0^1 e^{x+y}\,dx\,dy$ using composite Trapezoidal rule and composite Simpson's 1/3 rule with $h = k = 0.5$.
Appeared in:2082 Kartik2078 Bhadra

Solution of Ordinary Differential Equations (ODE)

5 Questions
#1Repeated 5 Times[8 Marks]Solution of Ordinary Differential Equations (ODE)
Using Runge-Kutta 4th Order (RK-4) method, solve the initial value problem $\frac{dy}{dx} = x^2 + y^2$ with $y(0) = 1$ to find $y(0.2)$ using step size $h = 0.1$.
Appeared in:2082 Kartik2080 Chaitra2078 Chaitra2075 Baisakh2073 Bhadra
#2Repeated 4 Times[8 Marks]Solution of Ordinary Differential Equations (ODE)
Solve the second order differential equation $\frac{d^2y}{dx^2} + x\frac{dy}{dx} + y = 0$ with initial conditions $y(0) = 1, y'(0) = 0$ for $x = 0.2$ using Runge-Kutta second-order (RK-2) method.
Appeared in:2080 Chaitra2076 Bhadra2074 Chaitra2071 Bhadra
#3Repeated 4 Times[8 Marks]Solution of Ordinary Differential Equations (ODE)
Solve the boundary value problem $\frac{d^2y}{dx^2} = y + x^2$ with boundary conditions $y(0) = 2$ and $y(2) = 5$ at $x = 0.5, 1.0, 1.5$ using the finite difference method.
Appeared in:2080 Chaitra2078 Chaitra2075 Baisakh2072 Ashwin
#4Repeated 2 Times[8 Marks]Solution of Ordinary Differential Equations (ODE)
Derive the classical fourth-order Runge-Kutta (RK-4) formula for solving $\frac{dy}{dx} = f(x, y)$ with $y(x_0) = y_0$. Solve $\frac{dy}{dx} = x + y^2, y(0) = 1$ to find $y(0.2)$ taking $h = 0.1$.
Appeared in:2082 Kartik2081 Chaitra
#5Repeated 2 Times[6 Marks]Solution of Ordinary Differential Equations (ODE)
Explain Milne's Predictor-Corrector method and Adams-Bashforth-Moulton method for solving initial value ODEs. How are initial starting points computed?
Appeared in:2080 Chaitra2078 Bhadra

Solution of Partial Differential Equations

3 Questions
#1Repeated 5 Times[8 Marks]Solution of Partial Differential Equations
Solve Laplace's equation $\nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$ on a square grid of size $8\text{ cm} \times 8\text{ cm}$ with given Dirichlet boundary temperatures using Liebmann's (Gauss-Seidel) iterative method.
Appeared in:2082 Kartik2080 Chaitra2078 Chaitra2075 Baisakh2073 Bhadra
#2Repeated 4 Times[8 Marks]Solution of Partial Differential Equations
Derive the Bender-Schmidt recurrence formula for solving the one-dimensional heat equation $\frac{\partial u}{\partial t} = c^2 \frac{\partial^2 u}{\partial x^2}$. State the condition for numerical stability and calculate temperatures for two time steps.
Appeared in:2082 Kartik2080 Chaitra2077 Chaitra2074 Chaitra
#3Repeated 2 Times[8 Marks]Solution of Partial Differential Equations
Solve the two-dimensional Laplace equation $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$ over a square domain $0 \le x, y \le 3$ with boundary conditions $u(0, y) = 0, u(3, y) = 100, u(x, 0) = 0, u(x, 3) = 100$ using the 5-point finite difference formula on a grid with $h = 1$.
Appeared in:2082 Kartik2080 Chaitra

Curriculum Syllabus & Course Topics

Sourced from TU curriculum portal
Full Syllabus (PDF)
Chapter-wise Units & Micro-Syllabus Topics (6 Units)

  1. 1. Solution of Non-Linear Equations

    • 1.1Errors and accuracy in numerical computations
    • 1.2Bisection method
    • 1.3Regula Falsi method and secant method
    • 1.4Newton Raphson method
    • 1.5Fixed point iteration method
    • 1.6Comparison of the methods (Bracketing vs open-ended methods and rates of convergence)
    • 1.7Solution of system of non-linear equations
    • 1.7.1Direct approach
    • 1.7.2Newton Raphson method

  2. 2. Solution of System of Linear Algebraic Equations

    • 2.1Direct methods
    • 2.1.1Gauss Jordan method
    • 2.1.2Gauss elimination method, pivoting strategies (Partial and complete)
    • 2.1.3Matrix inverse using Gauss Jordan and Gauss elimination methods
    • 2.1.4Factorization methods (Do-Little’s method and Crout’s method)
    • 2.2Iterative methods
    • 2.2.1Jacobi’s method
    • 2.2.2Gauss-Seidal method
    • 2.3Determination of largest and smallest Eigen values and corresponding vectors using the power method

  3. 3. Interpolation

    • 3.1Polynomial Interpolation
    • 3.1.1Finite differences (Forward, backward, central and divided differences)
    • 3.1.2Interpolation with equally spaced intervals: Newton’s forward and backward difference interpolation, Stirling’s and Bessel’s central difference interpolation
    • 3.1.3Interpolation with unequally spaced intervals: Newton’s divided difference interpolation, Lagrange interpolation
    • 3.2Least square method of curve fitting
    • 3.2.1Linear form and forms reducible to linear form
    • 3.2.2Quadratic form and forms reducible to quadratic form
    • 3.2.3Higher degree polynomials
    • 3.3Cubic spline interpolation
    • 3.3.1Equally spaced interval
    • 3.3.2Unequally spaced interval

  4. 4. Numerical Differentiation and Integration

    • 4.1Numerical differentiation
    • 4.1.1Differentiation using polynomial interpolation formulae for equally spaced intervals
    • 4.1.2Local maxima and minima from equally spaced data
    • 4.2Numerical integration
    • 4.2.1Newton Cote’s general quadrature formula
    • 4.2.2Trapezoidal rule, Simpson’s 1/3 and 3/8 rules, Boole’s rule, Weddle’s rule
    • 4.2.3Romberg integration
    • 4.2.4Gauss-Legendre integration (up to 3-point formula)

  5. 5. Solution of Ordinary Differential Equations (ODE)

    • 5.1Initial value problems
    • 5.1.1Solution of first order equations: Taylor’s series method, Euler’s method, Runge-Kutta methods (Second and fourth order)
    • 5.1.2Solution of system of first order ODEs via Runge-Kutta methods
    • 5.1.3Solution of second order ODEs via Runge-Kutta methods
    • 5.2Two-point boundary value problems
    • 5.2.1Shooting method
    • 5.2.2Finite difference method

  6. 6. Solution of Partial Differential Equations

    • 6.1Introduction and classification
    • 6.2Finite difference approximations of partial derivatives
    • 6.3Solution of elliptic equations
    • 6.3.1Laplace equation
    • 6.3.2Poisson’s equation
    • 6.4Solution of parabolic and hyperbolic equations
    • 6.4.1One-dimensional heat equation: Bendre-Schmidt method, Crank-Nicolson method
    • 6.4.2Solution of wave equation

Examination Scheme & Marks Distribution

Evaluation Structure

  • Final Board Theory Exam: 60 Marks (Pass mark: 24)
  • Internal Assessment: 40 Marks (Pass mark: 16)
  • Practical / Lab Exam: 25 or 50 Marks (Continuous lab evaluation + viva, where applicable)

* This is the general current IOE 60/40 scheme; verify course-specific details in the syllabus above.

Exam Preparation Guidelines

  • Review the 4 available past examination papers to identify recurring patterns, core problem types, and chapter weightage.
  • Practice numerical problems step-by-step with clean formula derivations, clear units, and standard assumptions.
  • Structure answers with labeled diagrams, concise bullet points, and highlight final answers in numerical solutions.

Frequently Asked Questions (Numerical Methods)

Q: How can I download Numerical Methods past question papers?

You can preview or download the Numerical Methods question papers (PDF) directly using the built-in viewer on this page with zero redirects or paywalls.

Q: What is the pass mark for Numerical Methods?

The general current scheme is a 60-mark final theory exam and a 40-mark internal assessment, with pass marks of 24 and 16. Verify the course-specific syllabus above.

Q: Where can I find the complete syllabus for this subject?

The available chapter-wise syllabus and topic breakdown is indexed in the Syllabus section above, with links to the curriculum PDF source.

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