Differential Equations

JKPSC Assistant Professor (Mathematics) · Mathematics · 49 practice questions with answers and explanations: 28 Easy, 20 Medium, 1 Hard.

Sample questions

Sample Easy question 1

The Laplace transform of sin(at) is:

  1. s/(s² + a²)
  2. a/(s² + a²)
  3. a/(s² − a²)
  4. 1/(s² + a²)
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Answer: B. It is a standard transform pair.

Sample Easy question 2

The general solution of dy/dx = 2y is:

  1. y = Ce^(2x)
  2. y = 2x + C
  3. y = Ce^(−2x)
  4. y = C/2x
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Answer: A. Separate variables: dy/y = k dx, so ln y = kx + c.

Sample Easy question 3

The roots of the characteristic equation of y″ − 4y′ + 3y = 0 are:

  1. 4 and 3
  2. −3 and −1
  3. 3 and 1
  4. 3 and −1
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Answer: C. m² − 4m + 3 = (m − 3)(m − 1) = 0.

Sample Medium question 4

A radioactive substance has a half-life of 4 years. What fraction remains after 12 years (as 1/x, give x)?

  1. 9
  2. 16
  3. 8
  4. 6
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Answer: C. After 3 half-lives the fraction is (1/2)³ = 1/8.

Sample Medium question 5

A radioactive substance has a half-life of 3 years. What fraction remains after 9 years (as 1/x, give x)?

  1. 6
  2. 9
  3. 8
  4. 16
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Answer: C. After 3 half-lives the fraction is (1/2)³ = 1/8.

Sample Medium question 6

A radioactive substance has a half-life of 2 years. What fraction remains after 4 years (as 1/x, give x)?

  1. 3
  2. 4
  3. 2
  4. 8
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Answer: B. After 2 half-lives the fraction is (1/2)² = 1/4.

Sample Hard question 7

The Cauchy problem for a first-order PDE specifies the solution on:

  1. A single point
  2. The whole plane
  3. An initial curve
  4. The boundary of a disc
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Answer: C. The characteristics are then used to extend it.

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