Numerical Methods

Question 1

Q1. Truncation error occurs when ___.
A) We stop a series expansion early B) Calculator rounding C) Wrong formula D) Typing error

Answer 1

A – Series cutoff → truncation error.

Question 2

Q2. Round-off error results from ___.
A) Limited significant digits B) Series cutoff C) Operator bias D) Measuring tape length

Answer 2

A – Rounding of digits causes round-off.

Question 3

Q3.True error = ?
A) Approx – True B) True – Approx C) (True – Approx)/True D) None

Answer 3

B – True – Approx gives signed error.

Question 4

Q4. Absolute error = ?
A) |True – Approx| B) (True – Approx)/True C) True × Approx D) 0

Answer 4

A – Absolute error = magnitude difference.

Question 5

Q5. Relative error = ?
A) (True – Approx)/True B) True – Approx C) Approx / True D) 1 – True

Answer 5

A – Relative = error / true value.

Question 6

Q6. Significant figures represent ___.
A) Reliable digits in a number B) Decimal places only C) Random digits D) Binary digits

Answer 6

A – Only reliable digits count.

Question 7

Q7. Bisection method solves ___.
A) f(x)=0 by interval halving B) Integration C) Differentiation D) Matrix inversion

Answer 7

A – Bisection halves interval each step.

Question 8

Q8. Newton-Raphson formula:
A) xₙ₊₁ = xₙ – f(xₙ)/f′(xₙ) B) xₙ₊₁ = f(xₙ)/f′(xₙ) C) xₙ = xₙ₊₁ f′ D) f′=f/x

Answer 8

A – Newton-Raphson standard root formula.

Question 9

Q9. Convergence means ___.
A) Successive approximations approach true value B) Diverge C) Alternate signs only D) Oscillate forever

Answer 9

A – Convergent sequence → stable approach.

Question 10

Q10. Gauss-Seidel method used for ___.
A) Nonlinear equations B) Simultaneous linear equations C) Integration D) Statistics

Answer 10

B – Used for linear systems.

Question 11

Q11. Jacobi vs Gauss-Seidel: difference?
A) Update immediately in Gauss-Seidel B) Same method C) Use pivoting D) No iteration

Answer 11

A – Immediate update → faster convergence.

Question 12

Q12. Simpson’s 1/3 rule formula:
A) ( \frac{h}{3}[y₀ + 4y₁ + y₂] ) B) ( \frac{h}{2}[y₀ + y₁] ) C) ( h[y₀ + y₂] ) D) ( 3h(y₀ + y₁) )

Answer 12

A – Classic Simpson’s 1/3 rule.

Question 13

Q13. Trapezoidal rule = ? **IMAGE
A) ( \frac{h}{2}[y₀ + yₙ + 2Σy_i] ) B) ( \frac{h}{3}[y₀ + 4y₁ + y₂] ) C) ( h[y₀ + yₙ] ) D) ( 3hΣy_i )

Answer 13

A – Trapezoidal area summation formula.

Question 14

Q14. h = (b – a)/n represents ___.
A) Sub-interval width B) Step height C) Sample variance D) Iteration count

Answer 14

A – Width of each sub-interval.

Question 15

Q15. Central difference formula approximates ___.
A) Derivative B) Integral C) Product D) Quotient

Answer 15

A – Finite difference gives slope estimate.

Question 16

Q16. Interpolation estimates ___.
A) Unknown between data points B) Beyond known range C) Exact value D) Constant slope

Answer 16

A – Estimate inside range.

Question 17

Q17. Extrapolation estimates ___.
A) Beyond known range B) Within data C) Midpoint D) Average

Answer 17

A – Estimate beyond data range.

Question 18

Q18. Lagrange interpolation fits ___.
A) Polynomial through given points B) Linear trend only C) Trig curve D) Random curve

Answer 18

A – Fits exact polynomial through all given points.

Question 19

Q19. Finite difference table used in ___.
A) Numerical differentiation/interpolation B) Statistics C) Structural analysis D) Simple algebra

Answer 19

A – Used in interpolation and differentiation tables.

Question 20

Q20. Euler’s method approximates ___.
A) Differential equation solution B) Integration C) Regression line D) Determinant

Answer 20

A – Numerical DE solver (Euler).

Question 21

Q21.Runge-Kutta (4th order) = ?
A) Improved Euler accuracy B) Exact solution C) Graphical estimate D) Monte-Carlo method

Answer 21

A – Higher-order Euler improvement.

Question 22

Q22. Iteration stops when ___.
A) |xₙ₊₁ – xₙ| < tolerance B) Error increases C) i > 5 D) CPU heats up 😂

Answer 22

A – Iteration stops when successive values stabilize.

Question 23

Q23. Matrix inversion (A⁻¹) satisfies ___.
A) A A⁻¹ = I B) A × A = I C) A⁻¹ × A⁻¹ = I D) A / A = I

Answer 23

A – Definition of matrix inverse.

Question 24

Q24. Determinant (Δ) ≠ 0 means ___.
A) System solvable B) No solution C) Singular matrix D) Dependent equations

Answer 24

A – Non-zero Δ → unique solution.

Question 25

Q25. Condition number indicates ___. A) Sensitivity of system to small changes B) Size of matrix C) Accuracy of calc D) Sign of determinant

Answer 25

A – Large value → unstable/sensitive system.