Numerical Methods

JKPSC Assistant Professor (Mathematics) · Mathematics · 48 practice questions with answers and explanations: 7 Easy, 37 Medium, 4 Hard.

Sample questions

Sample Easy question 1

The number of correct digits roughly doubles at every step in:

  1. A fixed-point iteration with slope 0.9
  2. Bisection
  3. Newton–Raphson
  4. Regula falsi
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Answer: C. Its convergence is quadratic.

Sample Easy question 2

The Newton–Raphson iteration for f(x) = 0 is:

  1. xₙ₊₁ = f(xₙ)/f′(xₙ)
  2. xₙ₊₁ = xₙ + f(xₙ)
  3. xₙ₊₁ = xₙ − f(xₙ)/f′(xₙ)
  4. xₙ₊₁ = xₙ f(xₙ)
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Answer: C. It uses the tangent line.

Sample Easy question 3

The order of convergence of Newton's method at a simple root is:

  1. 2
  2. 1
  3. 1.5
  4. 3
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Answer: A. It converges quadratically.

Sample Medium question 4

Gaussian elimination reduces a linear system to:

  1. Diagonal form always
  2. Lower triangular form only
  3. The identity matrix
  4. Upper triangular form
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Answer: D. The solution then follows by back substitution.

Sample Medium question 5

Partial pivoting in Gaussian elimination is used to:

  1. Find eigenvalues
  2. Make the matrix symmetric
  3. Increase the matrix size
  4. Reduce round-off error
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Answer: D. It avoids dividing by a small pivot.

Sample Medium question 6

The Jacobi and Gauss–Seidel methods converge if the coefficient matrix is:

  1. Singular
  2. Strictly diagonally dominant
  3. Skew-symmetric
  4. Upper triangular only
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Answer: B. This is a sufficient condition.

Sample Hard question 7

The classical fourth-order Runge–Kutta method evaluates f how many times per step?

  1. 4
  2. 2
  3. 3
  4. 6
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Answer: A. It combines the slopes k₁, k₂, k₃ and k₄.

Sample Hard question 8

Simpson's 1/3 rule is exact for polynomials up to degree:

  1. 2
  2. 3
  3. 1
  4. 4
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Answer: B. The cubic term integrates exactly because of symmetry.

Sample Hard question 9

The local truncation error of the classical fourth-order Runge–Kutta method is of order:

  1. h⁴
  2. h³
  3. h²
  4. h⁵
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Answer: D. The global error is of order h⁴.

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